time travel theory in physics

Can we time travel? A theoretical physicist provides some answers

Emeritus professor, Physics, Carleton University

Disclosure statement

Peter Watson received funding from NSERC. He is affiliated with Carleton University and a member of the Canadian Association of Physicists.

Carleton University provides funding as a member of The Conversation CA.

Carleton University provides funding as a member of The Conversation CA-FR.

View all partners

• Bahasa Indonesia

Time travel makes regular appearances in popular culture, with innumerable time travel storylines in movies, television and literature. But it is a surprisingly old idea: one can argue that the Greek tragedy Oedipus Rex , written by Sophocles over 2,500 years ago, is the first time travel story .

But is time travel in fact possible? Given the popularity of the concept, this is a legitimate question. As a theoretical physicist, I find that there are several possible answers to this question, not all of which are contradictory.

The simplest answer is that time travel cannot be possible because if it was, we would already be doing it. One can argue that it is forbidden by the laws of physics, like the second law of thermodynamics or relativity . There are also technical challenges: it might be possible but would involve vast amounts of energy.

There is also the matter of time-travel paradoxes; we can — hypothetically — resolve these if free will is an illusion, if many worlds exist or if the past can only be witnessed but not experienced. Perhaps time travel is impossible simply because time must flow in a linear manner and we have no control over it, or perhaps time is an illusion and time travel is irrelevant.

Laws of physics

Since Albert Einstein’s theory of relativity — which describes the nature of time, space and gravity — is our most profound theory of time, we would like to think that time travel is forbidden by relativity. Unfortunately, one of his colleagues from the Institute for Advanced Study, Kurt Gödel, invented a universe in which time travel was not just possible, but the past and future were inextricably tangled.

We can actually design time machines , but most of these (in principle) successful proposals require negative energy , or negative mass, which does not seem to exist in our universe. If you drop a tennis ball of negative mass, it will fall upwards. This argument is rather unsatisfactory, since it explains why we cannot time travel in practice only by involving another idea — that of negative energy or mass — that we do not really understand.

Mathematical physicist Frank Tipler conceptualized a time machine that does not involve negative mass, but requires more energy than exists in the universe .

Time travel also violates the second law of thermodynamics , which states that entropy or randomness must always increase. Time can only move in one direction — in other words, you cannot unscramble an egg. More specifically, by travelling into the past we are going from now (a high entropy state) into the past, which must have lower entropy.

This argument originated with the English cosmologist Arthur Eddington , and is at best incomplete. Perhaps it stops you travelling into the past, but it says nothing about time travel into the future. In practice, it is just as hard for me to travel to next Thursday as it is to travel to last Thursday.

Resolving paradoxes

There is no doubt that if we could time travel freely, we run into the paradoxes. The best known is the “ grandfather paradox ”: one could hypothetically use a time machine to travel to the past and murder their grandfather before their father’s conception, thereby eliminating the possibility of their own birth. Logically, you cannot both exist and not exist.

Read more: Time travel could be possible, but only with parallel timelines

Kurt Vonnegut’s anti-war novel Slaughterhouse-Five , published in 1969, describes how to evade the grandfather paradox. If free will simply does not exist, it is not possible to kill one’s grandfather in the past, since he was not killed in the past. The novel’s protagonist, Billy Pilgrim, can only travel to other points on his world line (the timeline he exists in), but not to any other point in space-time, so he could not even contemplate killing his grandfather.

The universe in Slaughterhouse-Five is consistent with everything we know. The second law of thermodynamics works perfectly well within it and there is no conflict with relativity. But it is inconsistent with some things we believe in, like free will — you can observe the past, like watching a movie, but you cannot interfere with the actions of people in it.

Could we allow for actual modifications of the past, so that we could go back and murder our grandfather — or Hitler ? There are several multiverse theories that suppose that there are many timelines for different universes. This is also an old idea: in Charles Dickens’ A Christmas Carol , Ebeneezer Scrooge experiences two alternative timelines, one of which leads to a shameful death and the other to happiness.

Time is a river

Roman emperor Marcus Aurelius wrote that:

“ Time is like a river made up of the events which happen , and a violent stream; for as soon as a thing has been seen, it is carried away, and another comes in its place, and this will be carried away too.”

We can imagine that time does flow past every point in the universe, like a river around a rock. But it is difficult to make the idea precise. A flow is a rate of change — the flow of a river is the amount of water that passes a specific length in a given time. Hence if time is a flow, it is at the rate of one second per second, which is not a very useful insight.

Theoretical physicist Stephen Hawking suggested that a “ chronology protection conjecture ” must exist, an as-yet-unknown physical principle that forbids time travel. Hawking’s concept originates from the idea that we cannot know what goes on inside a black hole, because we cannot get information out of it. But this argument is redundant: we cannot time travel because we cannot time travel!

Researchers are investigating a more fundamental theory, where time and space “emerge” from something else. This is referred to as quantum gravity , but unfortunately it does not exist yet.

So is time travel possible? Probably not, but we don’t know for sure!

• Time travel
• Stephen Hawking
• Albert Einstein
• Listen to this article
• Time travel paradox
• Arthur Eddington

PhD Scholarship

Senior Lecturer, HRM or People Analytics

Senior Research Fellow - Neuromuscular Disorders and Gait Analysis

Centre Director, Transformative Media Technologies

Postdoctoral Research Fellowship

Is Time Travel Possible?

We all travel in time! We travel one year in time between birthdays, for example. And we are all traveling in time at approximately the same speed: 1 second per second.

We typically experience time at one second per second. Credit: NASA/JPL-Caltech

NASA's space telescopes also give us a way to look back in time. Telescopes help us see stars and galaxies that are very far away . It takes a long time for the light from faraway galaxies to reach us. So, when we look into the sky with a telescope, we are seeing what those stars and galaxies looked like a very long time ago.

However, when we think of the phrase "time travel," we are usually thinking of traveling faster than 1 second per second. That kind of time travel sounds like something you'd only see in movies or science fiction books. Could it be real? Science says yes!

This image from the Hubble Space Telescope shows galaxies that are very far away as they existed a very long time ago. Credit: NASA, ESA and R. Thompson (Univ. Arizona)

How do we know that time travel is possible?

More than 100 years ago, a famous scientist named Albert Einstein came up with an idea about how time works. He called it relativity. This theory says that time and space are linked together. Einstein also said our universe has a speed limit: nothing can travel faster than the speed of light (186,000 miles per second).

Einstein's theory of relativity says that space and time are linked together. Credit: NASA/JPL-Caltech

What does this mean for time travel? Well, according to this theory, the faster you travel, the slower you experience time. Scientists have done some experiments to show that this is true.

For example, there was an experiment that used two clocks set to the exact same time. One clock stayed on Earth, while the other flew in an airplane (going in the same direction Earth rotates).

After the airplane flew around the world, scientists compared the two clocks. The clock on the fast-moving airplane was slightly behind the clock on the ground. So, the clock on the airplane was traveling slightly slower in time than 1 second per second.

Credit: NASA/JPL-Caltech

Can we use time travel in everyday life?

We can't use a time machine to travel hundreds of years into the past or future. That kind of time travel only happens in books and movies. But the math of time travel does affect the things we use every day.

For example, we use GPS satellites to help us figure out how to get to new places. (Check out our video about how GPS satellites work .) NASA scientists also use a high-accuracy version of GPS to keep track of where satellites are in space. But did you know that GPS relies on time-travel calculations to help you get around town?

GPS satellites orbit around Earth very quickly at about 8,700 miles (14,000 kilometers) per hour. This slows down GPS satellite clocks by a small fraction of a second (similar to the airplane example above).

GPS satellites orbit around Earth at about 8,700 miles (14,000 kilometers) per hour. Credit: GPS.gov

However, the satellites are also orbiting Earth about 12,550 miles (20,200 km) above the surface. This actually speeds up GPS satellite clocks by a slighter larger fraction of a second.

Here's how: Einstein's theory also says that gravity curves space and time, causing the passage of time to slow down. High up where the satellites orbit, Earth's gravity is much weaker. This causes the clocks on GPS satellites to run faster than clocks on the ground.

The combined result is that the clocks on GPS satellites experience time at a rate slightly faster than 1 second per second. Luckily, scientists can use math to correct these differences in time.

If scientists didn't correct the GPS clocks, there would be big problems. GPS satellites wouldn't be able to correctly calculate their position or yours. The errors would add up to a few miles each day, which is a big deal. GPS maps might think your home is nowhere near where it actually is!

In Summary:

Yes, time travel is indeed a real thing. But it's not quite what you've probably seen in the movies. Under certain conditions, it is possible to experience time passing at a different rate than 1 second per second. And there are important reasons why we need to understand this real-world form of time travel.

If you liked this, you may like:

• Table of Contents
• Random Entry
• Chronological
• Editorial Information
• About the SEP
• Editorial Board
• How to Cite the SEP
• Special Characters
• Advanced Tools
• Support the SEP
• PDFs for SEP Friends
• Make a Donation
• SEPIA for Libraries
• Entry Contents

Bibliography

Academic tools.

• Friends PDF Preview
• Author and Citation Info
• Back to Top

Time Travel

There is an extensive literature on time travel in both philosophy and physics. Part of the great interest of the topic stems from the fact that reasons have been given both for thinking that time travel is physically possible—and for thinking that it is logically impossible! This entry deals primarily with philosophical issues; issues related to the physics of time travel are covered in the separate entries on time travel and modern physics and time machines . We begin with the definitional question: what is time travel? We then turn to the major objection to the possibility of backwards time travel: the Grandfather paradox. Next, issues concerning causation are discussed—and then, issues in the metaphysics of time and change. We end with a discussion of the question why, if backwards time travel will ever occur, we have not been visited by time travellers from the future.

1.1 Time Discrepancy

1.2 changing the past, 2.1 can and cannot, 2.2 improbable coincidences, 2.3 inexplicable occurrences, 3.1 backwards causation, 3.2 causal loops, 4.1 time travel and time, 4.2 time travel and change, 5. where are the time travellers, other internet resources, related entries, 1. what is time travel.

There is a number of rather different scenarios which would seem, intuitively, to count as ‘time travel’—and a number of scenarios which, while sharing certain features with some of the time travel cases, seem nevertheless not to count as genuine time travel: [ 1 ]

Time travel Doctor . Doctor Who steps into a machine in 2024. Observers outside the machine see it disappear. Inside the machine, time seems to Doctor Who to pass for ten minutes. Observers in 1984 (or 3072) see the machine appear out of nowhere. Doctor Who steps out. [ 2 ] Leap . The time traveller takes hold of a special device (or steps into a machine) and suddenly disappears; she appears at an earlier (or later) time. Unlike in Doctor , the time traveller experiences no lapse of time between her departure and arrival: from her point of view, she instantaneously appears at the destination time. [ 3 ] Putnam . Oscar Smith steps into a machine in 2024. From his point of view, things proceed much as in Doctor : time seems to Oscar Smith to pass for a while; then he steps out in 1984. For observers outside the machine, things proceed differently. Observers of Oscar’s arrival in the past see a time machine suddenly appear out of nowhere and immediately divide into two copies of itself: Oscar Smith steps out of one; and (through the window) they see inside the other something that looks just like what they would see if a film of Oscar Smith were played backwards (his hair gets shorter; food comes out of his mouth and goes back into his lunch box in a pristine, uneaten state; etc.). Observers of Oscar’s departure from the future do not simply see his time machine disappear after he gets into it: they see it collide with the apparently backwards-running machine just described, in such a way that both are simultaneously annihilated. [ 4 ] Gödel . The time traveller steps into an ordinary rocket ship (not a special time machine) and flies off on a certain course. At no point does she disappear (as in Leap ) or ‘turn back in time’ (as in Putnam )—yet thanks to the overall structure of spacetime (as conceived in the General Theory of Relativity), the traveller arrives at a point in the past (or future) of her departure. (Compare the way in which someone can travel continuously westwards, and arrive to the east of her departure point, thanks to the overall curved structure of the surface of the earth.) [ 5 ] Einstein . The time traveller steps into an ordinary rocket ship and flies off at high speed on a round trip. When he returns to Earth, thanks to certain effects predicted by the Special Theory of Relativity, only a very small amount of time has elapsed for him—he has aged only a few months—while a great deal of time has passed on Earth: it is now hundreds of years in the future of his time of departure. [ 6 ] Not time travel Sleep . One is very tired, and falls into a deep sleep. When one awakes twelve hours later, it seems from one’s own point of view that hardly any time has passed. Coma . One is in a coma for a number of years and then awakes, at which point it seems from one’s own point of view that hardly any time has passed. Cryogenics . One is cryogenically frozen for hundreds of years. Upon being woken, it seems from one’s own point of view that hardly any time has passed. Virtual . One enters a highly realistic, interactive virtual reality simulator in which some past era has been recreated down to the finest detail. Crystal . One looks into a crystal ball and sees what happened at some past time, or will happen at some future time. (Imagine that the crystal ball really works—like a closed-circuit security monitor, except that the vision genuinely comes from some past or future time. Even so, the person looking at the crystal ball is not thereby a time traveller.) Waiting . One enters one’s closet and stays there for seven hours. When one emerges, one has ‘arrived’ seven hours in the future of one’s ‘departure’. Dateline . One departs at 8pm on Monday, flies for fourteen hours, and arrives at 10pm on Monday.

A satisfactory definition of time travel would, at least, need to classify the cases in the right way. There might be some surprises—perhaps, on the best definition of ‘time travel’, Cryogenics turns out to be time travel after all—but it should certainly be the case, for example, that Gödel counts as time travel and that Sleep and Waiting do not. [ 7 ]

In fact there is no entirely satisfactory definition of ‘time travel’ in the literature. The most popular definition is the one given by Lewis (1976, 145–6):

What is time travel? Inevitably, it involves a discrepancy between time and time. Any traveller departs and then arrives at his destination; the time elapsed from departure to arrival…is the duration of the journey. But if he is a time traveller, the separation in time between departure and arrival does not equal the duration of his journey.…How can it be that the same two events, his departure and his arrival, are separated by two unequal amounts of time?…I reply by distinguishing time itself, external time as I shall also call it, from the personal time of a particular time traveller: roughly, that which is measured by his wristwatch. His journey takes an hour of his personal time, let us say…But the arrival is more than an hour after the departure in external time, if he travels toward the future; or the arrival is before the departure in external time…if he travels toward the past.

This correctly excludes Waiting —where the length of the ‘journey’ precisely matches the separation between ‘arrival’ and ‘departure’—and Crystal , where there is no journey at all—and it includes Doctor . It has trouble with Gödel , however—because when the overall structure of spacetime is as twisted as it is in the sort of case Gödel imagined, the notion of external time (“time itself”) loses its grip.

Another definition of time travel that one sometimes encounters in the literature (Arntzenius, 2006, 602) (Smeenk and Wüthrich, 2011, 5, 26) equates time travel with the existence of CTC’s: closed timelike curves. A curve in this context is a line in spacetime; it is timelike if it could represent the career of a material object; and it is closed if it returns to its starting point (i.e. in spacetime—not merely in space). This now includes Gödel —but it excludes Einstein .

The lack of an adequate definition of ‘time travel’ does not matter for our purposes here. [ 8 ] It suffices that we have clear cases of (what would count as) time travel—and that these cases give rise to all the problems that we shall wish to discuss.

Some authors (in philosophy, physics and science fiction) consider ‘time travel’ scenarios in which there are two temporal dimensions (e.g. Meiland (1974)), and others consider scenarios in which there are multiple ‘parallel’ universes—each one with its own four-dimensional spacetime (e.g. Deutsch and Lockwood (1994)). There is a question whether travelling to another version of 2001 (i.e. not the very same version one experienced in the past)—a version at a different point on the second time dimension, or in a different parallel universe—is really time travel, or whether it is more akin to Virtual . In any case, this kind of scenario does not give rise to many of the problems thrown up by the idea of travelling to the very same past one experienced in one’s younger days. It is these problems that form the primary focus of the present entry, and so we shall not have much to say about other kinds of ‘time travel’ scenario in what follows.

One objection to the possibility of time travel flows directly from attempts to define it in anything like Lewis’s way. The worry is that because time travel involves “a discrepancy between time and time”, time travel scenarios are simply incoherent. The time traveller traverses thirty years in one year; she is 51 years old 21 years after her birth; she dies at the age of 100, 200 years before her birth; and so on. The objection is that these are straightforward contradictions: the basic description of what time travel involves is inconsistent; therefore time travel is logically impossible. [ 9 ]

There must be something wrong with this objection, because it would show Einstein to be logically impossible—whereas this sort of future-directed time travel has actually been observed (albeit on a much smaller scale—but that does not affect the present point) (Hafele and Keating, 1972b,a). The most common response to the objection is that there is no contradiction because the interval of time traversed by the time traveller and the duration of her journey are measured with respect to different frames of reference: there is thus no reason why they should coincide. A similar point applies to the discrepancy between the time elapsed since the time traveller’s birth and her age upon arrival. There is no more of a contradiction here than in the fact that Melbourne is both 800 kilometres away from Sydney—along the main highway—and 1200 kilometres away—along the coast road. [ 10 ]

Before leaving the question ‘What is time travel?’ we should note the crucial distinction between changing the past and participating in (aka affecting or influencing) the past. [ 11 ] In the popular imagination, backwards time travel would allow one to change the past: to right the wrongs of history, to prevent one’s younger self doing things one later regretted, and so on. In a model with a single past, however, this idea is incoherent: the very description of the case involves a contradiction (e.g. the time traveller burns all her diaries at midnight on her fortieth birthday in 1976, and does not burn all her diaries at midnight on her fortieth birthday in 1976). It is not as if there are two versions of the past: the original one, without the time traveller present, and then a second version, with the time traveller playing a role. There is just one past—and two perspectives on it: the perspective of the younger self, and the perspective of the older time travelling self. If these perspectives are inconsistent (e.g. an event occurs in one but not the other) then the time travel scenario is incoherent.

This means that time travellers can do less than we might have hoped: they cannot right the wrongs of history; they cannot even stir a speck of dust on a certain day in the past if, on that day, the speck was in fact unmoved. But this does not mean that time travellers must be entirely powerless in the past: while they cannot do anything that did not actually happen, they can (in principle) do anything that did happen. Time travellers cannot change the past: they cannot make it different from the way it was—but they can participate in it: they can be amongst the people who did make the past the way it was. [ 12 ]

What about models involving two temporal dimensions, or parallel universes—do they allow for coherent scenarios in which the past is changed? [ 13 ] There is certainly no contradiction in saying that the time traveller burns all her diaries at midnight on her fortieth birthday in 1976 in universe 1 (or at hypertime A ), and does not burn all her diaries at midnight on her fortieth birthday in 1976 in universe 2 (or at hypertime B ). The question is whether this kind of story involves changing the past in the sense originally envisaged: righting the wrongs of history, preventing subsequently regretted actions, and so on. Goddu (2003) and van Inwagen (2010) argue that it does (in the context of particular hypertime models), while Smith (1997, 365–6; 2015) argues that it does not: that it involves avoiding the past—leaving it untouched while travelling to a different version of the past in which things proceed differently.

2. The Grandfather Paradox

The most important objection to the logical possibility of backwards time travel is the so-called Grandfather paradox. This paradox has actually convinced many people that backwards time travel is impossible:

The dead giveaway that true time-travel is flatly impossible arises from the well-known “paradoxes” it entails. The classic example is “What if you go back into the past and kill your grandfather when he was still a little boy?”…So complex and hopeless are the paradoxes…that the easiest way out of the irrational chaos that results is to suppose that true time-travel is, and forever will be, impossible. (Asimov 1995 [2003, 276–7]) travel into one’s past…would seem to give rise to all sorts of logical problems, if you were able to change history. For example, what would happen if you killed your parents before you were born. It might be that one could avoid such paradoxes by some modification of the concept of free will. But this will not be necessary if what I call the chronology protection conjecture is correct: The laws of physics prevent closed timelike curves from appearing . (Hawking, 1992, 604) [ 14 ]

The paradox comes in different forms. Here’s one version:

If time travel was logically possible then the time traveller could return to the past and in a suicidal rage destroy his time machine before it was completed and murder his younger self. But if this was so a necessary condition for the time trip to have occurred at all is removed, and we should then conclude that the time trip did not occur. Hence if the time trip did occur, then it did not occur. Hence it did not occur, and it is necessary that it did not occur. To reply, as it is standardly done, that our time traveller cannot change the past in this way, is a petitio principii . Why is it that the time traveller is constrained in this way? What mysterious force stills his sudden suicidal rage? (Smith, 1985, 58)

The idea is that backwards time travel is impossible because if it occurred, time travellers would attempt to do things such as kill their younger selves (or their grandfathers etc.). We know that doing these things—indeed, changing the past in any way—is impossible. But were there time travel, there would then be nothing left to stop these things happening. If we let things get to the stage where the time traveller is facing Grandfather with a loaded weapon, then there is nothing left to prevent the impossible from occurring. So we must draw the line earlier: it must be impossible for someone to get into this situation at all; that is, backwards time travel must be impossible.

In order to defend the possibility of time travel in the face of this argument we need to show that time travel is not a sure route to doing the impossible. So, given that a time traveller has gone to the past and is facing Grandfather, what could stop her killing Grandfather? Some science fiction authors resort to the idea of chaperones or time guardians who prevent time travellers from changing the past—or to mysterious forces of logic. But it is hard to take these ideas seriously—and more importantly, it is hard to make them work in detail when we remember that changing the past is impossible. (The chaperone is acting to ensure that the past remains as it was—but the only reason it ever was that way is because of his very actions.) [ 15 ] Fortunately there is a better response—also to be found in the science fiction literature, and brought to the attention of philosophers by Lewis (1976). What would stop the time traveller doing the impossible? She would fail “for some commonplace reason”, as Lewis (1976, 150) puts it. Her gun might jam, a noise might distract her, she might slip on a banana peel, etc. Nothing more than such ordinary occurrences is required to stop the time traveller killing Grandfather. Hence backwards time travel does not entail the occurrence of impossible events—and so the above objection is defused.

A problem remains. Suppose Tim, a time-traveller, is facing his grandfather with a loaded gun. Can Tim kill Grandfather? On the one hand, yes he can. He is an excellent shot; there is no chaperone to stop him; the laws of logic will not magically stay his hand; he hates Grandfather and will not hesitate to pull the trigger; etc. On the other hand, no he can’t. To kill Grandfather would be to change the past, and no-one can do that (not to mention the fact that if Grandfather died, then Tim would not have been born). So we have a contradiction: Tim can kill Grandfather and Tim cannot kill Grandfather. Time travel thus leads to a contradiction: so it is impossible.

Note the difference between this version of the Grandfather paradox and the version considered above. In the earlier version, the contradiction happens if Tim kills Grandfather. The solution was to say that Tim can go into the past without killing Grandfather—hence time travel does not entail a contradiction. In the new version, the contradiction happens as soon as Tim gets to the past. Of course Tim does not kill Grandfather—but we still have a contradiction anyway: for he both can do it, and cannot do it. As Lewis puts it:

Could a time traveler change the past? It seems not: the events of a past moment could no more change than numbers could. Yet it seems that he would be as able as anyone to do things that would change the past if he did them. If a time traveler visiting the past both could and couldn’t do something that would change it, then there cannot possibly be such a time traveler. (Lewis, 1976, 149)

Lewis’s own solution to this problem has been widely accepted. [ 16 ] It turns on the idea that to say that something can happen is to say that its occurrence is compossible with certain facts, where context determines (more or less) which facts are the relevant ones. Tim’s killing Grandfather in 1921 is compossible with the facts about his weapon, training, state of mind, and so on. It is not compossible with further facts, such as the fact that Grandfather did not die in 1921. Thus ‘Tim can kill Grandfather’ is true in one sense (relative to one set of facts) and false in another sense (relative to another set of facts)—but there is no single sense in which it is both true and false. So there is no contradiction here—merely an equivocation.

Another response is that of Vihvelin (1996), who argues that there is no contradiction here because ‘Tim can kill Grandfather’ is simply false (i.e. contra Lewis, there is no legitimate sense in which it is true). According to Vihvelin, for ‘Tim can kill Grandfather’ to be true, there must be at least some occasions on which ‘If Tim had tried to kill Grandfather, he would or at least might have succeeded’ is true—but, Vihvelin argues, at any world remotely like ours, the latter counterfactual is always false. [ 17 ]

Return to the original version of the Grandfather paradox and Lewis’s ‘commonplace reasons’ response to it. This response engenders a new objection—due to Horwich (1987)—not to the possibility but to the probability of backwards time travel.

