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TOM HARTMAN: Welcome, everyone. [INAUDIBLE] Is the audio working properly? Everyone can hear me? Good. OK, so it's my pleasure to introduce today's [INAUDIBLE] speaker and Bethe lecturer.
But before I introduce the speaker, I want to say a few words about this lecture series, which is named in honor of Hans Bethe, who was a professor in the Cornell physics department for the remarkable timespan of nearly 70 years around 1935 to his death in 2005. Bethe was active in making discoveries throughout that entire time, and he left an incredible mark on the Cornell physics department through his science, his mentorship, and his leadership. I never met Bethe myself, but I felt his presence here because I know that many people have credited him with establishing the friendly atmosphere he inspired in this department.
So this is the first in the series of three lectures this week. The second lecture is a seminar tomorrow. The third lecture is a public lecture on Wednesday evening.
The public lecture is as public as can be these days. It's open, in-person, to all members of the Cornell community and virtually to the rest of the public. So I hope that you'll all join us then.
I want to point out that the public lecture, the Wednesday lecture, will not be in this room. It will be in Kennedy Hall, which is about 200 meters or so that way. It's a different building, different from usual.
So I'll say a few more words about Hans Bethe's life and science at the beginning of the Wednesday lecture. For today, I just will say that Bethe was the world's leading expert on nuclear physics for many decades. In 1938, he discovered the nuclear reactions that power the sun, burning hydrogen to helium. And he received the Nobel Prize in 1967 for that work.
But he said in his acceptance speech, he said, "You have given me the prize, I believe, for a lifetime of quiet work in physics rather than for any spectacular single contribution. I'm very proud and very happy with this distinction."
And indeed, Bethe was prolific. He had a hand in many of the great discoveries of his generation in astrophysics, nuclear physics, and quantum electrodynamics. So it's now my honor to introduce today's Bethe lecturer Andy Strominger. Andy is the York Professor of Physics and the Director of the Center for the Fundamental Laws of Nature at Harvard University. And he's made seminal contributions to relativity, quantum gravity, quantum field theory, and string theory.
His work on [INAUDIBLE] compactifications showed that string theory has the potential to be a unified theory of nature. And he showed that in string theory, objects called brains provide a microscopic description of black holes and are able to holographically restore information. This explained how string theory can evade the black hole information paradox and demonstrated that spacetime can emerge from something more fundamental.
I find it fitting that he's here to give the lectures named after Hans Bethe. Bethe discovered the microscopic theory of stars, which was the frontier at the time. And much of Andy's work has been devoted to understanding the microscopic theory of black holes. The frontier has moved, the scale has changed, but it's a parallel problem for the next century.
Recently, Andy discovered an exact equivalence unifying three different phenomena. Soft theorems in quantum field theory-- and these are in the title of the talk-- soft theorems in quantum field theory, asymptotic symmetries, and the memory effect. And this equivalence has deep implications for infrared phenomena, ranging from quantum electrodynamics to black hole information. So we'll hear more about that in today's colloquium.
Andy's been recognized by numerous prizes and awards, including the 2017 breakthrough prize in fundamental physics, the 2016 Feynman Prize, the 2014 Oskar Klein Medal, and 2014 Dirac Medal, and many more. What perhaps doesn't come across in this list of accomplishments is the outsized role that I think Andy has played in training and inspiring the next generation of physicists, and myself included-- I was his PhD student about 10 years ago and it was his work that inspired me to study black holes and quantum gravity.
So with that, let me now turn over the floor. The title of today's talk is Memory, Soft Theorems, and the Symmetries of Nature. And please join me in welcoming Andy.
[APPLAUSE]
And he said he's happy to take questions along the way. So if you're on Zoom, you can just use the Raise Hand and somebody will help you out with that. And we can also do questions here.
ANDREW STROMINGER: OK. Thank you, Tom, for the introduction. Before I get started, I'd like to say it is a special honor and privilege for me to be giving the Bethe Lecture here. I did a little bit, very early in my career, know Hans Bethe. He used to escape the Cornell winters in Santa Barbara for a month or two.