Think about correlated events in general. Whenever we see two things frequently occurring together, this is because one of them causes the other, or some third thing causes both. Horwich calls this the Principle of V-Correlation:

if events of type A and B are associated with one another, then either there is always a chain of events between them…or else we find an earlier event of type C that links up with A and B by two such chains of events. What we do not see is…an inverse fork—in which A and B are connected only with a characteristic subsequent event, but no preceding one. (Horwich, 1987, 97–8)

For example, suppose that two students turn up to class wearing the same outfits. That could just be a coincidence (i.e. there is no common cause, and no direct causal link between the two events). If it happens every week for the whole semester, it is possible that it is a coincidence, but this is extremely unlikely . Normally, we see this sort of extensive correlation only if either there is a common cause (e.g. both students have product endorsement deals with the same clothing company, or both slavishly copy the same influencer) or a direct causal link (e.g. one student is copying the other).

Now consider the time traveller setting off to kill her younger self. As discussed, no contradiction need ensue—this is prevented not by chaperones or mysterious forces, but by a run of ordinary occurrences in which the trigger falls off the time traveller’s gun, a gust of wind pushes her bullet off course, she slips on a banana peel, and so on. But now consider this run of ordinary occurrences. Whenever the time traveller contemplates auto-infanticide, someone nearby will drop a banana peel ready for her to slip on, or a bird will begin to fly so that it will be in the path of the time traveller’s bullet by the time she fires, and so on. In general, there will be a correlation between auto-infanticide attempts and foiling occurrences such as the presence of banana peels—and this correlation will be of the type that does not involve a direct causal connection between the correlated events or a common cause of both. But extensive correlations of this sort are, as we saw, extremely rare—so backwards time travel will happen about as often as you will see two people wear the same outfits to class every day of semester, without there being any causal connection between what one wears and what the other wears.

We can set out Horwich’s argument this way:

• If time travel were ever to occur, we should see extensive uncaused correlations.
• It is extremely unlikely that we should ever see extensive uncaused correlations.
• Therefore time travel is extremely unlikely to occur.

The conclusion is not that time travel is impossible, but that we should treat it the way we treat the possibility of, say, tossing a fair coin and getting heads one thousand times in a row. As Price (1996, 278 n.7) puts it—in the context of endorsing Horwich’s conclusion: “the hypothesis of time travel can be made to imply propositions of arbitrarily low probability. This is not a classical reductio, but it is as close as science ever gets.”

Smith (1997) attacks both premisses of Horwich’s argument. Against the first premise, he argues that backwards time travel, in itself, does not entail extensive uncaused correlations. Rather, when we look more closely, we see that time travel scenarios involving extensive uncaused correlations always build in prior coincidences which are themselves highly unlikely. Against the second premise, he argues that, from the fact that we have never seen extensive uncaused correlations, it does not follow that we never shall. This is not inductive scepticism: let us assume (contra the inductive sceptic) that in the absence of any specific reason for thinking things should be different in the future, we are entitled to assume they will continue being the same; still we cannot dismiss a specific reason for thinking the future will be a certain way simply on the basis that things have never been that way in the past. You might reassure an anxious friend that the sun will certainly rise tomorrow because it always has in the past—but you cannot similarly refute an astronomer who claims to have discovered a specific reason for thinking that the earth will stop rotating overnight.

Sider (2002, 119–20) endorses Smith’s second objection. Dowe (2003) criticises Smith’s first objection, but agrees with the second, concluding overall that time travel has not been shown to be improbable. Ismael (2003) reaches a similar conclusion. Goddu (2007) criticises Smith’s first objection to Horwich. Further contributions to the debate include Arntzenius (2006), Smeenk and Wüthrich (2011, §2.2) and Elliott (2018). For other arguments to the same conclusion as Horwich’s—that time travel is improbable—see Ney (2000) and Effingham (2020).

Return again to the original version of the Grandfather paradox and Lewis’s ‘commonplace reasons’ response to it. This response engenders a further objection. The autoinfanticidal time traveller is attempting to do something impossible (render herself permanently dead from an age younger than her age at the time of the attempts). Suppose we accept that she will not succeed and that what will stop her is a succession of commonplace occurrences. The previous objection was that such a succession is improbable . The new objection is that the exclusion of the time traveler from successfully committing auto-infanticide is mysteriously inexplicable . The worry is as follows. Each particular event that foils the time traveller is explicable in a perfectly ordinary way; but the inevitable combination of these events amounts to a ring-fencing of the forbidden zone of autoinfanticide—and this ring-fencing is mystifying. It’s like a grand conspiracy to stop the time traveler from doing what she wants to do—and yet there are no conspirators: no time lords, no magical forces of logic. This is profoundly perplexing. Riggs (1997, 52) writes: “Lewis’s account may do for a once only attempt, but is untenable as a general explanation of Tim’s continual lack of success if he keeps on trying.” Ismael (2003, 308) writes: “Considered individually, there will be nothing anomalous in the explanations…It is almost irresistible to suppose, however, that there is something anomalous in the cases considered collectively, i.e., in our unfailing lack of success.” See also Gorovitz (1964, 366–7), Horwich (1987, 119–21) and Carroll (2010, 86).

There have been two different kinds of defense of time travel against the objection that it involves mysteriously inexplicable occurrences. Baron and Colyvan (2016, 70) agree with the objectors that a purely causal explanation of failure—e.g. Tim fails to kill Grandfather because first he slips on a banana peel, then his gun jams, and so on—is insufficient. However they argue that, in addition, Lewis offers a non-causal—a logical —explanation of failure: “What explains Tim’s failure to kill his grandfather, then, is something about logic; specifically: Tim fails to kill his grandfather because the law of non-contradiction holds.” Smith (2017) argues that the appearance of inexplicability is illusory. There are no scenarios satisfying the description ‘a time traveller commits autoinfanticide’ (or changes the past in any other way) because the description is self-contradictory (e.g. it involves the time traveller permanently dying at 20 and also being alive at 40). So whatever happens it will not be ‘that’. There is literally no way for the time traveller not to fail. Hence there is no need for—or even possibility of—a substantive explanation of why failure invariably occurs, and such failure is not perplexing.

3. Causation

Backwards time travel scenarios give rise to interesting issues concerning causation. In this section we examine two such issues.

Earlier we distinguished changing the past and affecting the past, and argued that while the former is impossible, backwards time travel need involve only the latter. Affecting the past would be an example of backwards causation (i.e. causation where the effect precedes its cause)—and it has been argued that this too is impossible, or at least problematic. [ 18 ] The classic argument against backwards causation is the bilking argument . [ 19 ] Faced with the claim that some event A causes an earlier event B , the proponent of the bilking objection recommends an attempt to decorrelate A and B —that is, to bring about A in cases in which B has not occurred, and to prevent A in cases in which B has occurred. If the attempt is successful, then B often occurs despite the subsequent nonoccurrence of A , and A often occurs without B occurring, and so A cannot be the cause of B . If, on the other hand, the attempt is unsuccessful—if, that is, A cannot be prevented when B has occurred, nor brought about when B has not occurred—then, it is argued, it must be B that is the cause of A , rather than vice versa.

The bilking procedure requires repeated manipulation of event A . Thus, it cannot get under way in cases in which A is either unrepeatable or unmanipulable. Furthermore, the procedure requires us to know whether or not B has occurred, prior to manipulating A —and thus, it cannot get under way in cases in which it cannot be known whether or not B has occurred until after the occurrence or nonoccurrence of A (Dummett, 1964). These three loopholes allow room for many claims of backwards causation that cannot be touched by the bilking argument, because the bilking procedure cannot be performed at all. But what about those cases in which it can be performed? If the procedure succeeds—that is, A and B are decorrelated—then the claim that A causes B is refuted, or at least weakened (depending upon the details of the case). But if the bilking attempt fails, it does not follow that it must be B that is the cause of A , rather than vice versa. Depending upon the situation, that B causes A might become a viable alternative to the hypothesis that A causes B —but there is no reason to think that this alternative must always be the superior one. For example, suppose that I see a photo of you in a paper dated well before your birth, accompanied by a report of your arrival from the future. I now try to bilk your upcoming time trip—but I slip on a banana peel while rushing to push you away from your time machine, my time travel horror stories only inspire you further, and so on. Or again, suppose that I know that you were not in Sydney yesterday. I now try to get you to go there in your time machine—but first I am struck by lightning, then I fall down a manhole, and so on. What does all this prove? Surely not that your arrival in the past causes your departure from the future. Depending upon the details of the case, it seems that we might well be entitled to describe it as involving backwards time travel and backwards causation. At least, if we are not so entitled, this must be because of other facts about the case: it would not follow simply from the repeated coincidental failures of my bilking attempts.

Backwards time travel would apparently allow for the possibility of causal loops, in which things come from nowhere. The things in question might be objects—imagine a time traveller who steals a time machine from the local museum in order to make his time trip and then donates the time machine to the same museum at the end of the trip (i.e. in the past). In this case the machine itself is never built by anyone—it simply exists. The things in question might be information—imagine a time traveller who explains the theory behind time travel to her younger self: theory that she herself knows only because it was explained to her in her youth by her time travelling older self. The things in question might be actions. Imagine a time traveller who visits his younger self. When he encounters his younger self, he suddenly has a vivid memory of being punched on the nose by a strange visitor. He realises that this is that very encounter—and resignedly proceeds to punch his younger self. Why did he do it? Because he knew that it would happen and so felt that he had to do it—but he only knew it would happen because he in fact did it. [ 20 ]

One might think that causal loops are impossible—and hence that insofar as backwards time travel entails such loops, it too is impossible. [ 21 ] There are two issues to consider here. First, does backwards time travel entail causal loops? Lewis (1976, 148) raises the question whether there must be causal loops whenever there is backwards causation; in response to the question, he says simply “I am not sure.” Mellor (1998, 131) appears to claim a positive answer to the question. [ 22 ] Hanley (2004, 130) defends a negative answer by telling a time travel story in which there is backwards time travel and backwards causation, but no causal loops. [ 23 ] Monton (2009) criticises Hanley’s counterexample, but also defends a negative answer via different counterexamples. Effingham (2020) too argues for a negative answer.

Second, are causal loops impossible, or in some other way objectionable? One objection is that causal loops are inexplicable . There have been two main kinds of response to this objection. One is to agree but deny that this is a problem. Lewis (1976, 149) accepts that a loop (as a whole) would be inexplicable—but thinks that this inexplicability (like that of the Big Bang or the decay of a tritium atom) is merely strange, not impossible. In a similar vein, Meyer (2012, 263) argues that if someone asked for an explanation of a loop (as a whole), “the blame would fall on the person asking the question, not on our inability to answer it.” The second kind of response (Hanley, 2004, §5) is to deny that (all) causal loops are inexplicable. A second objection to causal loops, due to Mellor (1998, ch.12), is that in such loops the chances of events would fail to be related to their frequencies in accordance with the law of large numbers. Berkovitz (2001) and Dowe (2001) both argue that Mellor’s objection fails to establish the impossibility of causal loops. [ 24 ] Effingham (2020) considers—and rebuts—some additional objections to the possibility of causal loops.

4. Time and Change

Gödel (1949a [1990a])—in which Gödel presents models of Einstein’s General Theory of Relativity in which there exist CTC’s—can well be regarded as initiating the modern academic literature on time travel, in both philosophy and physics. In a companion paper, Gödel discusses the significance of his results for more general issues in the philosophy of time (Gödel 1949b [1990b]). For the succeeding half century, the time travel literature focussed predominantly on objections to the possibility (or probability) of time travel. More recently, however, there has been renewed interest in the connections between time travel and more general issues in the metaphysics of time and change. We examine some of these in the present section. [ 25 ]

The first thing that we need to do is set up the various metaphysical positions whose relationships with time travel will then be discussed. Consider two metaphysical questions:

• Are the past, present and future equally real?
• Is there an objective flow or passage of time, and an objective now?

We can label some views on the first question as follows. Eternalism is the view that past and future times, objects and events are just as real as the present time and present events and objects. Nowism is the view that only the present time and present events and objects exist. Now-and-then-ism is the view that the past and present exist but the future does not. We can also label some views on the second question. The A-theory answers in the affirmative: the flow of time and division of events into past (before now), present (now) and future (after now) are objective features of reality (as opposed to mere features of our experience). Furthermore, they are linked: the objective flow of time arises from the movement, through time, of the objective now (from the past towards the future). The B-theory answers in the negative: while we certainly experience now as special, and time as flowing, the B-theory denies that what is going on here is that we are detecting objective features of reality in a way that corresponds transparently to how those features are in themselves. The flow of time and the now are not objective features of reality; they are merely features of our experience. By combining answers to our first and second questions we arrive at positions on the metaphysics of time such as: [ 26 ]

• the block universe view: eternalism + B-theory
• the moving spotlight view: eternalism + A-theory
• the presentist view: nowism + A-theory
• the growing block view: now-and-then-ism + A-theory.

So much for positions on time itself. Now for some views on temporal objects: objects that exist in (and, in general, change over) time. Three-dimensionalism is the view that persons, tables and other temporal objects are three-dimensional entities. On this view, what you see in the mirror is a whole person. [ 27 ] Tomorrow, when you look again, you will see the whole person again. On this view, persons and other temporal objects are wholly present at every time at which they exist. Four-dimensionalism is the view that persons, tables and other temporal objects are four-dimensional entities, extending through three dimensions of space and one dimension of time. On this view, what you see in the mirror is not a whole person: it is just a three-dimensional temporal part of a person. Tomorrow, when you look again, you will see a different such temporal part. Say that an object persists through time if it is around at some time and still around at a later time. Three- and four-dimensionalists agree that (some) objects persist, but they differ over how objects persist. According to three-dimensionalists, objects persist by enduring : an object persists from t 1 to t 2 by being wholly present at t 1 and t 2 and every instant in between. According to four-dimensionalists, objects persist by perduring : an object persists from t 1 to t 2 by having temporal parts at t 1 and t 2 and every instant in between. Perduring can be usefully compared with being extended in space: a road extends from Melbourne to Sydney not by being wholly located at every point in between, but by having a spatial part at every point in between.

It is natural to combine three-dimensionalism with presentism and four-dimensionalism with the block universe view—but other combinations of views are certainly possible.

Gödel (1949b [1990b]) argues from the possibility of time travel (more precisely, from the existence of solutions to the field equations of General Relativity in which there exist CTC’s) to the B-theory: that is, to the conclusion that there is no objective flow or passage of time and no objective now. Gödel begins by reviewing an argument from Special Relativity to the B-theory: because the notion of simultaneity becomes a relative one in Special Relativity, there is no room for the idea of an objective succession of “nows”. He then notes that this argument is disrupted in the context of General Relativity, because in models of the latter theory to date, the presence of matter does allow recovery of an objectively distinguished series of “nows”. Gödel then proposes a new model (Gödel 1949a [1990a]) in which no such recovery is possible. (This is the model that contains CTC’s.) Finally, he addresses the issue of how one can infer anything about the nonexistence of an objective flow of time in our universe from the existence of a merely possible universe in which there is no objectively distinguished series of “nows”. His main response is that while it would not be straightforwardly contradictory to suppose that the existence of an objective flow of time depends on the particular, contingent arrangement and motion of matter in the world, this would nevertheless be unsatisfactory. Responses to Gödel have been of two main kinds. Some have objected to the claim that there is no objective flow of time in his model universe (e.g. Savitt (2005); see also Savitt (1994)). Others have objected to the attempt to transfer conclusions about that model universe to our own universe (e.g. Earman (1995, 197–200); for a partial response to Earman see Belot (2005, §3.4)). [ 28 ]

Earlier we posed two questions:

Gödel’s argument is related to the second question. Let’s turn now to the first question. Godfrey-Smith (1980, 72) writes “The metaphysical picture which underlies time travel talk is that of the block universe [i.e. eternalism, in the terminology of the present entry], in which the world is conceived as extended in time as it is in space.” In his report on the Analysis problem to which Godfrey-Smith’s paper is a response, Harrison (1980, 67) replies that he would like an argument in support of this assertion. Here is an argument: [ 29 ]

A fundamental requirement for the possibility of time travel is the existence of the destination of the journey. That is, a journey into the past or the future would have to presuppose that the past or future were somehow real. (Grey, 1999, 56)

Dowe (2000, 442–5) responds that the destination does not have to exist at the time of departure: it only has to exist at the time of arrival—and this is quite compatible with non-eternalist views. And Keller and Nelson (2001, 338) argue that time travel is compatible with presentism:

There is four-dimensional [i.e. eternalist, in the terminology of the present entry] time-travel if the appropriate sorts of events occur at the appropriate sorts of times; events like people hopping into time-machines and disappearing, people reappearing with the right sorts of memories, and so on. But the presentist can have just the same patterns of events happening at just the same times. Or at least, it can be the case on the presentist model that the right sorts of events will happen, or did happen, or are happening, at the rights sorts of times. If it suffices for four-dimensionalist time-travel that Jennifer disappears in 2054 and appears in 1985 with the right sorts of memories, then why shouldn’t it suffice for presentist time-travel that Jennifer will disappear in 2054, and that she did appear in 1985 with the right sorts of memories?

Sider (2005) responds that there is still a problem reconciling presentism with time travel conceived in Lewis’s way: that conception of time travel requires that personal time is similar to external time—but presentists have trouble allowing this. Further contributions to the debate whether presentism—and other versions of the A-theory—are compatible with time travel include Monton (2003), Daniels (2012), Hall (2014) and Wasserman (2018) on the side of compatibility, and Miller (2005), Slater (2005), Miller (2008), Hales (2010) and Markosian (2020) on the side of incompatibility.

Leibniz’s Law says that if x = y (i.e. x and y are identical—one and the same entity) then x and y have exactly the same properties. There is a superficial conflict between this principle of logic and the fact that things change. If Bill is at one time thin and at another time not so—and yet it is the very same person both times—it looks as though the very same entity (Bill) both possesses and fails to possess the property of being thin. Three-dimensionalists and four-dimensionalists respond to this problem in different ways. According to the four-dimensionalist, what is thin is not Bill (who is a four-dimensional entity) but certain temporal parts of Bill; and what is not thin are other temporal parts of Bill. So there is no single entity that both possesses and fails to possess the property of being thin. Three-dimensionalists have several options. One is to deny that there are such properties as ‘thin’ (simpliciter): there are only temporally relativised properties such as ‘thin at time t ’. In that case, while Bill at t 1 and Bill at t 2 are the very same entity—Bill is wholly present at each time—there is no single property that this one entity both possesses and fails to possess: Bill possesses the property ‘thin at t 1 ’ and lacks the property ‘thin at t 2 ’. [ 30 ]

Now consider the case of a time traveller Ben who encounters his younger self at time t . Suppose that the younger self is thin and the older self not so. The four-dimensionalist can accommodate this scenario easily. Just as before, what we have are two different three-dimensional parts of the same four-dimensional entity, one of which possesses the property ‘thin’ and the other of which does not. The three-dimensionalist, however, faces a problem. Even if we relativise properties to times, we still get the contradiction that Ben possesses the property ‘thin at t ’ and also lacks that very same property. [ 31 ] There are several possible options for the three-dimensionalist here. One is to relativise properties not to external times but to personal times (Horwich, 1975, 434–5); another is to relativise properties to spatial locations as well as to times (or simply to spacetime points). Sider (2001, 101–6) criticises both options (and others besides), concluding that time travel is incompatible with three-dimensionalism. Markosian (2004) responds to Sider’s argument; [ 32 ] Miller (2006) also responds to Sider and argues for the compatibility of time travel and endurantism; Gilmore (2007) seeks to weaken the case against endurantism by constructing analogous arguments against perdurantism. Simon (2005) finds problems with Sider’s arguments, but presents different arguments for the same conclusion; Effingham and Robson (2007) and Benovsky (2011) also offer new arguments for this conclusion. For further discussion see Wasserman (2018) and Effingham (2020). [ 33 ]

We have seen arguments to the conclusions that time travel is impossible, improbable and inexplicable. Here’s an argument to the conclusion that backwards time travel simply will not occur. If backwards time travel is ever going to occur, we would already have seen the time travellers—but we have seen none such. [ 34 ] The argument is a weak one. [ 35 ] For a start, it is perhaps conceivable that time travellers have already visited the Earth [ 36 ] —but even granting that they have not, this is still compatible with the future actuality of backwards time travel. First, it may be that time travel is very expensive, difficult or dangerous—or for some other reason quite rare—and that by the time it is available, our present period of history is insufficiently high on the list of interesting destinations. Second, it may be—and indeed existing proposals in the physics literature have this feature—that backwards time travel works by creating a CTC that lies entirely in the future: in this case, backwards time travel becomes possible after the creation of the CTC, but travel to a time earlier than the time at which the CTC is created is not possible. [ 37 ]

• Adams, Robert Merrihew, 1997, “Thisness and time travel”, Philosophia , 25: 407–15.
• Arntzenius, Frank, 2006, “Time travel: Double your fun”, Philosophy Compass , 1: 599–616. doi:10.1111/j.1747-9991.2006.00045.x
• Asimov, Isaac, 1995 [2003], Gold: The Final Science Fiction Collection , New York: Harper Collins.
• Baron, Sam and Colyvan, Mark, 2016, “Time enough for explanation”, Journal of Philosophy , 113: 61–88.
• Belot, Gordon, 2005, “Dust, time and symmetry”, British Journal for the Philosophy of Science , 56: 255–91.
• Benovsky, Jiri, 2011, “Endurance and time travel”, Kriterion , 24: 65–72.
• Berkovitz, Joseph, 2001, “On chance in causal loops”, Mind , 110: 1–23.
• Black, Max, 1956, “Why cannot an effect precede its cause?”, Analysis , 16: 49–58.
• Brier, Bob, 1973, “Magicians, alarm clocks, and backward causation”, Southern Journal of Philosophy , 11: 359–64.
• Carlson, Erik, 2005, “A new time travel paradox resolved”, Philosophia , 33: 263–73.
• Carroll, John W., 2010, “Context, conditionals, fatalism, time travel, and freedom”, in Time and Identity , Joseph Keim Campbell, Michael O’Rourke, and Harry S. Silverstein, eds., Cambridge MA: MIT Press, 79–93.
• Craig, William L., 1997, “Adams on actualism and presentism”, Philosophia , 25: 401–5.
• Daniels, Paul R., 2012, “Back to the present: Defending presentist time travel”, Disputatio , 4: 469–84.
• Deutsch, David and Lockwood, Michael, 1994, “The quantum physics of time travel”, Scientific American , 270(3): 50–6.
• Dowe, Phil, 2000, “The case for time travel”, Philosophy , 75: 441–51.
• –––, 2001, “Causal loops and the independence of causal facts”, Philosophy of Science , 68: S89–S97.
• –––, 2003, “The coincidences of time travel”, Philosophy of Science , 70: 574–89.
• Dummett, Michael, 1964, “Bringing about the past”, Philosophical Review , 73: 338–59.
• Dwyer, Larry, 1977, “How to affect, but not change, the past”, Southern Journal of Philosophy , 15: 383–5.
• Earman, John, 1995, Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes , New York: Oxford University Press.
• Effingham, Nikk, 2020, Time Travel: Probability and Impossibility , Oxford: Oxford University Press.
• Effingham, Nikk and Robson, Jon, 2007, “A mereological challenge to endurantism”, Australasian Journal of Philosophy , 85: 633–40.
• Ehring, Douglas, 1997, “Personal identity and time travel”, Philosophical Studies , 52: 427–33.
• Elliott, Katrina, 2019, “How to Know That Time Travel Is Unlikely Without Knowing Why”, Pacific Philosophical Quarterly , 100: 90–113.
• Fulmer, Gilbert, 1980, “Understanding time travel”, Southwestern Journal of Philosophy , 11: 151–6.
• Gilmore, Cody, 2007, “Time travel, coinciding objects, and persistence”, in Oxford Studies in Metaphysics , Dean W. Zimmerman, ed., Oxford: Clarendon Press, vol. 3, 177–98.
• Goddu, G.C., 2003, “Time travel and changing the past (or how to kill yourself and live to tell the tale)”, Ratio , 16: 16–32.
• –––, 2007, “Banana peels and time travel”, Dialectica , 61: 559–72.
• Gödel, Kurt, 1949a [1990a], “An example of a new type of cosmological solutions of Einstein’s field equations of gravitation”, in Kurt Gödel: Collected Works (Volume II), Solomon Feferman, et al. (eds.), New York: Oxford University Press, 190–8; originally published in Reviews of Modern Physics , 21 (1949): 447–450.
• –––, 1949b [1990b], “A remark about the relationship between relativity theory and idealistic philosophy”, in Kurt Gödel: Collected Works (Volume II), Solomon Feferman, et al. (eds.), New York: Oxford University Press, 202–7; originally published in P. Schilpp (ed.), Albert Einstein: Philosopher-Scientist , La Salle: Open Court, 1949, 555–562.
• Godfrey-Smith, William, 1980, “Travelling in time”, Analysis , 40: 72–3.
• Gorovitz, Samuel, 1964, “Leaving the past alone”, Philosophical Review , 73: 360–71.
• Grey, William, 1999, “Troubles with time travel”, Philosophy , 74: 55–70.
• Hafele, J. C. and Keating, Richard E., 1972a, “Around-the-world atomic clocks: Observed relativistic time gains”, Science , 177: 168–70.
• –––, 1972b, “Around-the-world atomic clocks: Predicted relativistic time gains”, Science , 177: 166–8.
• Hales, Steven D., 2010, “No time travel for presentists”, Logos & Episteme , 1: 353–60.
• Hall, Thomas, 2014, “In Defense of the Compossibility of Presentism and Time Travel”, Logos & Episteme , 2: 141–59.
• Hanley, Richard, 2004, “No end in sight: Causal loops in philosophy, physics and fiction”, Synthese , 141: 123–52.
• Harrison, Jonathan, 1980, “Report on analysis ‘problem’ no. 18”, Analysis , 40: 65–9.
• Hawking, S.W., 1992, “Chronology protection conjecture”, Physical Review D , 46: 603–11.
• Holt, Dennis Charles, 1981, “Time travel: The time discrepancy paradox”, Philosophical Investigations , 4: 1–16.
• Horacek, David, 2005, “Time travel in indeterministic worlds”, Monist (Special Issue on Time Travel), 88: 423–36.
• Horwich, Paul, 1975, “On some alleged paradoxes of time travel”, Journal of Philosophy , 72: 432–44.
• –––, 1987, Asymmetries in Time: Problems in the Philosophy of Science , Cambridge MA: MIT Press.
• Ismael, J., 2003, “Closed causal loops and the bilking argument”, Synthese , 136: 305–20.
• Keller, Simon and Nelson, Michael, 2001, “Presentists should believe in time-travel”, Australasian Journal of Philosophy , 79: 333–45.
• Kiourti, Ira, 2008, “Killing baby Suzy”, Philosophical Studies , 139: 343–52.
• Le Poidevin, Robin, 2003, Travels in Four Dimensions: The Enigmas of Space and Time , Oxford: Oxford University Press.
• –––, 2005, “The Cheshire Cat problem and other spatial obstacles to backwards time travel”, Monist (Special Issue on Time Travel), 88: 336–52.
• Lewis, David, 1976, “The paradoxes of time travel”, American Philosophical Quarterly , 13: 145–52.
• Loss, Roberto, 2015, “How to Change the Past in One-Dimensional Time”, Pacific Philosophical Quarterly , 96: 1–11.
• Luminet, Jean-Pierre, 2011, “Time, topology, and the twin paradox”, in The Oxford Handbook of Philosophy of Time , Craig Callender (ed.), Oxford: Oxford University Press. doi:10.1093/oxfordhb/9780199298204.003.0018
• Markosian, Ned, 2004, “Two arguments from Sider’s Four-Dimensionalism ”, Philosophy and Phenomenological Research , 68: 665–73.
• Markosian, Ned, 2020, “The Dynamic Theory of Time and Time Travel to the Past”, Disputatio , 12: 137–65.
• Maudlin, Tim, 2012, Philosophy of Physics: Space and Time , Princeton: Princeton University Press.
• Meiland, Jack W., 1974, “A two-dimensional passage model of time for time travel”, Philosophical Studies , 26: 153–73.
• Mellor, D.H., 1998, Real Time II , London: Routledge.
• Meyer, Ulrich, 2012, “Explaining causal loops”, Analysis , 72: 259–64.
• Miller, Kristie, 2005, “Time travel and the open future”, Disputatio , 1: 223–32.
• –––, 2006, “Travelling in time: How to wholly exist in two places at the same time”, Canadian Journal of Philosophy , 36: 309–34.
• –––, 2008, “Backwards causation, time, and the open future”, Metaphysica , 9: 173–91.
• Monton, Bradley, 2003, “Presentists can believe in closed timelike curves”, Analysis , 63: 199–202.
• –––, 2009, “Time travel without causal loops”, Philosophical Quarterly , 59: 54–67.
• Nerlich, Graham, 1981, “Can time be finite?”, Pacific Philosophical Quarterly , 62: 227–39.
• Ney, S.E., 2000, “Are grandfathers an endangered species?”, Journal of Philosophical Research , 25: 311–21.
• Price, Huw, 1996, Time’s Arrow & Archimedes’ Point: New Directions for the Physics of Time , New York: Oxford University Press.
• Putnam, Hilary, 1975, “It ain’t necessarily so”, in Mathematics, Matter and Method , Cambridge: Cambridge University Press, vol. 1 of Philosophical Papers , 237–49.
• Reinganum, Marc R., 1986, “Is time travel impossible? A financial proof”, Journal of Portfolio Management , 13: 10–2.
• Riggs, Peter J., 1991, “A critique of Mellor’s argument against ‘backwards’ causation”, British Journal for the Philosophy of Science , 42: 75–86.
• –––, 1997, “The principal paradox of time travel”, Ratio , 10: 48–64.
• Savitt, Steven, 1994, “The replacement of time”, Australasian Journal of Philosophy , 74: 463–73.
• –––, 2005, “Time travel and becoming”, Monist (Special Issue on Time Travel), 88: 413–22.
• Sider, Theodore, 2001, Four-Dimensionalism: An Ontology of Persistence and Time , Oxford: Clarendon Press.
• –––, 2002, “Time travel, coincidences and counterfactuals”, Philosophical Studies , 110: 115–38.
• –––, 2004, “Replies to Gallois, Hirsch and Markosian”, Philosophy and Phenomenological Research , 68: 674–87.
• –––, 2005, “Traveling in A- and B- time”, Monist (Special Issue on Time Travel), 88: 329–35.
• Simon, Jonathan, 2005, “Is time travel a problem for the three-dimensionalist?”, Monist (Special Issue on Time Travel), 88: 353–61.
• Slater, Matthew H., 2005, “The necessity of time travel (on pain of indeterminacy)”, Monist (Special Issue on Time Travel), 88: 362–9.
• Smart, J.J.C., 1963, “Is time travel possible?”, Journal of Philosophy , 60: 237–41.
• Smeenk, Chris and Wüthrich, Christian, 2011, “Time travel and time machines”, in The Oxford Handbook of Philosophy of Time , Craig Callender (ed.), Oxford: Oxford University Press, online ed. doi:10.1093/oxfordhb/9780199298204.003.0021
• Smith, Joseph Wayne, 1985, “Time travel and backward causation”, Cogito , 3: 57–67.
• Smith, Nicholas J.J., 1997, “Bananas enough for time travel?”, British Journal for the Philosophy of Science , 48: 363–89.
• –––, 1998, “The problems of backward time travel”, Endeavour , 22(4): 156–8.
• –––, 2004, “Review of Robin Le Poidevin Travels in Four Dimensions: The Enigmas of Space and Time ”, Australasian Journal of Philosophy , 82: 527–30.
• –––, 2005, “Why would time travellers try to kill their younger selves?”, Monist (Special Issue on Time Travel), 88: 388–95.
• –––, 2011, “Inconsistency in the A-theory”, Philosophical Studies , 156: 231–47.
• –––, 2015, “Why time travellers (still) cannot change the past”, Revista Portuguesa de Filosofia , 71: 677–94.
• –––, 2017, “I’d do anything to change the past (but I can’t do ‘that’)”, American Philosophical Quarterly , 54: 153–68.
• van Inwagen, Peter, 2010, “Changing the past”, in Oxford Studies in Metaphysics (Volume 5), Dean W. Zimmerman (ed.), Oxford: Oxford University Press, 3–28.
• Vihvelin, Kadri, 1996, “What time travelers cannot do”, Philosophical Studies , 81: 315–30.
• Vranas, Peter B.M., 2005, “Do cry over spilt milk: Possibly you can change the past”, Monist (Special Issue on Time Travel), 88: 370–87.
• –––, 2009, “Can I kill my younger self? Time travel and the retrosuicide paradox”, Pacific Philosophical Quarterly , 90: 520–34.
• –––, 2010, “What time travelers may be able to do”, Philosophical Studies , 150: 115–21.
• Wasserman, Ryan, 2018, Paradoxes of Time Travel , Oxford: Oxford University Press.
• Williams, Donald C., 1951, “The myth of passage”, Journal of Philosophy , 48: 457–72.
• Wright, John, 2006, “Personal identity, fission and time travel”, Philosophia , 34: 129–42.
• Yourgrau, Palle, 1999, Gödel Meets Einstein: Time Travel in the Gödel Universe , Chicago: Open Court.

How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

• Time Travel , entry by Joel Hunter (Truckee Meadows Community College) in the Internet Encyclopedia of Philosophy .

causation: backward | free will: divine foreknowledge and | identity: over time | location and mereology | temporal parts | time | time machines | time travel: and modern physics

Copyright © 2024 by Nicholas J.J. Smith < nicholas . smith @ sydney . edu . au >

• Accessibility

Support SEP

Mirror sites.

View this site from another server:

• Info about mirror sites

The Stanford Encyclopedia of Philosophy is copyright © 2024 by The Metaphysics Research Lab , Department of Philosophy, Stanford University

Library of Congress Catalog Data: ISSN 1095-5054

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

• View all journals
• My Account Login
• Explore content
• About the journal
• Publish with us
• Sign up for alerts
• Open access
• Published: 27 June 2024

From square plaquettes to triamond lattices for SU(2) gauge theory

• Ali H. Z. Kavaki 1 &
• Randy Lewis   ORCID: orcid.org/0009-0005-4869-7155 1  

Communications Physics volume  7 , Article number:  208 ( 2024 ) Cite this article

Metrics details

• Quantum simulation
• Theoretical nuclear physics
• Theoretical particle physics

Lattice gauge theory should be able to address significant new scientific questions when implemented on quantum computers. In practice, error-mitigation techniques have already allowed encouraging progress on small lattices. In this work we focus on a truncated version of SU(2) gauge theory, which is a familiar non-Abelian step toward quantum chromodynamics. First, we demonstrate effective error mitigation for imaginary time evolution on a lattice having two square plaquettes, obtaining the ground state using an IBM quantum computer and observing that this would have been impossible without error mitigation. Then we propose the triamond lattice as an expedient approach to lattice gauge theories in three spatial dimensions and we derive the Hamiltonian. Finally, error-mitigated imaginary time evolution is applied to the three-dimensional triamond unit cell, and its ground state is obtained from an IBM quantum computer. Future work will want to relax the truncation on the gauge fields, and the triamond lattice is increasingly valuable for such studies.

Similar content being viewed by others

Quantum error correction with gauge symmetries

Lattice quantum electrodynamics in (3+1)-dimensions at finite density with tensor networks

Floquet approach to ℤ2 lattice gauge theories with ultracold atoms in optical lattices

Introduction.

Quantum field theory combines quantum mechanics with special relativity and is of widespread importance in high-energy physics, nuclear physics, and condensed matter physics. Enforcing a spatially local symmetry leads to the special case of a gauge theory. The standard model of particle physics is a collection of three gauge theories describing the strong, weak, and electromagnetic forces. Because perturbation theory cannot be applied to a large coupling constant, many aspects of the strong force are best described by a numerical approach called lattice gauge theory.

Lattice gauge theory has been of central importance to quantum chromodynamics for several decades and has become a precision tool for practitioners 1 , 2 . It provides rigorous calculations of static properties, such as the masses and form factors of hadrons, directly from the gauge theory of quarks and gluons. Such calculations are crucial in the ongoing quest to understand hadron physics, such as the tetraquarks and pentaquarks that have been discovered experimentally in recent years 3 , 4 . Lattice gauge theory calculations also provide necessary input for experiments seeking new physics beyond the standard model, where a famous recent example is the anomalous magnetic moment of the muon 5 .

Quantum computers offer a prospective way to calculate more than just static properties. Traditional lattice gauge theory uses Monte Carlo methods that would have sign problems if applied straightforwardly to situations involving dynamics, nonzero density, or some topological interactions 6 , 7 . These sign problems can be avoided by using a Hamiltonian approach, where the exponentially large Hilbert space can fit naturally onto a quantum computer. Now that quantum computing hardware is becoming a reality, this powerful new approach to lattice gauge theory is being developed by a broad community of researchers 7 , 8 , 9 .

With quantum chromodynamics as a motivating long-range goal, the present work is focused on the simplest non-Abelian gauge theory. Being non-Abelian means that gauge fields carry the relevant charge directly and can therefore interact among themselves, like the gluons of quantum chromodynamics and in contrast to the photons of quantum electrodynamics. Specifically our work will consider SU(2) gauge theory in the absence of fermions, and we will truncate the gauge field to fit on a small number of qubits. Several research groups have already carried out exploratory studies of non-Abelian gauge theories on quantum computing hardware 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 . There have also been many theoretical studies laying the groundwork for anticipated implementations on larger quantum computers 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 , 65 , 66 , 67 , 68 , 69 . In the present work, we address two important issues: an effective way to mitigate errors and a practical way to extend lattices into three spatial dimensions.

Future quantum computers are expected to have smaller error rates and robust methods for correcting those errors, but calculations in the present era face substantial error rates and typically have insufficient resources to implement true error correction. Instead, error mitigation methods have been devised 70 , and these can provide significant improvements for computations performed on today’s hardware. Our study demonstrates the use of an existing method called self-mitigation 17 but in a new context: evolution of a quantum state through imaginary time. As is well known, after sufficiently many steps of imaginary time the excited state contributions to a generic initial state will become exponentially suppressed relative to the ground state, thus providing a way to create the ground state and to determine its eigenvalue. Our results will demonstrate that self-mitigation correctly finds and sustains the true ground state of a two-plaquette lattice even though the unmitigated data points are moving ever farther from the true result with each additional time step.

With successful mitigation in hand, we then turn our attention to lattice gauge theory in three spatial dimensions. While this can be done with a standard cubic lattice, where six gauge links touch each lattice site, there is an important issue to address. Specifically, for any lattice where more than three gauge links meet at a lattice site, the quantum numbers of the gauge links themselves are insufficient to fully define the state of the gauge field across the lattice 36 , 40 , 57 , 62 . To understand why, consider a two-dimensional square lattice where four SU(2) gauge links meet at every site. At one particular site, suppose each of the four gauge links happen to have quantum number \(j=\frac{1}{2}\) , where j is notation familiar from quantized angular momentum though here it refers to the gauge degrees of freedom. One pair of these links could sum to j pair  = 0 or 1. The other pair must have the same sum because Gauss’s law requires the total of all four to be j site  = 0. Therefore a full description of the state requires not just the j values of individual links, but also the value of j pair for some pair selected by the user to define that site.

One way to handle this issue is to assign a sufficient number of additional qubits at each lattice site, being careful to define a convention at each site, thereby completing the state’s definition 36 , 40 , 57 , 62 . The number of additional qubits needed to define j pair grows logarithmically with the user’s choice for \({j}_{\max }\) . Moreover, a three-dimensional lattice needs more than a single j pair at each site. In the present work we propose an alternative based on a structure from crystallography 71 , 72 , 73 , 74 , 75 , 76 . Having only three links at each site, our choice will avoid the need for any quantum numbers beyond the individual link values.

Consider a lattice where each site is touched by exactly three gauge links that have equal lengths, lie in a single plane, and are placed at equal angles around the site, i.e., 2 π /3. This forms a three-dimensional lattice that has a high degree of symmetry and needs no additional qubits beyond the links themselves. Because of its similarity to a diamond crystal of carbon atoms but with three links per site instead of more, it has been called the triamond lattice 74 , 75 . It is also known as the Laves lattice 72 or the K 4 lattice 73 , 76 . In this work, we provide an introduction to lattice gauge theory on the triamond framework. We derive the SU(2) Hamiltonian for a triamond lattice and demonstrate its use on a noiseless simulator. Then we employ our method for error-mitigated imaginary time evolution to create the ground state of the three-dimensional triamond unit cell on an IBM quantum computer 77 .

Imaginary time evolution on square plaquettes

Finding the ground state for a given Hamiltonian is an important ingredient of many scientific studies. Two common approaches are variational methods and imaginary time evolution. Variational methods rely on an ansatz chosen by the user, and can only get as close to the true ground state as the ansatz will allow. In contrast, imaginary time evolution always converges to the true ground state unless the initial state is perfectly orthogonal to it.

Imaginary time evolution is used routinely in traditional lattice gauge theory on classical computers. Our motivation to study it now on a quantum computer is not to compete with successful classical methods, but rather to imagine that ground state preparation will become the first step in a larger computation that truly does require quantum computing 78 , 79 , 80 . Imaginary time evolution has recently been applied to Z 2 gauge theory as well 81 .

Our first computation uses a lattice comprising two side-by-side square plaquettes that share a single gauge link, thus forming a left path, a center path, and a right path as shown in Fig.  1 . Each of the seven gauge links on this lattice is a superposition of SU(2) representations, and Gauss’s law requires the three on the left path to equal one another and similarly the three on the right path must equal one another. For our calculation, we truncate the basis and retain only the two lowest states for each gauge link, j  = 0 and \(j=\frac{1}{2}\) .

This is the first lattice used to demonstrate self-mitigation for quantum imaginary time evolution. The text refers to the 3 paths between sites A and B as the left path (three links), the center path (one link), and the right path (three links).

Any state of this system can be described by two qubits, one for the left path and one for the right path because Gauss’s law then determines the center path uniquely. Such a tiny system is a valuable place to study error mitigation because a single time step requires only a simple circuit while more and more time steps will require more and more entangling gates. The real-time evolution of this system was studied previously as the first example of self-mitigation 17 . The Hamiltonian, in units of g 2 /2, is

where x  = 2/ g 4 and g is the coupling constant, i.e., the strength of the strong interaction for this physics theory. The Pauli gates acting on qubit i are \({\hat{X}}_{i},{\hat{Y}}_{i}\) and \({\hat{Z}}_{i}\) .

Imaginary time evolution is not a unitary operation, so how can it be implemented on a quantum computer? We use an algorithm developed by Motta et al., called quantum imaginary time evolution (QITE) 82 , which can be described as follows. The evolution we seek is

where τ is the real-valued Euclidean time parameter. We can define a normalized state \(\left\vert \psi \right\rangle\) such that \(\left\vert \Psi (0)\right\rangle =r\left\vert \psi \right\rangle\) for positive r . The evolution can always be expressed as

for some unitary operator \({e}^{-i\tau \hat{A}}\) and positive \({r}^{{\prime} }\) . An immediate consequence is

The operator \(\hat{A}\) can be determined from a state tomography of \(\left\vert \psi \right\rangle\) that reflects the specific qubit connectivities of \(\hat{H}\) .

For our physics example, Eq. ( 1 ) confirms that the Hamiltonian is purely real, meaning that we can restrict ourselves to real-valued basis states as well. Together with Eqs. ( 2 ) and ( 3 ), this realness means \(\hat{A}\) must be purely imaginary, leading to a general expression with odd powers of \({\hat{Y}}_{i}\) gates,

The state tomography that determines the values of the coefficients a j k is presented in Methods. With those values in hand, and with \({r}^{{\prime} }\) obtained from the calculation of Eq. ( 4 ), the state at a small value of τ can be computed from Eq. ( 3 ). In particular, we use

where C X j k is a controlled not (CNOT) gate with qubit j as control and qubit k as target.

For sufficiently small τ , the operator \({e}^{-i\theta \hat{A}}\) can be approximated by a product of six factors, one for each term in \(\hat{A}\) . For a better approximation, that first-order Trotter expression can be replaced by the second-order Trotter expression represented by the circuit shown in Fig.  2 . To reach a larger value of τ , we can perform a sequence of small steps where the circuit is several end-to-end copies of Fig.  2 . Notice that the tomography needs to be computed separately for each step, using the state at the previous step as input.

This circuit is used for one second-order Trotter step of imaginary time evolution for SU(2) gauge theory on a lattice comprising two square plaquettes. τ is the real-valued Euclidean time parameter. The coefficients a j k define time evolution as shown in Eq. ( 5 ).

Because the two-qubit CNOT gate is noisier than a single-qubit gate on quantum hardware, the gates in Fig.  2 have been ordered in a way that minimizes the number of CNOT gates. Also, the CNOT gate at the end of one Trotter step will cancel the CNOT gate at the beginning of the next step. As well, the very first CNOT gate in the circuit can be omitted because the initial control qubit is always off. Nevertheless, the error bars without symbols in Fig.  3 show that only the first two or three time steps approach the true ground state energy and then subsequent time steps move further and further away. The quantum computation is overwhelmed by hardware noise without ever reaching the correct result.

These results are for SU(2) gauge theory with \({j}_{\max }=\frac{1}{2}\) on a lattice comprising two square plaquettes. The gauge coupling is x  = 1 and the time step is Δ τ  = 0.1 in units of 2/ g 2 . Each quantum circuit uses 50 randomized compilings with 10 4 shots per compiling. All quantum computations were performed on ibm_lagos . Black data points use self-mitigation and blue data points do not. Error bars are 95% confidence intervals. The true ground state is shown as a red dashed line. The pure noise limit is shown as a blue dotted line.

Self-mitigation is able to extract the true result from the computed data. The original implementation 17 , which was inspired by the work of ref. 83 , has subsequently been used and extended in various ways 18 , 19 , 21 , 60 , 79 , 84 , 85 , 86 , 87 , 88 . The basic idea is to run a mitigation circuit that is identical to the physics circuit except that τ  → − τ in the second half of the circuit. If your circuit has an odd number of second-order Trotter steps, use 2 τ  → 0 in the center gate (see, for example, Fig.  2) and insert a barrier to prevent CNOT cancellation.

The true outcome of the mitigation circuit should be identical to the initial state, i.e., all qubits should be in the off position. Comparison of the computed results with this known true result provides a measurement of hardware errors. The physics circuit will have similar hardware errors because the two circuits are identical up to the sign of τ in the latter half.

More precisely, randomized compiling (see Methods) is used to convert CNOT errors into incoherent noise that is well described by the depolarizing noise model 83 . For our example of imaginary time evolution, this leads to a pair of ratios,

where \({\hat{P}}_{j}\in \{{\hat{X}}_{j},{\hat{Y}}_{j},{\hat{Z}}_{j}\}\) . These equations give the true expectation value of a Pauli string as the ratio of results computed by a physics run and a mitigation run. They are the counterparts of Eq. ( 8 ) in the original paper 17 but look slightly different because the previous work studied probabilities that equalled \(\frac{1}{2}\) in the pure noise limit whereas the present work employs expectation values of Pauli strings that are zero for pure noise.

The filled data points of Fig.  3 were obtained by computing each term of Eq. ( 1 ) from the ratios in Eqs. ( 9 ) and ( 10 ). The physics runs and mitigation runs (with 50 randomized compilings for each) and the readout error mitigation runs were all completed within a single q iskit job 89 to ensure they were experiencing the same hardware conditions. Whereas the unmitigated data in Fig.  3 approach the pure noise limit of \(\frac{21}{8}\) , where only the first term in Eq. ( 1 ) makes a nonzero contribution, the self-mitigated data points in Fig.  3 remain consistent with the true ground state as τ increases. Error bars grow as the circuit gets longer, but meaningful results are still obtained with more than 400 CNOT gates. This success provides an incentive to move toward larger lattices, including novel three-dimensional approaches like the triamond lattice.

The SU(2) Hamiltonian for a triamond lattice

Figure  4 shows the locations of lattice sites and gauge links on a triamond lattice. See also our 2-min video presentation ( https://vimeo.com/904585432 ), which provides views of the triamond lattice from a camera moving around it.

a shows two views of the unit cell, which always contains 8 sites and 12 gauge links. b shows a lattice having 3 unit cells along each direction. In both panels, the color of each link defines its direction according to Eqs. ( 11 ) and ( 12 ).

To introduce the triamond lattice quantitatively, we will first define six unit vectors labeled as colors (red, green, blue, cyan, magenta, yellow) to aid the discussion,

where \(\hat{i},\hat{j}\) and \(\hat{k}\) are the standard orthonormal vectors. Every gauge link in the triamond lattice is along one of these color directions. As an aside, we note that the colors defined here have nothing to do with the colors of quantum chromodynamics.

Each lattice site is touched by three gauge links that lie in a plane. The complete lattice has just four sets of parallel planes, so we color each site white, red, green, or blue to represent which plane is at that site. The links and sites form a color scheme that is motivated by the well-known RGB model of colors: each white site is touched by red, green, and blue links, each red site is touched by red, magenta, and yellow links, each green site is touched by green, cyan and yellow links, and each blue site is touched by blue, cyan and magenta links.

The angle between any two planes is \(\arccos (\frac{1}{3})\) , which is familiar because this same angle lies between any pair of lines joining the center of a cube to its corners. Because the unit vector orthogonal to a plane is only unique up to its sign, the triamond lattice is chiral. Specifically, the mirror image of Fig.  4 is a distinct but equally valid triamond lattice.

Notice that the white sites of a triamond lattice form a body-centered cubic (bcc) structure. It is useful to compare the options of defining a lattice gauge theory on a triamond lattice, on a bcc lattice, or on a simple cubic lattice. Relative to a simple cubic lattice of the same volume, a bcc lattice has twice as many lattice sites, gauge links that are shorter by a factor of \(\sqrt{2}\) , and each bcc site has 8 nearest neighbors compared to 6 for the simple cubic lattice. This suggests that a bcc lattice could be valuable for efficient discussions of the continuum limit, perhaps alongside simple cubic studies. Importantly, the bcc and simple cubic lattices both have more than three gauge links touching each lattice site, meaning that extra qubits would be needed within each lattice site (and a convention must be chosen) to fully define the quantum state of the lattice 36 , 40 , 57 . In a sense, the triamond lattice can be viewed as a bcc lattice where those implicit qubits have been made explicit and unambiguous by creating the red, green, and blue sites.

Although the white sites do form a bcc lattice, the triamond lattice is more economical than merely being a clarified bcc lattice. Imagine, for example, shrinking the red, green, and blue links into their white sites. The remaining lattice would not be bcc because it would have only 6 links per unit cell (2 cyan, 2 magenta, and 2 yellow) whereas a bcc lattice would require 8 links per unit cell.

The comparison to bcc is encouraging, but our primary motivation for proposing the triamond lattice is the fact that it has exactly three links touching each site, so the j values of each gauge link are sufficient to define basis states for the lattice. For planar physics, a hexagonal lattice succeeds in having three links touching each site, and SU(2) gauge theory has recently been implemented on a hexagonal lattice 62 , 65 . Notice that the shortest closed paths on a hexagonal lattice have six gauge links.

The shortest closed paths on a triamond lattice have 10-gauge links. We refer to these as elementary plaquettes. Among the 10 gauge links comprising any elementary plaquette, one of the 6 colors is absent and the other 5 colors always occur twice. The lattice has only 6 orientations of plaquettes (the non-red plaquette, the non-green plaquette, etc.), with copies of them translated spatially throughout the lattice.

To perform any quantum computation on a triamond lattice, we need a Hamiltonian. For SU(2) gauge theory, the Hamiltonian is the non-Abelian generalization of quantum electrodynamics and is written as a sum of electric terms and magnetic terms 22 . In the continuum limit, the Hamiltonian is

where each trace is over the SU(2) indices of the electric field \(\overrightarrow{E}\) or field strength tensor F μ ν . (Other normalization conventions are sometimes used for \(\overrightarrow{E}\) and F μ ν 62 .)

On a lattice, each gauge link is an element of SU(2),

where a is the lattice spacing. The component of the gauge field at site \(\overrightarrow{n}\) that points in direction \(\hat{s}\) is an element of the SU(2) algebra,

Our computations will be performed in the electric basis, where the magnetic terms are best expressed as a sum of all elementary plaquettes in this lattice and each plaquette is the trace of the ordered product of the 10 gauge links comprising that plaquette. The somewhat tedious derivation of this lattice Hamiltonian is outlined in Methods. The result is

where the lattice spacing a is defined to be the distance between nearest-neighbor sites, i.e., the length of each gauge link on the lattice. The sum in H B over 6 plaquettes, \({{{{{{{{\mathcal{P}}}}}}}}}_{s}(\overrightarrow{w})\) , for each white lattice site includes all plaquettes on the lattice. Specifically, there is one of each type of plaquette (non-red, non-green, etc.) associated with each white site, and we use the subscript s to identify the color that is absent from the plaquette.

Any eigenstate of H E is fully defined by providing the quantum numbers for every gauge link on the triamond lattice, j 1 ,  j 2 ,  j 3 , …, j N . Our shorthand for an eigenstate is \(\left\vert \{j\}\right\rangle\) . The diagonal matrix elements of the Hamiltonian are

and the off-diagonal matrix elements from initial state \(\left\vert \{j\}\right\rangle\) to final state \(\left\vert \{J\}\right\rangle\) come from Eq. ( 20 ) with

where the right-hand side has a standard 6 j symbol and the product includes the 10 sites in order around the plaquette. For SU(2), either direction around the plaquette gives the same result. There are three links at each site, one external to the plaquette (subscript e ), one in the forward direction around the plaquette (subscript f ), and one in the backward direction around the plaquette (subscript b ). Eq. ( 22 ) is not specific to the triamond lattice and has been obtained previously by other authors 57 , 58 .

Computations on the triamond unit cell

As shown in Fig.  4 , a unit cell contains 12 gauge links. By using periodic boundary conditions, a single unit cell becomes a viable three-dimensional lattice having exactly three gauge links touching each site. If each SU(2) gauge link is truncated to just two basis states, j  = 0 and \(j=\frac{1}{2}\) , the unit cell accommodates 2 12  = 4096 basis states.

A simple way to begin exploring this theory is to use the variational method, which will find the lowest energy state among all states that couple to a selected ansatz. Figure  5 shows three energy eigenstates obtained from three different ansatze, each having just one variational parameter. (We name the parameter θ in every case, but they are three different parameters.) Each of these examples is successful in finding a true eigenstate of the Hamiltonian. The 12-qubit quantum circuits that produced these results can be found in Methods.

a , b , c correspond to the third excited state, the first excited state, and the ground state respectively. The gauge coupling is g  = 1 and periodic boundary conditions are used in all directions. Each data point was obtained by running qiskit code with 10 4 shots on the ibm_qasm noiseless simulator. Error bars are standard deviations. For comparison, red dashed lines show the true eigenvalues.

To gain more intuition, notice that there is no 10-sided path inside a unit cell. This means any plaquette operator will wrap around the lattice and touch one of the gauge links twice, thus changing that value of j twice. With our truncation to \(j\in \{0,\frac{1}{2}\}\) , this means only 8 gauge links are changed when a plaquette operator is applied to the single-cell lattice.

For example, begin with j  = 0 on all 12 links (called the bare vacuum) and then apply a non-red plaquette. The non-red plaquette touches one of the cyan links twice while avoiding the other cyan link, thus producing a state with j  = 0 for all red and cyan links but \(j=\frac{1}{2}\) for the 8 other links. If we had applied a non-cyan plaquette instead, the result would have been exactly the same because it touches one of the red links twice. Similarly, the non-green and non-magenta plaquettes are identical to each other ( j  = 0 for green and magenta but \(j=\frac{1}{2}\) for the 8 other links), and the non-blue and non-yellow plaquettes are identical to each other ( j  = 0 for blue and yellow but \(j=\frac{1}{2}\) for the 8 other links).

In fact, any number of plaquette operators can be applied to the bare vacuum one after the other in any order, and the final state will always be one of these four basis states: the bare vacuum, the non-red non-cyan plaquette, the non-green non-magenta plaquette, and the non-blue non-yellow plaquette. The part of the Hamiltonian matrix that governs these four states is

Of the 4096 basis states for the periodic unit cell, only 32 obey Gauss’s law. The requirement of Gauss’s law is that each site has either all three gauge links with j  = 0 or exactly one gauge link with j  = 0. Because the triamond lattice is symmetric under color interchanges, the 32 × 32 Hamiltonian for Gauss-compliant states can be block diagonalized into eight 4 × 4 blocks, including the one shown explicitly in Eq. ( 23 ). Each of the 32 states has exactly 0, 4, 6, or 8 of the gauge links with \(j=\frac{1}{2}\) . From top to bottom, the three examples in Fig.  5 are dominated by a state where the number of gauge links with \(j=\frac{1}{2}\) is 6, 4, and 0 respectively. In particular, the bottom panel of Fig.  5 is showing the true ground state of the theory, which is built from four basis states.

Block diagonalizing the Hamiltonian is simple enough for such a small lattice, but phenomenological applications will require lattices with more than a single unit cell, and this is where the Hilbert space grows rapidly. A lattice with N unit cells will have 12 N gauge links. Truncating each link to \(j\in \{0,\frac{1}{2}\}\) leads to 2 12 N basis states, and the ground state is built from 2 4 N −2 of them. Larger lattices are where the variational method, imaginary time evolution, and other algorithms implemented with error mitigation on quantum computers have the potential to be of great value.

Triamond imaginary time evolution

When future computations are performed on triamond lattices, a typical first step will be to prepare the interacting vacuum. Imaginary time evolution is a reliable method for doing so, but it is only practical if error mitigation methods are able to account for the hardware errors. As a first confirmation of self-mitigation for triamond lattices, consider a single-unit cell with periodic boundary conditions. The Hamiltonian matrix is Eq. ( 23 ), which can be written as

This is similar to the Hamiltonian for square plaquettes given in Eq. ( 1 ) but with different coefficients and additional terms. We adapted our code from the square plaquette study to handle the triamond case, obtaining the results displayed in Fig.  6 . Perhaps surprisingly, the unmitigated results are moving away from the true value already at the very first step of imaginary time. Notice, however, that we are performing a particularly difficult computation: the pure noise limit is at a height of 6 (far beyond the top of the graph) while the true result is only slightly below zero. This means the true result contains a delicate cancellation between the large first term in Eq. ( 24 ) and the sum of all other terms.

These results are for SU(2) gauge theory with \({j}_{\max }=\frac{1}{2}\) on a triamond unit cell with periodic boundaries in all three spatial directions. The gauge coupling is g  = 1 and the time step is Δ τ  = 0.05 in units of \(a/(2\sqrt{2}{g}^{2})\) . Each quantum circuit uses between 32 and 50 randomized compilings with 2000 shots per compiling. All quantum computations were performed on ibm_brisbane . Black data points use self-mitigation and blue data points do not. Error bars are 95% confidence intervals. The true ground state is shown as a red dashed line.

As expected, the unmitigated data points in Fig.  6 are approaching the pure noise limit as imaginary time increases. Fluctuations in that approach are observed and were expected because the computation at each time step was usually performed hours or days after the previous time step, and was run on whichever qubits were performing best at that time. In contrast, the self-mitigated results arrive at the correct value after two steps of imaginary time and remain in agreement for the duration of our computations. Even the fluctuations evident in the unmitigated results are handled nicely by self-mitigation, with no visible effect remaining in the self-mitigated data.

For τ  < 0.7 in the units of Fig.  6 , we used 50 randomized compilings. As τ increased beyond 0.7, the number of randomized compilings was reduced linearly to just 32 randomized compilings at τ  = 1.1. No sensitivity to this choice is observed in the results.

It is also worth noticing a significant difference between the two quantum computers we have used in this work. The square plaquette results in Fig.  3 ran on ibm_lagos where the CNOT gate is a native gate, but our triamond results in Fig.  6 ran on ibm_brisbane where CNOT gates are not native and are constructed from the echoed cross-resonance (ECR) gate. Self-mitigation has been tailored specifically to an ECR device by other authors 86 but our code is written in terms of CNOT gates (see Table  1 ), and it is encouraging to observe that a CNOT-based self-mitigation code performs well on the ECR device also.

In this work, SU(2) gauge theory has been truncated to \(j\in \{0,\frac{1}{2}\}\) and studied on small lattices.

Self-mitigation has allowed the QITE algorithm 82 to find the ground state of a lattice with two square plaquettes on an IBM quantum computer. As shown in Fig.  3 , the computation without self-mitigation did not attain the correct ground state. The basic effect of self-mitigation is a rescaling of expectation values, where raw results from the physics circuit are divided by results from a mitigation circuit according to Eqs. ( 9 ) and ( 10 ). The Hamiltonian’s expectation value is a linear combination of the individual rescaled expectation values, leading to the large improvement observed in Fig.  3 where, from an initial value of zero, the self-mitigated and unmitigated results ultimately move in opposite directions. The unmitigated result is approaching the pure noise limit while the self-mitigated result finds the true value quickly and remains consistent with it.

The error bars in Fig.  3 are statistical only, and self-mitigation has clearly handled the dominant systematic error. The mitigated result at τ  = 1.3 happens to be furthest from the true ground state, being several standard deviations away, but subsequent time steps agree nicely with the true ground state. This demonstrates a valuable feature of the QITE algorithm. Output from one time step is needed as input for the following step, and yet the algorithm recovers quickly from an outlier.

The application of QITE to a larger lattice will require more terms in the expression for \(\hat{A}\) in Eq. ( 5 ), leading to more expectation values that need to be computed. However, the expression for \(\hat{A}\) will generally not require all possible Pauli strings but is instead governed by the correlation length of the physics under consideration 82 . It will be interesting to see future studies that use self-mitigated QITE to examine non-Abelian gauge theories on larger lattices.

The triamond crystal offers a systematic way to define three-dimensional lattices. Because there are only three gauge links touching each lattice site, the quantum numbers of the links themselves are sufficient to fully define any basis state of the lattice. This is not true for a simple cubic lattice 62 . In fact, the triamond lattice is the only highly symmetric lattice that achieves this goal in three dimensions. To be more precise, the triamond lattice is the only three-dimensional lattice that is strongly isotropic and has three gauge links per site 73 . Strong isotropy refers to the fact that rotation (or reflection) of the lattice around any site can interchange any pair of links at that site while leaving the physical structure of the entire lattice invariant 73 . In particular, a simple cubic lattice is not strongly isotropic 73 , 76 .

Another interesting glimpse into the symmetries of the triamond lattice comes from noticing that the six vectors comprising a triamond lattice form a regular tetrahedron. Begin with one gauge link of each color (recall Eqs. ( 11 ) and ( 12 )) and translate each one spatially without any rotations. The red, green and blue links will form the equilateral triangular base of the tetrahedron. The cyan, magenta, and yellow links rise from the corners of that base to meet at the top of the tetrahedron. As might be expected from the sentences preceding Eq. ( 23 ), the cyan link of the tetrahedron touches all links except the red one, the magenta link touches all but green, and the yellow link touches all but blue.

For eyes used to seeing cubic lattices, the triamond might seem difficult to visualize. However, our derivation shows that the SU(2) Hamiltonian has the familiar form of a sum over just a few plaquette orientations, in this case six, and our quantum circuits for obtaining ground and excited states (see Figs.  7 , 8) are quite simple due to the triamond symmetries. The triamond lattice has only the expected parameters, namely the gauge coupling and the lattice volume, but it provides extra gauge links inside a unit cell, representing a smaller lattice spacing and a new trajectory for approaching the continuum limit of a gauge theory.

The 12 qubits are labeled by their colors, with two copies of each color appearing in the list. A constant angle is defined by \(\phi =\arccos (1/\sqrt{3})\) . The ground state eigenvalue for one unit cell of the triamond lattice is displayed in Fig.  5c .

Figure  6 shows the successful application of imaginary time evolution to a lattice in three spatial dimensions. The result is a robust preparation of the ground state. The triamond framework offers a systematic path toward larger lattices and relaxed truncations of the gauge theory. Future work in this direction can continue the progress toward a practical implementation of lattice gauge theories on quantum computers.

Tomography for the QITE algorithm

The QITE procedure 82 needs optimal values for the coefficients a j k in Eq. ( 5 ). For notational convenience the subscripts can be combined into a single subscript, so a j k  ≡  a J with J running from 1 to 6. The a J values can be obtained by minimizing Trotter errors, which means minimizing the difference between

It is more convenient to work with a scalar function rather than states, so we actually minimize

where the 6-component vector b and the 6 × 6 matrix S are real-valued. Minimization results in

and the only nonzero entries in S  +  S T are

The commutators defining the elements of b use the Hamiltonian from Eq. ( 1 ) for square plaquettes or Eq. ( 23 ) for the triamond unit cell. The numerical computation of b ,  S  +  S T and \({r}^{{\prime} }/r\) (from Eq. ( 4 )) requires nine expectation values. Three of them can be obtained by preparing the state \(\left\vert \psi \right\rangle\) in a quantum computer and measuring each qubit, which gives

where P j is the probability that qubit j is 1, and P 1 ⊕ 0 is the probability that both qubits are 1. Three other expectation values are obtained by preparing the state \(\left\vert {\psi }^{{\prime} }\right\rangle =R{Y}_{1}(-\frac{\pi }{2})R{Y}_{0}(-\frac{\pi }{2})\left\vert \psi \right\rangle\) in a quantum computer and measuring each qubit, which gives

The remaining three expectation values are obtained similarly: Preparing \(\left\vert {\psi }^{{\prime\prime} }\right\rangle =R{X}_{1}\left(\frac{\pi }{2}\right)R{X}_{0}\left(\frac{\pi }{2}\right)\left\vert \psi \right\rangle\) gives

Preparing \(\left\vert {\psi }^{\prime\prime\prime}\right\rangle =R{Y}_{0}(-\frac{\pi }{2})\left\vert \psi \right\rangle\) gives

Preparing \(\left\vert {\psi }^{\prime\prime\prime\prime}\right\rangle =R{Y}_{1}(-\frac{\pi }{2})\left\vert \psi \right\rangle\) gives

Our computations create and measure these five circuits separately for each time step. This means each one is self-mitigated, with 50 randomized compilings of its physics circuit and 50 of its mitigation circuit.

It should be noted that the QITE procedure does not require S  +  S T to be an invertible matrix. For all practical purposes, \({(S+{S}^{T})}^{-1}\) can be interpreted as the Moore-Penrose pseudo-inverse 90 , 91 and this is what we use for our calculations.

Randomized compiling for CNOT gates

The conversion of CNOT errors into incoherent noise is accomplished by randomizing the input to each CNOT gate in a circuit 17 , 83 . Two random Pauli gates are applied immediately before the CNOT gate, one to the control qubit and the other to the target qubit. Specifically, each gate is chosen randomly from the set { I ,  X ,  Y ,  Z }. Immediately after the CNOT gate, two Pauli gates are applied to ensure that the combined effect of Pauli gates would not change the circuit’s output on error-free hardware. The 16 options are shown explicitly in Table  1 .

Deriving the triamond SU(2) Hamiltonian

This section outlines the derivation that begins with gauge links in the form of Eq. ( 16 ) on a triamond lattice and arrives at the Hamiltonian in Eqs. ( 18 – 20 ). The coefficients in Eqs. ( 18 – 20 ) are defined by their need to agree with continuum SU(2) gauge theory, so our first step will be to expand the sum over plaquettes in powers of the lattice spacing.

Consider a general gauge link, \(U(\overrightarrow{w}+\overrightarrow{t},\hat{s})\) , where \(\overrightarrow{w}\) is a white site on the lattice. The lattice spacing a enters through the vectors \(\overrightarrow{t}\) and \(\hat{s}\) . As a specific example, the gauge link that begins at the nearest green site and points toward the neighboring yellow site is

Although not displayed here, terms at O ( a 3 ) and O ( a 4 ) must also be retained. Performing 10 such expansions provides an expression for the non-blue plaquette, which is

and can be expanded to become

where all fields are evaluated at \(\overrightarrow{w}\) . The expression for the non-yellow plaquette is similar to this non-blue result and summing the two of them removes some cross terms,

The sum of non-green and non-magenta plaquettes can be obtained from this by relabeling colors and x ,  y ,  z subscripts. The same is true for the sum of non-red and non-cyan. Finally, the sum of all 6 plaquettes is

where the field strength tensor is F μ ν  = ∂ μ A ν  − ∂ ν A μ  +  i [ A μ ,  A ν ].

The continuum expression for the magnetic Hamiltonian, Eq. ( 15 ), can easily be converted from an integral to a sum,

where V is the volume per white site. There are two white sites in each unit cell, and the volume of a unit cell is \({(2\sqrt{2}a)}^{3}\) , so \(V=16\sqrt{2}{a}^{3}\) . Therefore the magnetic Hamiltonian becomes

The constant term has no effect on dynamics so it can be dropped and we are left with the expression given in Eq. ( 20 ).

Converting the electric Hamiltonian from the continuum to the triamond lattice is straightforward. An important detail is that instead of the volume per white site, V , we now need the volume per link which is V /6. Begin with Eq. ( 14 ) and discretize to find

in agreement with Eq. ( 19 ).

Hamiltonian and circuits for triamond unit cell

To obtain the results in Fig.  5 , we begin with the Hamiltonian of SU(2) gauge theory truncated to \(j\in \{0,\frac{1}{2}\}\) on a unit cell of the triamond lattice with periodic boundary conditions. The general expression was obtained in Results and here it is written in terms of Pauli operators.

The Hamiltonian is the sum of an electric part and three magnetic parts,

The electric part is a sum of contributions from all gauge links as in Eq. ( 21 ),

The magnetic terms correspond respectively to application of the non-red non-cyan plaquette,

application of the non-green non-magenta plaquette,

and application of the non-blue non-yellow plaquette,

Notice that each magnetic term has 8 Pauli X gates and a projector acting on the other 4 qubits.

The next thing needed to obtain the results in Fig.  5 is a quantum circuit to prepare the ansatz containing the variational parameter. Based on the discussion surrounding Eq. ( 23 ), we anticipate that the physical ground state will have a contribution from the bare ground state as well as a triply degenerate contribution from three plaquette terms. Our variational parameter θ will be used to adjust the relative strengths of these two contributions. Our circuit is displayed in Fig.  7 and is responsible for the bottom panel of Fig.  5 . This particular ansatz is quite simple, with two qubits at its core and extra CNOT gates adjusting the other qubits in a straightforward way. The circuit has been designed such that it requires minimal connectivity among the 12 qubits and is suitable, for example, to run on IBM’s heavy hex architecture without additional swap gates. The center panel of Fig.  5 uses the trial state obtained from the circuit in Fig.  8 . The top panel of Fig.  5 uses a trial state that differs from Fig.  8 simply by appending a Pauli X gate at the end of the circuit for each cyan, magenta, and yellow qubit.

The 12 qubits are labeled by their colors, with two copies of each color appearing in the list. The first excited state eigenvalue for one unit cell of the triamond lattice is displayed in Fig.  5b .

The final step to obtain the results in Fig.  5 is to measure the expectation value of each term in the Hamiltonian. Each term is a product of X and Z gates. Expectation values involving only Z gates can be obtained directly from measurements of the state itself. Expectation values involving X gates can be obtained by rotating into the computational basis with a pair of Hadamard gates, X  =  H Z H . Each data point in Fig.  5 comes from the linear combination of output from 4 separate runs of the circuit, one for H E , one for \({H}_{{\bar{r}}{\bar{c}}}\) , one for \({H}_{{\bar{g}}{\bar{m}}}\) , and one for \({H}_{{\bar{b}}{\bar{y}}}\) .

Data availability

The numerical data for generating the figures are available from the authors upon request.

Code availability

Code for calculating self-mitigated imaginary time evolution is provided at https://github.com/randylewis/SelfMitigatedQITE .

Gattringer, C. & Lang, C. B. Quantum chromodynamics on the lattice. Lect. Notes Phys. 788 , 1–343 (2010).

Article   ADS   MathSciNet   Google Scholar  

Knechtli, F., Günther, M. & Peardon, M. Lattice Quantum Chromodynamics: Practical Essentials (Springer, 2017).

Brambilla, N. et al. The X Y Z states: experimental and theoretical status and perspectives. Phys. Rep. 873 , 1–154 (2020).

Article   ADS   Google Scholar  

Bicudo, P. Tetraquarks and pentaquarks in lattice QCD with light and heavy quarks. Phys. Rep. 1039 , 1–49 (2023).

Aoyama, T. et al. The anomalous magnetic moment of the muon in the Standard Model. Phys. Rep. 887 , 1–166 (2020).

Nagata, K. Finite-density lattice QCD and sign problem: Current status and open problems. Prog. Part. Nucl. Phys. 127 , 103991 (2022).

Article   Google Scholar  

Funcke, L., Hartung, T., Jansen, K. & Kühn, S. Review on Quantum Computing for Lattice Field Theory. Proceedings of Science (LATTICE2022) , 228 (2023).

Bauer, C. W. et al. Quantum simulation for high-energy physics. PRX Quantum 4 , 027001 (2023).

Di Meglio, A. et al. Quantum computing for high-energy physics: state of the art and challenges. Summary of the QC4HEP working group. [arXiv:2307.03236 [quant-ph]].

Banerjee, D. et al. Atomic quantum simulation of U(N) and SU(N) non-Abelian lattice gauge theories. Phys. Rev. Lett. 110 , 125303 (2013).

Klco, N., Stryker, J. R. & Savage, M. J. SU(2) non-Abelian gauge field theory in one dimension on digital quantum computers. Phys. Rev. D 101 , 074512 (2020).

Ciavarella, A., Klco, N. & Savage, M. J. Trailhead for quantum simulation of SU(3) Yang-Mills lattice gauge theory in the local multiplet basis. Phys. Rev. D 103 , 094501 (2021).

Atas, Y. Y. et al. SU(2) hadrons on a quantum computer via a variational approach. Nat. Commun. 12 , 6499 (2021).

Rahman, S. A., Lewis, R., Mendicelli, E. & Powell, S. SU(2) lattice gauge theory on a quantum annealer. Phys. Rev. D 104 , 034501 (2021).

Ciavarella, A. N. & Chernyshev, I. A. Preparation of the SU(3) lattice Yang-Mills vacuum with variational quantum methods. Phys. Rev. D 105 , 074504 (2022).

Illa, M. & Savage, M. J. Basic elements for simulations of standard-model physics with quantum annealers: multigrid and clock states. Phys. Rev. A 106 , 052605 (2022).

Rahman, S. A., Lewis, R., Mendicelli, E. & Powell, S. Self-mitigating Trotter circuits for SU(2) lattice gauge theory on a quantum computer. Phys. Rev. D 106 , 074502 (2022).

Farrell, R. C. et al. Preparations for quantum simulations of quantum chromodynamics in 1+1 dimensions. I. Axial gauge. Phys. Rev. D 107 , 054512 (2023).

Atas, Y. Y. et al. Simulating one-dimensional quantum chromodynamics on a quantum computer: real-time evolutions of tetra- and pentaquarks. Phys. Rev. Res. 5 , 033184 (2023).

Farrell, R. C. et al. Preparations for quantum simulations of quantum chromodynamics in 1+1 dimensions. II. Single-baryon β -decay in real time. Phys. Rev. D 107 , 054513 (2023).

Ciavarella, A. N. Quantum simulation of lattice QCD with improved Hamiltonians. Phys. Rev. D 108 , 094513 (2023).

Kogut, J. B. & Susskind, L. Hamiltonian formulation of Wilson’s lattice gauge theories. Phys. Rev. D 11 , 395–408 (1975).

Chandrasekharan, S. & Wiese, U. J. Quantum link models: a discrete approach to gauge theories. Nucl. Phys. B 492 , 455–474 (1997).

Mathur, M. Harmonic oscillator prepotentials in SU(2) lattice gauge theory. J. Phys. A 38 , 10015–10026 (2005).

Byrnes, T. & Yamamoto, Y. Simulating lattice gauge theories on a quantum computer. Phys. Rev. A 73 , 022328 (2006).

Tagliacozzo, L., Celi, A., Orland, P. & Lewenstein, M. Simulations of non-Abelian gauge theories with optical lattices. Nat. Commun. 4 , 2615 (2013).

Zohar, E., Cirac, J. I. & Reznik, B. Cold-atom quantum simulator for SU(2) Yang-Mills lattice gauge theory. Phys. Rev. Lett. 110 , 125304 (2013).

Zohar, E., Cirac, J. I. & Reznik, B. Quantum simulations of gauge theories with ultracold atoms: local gauge invariance from angular momentum conservation. Phys. Rev. A 88 , 023617 (2013).

Stannigel, K. et al. Constrained dynamics via the Zeno effect in quantum simulation: Implementing non-Abelian lattice gauge theories with cold atoms. Phys. Rev. Lett. 112 , 120406 (2014).

Anishetty, R. & Raychowdhury, I. SU(2) lattice gauge theory: Local dynamics on nonintersecting electric flux loops. Phys. Rev. D 90 , 114503 (2014).

Zohar, E. & Burrello, M. Formulation of lattice gauge theories for quantum simulations. Phys. Rev. D 91 , 054506 (2015).

Mezzacapo, A. et al. Non-Abelian S U (2) lattice gauge theories in superconducting circuits. Phys. Rev. Lett. 115 , 240502 (2015).

Silvi, P., Rico, E., Dalmonte, M., Tschirsich, F. & Montangero, S. Finite-density phase diagram of a (1+1)-d non-abelian lattice gauge theory with tensor networks. Quantum 1 , 9 (2017).

Bañuls, M. C., Cichy, K., Cirac, J. I., Jansen, K. & Kühn, S. Efficient basis formulation for 1+1 dimensional SU(2) lattice gauge theory: Spectral calculations with matrix product states. Phys. Rev. X 7 , 041046 (2017).

Google Scholar  

Banerjee, D., Jiang, F. J., Olesen, T. Z., Orland, P. & Wiese, U. J. From the S U (2) quantum link model on the honeycomb lattice to the quantum dimer model on the kagomé lattice: Phase transition and fractionalized flux strings. Phys. Rev. B 97 , 205108 (2018).

Raychowdhury, I. Low energy spectrum of SU(2) lattice gauge theory: an alternate proposal via loop formulation. Eur. Phys. J. C 79 , 235 (2019).

Sala, P. et al. Variational study of U(1) and SU(2) lattice gauge theories with Gaussian states in 1+1 dimensions. Phys. Rev. D 98 , 034505 (2018).

Raychowdhury, I. & Stryker, J. R. Solving Gauss’s law on digital quantum computers with loop-string-hadron digitization. Phys. Rev. Res. 2 , 033039 (2020).

Zohar, E. & Cirac, J. I. Removing staggered fermionic matter in U ( N ) and S U ( N ) lattice gauge theories. Phys. Rev. D 99 , 114511 (2019).

Raychowdhury, I. & Stryker, J. R. Loop, string, and hadron dynamics in SU(2) Hamiltonian lattice gauge theories. Phys. Rev. D 101 , 114502 (2020).

Ji, Y. et al. Gluon field digitization via group space decimation for quantum computers. Phys. Rev. D 102 , 114513 (2020).

Kasper, V., Juzeliunas, G., Lewenstein, M., Jendrzejewski, F. & Zohar, E. From the Jaynes–Cummings model to non-abelian gauge theories: a guided tour for the quantum engineer. New J. Phys. 22 , 103027 (2020).

Davoudi, Z., Raychowdhury, I. & Shaw, A. Search for efficient formulations for Hamiltonian simulation of non-Abelian lattice gauge theories. Phys. Rev. D 104 , 074505 (2021).

Dasgupta, R. & Raychowdhury, I. Cold-atom quantum simulator for string and hadron dynamics in non-Abelian lattice gauge theory. Phys. Rev. A 105 , 023322 (2022).

Kasper, V., Zache, T. V., Jendrzejewski, F., Lewenstein, M. & Zohar, E. Non-Abelian gauge invariance from dynamical decoupling. Phys. Rev. D 107 , 014506 (2023).

Kan, A. et al. Investigating a (3+1)D topological θ -term in the Hamiltonian formulation of lattice gauge theories for quantum and classical simulations. Phys. Rev. D 104 , 034504 (2021).

Zohar, E. Quantum simulation of lattice gauge theories in more than one space dimension—requirements, challenges and methods. Phil. Trans. A. Math. Phys. Eng. Sci. 380 , 20210069 (2021).

Halimeh, J. C., Lang, H. & Hauke, P. Gauge protection in non-Abelian lattice gauge theories. New J. Phys . 24 , 033015 (2022).

Raychowdhury, I. Toward quantum simulating non-Abelian gauge theories. Indian J. Phys. 95 , 1681–1690 (2021).

Klco, N., Roggero, A. & Savage, M. J. Standard model physics and the digital quantum revolution: thoughts about the interface. Rep. Prog. Phys. 85 , 064301 (2022).

Mildenberger, J., Mruczkiewicz, W., Halimeh, J. C., Jiang, Z. & Hauke, P. [arXiv:2203.08905 [quant-ph]].

González-Cuadra, D., Zache, T. V., Carrasco, J., Kraus, B. & Zoller, P. Hardware efficient quantum simulation of non-Abelian gauge theories with qudits on Rydberg Platforms. Phys. Rev. Lett. 129 , 160501 (2022).

Carena, M., Gustafson, E. J., Lamm, H., Li, Y. Y. & Liu, W. Gauge theory couplings on anisotropic lattices. Phys. Rev. D 106 , 114504 (2022).

Davoudi, Z., Shaw, A. F. & Stryker, J. R. General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory. Quantum 7 , 1213 (2023).

Yao, X. SU(2) gauge theory in 2+1 dimensions on a plaquette chain obeys the eigenstate thermalization hypothesis. Phys. Rev. D 108 , L031504 (2023).

Jakobs, T. et al. Canonical momenta in digitized Su(2) lattice gauge theory: definition and free theory. Eur. Phys. J. C 83 , 669 (2023).

Zache, T. V., González-Cuadra, D. & Zoller, P. Quantum and Classical Spin-Network Algorithms for q-Deformed Kogut-Susskind Gauge Theories. Phys. Rev. Lett. 131 , 171902 (2023).

Hayata, T. & Hidaka, Y. String-net formulation of Hamiltonian lattice Yang-Mills theories and quantum many-body scars in a nonabelian gauge theory. JHEP 09 , 126 (2023).

Halimeh, J. C., Homeier, L., Bohrdt, A. & Grusdt, F. [arXiv:2305.06373 [cond-mat.quant-gas]].

Chan, A., Shi, Z., Dellantonio, L., Dür, W. & Muschik, C. A. Hybrid variational quantum eigensolvers: merging computational models. [arXiv:2305.19200 [quant-ph]].

Bauer, C. W., Davoudi, Z., Klco, N. & Savage, M. J. Quantum simulation of fundamental particles and forces. Nat. Rev. Phys. 5 , 420–432 (2023).

Müller, B. & Yao, X. Simple Hamiltonian for quantum simulation of strongly coupled (2+1)D SU(2) lattice gauge theory on a honeycomb lattice. Phys. Rev. D 108 , 094505 (2023).

Cataldi, G., Magnifico, G., Silvi, P. and Montangero, S. [arXiv:2307.09396 [hep-lat]].

Bauer, C. W., D’Andrea, I., Freytsis, M. & Grabowska, D. M. A new basis for Hamiltonian SU(2) simulations. [arXiv:2307.11829 [hep-ph]].

Ebner, L., Müller, B., Schäfer, A., Seidl, C. & Yao, X. Eigenstate thermalization in (2+1)-dimensional SU(2) lattice gauge theory. Phys. Rev. D 109 , 14504.

Gustafson, E. J., Lamm, H. & Lovelace, F. Primitive Quantum Gates for an S U (2) Discrete Subgroup: Binary Octahedral. [arXiv:2312.10285 [hep-lat]].

Ebner, L., Schäfer, A., Seidl, C., Müller, B. and Yao, X. [arXiv:2401.15184 [hep-lat]].

Turro, F., Ciavarella, A. & Yao, X. Classical and quantum computing of shear viscosity for (2+ 1) D SU (2) gauge theory. Phys. Rev. D 109 , 114511 (2024).

Illa, M., Robin, C. E. P. & Savage, M. J. [arXiv:2403.14537 [quant-ph]].

Endo, S., Cai, Z., Benjamin, S. C. & Yuan, X. Hybrid quantum-classical algorithms and quantum error mitigation. J. Phys. Soc. Jap. 90 , 032001 (2021).

Laves, F. Zur Klassifikation der Silikate. Geometrische Untersuchungen möglicher Silicium-Sauerstoff- Verbände als Verknüpfungsmöglichkeiten regulärer Tetraeder. Zeitschrift für Kristallographie, 82 , 1 (1932).

Coxeter, H. S. M. On Laves’ Graph Of Girth Ten. Can. J. Math. 7 , 18 (1955).

Article   MathSciNet   Google Scholar  

Sunada, T. Crystals that nature might miss creating. Not. Am. Math. Soc. 55 , 208 (2008).

MathSciNet   Google Scholar  

Séquin, C. H. Intricate Isohedral Tilings of 3D Euclidean Space. in Bridges Leeuwarden: Mathematics, Music, Art, Architecture, Culture (Tarquin Publications, London, 2008) ISBN 9780966520194. http://archive.bridgesmathart.org/2008/bridges2008-139.html .

Conway, J. H., Burgiel, H., Goodman-Strauss, C. The Symmetries of Things (A. K. Peters Ltd, Wellesley, 2008) ISBN 9781568812205.

Suizu, R. & Awaga, K. Line graph theory reveals hidden spin frustration and bond frustration in molecular crystals with strong isotropy. J. Mater. Chem, 10 , 1196 (2022).

IBM Quantum, https://quantum-computing.ibm.com .

Chai, Y. et al. Entanglement production from scattering of fermionic wave packets: a quantum computing approach. [arXiv:2312.02272 [quant-ph]].

Farrell, R. C., Illa, M., Ciavarella, A. N. & Savage, M. J. Quantum Simulations of hadron dynamics in the Schwinger model using 112 Qubits. Phys. Rev. D 109 , 114510 (2024).

Davoudi, Z., Hsieh, C. C. & Kadam, S. V. [arXiv:2402.00840 [quant-ph]].

Davoudi, Z., Mueller, N. & Powers, C. Towards quantum computing phase diagrams of gauge theories with thermal pure quantum states. Phys. Rev. Lett . 131 , 081901 (2023).

Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nat. Phys. 16 , 205–210 (2019).

Urbanek, M. et al. Mitigating depolarizing noise on quantum computers with noise-estimation circuits. Phys. Rev. Lett. 127 , 270502 (2021).

Charles, C. et al. Simulating \({{\mathbb{Z}}}_{2}\) lattice gauge theory on a quantum computer. [arXiv:2305.02361 [hep-lat]].

Farrell, R. C., Illa, M., Ciavarella, A. N. & Savage, M. J. Scalable circuits for preparing ground states on digital quantum computers: the Schwinger model vacuum on 100 qubits. PRX Quantum . 5 , 020315 (2024).

Asaduzzaman, M., Jha, R. G. & Sambasivam, B. Sachdev-Ye-Kitaev model on a noisy quantum computer. Phys. Rev. D 109 , 105002 (2024).

Hidalgo, L. & Draper, P. Quantum simulations for strong-field QED. Phys. Rev. D 109 , 076004 (2024).

Kiss, O., Grossi, M. & Roggero, A. Quantum error mitigation for Fourier moment computation. [arXiv:2401.13048 [quant-ph]].

Anis, M. S. et al. Qiskit: an open-source framework for quantum computing. https://doi.org/10.5281/zenodo.2573505 (2021).

Moore, E. H. On the reciprocal of the general algebraic matrix. Bull. Am. Math. Soc. 26 , 394–395 (1920).

Penrose, R. A generalized inverse for matrices. Proc. Cambridge Phil. Soc. 51 , 406–413 (1955).

Download references

Acknowledgements

The authors thank R. M. Woloshyn for suggesting that self-mitigation be applied to QITE, W. G. Parrott for discussions about quantum imaginary time evolution, and M. Tsang for discussions about triamond crystals. We thank all three for reading a draft of the manuscript. We also acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Quantum team. This work is affiliated with the QC4HEP Working Group and supported by Canada’s Natural Sciences and Engineering Research Council.

Author information

Authors and affiliations.

Department of Physics and Astronomy, York University, Toronto, ON, M3J 1P3, Canada

Ali H. Z. Kavaki & Randy Lewis

You can also search for this author in PubMed   Google Scholar

Contributions

A.H.Z.K. and R.L. contributed to the analytical derivations, numerical computations, and creation of the manuscript.

Corresponding authors

Correspondence to Ali H. Z. Kavaki or Randy Lewis .

Ethics declarations

Competing interests.

The authors declare no competing interests.

Peer review

Peer review information.

Communications Physics thanks Anthony Ciavarella and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ .

Reprints and permissions

About this article

Cite this article.

Kavaki, A.H.Z., Lewis, R. From square plaquettes to triamond lattices for SU(2) gauge theory. Commun Phys 7 , 208 (2024). https://doi.org/10.1038/s42005-024-01697-4

Download citation

Received : 31 January 2024

Accepted : 17 June 2024

Published : 27 June 2024

DOI : https://doi.org/10.1038/s42005-024-01697-4

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

By submitting a comment you agree to abide by our Terms and Community Guidelines . If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Quick links

• Explore articles by subject
• Guide to authors
• Editorial policies

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Subscribe or renew today

Every print subscription comes with full digital access

Science News

Can time travel survive a theory of everything.

It’s complicated. But studying ways to visit the past could help us understand the cosmos

By Tom Siegfried

Contributing Correspondent

September 20, 2019 at 8:00 am

The TARDIS and other fictional time machines make time travel look relatively easy (Peter Capaldi and Jenna Coleman are shown promoting the BBC’s Doctor Who ). But just because it’s permitted by the laws of physics, doesn’t mean it’s practical.

Twocoms/Shutterstock

Share this:

In many universes, typically those on TV shows or in movies, time travel is not much more difficult than driving downtown in any major city during rush hour. Sure, the traffic can get gnarly, but no law of physics prevents you from reaching your destination eventually.

In real life, time travel isn’t so easy. In fact, it’s probably impossible, a fantasy more farfetched than visiting Alice’s Wonderland, finding gold at the end of a rainbow or cleansing all the hate speech off of Facebook.

Yet time travel does not necessarily violate the laws of physics. In Einstein’s theory of gravity — general relativity — space and time are merged as spacetime, which allows for the possibility of pathways that could bend back to the past and loop back to the future.

Such paths are known as closed timelike curves. They’re a little like great circles around the surface of the Earth — if you start out in one direction and keep going straight, eventually you come back to where you started from. In that case the Earth’s curvature guides you back to your previous point in space; with closed timelike curves, the geometry of spacetime guides you back to an earlier moment in time.

Nobody thinks that general relativity’s time loops would be practical for time travel even if they are possible. For one thing, they might exist only under certain circumstances — the universe would have to be rotating, and not expanding — as the mathematician Kurt Gödel showed in the 1940s. But the universe is expanding, and probably isn’t rotating, so that dampens the prospects for revisiting the Stone Age or acquiring a pet dinosaur.

Besides, even if such pathways did exist, building a ship to traverse them would cost more than all the DeLoreans (and all other transportation vehicles) ever made. It would need a cruising speed of 140,000 miles per second. And with no place to stop for gas (or whatever), the fuel tank would have to be more than a trillion times the size of an oil tanker.

So for practical purposes, time travel’s time has not yet arrived. But even if it’s possible only in principle, the potential ramifications for the basic physics of the universe might make it worth the time to investigate it. Time loops might not enable you to traverse the cosmos in a TARDIS, but perhaps could still help you understand the cosmos more deeply.

A first step would be to attempt to figure out exactly what the relevant laws of physics really are. Einstein’s general relativity is great, but indubitably not the last word about the physics of the universe. After all, it coexists uneasily with quantum mechanics, which rules the subatomic world and presumably, since everything is made of subatomic stuff, the rest of the universe as well. Whether the quantum–general relativity combo truly permits time travel might depend on what the ultimate correct theory combining the two turns out to be.

Several candidate theories have been developed for merging general relativity and quantum mechanics into a unified theory. It’s an open question whether these candidates would allow time travel in something like the way general relativity does, philosopher Christian Wüthrich of the University of Geneva notes in a new paper.

It’s possible, he says, that a theory that supersedes general relativity might still in some way include the equivalent of general relativity’s timelike loops. And even if the basic theory does not include such loops, they still might emerge in practice.

“Although the fundamental theory would then remain inhospitable to time travel itself, it would tolerate the possibility of time travel at some other, less fundamental, scale,” Wüthrich writes in his paper , posted online in June. “Depending on what the relationship between the fundamental theory and emergent spacetime may be in each case, we may find that the emergent, macroscopic spacetime structure permits time travel.”

Reviewing the major proposals for quantum gravity theories does not provide a lot of hope, though. One approach, known as causal set theory, requires sets of events to be ordered in a proper cause-and-effect relationship. So its central idea seems to rule out closed timelike curves.

Another popular approach, known as loop quantum gravity, envisions space to be constructed of fundamental loops (kind of like “atoms of space”). This view has encountered technical difficulties, one of which is how to work time into the picture with space. “Thus, we seem to be faced with a temporally innocuous structure in which no meaningful sense of time travel is permitted,” Wüthrich writes.

It’s possible that the networks of these “atoms of space” could produce high-level spacetime that did contain closed timelike curves. But analysis of the details at this stage of loop quantum gravity’s development does not offer much reason for optimism, Wüthrich concludes.

Time travel’s future might look a little brighter if the correct approach to quantum gravity turns out to be string theory, currently the most popular contender. In string theory, matter’s basic particles are tiny vibrating snippets of energy, called “strings” because they extend in one dimension. Multiple versions of string theory have been constructed, suggesting that they are different manifestations of a more fundamental master theory known as M-theory.

“As M-theory does not yet exist, it is impossible to determine its verdict on time travel,” writes Wüthrich. But investigations of various string theory scenarios do suggest that the ultimate theory would, in fact, naturally incorporate closed timelike curves.

Even if time loops exist in the fundamental theory, though, there’s still no guarantee that they would be preserved in the emergent large-scale spacetime that would be relevant in real life. For that matter, Wüthrich points out, predicting the existence of time travel loops might be taken as evidence against the theory, considering the serious likelihood that time travel really isn’t possible at all.

So whether general relativity’s time loops will survive in a deeper theory remains an open question. “A more fundamental theory may well admit structures amounting to closed timelike curves and thus permit time travel,” Wüthrich asserts. “This clearly remains a live option at the present stage of knowledge.”

In any case, investigating whether quantum gravity theories retain general relativity’s time travel loophole can illuminate many tough questions that must be answered to develop a successful theory and understand how it relates to general relativity. “For this reason alone,” Wüthrich writes, “the question of time travel beyond general relativity is worth our while.”

More Stories from Science News on Physics

Something weird is happening to Earth’s inner core

A black hole made from pure light is impossible, thanks to quantum physics 

Physicists measured Earth’s rotation using quantum entanglement

The second law of thermodynamics underlies nearly everything. But is it inviolable?

Scientists propose a hunt for never-before-seen ‘tauonium’ atoms 

Two real-world tests of quantum memories bring a quantum internet closer to reality

Here’s how ice may get so slippery 

The neutrino’s quantum fuzziness is beginning to come into focus

Subscribers, enter your e-mail address for full access to the Science News archives and digital editions.

Not a subscriber? Become one now .

Time travel: Is it possible?

Science says time travel is possible, but probably not in the way you're thinking.

Albert Einstein's theory

• General relativity and GPS
• Wormhole travel
• Alternate theories

Science fiction

Is time travel possible? Short answer: Yes, and you're doing it right now — hurtling into the future at the impressive rate of one second per second. 

You're pretty much always moving through time at the same speed, whether you're watching paint dry or wishing you had more hours to visit with a friend from out of town. 

But this isn't the kind of time travel that's captivated countless science fiction writers, or spurred a genre so extensive that Wikipedia lists over 400 titles in the category "Movies about Time Travel." In franchises like " Doctor Who ," " Star Trek ," and "Back to the Future" characters climb into some wild vehicle to blast into the past or spin into the future. Once the characters have traveled through time, they grapple with what happens if you change the past or present based on information from the future (which is where time travel stories intersect with the idea of parallel universes or alternate timelines). 

Related: The best sci-fi time machines ever

Although many people are fascinated by the idea of changing the past or seeing the future before it's due, no person has ever demonstrated the kind of back-and-forth time travel seen in science fiction or proposed a method of sending a person through significant periods of time that wouldn't destroy them on the way. And, as physicist Stephen Hawking pointed out in his book " Black Holes and Baby Universes" (Bantam, 1994), "The best evidence we have that time travel is not possible, and never will be, is that we have not been invaded by hordes of tourists from the future."

Science does support some amount of time-bending, though. For example, physicist Albert Einstein 's theory of special relativity proposes that time is an illusion that moves relative to an observer. An observer traveling near the speed of light will experience time, with all its aftereffects (boredom, aging, etc.) much more slowly than an observer at rest. That's why astronaut Scott Kelly aged ever so slightly less over the course of a year in orbit than his twin brother who stayed here on Earth. 

Related: Controversially, physicist argues that time is real

There are other scientific theories about time travel, including some weird physics that arise around wormholes , black holes and string theory . For the most part, though, time travel remains the domain of an ever-growing array of science fiction books, movies, television shows, comics, video games and more. 

Einstein developed his theory of special relativity in 1905. Along with his later expansion, the theory of general relativity , it has become one of the foundational tenets of modern physics. Special relativity describes the relationship between space and time for objects moving at constant speeds in a straight line. 

The short version of the theory is deceptively simple. First, all things are measured in relation to something else — that is to say, there is no "absolute" frame of reference. Second, the speed of light is constant. It stays the same no matter what, and no matter where it's measured from. And third, nothing can go faster than the speed of light.

From those simple tenets unfolds actual, real-life time travel. An observer traveling at high velocity will experience time at a slower rate than an observer who isn't speeding through space. 

While we don't accelerate humans to near-light-speed, we do send them swinging around the planet at 17,500 mph (28,160 km/h) aboard the International Space Station . Astronaut Scott Kelly was born after his twin brother, and fellow astronaut, Mark Kelly . Scott Kelly spent 520 days in orbit, while Mark logged 54 days in space. The difference in the speed at which they experienced time over the course of their lifetimes has actually widened the age gap between the two men.

"So, where[as] I used to be just 6 minutes older, now I am 6 minutes and 5 milliseconds older," Mark Kelly said in a panel discussion on July 12, 2020, Space.com previously reported . "Now I've got that over his head."

General relativity and GPS time travel

The difference that low earth orbit makes in an astronaut's life span may be negligible — better suited for jokes among siblings than actual life extension or visiting the distant future — but the dilation in time between people on Earth and GPS satellites flying through space does make a difference. 

Read more: Can we stop time?

The Global Positioning System , or GPS, helps us know exactly where we are by communicating with a network of a few dozen satellites positioned in a high Earth orbit. The satellites circle the planet from 12,500 miles (20,100 kilometers) away, moving at 8,700 mph (14,000 km/h). 

According to special relativity, the faster an object moves relative to another object, the slower that first object experiences time. For GPS satellites with atomic clocks, this effect cuts 7 microseconds, or 7 millionths of a second, off each day, according to the American Physical Society publication Physics Central .  

Read more: Could Star Trek's faster-than-light warp drive actually work?

Then, according to general relativity, clocks closer to the center of a large gravitational mass like Earth tick more slowly than those farther away. So, because the GPS satellites are much farther from the center of Earth compared to clocks on the surface, Physics Central added, that adds another 45 microseconds onto the GPS satellite clocks each day. Combined with the negative 7 microseconds from the special relativity calculation, the net result is an added 38 microseconds. 

This means that in order to maintain the accuracy needed to pinpoint your car or phone — or, since the system is run by the U.S. Department of Defense, a military drone — engineers must account for an extra 38 microseconds in each satellite's day. The atomic clocks onboard don’t tick over to the next day until they have run 38 microseconds longer than comparable clocks on Earth.

Given those numbers, it would take more than seven years for the atomic clock in a GPS satellite to un-sync itself from an Earth clock by more than a blink of an eye. (We did the math: If you estimate a blink to last at least 100,000 microseconds, as the Harvard Database of Useful Biological Numbers does, it would take thousands of days for those 38 microsecond shifts to add up.) 

This kind of time travel may seem as negligible as the Kelly brothers' age gap, but given the hyper-accuracy of modern GPS technology, it actually does matter. If it can communicate with the satellites whizzing overhead, your phone can nail down your location in space and time with incredible accuracy. 

Can wormholes take us back in time?

General relativity might also provide scenarios that could allow travelers to go back in time, according to NASA . But the physical reality of those time-travel methods is no piece of cake. 

Wormholes are theoretical "tunnels" through the fabric of space-time that could connect different moments or locations in reality to others. Also known as Einstein-Rosen bridges or white holes, as opposed to black holes, speculation about wormholes abounds. But despite taking up a lot of space (or space-time) in science fiction, no wormholes of any kind have been identified in real life. 

Related: Best time travel movies

"The whole thing is very hypothetical at this point," Stephen Hsu, a professor of theoretical physics at the University of Oregon, told Space.com sister site Live Science . "No one thinks we're going to find a wormhole anytime soon."

Primordial wormholes are predicted to be just 10^-34 inches (10^-33 centimeters) at the tunnel's "mouth". Previously, they were expected to be too unstable for anything to be able to travel through them. However, a study claims that this is not the case, Live Science reported . 

The theory, which suggests that wormholes could work as viable space-time shortcuts, was described by physicist Pascal Koiran. As part of the study, Koiran used the Eddington-Finkelstein metric, as opposed to the Schwarzschild metric which has been used in the majority of previous analyses.

In the past, the path of a particle could not be traced through a hypothetical wormhole. However, using the Eddington-Finkelstein metric, the physicist was able to achieve just that.

Koiran's paper was described in October 2021, in the preprint database arXiv , before being published in the Journal of Modern Physics D.

Alternate time travel theories

While Einstein's theories appear to make time travel difficult, some researchers have proposed other solutions that could allow jumps back and forth in time. These alternate theories share one major flaw: As far as scientists can tell, there's no way a person could survive the kind of gravitational pulling and pushing that each solution requires.

Infinite cylinder theory

Astronomer Frank Tipler proposed a mechanism (sometimes known as a Tipler Cylinder ) where one could take matter that is 10 times the sun's mass, then roll it into a very long, but very dense cylinder. The Anderson Institute , a time travel research organization, described the cylinder as "a black hole that has passed through a spaghetti factory."

After spinning this black hole spaghetti a few billion revolutions per minute, a spaceship nearby — following a very precise spiral around the cylinder — could travel backward in time on a "closed, time-like curve," according to the Anderson Institute. 

The major problem is that in order for the Tipler Cylinder to become reality, the cylinder would need to be infinitely long or be made of some unknown kind of matter. At least for the foreseeable future, endless interstellar pasta is beyond our reach.

Time donuts

Theoretical physicist Amos Ori at the Technion-Israel Institute of Technology in Haifa, Israel, proposed a model for a time machine made out of curved space-time — a donut-shaped vacuum surrounded by a sphere of normal matter.

"The machine is space-time itself," Ori told Live Science . "If we were to create an area with a warp like this in space that would enable time lines to close on themselves, it might enable future generations to return to visit our time."

Amos Ori is a theoretical physicist at the Technion-Israel Institute of Technology in Haifa, Israel. His research interests and publications span the fields of general relativity, black holes, gravitational waves and closed time lines.

There are a few caveats to Ori's time machine. First, visitors to the past wouldn't be able to travel to times earlier than the invention and construction of the time donut. Second, and more importantly, the invention and construction of this machine would depend on our ability to manipulate gravitational fields at will — a feat that may be theoretically possible but is certainly beyond our immediate reach.

Time travel has long occupied a significant place in fiction. Since as early as the "Mahabharata," an ancient Sanskrit epic poem compiled around 400 B.C., humans have dreamed of warping time, Lisa Yaszek, a professor of science fiction studies at the Georgia Institute of Technology in Atlanta, told Live Science .  

Every work of time-travel fiction creates its own version of space-time, glossing over one or more scientific hurdles and paradoxes to achieve its plot requirements. 

Some make a nod to research and physics, like " Interstellar ," a 2014 film directed by Christopher Nolan. In the movie, a character played by Matthew McConaughey spends a few hours on a planet orbiting a supermassive black hole, but because of time dilation, observers on Earth experience those hours as a matter of decades. 

Others take a more whimsical approach, like the "Doctor Who" television series. The series features the Doctor, an extraterrestrial "Time Lord" who travels in a spaceship resembling a blue British police box. "People assume," the Doctor explained in the show, "that time is a strict progression from cause to effect, but actually from a non-linear, non-subjective viewpoint, it's more like a big ball of wibbly-wobbly, timey-wimey stuff." 

Long-standing franchises like the "Star Trek" movies and television series, as well as comic universes like DC and Marvel Comics, revisit the idea of time travel over and over. 

Related: Marvel movies in order: chronological & release order

Here is an incomplete (and deeply subjective) list of some influential or notable works of time travel fiction:

Books about time travel:

• Rip Van Winkle (Cornelius S. Van Winkle, 1819) by Washington Irving
• A Christmas Carol (Chapman & Hall, 1843) by Charles Dickens
• The Time Machine (William Heinemann, 1895) by H. G. Wells
• A Connecticut Yankee in King Arthur's Court (Charles L. Webster and Co., 1889) by Mark Twain
• The Restaurant at the End of the Universe (Pan Books, 1980) by Douglas Adams
• A Tale of Time City (Methuen, 1987) by Diana Wynn Jones
• The Outlander series (Delacorte Press, 1991-present) by Diana Gabaldon
• Harry Potter and the Prisoner of Azkaban (Bloomsbury/Scholastic, 1999) by J. K. Rowling
• Thief of Time (Doubleday, 2001) by Terry Pratchett
• The Time Traveler's Wife (MacAdam/Cage, 2003) by Audrey Niffenegger
• All You Need is Kill (Shueisha, 2004) by Hiroshi Sakurazaka

Movies about time travel:

• Planet of the Apes (1968)
• Superman (1978)
• Time Bandits (1981)
• The Terminator (1984)
• Back to the Future series (1985, 1989, 1990)
• Star Trek IV: The Voyage Home (1986)
• Bill & Ted's Excellent Adventure (1989)
• Groundhog Day (1993)
• Galaxy Quest (1999)
• The Butterfly Effect (2004)
• 13 Going on 30 (2004)
• The Lake House (2006)
• Meet the Robinsons (2007)
• Hot Tub Time Machine (2010)
• Midnight in Paris (2011)
• Looper (2012)
• X-Men: Days of Future Past (2014)
• Edge of Tomorrow (2014)
• Interstellar (2014)
• Doctor Strange (2016)
• A Wrinkle in Time (2018)
• The Last Sharknado: It's About Time (2018)
• Avengers: Endgame (2019)
• Tenet (2020)
• Palm Springs (2020)
• Zach Snyder's Justice League (2021)
• The Tomorrow War (2021)

Television about time travel:

• Doctor Who (1963-present)
• The Twilight Zone (1959-1964) (multiple episodes)
• Star Trek (multiple series, multiple episodes)
• Samurai Jack (2001-2004)
• Lost (2004-2010)
• Phil of the Future (2004-2006)
• Steins;Gate (2011)
• Outlander (2014-2023)
• Loki (2021-present)

Games about time travel:

• Chrono Trigger (1995)
• TimeSplitters (2000-2005)
• Kingdom Hearts (2002-2019)
• Prince of Persia: Sands of Time (2003)
• God of War II (2007)
• Ratchet and Clank Future: A Crack In Time (2009)
• Sly Cooper: Thieves in Time (2013)
• Dishonored 2 (2016)
• Titanfall 2 (2016)
• Outer Wilds (2019)

Additional resources

Explore physicist Peter Millington's thoughts about Stephen Hawking's time travel theories at The Conversation . Check out a kid-friendly explanation of real-world time travel from NASA's Space Place . For an overview of time travel in fiction and the collective consciousness, read " Time Travel: A History " (Pantheon, 2016) by James Gleik. 

Join our Space Forums to keep talking space on the latest missions, night sky and more! And if you have a news tip, correction or comment, let us know at: [email protected].

Get the Space.com Newsletter

Breaking space news, the latest updates on rocket launches, skywatching events and more!

Vicky Stein is a science writer based in California. She has a bachelor's degree in ecology and evolutionary biology from Dartmouth College and a graduate certificate in science writing from the University of California, Santa Cruz (2018). Afterwards, she worked as a news assistant for PBS NewsHour, and now works as a freelancer covering anything from asteroids to zebras. Follow her most recent work (and most recent pictures of nudibranchs) on Twitter. 

Is there liquid water on Mars today? Marsquake data could tell us

Ed Stone, who led NASA's iconic Voyager project for 50 years, dies at 88

SpaceX unveils new Starlink Mini antenna for internet users on the go

Most Popular

• 2 Why is mystery object Cygnus X-3 so bright? Astronomers may now have the answer
• 3 Powerful GOES-U weather satellite launches to orbit atop SpaceX Falcon Heavy rocket (video)
• 4 ISS astronauts conduct 'spacewalk review' after spacesuit coolant leak
• 5 Europe's new Ariane 6 rocket on track for long-awaited 1st launch on July 9

Precision instrument bolsters efforts to find elusive dark energy

Experiment captures atoms in free fall to look for gravitational anomalies caused by universe's missing energy.

Dark energy -- a mysterious force pushing the universe apart at an ever-increasing rate -- was discovered 26 years ago, and ever since, scientists have been searching for a new and exotic particle causing the expansion.

Pushing the boundaries of this search, University of California, Berkeley physicists have now built the most precise experiment yet to look for minor deviations from the accepted theory of gravity that could be evidence for such a particle, which theorists have dubbed a chameleon or symmetron.

The experiment, which combines an atom interferometer for precise gravity measurements with an optical lattice to hold the atoms in place, allowed the researchers to immobilize free-falling atoms for seconds instead of milliseconds to look for gravitational effects, besting the current most precise measurement by a factor of five.

Though the researchers found no deviation from what is predicted by the theory spelled out by Isaac Newton 400 years ago, expected improvements in the precision of the experiment could eventually turn up evidence that supports or disproves theories of a hypothetical fifth force mediated by chameleons or symmetrons.

The ability of the lattice atom interferometer to hold atoms for up to 70 seconds -- and potentially 10 times longer -- also opens up the possibility of probing gravity at the quantum level, said Holger Müller, UC Berkeley professor of physics. While physicists have well-tested theories describing the quantum nature of three of the four forces of nature -- electromagnetism and the strong and weak forces -- the quantum nature of gravity has never been demonstrated.

"Most theorists probably agree that gravity is quantum. But nobody has ever seen an experimental signature of that," Müller said. "It's very hard to even know whether gravity is quantum, but if we could hold our atoms 20 or 30 times longer than anyone else, because our sensitivity increases with the second or fourth power of the hold time, we could have a 400 to 800,000 times better chance of finding experimental proof that gravity is indeed quantum mechanical."

Aside from precision measurements of gravity, other applications of the lattice atom interferometer include quantum sensing.

"Atom interferometry is particularly sensitive to gravity or inertial effects. You can build gyroscopes and accelerometers," said UC Berkeley postdoctoral fellow Cristian Panda, who is first author of a paper about the gravity measurements set to be published this week in the journal Nature and is co-authored by Müller. "But this gives a new direction in atom interferometry, where quantum sensing of gravity, acceleration and rotation could be done with atoms held in optical lattices in a compact package that is resilient to environmental imperfections or noise."

Because the optical lattice holds atoms rigidly in place, the lattice atom interferometer could even operate at sea, where sensitive gravity measurements are employed to map the geology of the ocean floor.

Screened forces can hide in plain sight

Dark energy was discovered in 1998 by two teams of scientists: a group of physicists based at Lawrence Berkeley National Laboratory, led by Saul Perlmutter, now a UC Berkeley professor of physics, and a group of astronomers that included UC Berkeley postdoctoral fellow Adam Riess. The two shared the 2011 Nobel Prize in Physics for the discovery.

The realization that the universe was expanding more rapidly than it should came from tracking distant supernovas and using them to measure cosmic distances. Despite much speculation by theorists about what's actually pushing space apart, dark energy remains an enigma -- a large enigma, since about 70% of the entire matter and energy of the universe is in the form of dark energy.

One theory is that dark energy is merely the vacuum energy of space. Another is that it is an energy field called quintessence, which varies over time and space.

Another proposal is that dark energy is a fifth force much weaker than gravity and mediated by a particle that exerts a repulsive force that varies with the density of surrounding matter. In the emptiness of space, it would exert a repulsive force over long distances, able to push space apart. In a laboratory on Earth, with matter all around to shield it, the particle would have an extremely small reach.

This particle has been dubbed a chameleon, as if it's hiding in plain sight.

In 2015, Müller adapted an atom interferometer to search for evidence of chameleons using cesium atoms launched into a vacuum chamber, which mimics the emptiness of space. During the 10 to 20 milliseconds it took the atoms to rise and fall above a heavy aluminum sphere, he and his team detected no deviation from what would be expected from the normal gravitational attraction of the sphere and Earth.

The key to using free-falling atoms to test gravity is the ability to excite each atom into a quantum superposition of two states, each with a slightly different momentum that carries them different distances from a heavy tungsten weight hanging overhead. The higher momentum, higher elevation state experiences more gravitational attraction to the tungsten, changing its phase. When the atom's wave function collapses, the phase difference between the two parts of the matter wave reveals the difference in gravitational attraction between them.

"Atom interferometry is the art and science of using the quantum properties of a particle, that is, the fact that it's both a particle and a wave. We split the wave up so that the particle is taking two paths at the same time and then interfere them at the end," Müller said. "The waves can either be in phase and add up, or the waves can be out of phase and cancel each other out. The trick is that whether they are in phase or out of phase depends very sensitively on some quantities that you might want to measure, such as acceleration, gravity, rotation or fundamental constants."

In 2019, Müller and his colleagues added an optical lattice to keep the atoms close to the tungsten weight for a much longer time -- an astounding 20 seconds -- to increase the effect of gravity on the phase. The optical lattice employs two crossed laser beams that create a lattice-like array of stable places for atoms to congregate, levitating in the vacuum. But was 20 seconds the limit, he wondered?

During the height of the COVID-19 pandemic, Panda worked tirelessly to extend the hold time, systematically fixing a list of 40 possible roadblocks until establishing that the wiggling tilt of the laser beam, caused by vibrations, was a major limitation. By stabilizing the beam within a resonant chamber and tweaking the temperature to be a bit colder -- in this case less than a millionth of a Kelvin above absolute zero, or a billion times colder than room temperature -- he was able to extend the hold time to 70 seconds.

He and Müller published those results in the June 11, 2024, issue of Nature Physics .

Gravitational entanglement

In the newly reported gravity experiment, Panda and Müller traded a shorter time, 2 seconds, for a greater separation of the wave packets to several microns, or several thousandths of a millimeter. There are about 10,000 cesium atoms in the vacuum chamber for each experiment -- too sparsely distributed to interact with one another -- dispersed by the optical lattice into clouds of about 10 atoms each.

"Gravity is trying to push them down with a force a billion times stronger than their attraction to the tungsten mass, but you have the restoring force from the optical lattice that's holding them, kind of like a shelf," Panda said. "We then take each atom and split it into two wave packets, so now it's in a superposition of two heights. And then we take each one of those two wave packets and load them in a separate lattice site, a separate shelf, so it looks like a cupboard. When we turn off the lattice, the wave packets recombine, and all the quantum information that was acquired during the hold can be read out."

Panda plans to build his own lattice atom interferometer at the University of Arizona, where he was just appointed an assistant professor of physics. He hopes to use it to, among other things, more precisely measure the gravitational constant that links the force of gravity with mass.

Meanwhile, Müller and his team are building from scratch a new lattice atom interferometer with better vibration control and a lower temperature. The new device could produce results that are 100 times better than the current experiment, sensitive enough to detect the quantum properties of gravity. The planned experiment to detect gravitational entanglement, if successful, would be akin to the first demonstration of quantum entanglement of photons performed at UC Berkeley in 1972 by the late Stuart Freedman and former postdoctoral fellow John Clauser. Clauser shared the 2022 Nobel Prize in Physics for that work.

Other co-authors of the gravity paper are graduate student Matthew Tao and former undergraduate student Miguel Ceja of UC Berkeley, Justin Khoury of the University of Pennsylvania in Philadelphia and Guglielmo Tino of the University of Florence in Italy. The work is supported by the National Science Foundation (1708160, 2208029), Office of Naval Research (N00014-20-1-2656) and Jet Propulsion Laboratory (1659506, 1669913).

• Black Holes
• Dark Matter
• Astrophysics
• Space Telescopes
• Quantum Physics
• Albert Einstein
• Dark energy
• Ultimate fate of the universe
• Particle physics
• Dark matter
• Subatomic particle
• Ionizing radiation
• Gravitation

Story Source:

Materials provided by University of California - Berkeley . Original written by Robert Sanders. Note: Content may be edited for style and length.

Journal Reference :

• Cristian D. Panda, Matthew J. Tao, Miguel Ceja, Justin Khoury, Guglielmo M. Tino, Holger Müller. Measuring gravitational attraction with a lattice atom interferometer . Nature , 2024; DOI: 10.1038/s41586-024-07561-3

Cite This Page :

Explore More

• Bird Flu Stable On Milking Equipment One Hour
• How Comb Jellies Have Adapted to Deep Sea
• Micro-Robot-Packed Pill Treats IBD
• Kirigami-Inspired Mechanical Computer
• The Density Difference of Sub-Neptunes
• Antarctic Ice Shelves Hold Twice the Meltwater
• Last Surviving Woolly Mammoths Slightly Inbred
• Common Plastics Passively Cool, Heat, Buildings
• Why U.S. Has So Many Tornadoes
• Wolves Reintroduced to Isle Royale

Trending Topics

Strange & offbeat.

• Collections

• APS Journals

Measuring Qubits with “Time Travel” Protocol

Quantum sensing can outperform classical sensing by placing the sensor in an initial state that optimally measures the target. However, choosing this optimal state requires having some preknowledge, such as knowing the orientation of a magnetic field in order to measure its strength. A new experiment overcomes this limitation using two entangled quantum bits (qubits), which are manipulated in a way that is equivalent to a qubit traveling back in time [ 1 ]. Through this “time travel,” the qubits can be placed in an optimal state without any preknowledge.

“Our work addresses a specific kind of problem that plagues many sensing setups: you have to know which direction to point the sensor,” explains Kater Murch from Washington University in St. Louis. When measuring a magnetic field with a spin qubit, for example, the spin’s rotation will return information about the field strength only if you point it in the optimal direction. Point it in a nonoptimal direction and you’ll get zero information about the field, wasting the measurement.

Murch and his colleagues have devised a protocol in which the probe qubit is entangled with a second qubit, called the ancilla. Following previous work, they show that the entanglement is mathematically equivalent to the ancilla traveling back in time to place the probe in an optimal state [ 2 ]. They further show that measuring the ancilla and the probe in a particular sequence can recover information about the field strength in all cases—so no measurement data are wasted as they can be in other protocols. The researchers foresee using this entanglement scheme in situations where a field—or another observable—is changing over time.

–Michael Schirber

Michael Schirber is a Corresponding Editor for  Physics Magazine based in Lyon, France.

• X. Song et al. , “Agnostic phase estimation,” Phys. Rev. Lett. 132 , 260801 (2024) .
• D. R.  M. Arvidsson-Shukur et al. , “Nonclassical advantage in metrology established via quantum simulations of hypothetical closed timelike curves,” Phys. Rev. Lett. 131 (2023) .

Agnostic Phase Estimation

Xingrui Song, Flavio Salvati, Chandrashekhar Gaikwad, Nicole Yunger Halpern, David R. M. Arvidsson-Shukur, and Kater Murch

Phys. Rev. Lett. 132 , 260801 (2024)

Published June 27, 2024

Subject Areas

Related articles.

Mechanical Coupling to Spin Qubits

A vibrating nanobeam could be used to share information between distant solid-state spin qubits, potentially allowing use of these qubits in complex computations. Read More »

A Simple Electronic Circuit Manifests a Complex Physical Effect

Using a single set of measurements of an electronic circuit, researchers have characterized the properties of the topologically protected edge states of a quantum Hall system. Read More »

Informing Potential Remedies for Quasiparticle Poisoning

Measurements of the temperature distribution of quasiparticles in superconducting circuits reveal behavior that could inform strategies for mitigating quasiparticle-induced errors in superconducting qubits. Read More »

Sign up to receive weekly email alerts from Physics Magazine .

Accessibility Links

• Skip to content
• Skip to search IOPscience
• Skip to Journals list
• Accessibility help
• Accessibility Help

Click here to close this panel.

The physics of time travel

Jonathan Z Simon

Published under licence by IOP Publishing Ltd Physics World , Volume 7 , Number 12 Citation Jonathan Z Simon 1994 Phys. World 7 (12) 27 DOI 10.1088/2058-7058/7/12/27

Article metrics

160 Total downloads

Permissions

Get permission to re-use this article

Share this article

Time travel has traditionally been the domain of science fiction, not physics. Fortunately, however, at least within Einstein's theories of relativity, discussions of time travel are open to physicists as well. Special relativity unifies the concepts of time and space. General relativity goes beyond unification and allows time and space to warp together in the presence of matter. General relativity even permits sufficient warping to allow "closed time-like curves". These seemingly perverse trajectories describe paths through space-time that always move forward in local time (i.e. an observer's watch always runs forward), but eventually end up back where and when they started. A space-time that contains closed time-like curves, localized in one region, can be said to have a "time machine".

Export citation and abstract BibTeX RIS

Advertisement

Maxwell’s demon charges quantum batteries inside of a quantum computer

A technique to charge a battery inside a quantum computer relies on sorting qubits in an imitation of Maxwell’s demon, a 19th-century thought experiment once thought to break the laws of physics

By Karmela Padavic-Callaghan

26 June 2024

Splitting qubits inside a quantum computer into high and low-energy groups can charge a battery

Shutterstock / Pavel Chukhov

A 19th-century thought experiment , considered for decades to break the laws of thermodynamics, has been brought to life inside a quantum computer and used to charge a quantum battery.

Physicist James Clerk Maxwell imagined his demon in 1867 while thinking about how to cheat the laws of thermodynamics. He considered two boxes of gas separated by a weightless door and a tiny demon that controls which particles can go through it. The demon uses this control to make one box hotter and the other cooler, contradicting the thermodynamic edict that heat must flow from the hotter to the colder box until they eventually even out.

Quantum experiments add weight to a fringe theory of consciousness

Later, physicists realised that the demon could not break thermodynamic laws “for free” because it would spend energy during its particle selection process, but the idea remained of interest because it can naturally occur in biology and has uses in chemistry .

“The exploration of Maxwell’s demon in a quantum setting forces us to think deeply about what’s behind the fundamental laws of quantum information, thermodynamics and especially their combination – quantum thermodynamics,” says Bill Munro at the Okinawa Institute of Science and Technology in Japan.

He and his colleagues used a quantum computer comprising 62 quantum bits, or qubits, made from superconducting circuits to explore such “demonic effects” – more qubits than have ever been used to implement Maxwell’s demon before.

Sign up to our Lost in Space-Time newsletter

Untangle the weirdness of reality with our monthly newsletter.

Munro and his colleagues divided the qubits into two groups within a quantum computer, with each group representing one of Maxwell’s boxes. Then they implemented a demon-like procedure that used pulses of microwaves to force one group to contain more energetic qubits, and the other to contain far less energetic ones. In this way, the researchers effectively built a quantum battery , or a device that uses quantum processes to fill up with energy.

Quantum batteries are thought to be a promising, fast-charging energy technology of the future , but have so far only been explored in theory and modest proof-of-concept experiments. Here, the researchers could evaluate the effect of the demon on their actual battery. They found that the demon was much faster at changing the temperature – which points to a change in energy – of the two subsystems than a more conventional battery charging protocol.

Quantum physicists just got more certain about quantum uncertainty

They also verified that their experiment followed a modified version of the second law of quantum thermodynamics that explicitly accounts for the qubits’ quantum nature. This quantumness is the key novelty of the experiment, says Mauro Paternostro at Queen’s University Belfast in the UK. The experiment included enough qubits to exhibit so-called quantum many-body effects , which are thought to fundamentally affect how qubits can, or cannot, reach a state of equilibrium temperature.

The other exciting feature, he says, is that this version of Maxwell’s demon performs quantum measurements in order to sort qubits, and “the act of measuring something quantum mechanically is so violent, so strong, that you really fundamentally affect its state”. In other words, the new demon does not just measure qubits to sort them, but changes their states in the process, which improves its ability to charge a quantum battery.

“This was not anticipated by James Maxwell back in the 19th century,” says Munro.

Journal reference:

Physical Review A DOI: 10.1103/PhysRevA.109.062614

• quantum computing /
• quantum physics /
• quantum theory

Sign up to our weekly newsletter

Receive a weekly dose of discovery in your inbox! We'll also keep you up to date with New Scientist events and special offers.

More from New Scientist

Explore the latest news, articles and features

Is it possible to fully understand the universe while living in it?

Subscriber-only

Are space and time illusions? The answer could lie in black holes

Quantum ‘super behaviour’ could create energy seemingly from nothing

Time crystals may make quantum computers more reliable

Popular articles.

Trending New Scientist articles

A beginner's guide to time travel

Learn exactly how Einstein's theory of relativity works, and discover how there's nothing in science that says time travel is impossible.

Everyone can travel in time . You do it whether you want to or not, at a steady rate of one second per second. You may think there's no similarity to traveling in one of the three spatial dimensions at, say, one foot per second. But according to Einstein 's theory of relativity , we live in a four-dimensional continuum — space-time — in which space and time are interchangeable.

Einstein found that the faster you move through space, the slower you move through time — you age more slowly, in other words. One of the key ideas in relativity is that nothing can travel faster than the speed of light — about 186,000 miles per second (300,000 kilometers per second), or one light-year per year). But you can get very close to it. If a spaceship were to fly at 99% of the speed of light, you'd see it travel a light-year of distance in just over a year of time. 

That's obvious enough, but now comes the weird part. For astronauts onboard that spaceship, the journey would take a mere seven weeks. It's a consequence of relativity called time dilation , and in effect, it means the astronauts have jumped about 10 months into the future. 

Traveling at high speed isn't the only way to produce time dilation. Einstein showed that gravitational fields produce a similar effect — even the relatively weak field here on the surface of Earth . We don't notice it, because we spend all our lives here, but more than 12,400 miles (20,000 kilometers) higher up gravity is measurably weaker— and time passes more quickly, by about 45 microseconds per day. That's more significant than you might think, because it's the altitude at which GPS satellites orbit Earth, and their clocks need to be precisely synchronized with ground-based ones for the system to work properly. 

The satellites have to compensate for time dilation effects due both to their higher altitude and their faster speed. So whenever you use the GPS feature on your smartphone or your car's satnav, there's a tiny element of time travel involved. You and the satellites are traveling into the future at very slightly different rates.

But for more dramatic effects, we need to look at much stronger gravitational fields, such as those around black holes , which can distort space-time so much that it folds back on itself. The result is a so-called wormhole, a concept that's familiar from sci-fi movies, but actually originates in Einstein's theory of relativity. In effect, a wormhole is a shortcut from one point in space-time to another. You enter one black hole, and emerge from another one somewhere else. Unfortunately, it's not as practical a means of transport as Hollywood makes it look. That's because the black hole's gravity would tear you to pieces as you approached it, but it really is possible in theory. And because we're talking about space-time, not just space, the wormhole's exit could be at an earlier time than its entrance; that means you would end up in the past rather than the future.

Trajectories in space-time that loop back into the past are given the technical name "closed timelike curves." If you search through serious academic journals, you'll find plenty of references to them — far more than you'll find to "time travel." But in effect, that's exactly what closed timelike curves are all about — time travel

This article is brought to you by  How It Works .

How It Works is the action-packed magazine that's bursting with exciting information about the latest advances in science and technology, featuring everything you need to know about how the world around you — and the universe — works.

There's another way to produce a closed timelike curve that doesn't involve anything quite so exotic as a black hole or wormhole: You just need a simple rotating cylinder made of super-dense material. This so-called Tipler cylinder is the closest that real-world physics can get to an actual, genuine time machine. But it will likely never be built in the real world, so like a wormhole, it's more of an academic curiosity than a viable engineering design.

Yet as far-fetched as these things are in practical terms, there's no fundamental scientific reason — that we currently know of — that says they are impossible. That's a thought-provoking situation, because as the physicist Michio Kaku is fond of saying, "Everything not forbidden is compulsory" (borrowed from T.H. White's novel, "The Once And Future King"). He doesn't mean time travel has to happen everywhere all the time, but Kaku is suggesting that the universe is so vast it ought to happen somewhere at least occasionally. Maybe some super-advanced civilization in another galaxy knows how to build a working time machine, or perhaps closed timelike curves can even occur naturally under certain rare conditions.

This raises problems of a different kind — not in science or engineering, but in basic logic. If time travel is allowed by the laws of physics, then it's possible to envision a whole range of paradoxical scenarios . Some of these appear so illogical that it's difficult to imagine that they could ever occur. But if they can't, what's stopping them? 

Thoughts like these prompted Stephen Hawking , who was always skeptical about the idea of time travel into the past, to come up with his "chronology protection conjecture" — the notion that some as-yet-unknown law of physics prevents closed timelike curves from happening. But that conjecture is only an educated guess, and until it is supported by hard evidence, we can come to only one conclusion: Time travel is possible.

A party for time travelers 

Hawking was skeptical about the feasibility of time travel into the past, not because he had disproved it, but because he was bothered by the logical paradoxes it created. In his chronology protection conjecture, he surmised that physicists would eventually discover a flaw in the theory of closed timelike curves that made them impossible. 

In 2009, he came up with an amusing way to test this conjecture. Hawking held a champagne party (shown in his Discovery Channel program), but he only advertised it after it had happened. His reasoning was that, if time machines eventually become practical, someone in the future might read about the party and travel back to attend it. But no one did — Hawking sat through the whole evening on his own. This doesn't prove time travel is impossible, but it does suggest that it never becomes a commonplace occurrence here on Earth.

The arrow of time 

One of the distinctive things about time is that it has a direction — from past to future. A cup of hot coffee left at room temperature always cools down; it never heats up. Your cellphone loses battery charge when you use it; it never gains charge. These are examples of entropy , essentially a measure of the amount of "useless" as opposed to "useful" energy. The entropy of a closed system always increases, and it's the key factor determining the arrow of time.

It turns out that entropy is the only thing that makes a distinction between past and future. In other branches of physics, like relativity or quantum theory, time doesn't have a preferred direction. No one knows where time's arrow comes from. It may be that it only applies to large, complex systems, in which case subatomic particles may not experience the arrow of time.

Time travel paradox 

If it's possible to travel back into the past — even theoretically — it raises a number of brain-twisting paradoxes — such as the grandfather paradox — that even scientists and philosophers find extremely perplexing.

Killing Hitler

A time traveler might decide to go back and kill him in his infancy. If they succeeded, future history books wouldn't even mention Hitler — so what motivation would the time traveler have for going back in time and killing him?

Killing your grandfather

Instead of killing a young Hitler, you might, by accident, kill one of your own ancestors when they were very young. But then you would never be born, so you couldn't travel back in time to kill them, so you would be born after all, and so on … 

A closed loop

Suppose the plans for a time machine suddenly appear from thin air on your desk. You spend a few days building it, then use it to send the plans back to your earlier self. But where did those plans originate? Nowhere — they are just looping round and round in time.

Sign up for the Live Science daily newsletter now

Get the world’s most fascinating discoveries delivered straight to your inbox.

Andrew May holds a Ph.D. in astrophysics from Manchester University, U.K. For 30 years, he worked in the academic, government and private sectors, before becoming a science writer where he has written for Fortean Times, How It Works, All About Space, BBC Science Focus, among others. He has also written a selection of books including Cosmic Impact and Astrobiology: The Search for Life Elsewhere in the Universe, published by Icon Books.

NASA awards SpaceX $843 million contract to destroy the International Space Station

James Webb Space Telescope spies strange shapes above Jupiter's Great Red Spot

Shattered Russian satellite forces ISS astronauts to take shelter in stricken Starliner capsule

Most Popular

• 2 China rover returns historic samples from far side of the moon — and they may contain secrets to Earth's deep past
• 3 Earth's rotating inner core is starting to slow down — and it could alter the length of our days
• 4 'The early universe is nothing like we expected': James Webb telescope reveals 'new understanding' of how galaxies formed at cosmic dawn
• 5 Butterflies cross Atlantic ocean on 2,600-mile non-stop flight never recorded in any insect before
• 2 'Exceptional' discovery reveals more than 30 ancient Egyptian tombs built into hillside

To revisit this article, visit My Profile, then View saved stories .

• Backchannel
• Newsletters
• WIRED Insider
• WIRED Consulting

If you buy something using links in our stories, we may earn a commission. Learn more.

Steve Nadis

What Came Before the Big Bang?

The original version of this story appeared in Quanta Magazine .

About 13.8 billion years ago, the entire cosmos consisted of a tiny, hot, dense ball of energy that suddenly exploded.

That’s how everything began, according to the standard scientific story of the Big Bang, a theory that first took shape in the 1920s. The story has been refined over the decades, most notably in the 1980s, when many cosmologists came to believe that in its first moments, the universe went through a brief period of extraordinarily fast expansion called inflation before settling into a lower gear.

That brief period is thought to have been caused by a peculiar form of high-energy matter that throws gravity into reverse, “inflating” the fabric of the universe exponentially quickly and causing it to grow by a factor of a million billion billion in less than a billionth of a billionth of a billionth of a second. Inflation explains why the universe appears to be so smooth and homogeneous when astronomers examine it at large scales.

But if inflation is responsible for all that can be seen today, that raises the question: What, if anything, came before?

No experiment has yet been devised that can observe what happened before inflation. However, mathematicians can sketch out some possible scenarios. The strategy is to apply Einstein’s general theory of relativity—a theory that equates gravity with the curvature of space-time—as far back into time as it can go.

That’s the hope of three researchers: Ghazal Geshnizjani of the Perimeter Institute, Eric Ling of the University of Copenhagen, and Jerome Quintin of the University of Waterloo. The trio recently published a paper in the Journal of High Energy Physics in which, Ling said, “We mathematically showed that there might be a way to see beyond our universe.”

Working together with Jerome Quintin and Eric Ling, Ghazal Geshnizjani of the Perimeter Institute examined ways in which space-time might be extended beyond the Big Bang.

By Kelly Clancy

By Jaina Grey

By David Gilbert

By Kate Knibbs

Robert Brandenberger, a physicist at McGill University who was not involved with the study, said the new paper “sets a new standard of rigor for the analysis” of the mathematics of the beginning of time. In some cases, what appears at first to be a singularity—a point in space-time where mathematical descriptions lose their meaning—may in fact be an illusion.

A Taxonomy of Singularities

The central issue confronting Geshnizjani, Ling, and Quintin is whether there is a point prior to inflation at which the laws of gravity break down in a singularity. The simplest example of a mathematical singularity is what happens to the function 1/ x as x approaches zero. The function takes a number x as an input, and outputs another number. As x gets smaller and smaller, 1/ x gets larger and larger, approaching infinity. If x is zero, the function is no longer well defined: It can’t be relied upon as a description of reality.

“We mathematically showed that there might be a way to see beyond our universe,” said Eric Ling of the University of Copenhagen.

Sometimes, however, mathematicians can get around a singularity. For example, consider the prime meridian, which passes through Greenwich, England, at longitude zero. If you had a function of 1/longitude, it would go berserk in Greenwich. But there’s not actually anything physically special about suburban London: You could easily redefine zero longitude to pass through some other place on Earth, and then your function would behave perfectly normally when approaching the Royal Observatory in Greenwich.

Something similar happens at the boundary of mathematical models of black holes. The equations that describe spherical nonrotating black holes, worked out by the physicist Karl Schwarzschild in 1916, have a term whose denominator goes to zero at the event horizon of the black hole—the surface surrounding a black hole beyond which nothing can escape. That led physicists to believe that the event horizon was a physical singularity. But eight years later the astronomer Arthur Eddington showed that if a different set of coordinates is used, the singularity disappears. Like the prime meridian, the event horizon is an illusion: a mathematical artifact called a coordinate singularity, which only arises because of the choice of coordinates.

At a black hole’s center, by contrast, the density and curvature go to infinity in a way that can’t be eliminated by using a different coordinate system. The laws of general relativity start spewing out gibberish. This is called a curvature singularity. It implies that something is taking place that’s beyond the ability of current physical and mathematical theories to describe.

Geshnizjani, Ling, and Quintin studied whether the onset of the Big Bang is more like the center of a black hole, or more like an event horizon. Their investigation builds upon a theorem proved in 2003 by Arvind Borde, Alan Guth (one of the first people to propose the idea of inflation), and Alexander Vilenkin. This theorem, known by the authors’ initials as BGV, says that inflation must have had a beginning—it can’t have been going on ceaselessly into the past. There must have been a singularity to kick things off. BGV establishes the existence of this singularity, without saying what kind of singularity it is.

As Quintin puts it, he and his colleagues have worked to figure out if that singularity is a brick wall—a curvature singularity—or a curtain that can be pulled back—a coordinate singularity. Eric Woolgar, a mathematician at the University of Alberta who was not involved in the study, said that it clarifies our picture of the Big Bang singularity. “They can say whether the curvature is infinite at the initial singularity or whether the singularity is milder, which might allow us to extend our model of the universe to times before the Big Bang.”

“Light rays can actually go through the boundary,” said Jerome Quintin of the University of Waterloo.

To classify possible pre-inflationary scenarios, the three researchers used a parameter called the scale factor that describes how the distance between objects has changed over time as the universe expands. By definition, the Big Bang is the time when the scale factor was zero—everything was squeezed into a dimensionless point.

During inflation, the scale factor increased with exponential speed. Before inflation, the scale factor could have varied in any number of ways. The new paper provides a taxonomy of singularities for different scale-factor scenarios. “We show that under certain conditions, the scale factor will produce a curvature singularity, and under other conditions it does not,” Ling said.

Researchers already knew that in a universe with so-called dark energy , but without matter, the start of inflation identified in the BGV theorem is a coordinate singularity that can be eliminated. But the real universe has matter, of course. Might mathematical tricks also make it possible to get around its singularity? The researchers showed that if the amount of matter is negligible compared to the amount of dark energy, then the singularity can be eliminated. “Light rays can actually go through the boundary,” Quintin said. “And in that sense, you can see beyond the boundary; it’s not like a brick wall.” The universe’s history would extend beyond the Big Bang.

However, cosmologists think that the early universe had more matter than energy. In this case, the new work shows that the BGV singularity would be a real physical curvature singularity, at which the laws of gravity stop making sense.

A singularity hints at the fact that general relativity can’t be a complete description of the basic rules of physics. Efforts to form such a description, which would require reconciling general relativity with quantum mechanics, are ongoing. Ling said he sees the new paper as a stepping stone to such a theory. In order to make sense of the universe at the highest energy levels, he said, “we first need to understand classical physics as well as we can.”

Original story reprinted with permission from Quanta Magazine , an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.

You Might Also Like …

In your inbox: Will Knight's Fast Forward explores advances in AI

Inside the biggest FBI sting operation in history

The WIRED AI Elections Project : Tracking more than 60 global elections

Ecuador is literally powerless in the face of drought

Rest assured: Here are the best mattresses you can buy online

Lyndie Chiou

Amos Zeeberg

João Medeiros

Sebastian Wieczorek

Rhett Allain

Victoria St. Martin

Jonathan O'Callaghan

October 21, 1999

According to current physical theory, is it possible for a human being to travel through time?

As several respondents noted, we constantly travel through time--just forward, and all at the same rate. But seriously, time travel is more than mere fantasy, as noted by Gary T. Horowitz, a professor of physics at the University of California at Santa Barbara:

"Perhaps surprisingly, this turns out to be a subtle question. It is not obviously ruled out by our current laws of nature. Recent investigations into this question have provided some evidence that the answer is no, but it has not yet been proven to be impossible."

Even the slight possibility of time travel exerts such fascination that many physicists continue to study not only whether it may be possible but also how one might do it.

On supporting science journalism

If you're enjoying this article, consider supporting our award-winning journalism by subscribing . By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.

One of the leading researchers in this area is William A. Hiscock, a professor of physics at Montana State University. Here are his thoughts on the matter:

"Is it possible to travel through time? To answer this question, we must be a bit more specific about what we mean by traveling through time. Discounting the everyday progression of time, the question can be divided into two parts: Is it possible, within a short time (less than a human life span), to travel into the distant future? And is it possible to travel into the past?

"Our current understanding of fundamental physics tells us that the answer to the first question is a definite yes, and to the second, maybe.

"The mechanism for traveling into the distant future is to use the time-dilation effect of Special Relativity, which states that a moving clock appears to tick more slowly the closer it approaches the speed of light. This effect, which has been overwhelmingly supported by experimental tests, applies to all types of clocks, including biological aging.

"If one were to depart from the earth in a spaceship that could accelerate continuously at a comfortable one g (an acceleration that would produce a force equal to the gravity at the earth's surface), one would begin to approach the speed of light relative to the earth within about a year. As the ship continued to accelerate, it would come ever closer to the speed of light, and its clocks would appear to run at an ever slower rate relative to the earth. Under such circumstances, a round trip to the center of our galaxy and back to the earth--a distance of some 60,000 light-years--could be completed in only a little more than 40 years of ship time. Upon arriving back at the earth, the astronaut would be only 40 years older, while 60,000 years would have passed on the earth. (Note that there is no 'twin paradox,' because it is unambiguous that the space traveler has felt the constant acceleration for 40 years, while a hypothetical twin left behind on a spaceship circling the earth has not.)

"Such a trip would pose formidable engineering problems: the amount of energy required, even assuming a perfect conversion of mass into energy, is greater than a planetary mass. But nothing in the known laws of physics would prevent such a trip from occurring.

"Time travel into the past, which is what people usually mean by time travel, is a much more uncertain proposition. There are many solutions to Einstein's equations of General Relativity that allow a person to follow a timeline that would result in her (or him) encountering herself--or her grandmother--at an earlier time. The problem is deciding whether these solutions represent situations that could occur in the real universe, or whether they are mere mathematical oddities incompatible with known physics. No experiment or observation has ever indicated that time travel is occurring in our universe. Much work has been done by theoretical physicists in the past decade to try to determine whether, in a universe that is initially without time travel, one can build a time machine--in other words, if it is possible to manipulate matter and the geometry of space-time in such a way as to create new paths that circle back in time.

"How could one build a time machine? The simplest way currently being discussed is to take a wormhole (a tunnel connecting spatially separated regions of space-time) and give one mouth of the wormhole a substantial velocity with respect to the other. Passage through the wormhole would then allow travel to the past.

"Easily said--but where does one obtain a wormhole? Although the theoretical properties of wormholes have been extensively studied over the past decade, little is known about how to form a macroscopic wormhole, large enough for a human or a spaceship to pass through. Some speculative theories of quantum gravity tell us that space-time has a complicated, foamlike structure of wormholes on the smallest scales--10^-33 centimeter, or a billion billion times smaller than an electron. Some physicists believe it may be possible to grab one of these truly microscopic wormholes and enlarge it to usable size, but at present these ideas are all very hypothetical.

"Even if we had a wormhole, would nature allow us to convert it into a time machine? Stephen Hawking has formulated a "Chronology Protection Conjecture," which states that the laws of nature prevent the creation of a time machine. At the moment, however, this is just a conjecture, not proven.

"Theoretical physicists have studied various aspects of physics to determine whether this law or that might protect chronology and forbid the building of a time machine. In all the searching, however, only one bit of physics has been found that might prohibit using a wormhole to travel through time. In 1982, Deborah A. Konkowski of the U.S. Naval Academy and I showed that the energy in the vacuum state of a massless quantized field (such as the photon) would grow without bound as a time machine is being turned on, effectively preventing it from being used. Later studies by Hawking and Kip S. Thorne of Caltech have shown that it is unclear whether the growing energy would change the geometry of space-time rapidly enough to stop the operation of the time machine. Recent work by Tsunefumi Tanaka of Montana State University and myself, along with independent research by David Boulware of the University of Washington, has shown that the energy in the vacuum state of a field having mass (such as the electron) does not grow to unbounded levels; this finding indicates there may be a way to engineer the particle physics to allow a time machine to work.

"Perhaps the biggest surprise of the work of the past decade is that it is not obvious that the laws of physics forbid time travel. It is increasingly clear that the question may not be settled until scientists develop an adequate theory of quantum gravity."

John L. Friedman of the physics department at the University of Wisconsin at Milwaukee has also given this subject a great deal of consideration:

"Special relativity implies that people or clocks at rest (or not accelerating) age more quickly than partners traveling on round-trips in which one changes direction to return to one's partner. In the world's particle accelerators, this prediction is tested daily: Particles traveling in circles at nearly the speed of light decay more slowly than those at rest, and the decay time agrees with theory to the high precision of the measurements.

"Within the framework of Special Relativity, the fact that particles cannot move faster than light prevents one from returning after a high-speed trip to a time earlier than the time of departure. Once gravity is included, however, spacetime is curved, so there are solutions to the equations of General Relativity in which particles can travel in paths that take them back to earlier times. Other features of the geometries that solve the equations of General Relativity include gravitational lenses, gravitational waves and black holes; the dramatic explosion of discoveries in radio and X-ray astronomy during the past two decades has led to the observation of gravitational lenses and gravitational waves, as well as to compelling evidence for giant black holes in the centers of galaxies and stellar-sized black holes that arise from the collapse of dying stars. But there do not appear to be regions of spacetime that allow time travel, raising the fundamental question of what forbids them--or if they really are forbidden.

"A recent surprise is that one can circumvent the 'grandfather paradox,' the idea that it is logically inconsistent for particle paths to loop back to earlier times, because, for example, a granddaughter could go back in time to do away with her grandfather. For several simple physical systems, solutions to the equations of physics exist for any starting condition. In these model systems, something always intervenes to prevent inconsistency analogous to murdering one's grandfather.

"Then why do there seem to be no time machines? Two different answers are consistent with our knowledge. The first is simply that the classical theory has a much broader set of solutions than the correct theory of quantum gravity. It is not implausible that causal structure enters in a fundamental way in quantum gravity and that classical spacetimes with time loops are spurious--in other words, that they do not approximate any states of the complete theory. A second possible answer is provided by recent results that go by the name chronology protection: One supposes that quantum gravity allows microscopic structures that violate causality, and one shows that the character of macroscopic matter forbids the existence of regions with macroscopically large time loops. To create a time machine would require negative energy, and quantum mechanics appears to allow only extremely small regions of negative energy. And the forces needed to create an ordinary-sized region with time loops appear to be extremely large.

"To summarize: It is very likely that the laws of physics rule out macroscopic time machines, but possible that spacetime is filled with microscopic time loops.

• Science & Math

Enjoy fast, free delivery, exclusive deals, and award-winning movies & TV shows with Prime Try Prime and start saving today with fast, free delivery

Amazon Prime includes:

Fast, FREE Delivery is available to Prime members. To join, select "Try Amazon Prime and start saving today with Fast, FREE Delivery" below the Add to Cart button.

• Cardmembers earn 5% Back at Amazon.com with a Prime Credit Card.
• Unlimited Free Two-Day Delivery
• Streaming of thousands of movies and TV shows with limited ads on Prime Video.
• A Kindle book to borrow for free each month - with no due dates
• Listen to over 2 million songs and hundreds of playlists
• Unlimited photo storage with anywhere access

Important:  Your credit card will NOT be charged when you start your free trial or if you cancel during the trial period. If you're happy with Amazon Prime, do nothing. At the end of the free trial, your membership will automatically upgrade to a monthly membership.

Buy new: .savingPriceOverride { color:#CC0C39!important; font-weight: 300!important; } .reinventMobileHeaderPrice { font-weight: 400; } #apex_offerDisplay_mobile_feature_div .reinventPriceSavingsPercentageMargin, #apex_offerDisplay_mobile_feature_div .reinventPricePriceToPayMargin { margin-right: 4px; } $13.99 $ 13 . 99 FREE delivery Friday, July 5 on orders shipped by Amazon over $35 Ships from: Amazon.com Sold by: Amazon.com

Return this item for free.

We offer easy, convenient returns with at least one free return option: no shipping charges. All returns must comply with our returns policy.

• Go to your orders and start the return
• Select your preferred free shipping option
• Drop off and leave!

Return instructions

Save with used - good .savingpriceoverride { color:#cc0c39important; font-weight: 300important; } .reinventmobileheaderprice { font-weight: 400; } #apex_offerdisplay_mobile_feature_div .reinventpricesavingspercentagemargin, #apex_offerdisplay_mobile_feature_div .reinventpricepricetopaymargin { margin-right: 4px; } $13.07 $ 13 . 07 free delivery monday, july 8 on orders shipped by amazon over $35 ships from: amazon sold by: djarieke.

Download the free Kindle app and start reading Kindle books instantly on your smartphone, tablet, or computer - no Kindle device required .

Read instantly on your browser with Kindle for Web.

Using your mobile phone camera - scan the code below and download the Kindle app.

Image Unavailable

• To view this video download Flash Player

Follow the author

It's Really About Time: The Science of Time Travel Paperback – December 17, 2019

Purchase options and add-ons

• Print length 176 pages
• Language English
• Publication date December 17, 2019
• Dimensions 8 x 0.4 x 10 inches
• ISBN-10 1734264306
• ISBN-13 978-1734264302
• See all details

Frequently bought together

Similar items that may ship from close to you

Editorial Reviews

About the author, product details.

• Publisher ‏ : ‎ Tahilla Press (December 17, 2019)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 176 pages
• ISBN-10 ‏ : ‎ 1734264306
• ISBN-13 ‏ : ‎ 978-1734264302
• Item Weight ‏ : ‎ 13.8 ounces
• Dimensions ‏ : ‎ 8 x 0.4 x 10 inches
• #39 in Physics of Time (Books)
• #91 in Relativity Physics (Books)

About the author

John oliver ryan.

John Ryan was born in Tipperary, Ireland and attended University College Galway where he studied Physics and Mathematics. His interest in Special Relativity began at the age of 17, but it was only after developing a deeper understanding of the subject matter later in life that he decided to put it in writing. His first book “It’s Really About Time” was written to provide lay readers with a deep understanding of how and why special relativity allows for travel to a time that could be months, years or even centuries in the future.

After university, Ryan spent many years working as an electronics design engineer in the television and broadcasting industry. Ryan later co-founded a successful Silicon Valley-based technology company which is listed on the Nasdaq. He holds over 70 U.S. patents in the fields of television camera design and video encryption.

Customer reviews

Customer Reviews, including Product Star Ratings help customers to learn more about the product and decide whether it is the right product for them.

To calculate the overall star rating and percentage breakdown by star, we don’t use a simple average. Instead, our system considers things like how recent a review is and if the reviewer bought the item on Amazon. It also analyzed reviews to verify trustworthiness.

Reviews with images

• Sort reviews by Top reviews Most recent Top reviews

Top reviews from the United States

There was a problem filtering reviews right now. please try again later..

Top reviews from other countries

• Amazon Newsletter
• About Amazon
• Accessibility
• Sustainability
• Press Center
• Investor Relations
• Amazon Devices
• Amazon Science
• Sell on Amazon
• Sell apps on Amazon
• Supply to Amazon
• Protect & Build Your Brand
• Become an Affiliate
• Become a Delivery Driver
• Start a Package Delivery Business
• Advertise Your Products
• Self-Publish with Us
• Become an Amazon Hub Partner
• › See More Ways to Make Money
• Amazon Visa
• Amazon Store Card
• Amazon Secured Card
• Amazon Business Card
• Shop with Points
• Credit Card Marketplace
• Reload Your Balance
• Amazon Currency Converter
• Your Account
• Your Orders
• Shipping Rates & Policies
• Amazon Prime
• Returns & Replacements
• Manage Your Content and Devices
• Recalls and Product Safety Alerts
• Conditions of Use
• Privacy Notice
• Consumer Health Data Privacy Disclosure
• Your Ads Privacy Choices

• Aug 30, 2022

11 Widely Accepted Theories of Time Travel in Physics

Updated: Sep 7, 2022

Introducing 11 theories regarding time travelling in physics. From the neutron star theory to the Wormhole theory, you get an overview of all 11 theories about time travelling. Astrophysics #2 !

Figure 1: Time travel

Introduction

Time travel in science fiction has always been a plot device both of great mystery and delight for its audiences. We’ve also seen conflicting theories on time travel and how time works in our universe. For example, we’ve seen the idea of a fixed timeline in movies like Harry Potter and the Prisoner of Azkaban and The Terminator , a dynamic timeline in movies like Back to the Future , and a multiverse timeline in movies like Star Trek (2009) . With fixed timelines, all events remain at fixed points in time, and the actions of the traveller in the past have already become a part of history. In dynamic timelines, altered events in the past have definite impacts on the present, resulting in paradoxes like the grandfather paradox. Finally, in the multiverse interpretation of timelines, there’s an infinite number of parallel universes – travelling into the past causes a new divergent timeline from the first. While plenty of paradoxes arise, it may be particularly helpful to look into what physics has to say about how time travel works. While we’ve seen the concept of wormholes as a device to explain time travel in science-fiction, it isn’t the only way, theoretically.

Introduction to the Various Time - Travel Theories

According to theoretical physics, most physicists agree that there are at least 11 proposed time travel theories. In the anime Steins; Gate , the black hole theory is used most prominently to tell the exciting story of time travel. We will now explore these 11 theories below.

The Neutron Star Theory

A neutron star is the gravitationally collapsed core of a massive supergiant star, which had a total mass somewhere between 10 and 25 solar masses, possibly more if the star was especially metal-rich. To put that in perspective, these stars have a mass of around 466,000 times the mass of the earth, thus exerting a very high gravitational force (1.962×1012 m/s2, Earth's is 9.81 m/s2) although they have a diameter of 10-12 Km. According to Einstein’s special theory of relativity, time isn’t universal for every observer like a ticking universal clock that dictates all of the time. Rather, the larger the mass of an object, the stronger its gravitational field – the gravitational field really being the curvature of both space and time. The stronger the gravity, the more spacetime curves, and the slower time itself proceeds. The gravitational time dilation of a neutron star would pass 30% slower on the surface of such a star, meaning that if someone could travel to such a star and withstand this gravity, they could “travel” to the future. This theory only serves for time travel to the future, to the past wouldn’t be possible. Nevertheless, we actually experience this differing time dilation on earth – albeit to a significantly less degree.

The Black Hole Theory

Black holes, in addition to wormholes, are one of the most used resources in time-travelling novels, next to wormholes. The Kerr-black hole theory is the result of Roy Kerr’s calculations for relativity. A Kerr-black hole is a singularity that possesses mass and angular momentum but does not possess an electrical charge. This black hole spins around a central axis and has two event horizons, which contain a ring-formed singularity. Inside each of the two event horizons, time and space are reversed, so in a Kerr-black hole, this swapping occurs twice. It’s both possible to escape the ring-formed singularity as well as end up in “negative space” as you cross the singularity. Avoiding it would cause you to back in time while you are crossing the first event horizon.

The Light Speed Theory

Like the neutron star theory, the light speed theory, although unclear exactly what light speed theory refers to, makes several connections with other theories and some basic principles of Einstein’s theory of relativity. The first form of time travel that relates to light speed is when the matter is accelerated to close to light speed, causing time to slow down for it due to time dilation from Einstein’s relativity. This can only be used to travel to the future, not to the past. The neutron star theory demonstrated this example. While this first form of time travel is actually hard science and not science fiction, all it accomplishes is time travel to the future, not the past. The Light Speed theory, therefore, refers to a more broad definition of the other theories by which light speed travel is essential.

The Tachyon Theory

The Tachyon or a Tachyon particle is a hypothetical particle that always moves faster than light, Most physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. The existence of such particles, called tachyons, has not been totally ruled out, but several experiments have tried, without luck to detect them. If they did exist, and they interacted with ordinary matter, it would give us the means to communicate with the past. Tachyons could literally be sent outwards, bounce off a tachyonic mirror, and return before they were sent. Like most time travel theories, Tachyons also give rise to many logical problems – paradoxes.

The Wormhole Theory

A wormhole is a special solution to the equations describing Einstein's theory of general relativity that connects two distant points in space or time via a tunnel. Ideally, the length of this tunnel is shorter than the distance between those two points, making the wormhole a kind of shortcut. Though they are a staple of science fiction and have captured the popular imagination, wormholes are, as far as we know, only hypothetical. They are legitimate solutions to general relativity, but scientists have never figured out a way to maintain a stable wormhole in the real universe. To do so, you would need exotic matter – matter with negative mass – that can repel gravity, thus negating the supergravity of the wormhole that would otherwise collapse the wormhole as soon as it opened. The exotic matter, in itself, is also another proposed theory of time travel as we’ll see in the next theory.

The Exotic Matter Theory

Along with the way exotic matter can be used to stabilize a wormhole in order to make it traversable for time travel via wormholes, there are other methods of time travel using exotic matter. The most popularly discussed method is the Alcubierre warp drive, which requires exotic matter to create an area of space with negative mass, and achieves faster-than-light travel by warping the space-time continuum, contracting space in front of a spacecraft using the warp drive, and expanding space behind the spacecraft. Once the Alcubierre warp drive using exotic matter is achieved, faster-than-light travel can be used to go back in time, using the same mechanism as tachyons, as originally explained in Albert Einstein’s thought experiment of the tachyonic antitelephone that would use tachyons (subatomic particles that move faster than light speed) to send information back in time. The difference, however, is that you could actually build a spacecraft and it would produce an Alcubierre warp drive around it using exotic matter, in order to achieve physical time travel of ordinary matter back in time, including people inside a spacecraft and whatever else they want to bring along. So while tachyons would only be able to send data back in time, the Alcubierre warp drive could be used for physical time travel.

The Cosmic String Theory

The cosmic string theory for time travel was developed by the physicist J. Richard Gott is explained in the 2002 book How to Build a Time Machine by Paul Davies. As described in Gott’s book, a cosmic string is a string-shaped crack that has an extreme mass. You can think of the crack as something the width of an elementary particle, and at least the length of a galaxy. Because of their extreme mass, cosmic strings have the property of space-time distortion. Once you travel near the cosmic string, you experience a full rotation around the string in less than 360 degrees. In short, you can do something resembling a warp known as a space-time angular deficit. When you pass through an area of angular deficit, transit time becomes zero. Applying this, once the cosmic string moves approaching light speed, according to the theory of relativity, time will flow slower for the cosmic string in relation to its surroundings. Therefore, passing through the area of the angular deficit would cause the zero transit time to become negative. In other words, it will be the 'past' after transit. So, if you use two cosmic strings, you can do a space deficit jump. If you revolve back to your original location, you can return to the same time you started revolving. And that, roughly speaking, is time travel by means of cosmic string theory.

The Quantum Gravity Theory

In a rather short explanation compared to the previous theories, quantum gravity theory uses what’s known as Quantum jumps in the use of a “stale state” or a “dead state” where no movement is possible because of density.

The Cesium Laser Theory

At the NEC research institute in Princeton, Dr. Lijun Wang gave the world the first glimpse of multi-dimensional reality and the possibility of time travel in the future. Namely, Wang transmitted a pulse of light towards a chamber filled with specially treated cesium gas, and recorded its travel through the chamber at an accelerated speed of up to 300 times the speed of light. Before the pulse fully entered the chamber, Wang reported that it appeared at the same instant at a point 60 feet across the laboratory. In effect, it existed in two places at the same time. Thus Wang not only proved that objects can move at speeds exceeding the earlier prescribed limit of 186,000 miles per second, but he proved Einstein's theory that time slows when objects travel at a speed approaching (and exceeding) the speed of light. The implications of this are mind-boggling, hinting that time travel is quite possible.

The Elementary Particle Ring and Laser Theory

Elementary particle ring and laser theory is a time travel theory developed by Professor Ronald Mallett of the University of Connecticut in Storrs, Connecticut. It involves the use of a ring laser to generate a closed time-like curve or CTC for short. It is based on an older time travel theory called the Tipler cylinder except modified to be more practical, as Tipler cylinders would have to be infinitely long, so Ronald Mallett came up with a better way to generate CTCs using only a ring laser. The circulating light beams of a ring laser can theoretically twist spacetime and allow the dimension of time to be traversable just like the dimension of space is normally traversable. Then, one could theoretically walk through time in any direction, as one can do in space. It works by producing gamma rays or magnetic fields to warp time into CTCs. Theoretically, this could be used for physical time travel for human beings.

The Dirac Antiparticle Theory

Dirac antiparticle theory, more commonly known as Dirac hole theory, was the original theory to explain the existence of antimatter from 1929, postulated by Paul Dirac. It is derived from the Dirac equation formulated in 1928, a relativistic version of the Schrödinger equation that improved on the Schrödinger equation by making it compatible with Einstein’s relativity (which was rather important, since the purpose of the Schrödinger equation was to explain the behaviour of electrons, electrons typically move near the speed of light, and any equation describing things moving near the light speed that does not incorporate relativity produces erroneous results).

Dirac's antiparticle theory can also be used as an explanation for time travel. It is an extremely simple explanation. Antimatter is the same as matter, mathematically speaking, except if you mathematically negate time the values for time, and related quantities that use odd powers of time such as velocity, momentum, and angular momentum (quantities that use even powers of time like acceleration, force, work, and energy are unchanged since if you square a negative number you get a positive number). This explains things like how electrons have a spin of positive ½ but positrons (the antimatter equivalent to electrons) have a spin of negative ½, as spin is a measure of angular momentum which has an odd power of time, so if time is negated, spin is negated. Because of the way electric charge is derived from spin, electrons have a negative charge but positrons have a positive charge, which is what gives them the name “positrons”, short for “positively charged electrons”. antimatter travels backwards in time rather than forwards in time like matter, and actually, this is to be expected, since the laws of physics have many symmetries, and for time to only go in one direction rather than in both directions would break time symmetry, which is bad. This might seem to allow information to be sent to the past using antimatter

Although there are 11 widely accepted theories of time travel in the world, some are very impractical like the Dirac Antiparticle theory. It is, as far as modern science knows, impossible to send information back in time. In practice, this would mean, among matter particles, it is impossible for matter particles to send or receive information that comes from the future to the past, while among antimatter particles, it is impossible for antimatter particles to send or receive information that comes from the past to the future. Nevertheless, it’s still interesting to speculate on how scientists can achieve physical time travel. At least, for filmmakers, there are opportunities for even more time-warping and mind-knotting entertainment.

https://steins-gate.fandom.com/wiki/Time-travel_theories#:~:text=According%20to%20the%20theory%20of,at%20LA%20several%20years%20back.

https://www.space.com/21675-time-travel.html#:~:text=For%20example%2C%20physicist%20Albert%20Einstein,than%20an%20observer%20at%20rest.

https://www.livescience.com/what-are-wormholes

Hashtags: #science #scienceresearch #sciencebehindit #personalopinion #Blog #Interestingtopics #theories #NowScience #current #timetravelling #interestingfacts #astronomy #astrophysicsseries #physics

Recent Posts

Mathematical Proof That Black Holes Are Stable – What Does It Mean?

Giant Freakin Robot

Time Travel Equation Solved By Astrophysicist

Posted: March 25, 2024 | Last updated: March 25, 2024

After a lifetime of pursuing the idea, Physics Professor Ronald Mallett at the University of Connecticut has potentially figured out the theoretical aspects of time travel. Professor Mallett believes that black holes, rotating light, and gravitational pulls may hold the key to exploring time, but it’s all theoretical for now. There are still a lot of hurdles and limitations to handle before time travel can have real, practical applications.

A Life Spent Thinking About Time Travel

Love and loss pushed Professor Mallett into an obsession with time and space. When he was 10 years old, his father passed away from a heart attack. It was his father who nourished his love of science, but H.G. Wells’ book The Time Machine pushed him towards a focus on time travel.

He was hooked from the very first paragraph of the book, “Scientific people know very well that Time is only a kind of Space. And why cannot we move in Time as we move about in the other dimensions of Space?”

That paragraph never left him, and the professor let that time travel question guide him through school and into the Professor Emeritus of Physics position at the University of Connecticut.

Einstein And Black Holes

As he grew up, Professor Mallett spent much of his time on Albert Einstein’s theories about black holes. While his interest in time travel only continued to grow, a potential solution never showed itself. At least, not until the professor ended up in a hospital with a heart condition.

There, lying in the hospital bed, inspiration hit him. Black holes and the gravitational fields they created were the answer to time travel. These gravitational fields had the potential to lead to time loops, which then theoretically could allow people and objects to travel back in time.

Black Holes Manipulating Gravity

While this idea offered an ability to manipulate time, the other problem was how to use these time loops for time travel.

Professor Mallett found this time travel solution much easier than the first problem. Strong and continuous beams of light, like a ring of lasers, with a particular rotation could be used to manipulate gravity and mimic the distorting effects of a black hole.

Though the details are rather complicated, the big time travel picture is a lot simpler to grasp. The professor offers a comparison to help people understand. “Let’s say you have a cup of coffee in front of you. Start stirring the coffee with the spoon. It started to spin, right? That’s what a spinning black hole does. In Einstein’s theory, space and time are related to each other. That’s why it’s called space-time. So when the black hole spins, it will actually cause time to shift.”

Much To Figure Out

Professor Mallett may now have a theory on time travel and a machine to use to make it possible, but that doesn’t mean it will be here in the next few decades.

There’s still a lot to figure out to make such travel practical, such as where the insane amount of energy such a machine would require could come from, and how big the machine would need to be.

There’s also a major constraint on the machine. According to his theories, time travel would only be possible to the very beginning of when the machine was first built. In this way, it’s more like a one-way message service. You can potentially go forward quite a distance, but going back in time is limited by the machine’s creation.

Theoretical Aspects Of Time Travel

The professor has made a huge leap in figuring out the theoretical aspects of time travel, but there’s a lot more to discover and quite a few hurdles and paradoxes to figure out before scientists practically start messing around in time.

Still, the theory is a step in the right direction and does suggest that people can push past what science currently considers possible.

Source: Earth.com

More for You

Democrats Question Replacing Biden: Here’s How It Could Work

Garfield by Jim Davis

A Hawaii judge ordered the demolition of a house after a California woman bought a vacant plot, only to discover the building on her land

What Is the $1K Per Month in Retirement Rule?

Trevor Lawrence, wife announce they are expecting first child after $275 million contract extension

Common Blood Pressure Drug Increases Lifespan, Slows Aging in Animals

I quit sugar for 6 months and this is what it did to my face and body

How to clear the cache on your iPhone (and why you should)

Jamie Dimon says schools are making his job harder by not giving students the proper training to enter the working world

What Was The Last Year For The Harley-Davidson Factory Sidecar?

Flight attendant explains why you should always throw a bottle of water under hotel beds

Kamala Harris' Chances of Beating Donald Trump if She Replaces Biden: Polls

"It Blew Her Mind When I Told Her": Older Adults Are Sharing The "Common Knowledge" Practices From The Past That Have Become Obsolete Over Time

Bill Belichick’s House: An Inside Look at the Patriots Head Coach’s $1.9 Million Home

A Cracked Piece of Metal Healed Itself in an Experiment That Stunned Scientists

Boomers are stuck in their jobs because companies don’t want to hire them, but savvy CEOs who do can access an untapped oasis

Supreme Court sides with fishermen in landmark case deciding fate of the administrative state

The salary a single person needs to make ends meet in every U.S. state

Can You Find the Camouflaged Mountain Lion in This Viral Photo?

Gavin Newsom Strikes Back at Trump for 'Disgraceful' Debate Rumor