And I didn't talk to him so much. Our fields were pretty far apart, though as you'll see from our colloquium, they're perhaps closer now. But I do remember him bringing that he was friendly. He brought, what Tom was saying, the friendly ambience he created at Cornell-- he graced us with it in Santa Barbara for a few months every winter.
Also, I'm proud that Bethe is actually my academic grandfather. My advisor was his student, Roman [INAUDIBLE] And he was co-advised by Bethe and Ken Wilson, which I guess makes Paul Ginsparg, if he's here, my uncle, even though I was roommates with him in college. And then Tom is, of course, my advisee, my academic son, so I feel very connected to Cornell.
OK, so I am going to talk-- OK, now it's not moving. This thing? OK. OK. So I'm going to talk about a very basic question. And that is, what are all the symmetries of nature as we understand it so far?
So I'm not talking about symmetries that we might hope to find-- supersymmetry or grand unified symmetry or some kind of stringy symmetry-- but the symmetries of the laws of physics that we have already experimentally verified, namely the standard model and general relativity. And actually, this turns out to be a weirdly surprisingly subtle complex question about which there's been much disagreement over the last century. And we don't know the answer yet.
We know for sure that there are infinitely many. And this is a job for theorists, the experimentalists who have already found these laws for us. And so we are really way behind on our homework.
And as it goes on, I'll explain why this is such a subtle question. But as Tom alluded to, recent progress on this question has transpired through the discovery of an exact mathematical equivalence between three phenomena which have been studied since the '50s or before. So one of them is the so-called soft theorems in quantum field theory, which tell you about the behavior of a scattering amplitude when one of the external particles, massless particles, a photon or a graviton-- soft means the energy goes to zero-- and the amplitudes are equal to just some number times the amplitude with that particle absent.
But it's a very complicated number here, a function of the external momenta. And this was really started with Bloch and Nordsieck and Gell-Mann, Goldberger and Low a long time ago. But the way we think about it now using Feynman diagrams comes from a really important paper that Steve Weinberg wrote in '65.
So that is a huge subject. No experiment at the LHC-- can-- the results can't be calculated without taking into account the behavior controlled by the soft theorem. It's a huge subject. Soft physics is a huge subject in quantum field theory.
Then there's another subject, a smaller one, started by-- and we'll talk about all of these-- Bondi, Metzner, and Sachs, in which you look at the asymptotic. And in this talk, of course, there is a non-zero cosmological constant. But I'm going to-- and that brings in a whole other host of things we don't understand.
But in this talk, I'm going to be ignoring the cosmological constant. I'm going to think about asymptotically flat spaces and scattering in asymptotically flat spaces as we do at an accelerator. And in asymptotically flat spacetimes, there are symmetries associated with the asymptotic region. And there's been a whole school that started in the '50s and the '60s, mostly in the relativity community, that has studied-- general relativity community that has studied these things.
And then the third corner of this triangle is the so-called gravitational memory effect, which is first written about by Zeldovich and Komarov in the '70s. And it concerns the fact that if you have two-- and will talk more about this later-- if you have to inertial objects and a gravity wave passes by, they'll oscillate as the wave goes by. But then after the wave goes by, they will be at a different distance.
And there are very beautiful mathematical properties of understanding exactly how that distance is different. And that is called the gravitational memory effect. It's predicted by the Einstein equation.
And understanding that all these things are actually different descriptions of the same basic phenomena in the deep infrared has given us insights into other kinds of infrared phenomena. And also, enabled to generalize these phenomena and make predictions for LIGO, for LISE and for the Electron Ion Collider that was just funded at Brookhaven And hopefully more. So weirdly-- usually gravity is the most complicated thing-- but weirdly, in this case, the first discovered of these triangles-- and in some ways the simplest-- is the gravity triangle, which has the soft graviton theorem, so let's start with that.
All right, so the question we're posing now are what are the non-trivial symmetries of general relativity? Now, if you've had a course in general relativity-- and I've even said this in my course-- you might say the symmetry of general relativity is diffeomorphism invariance. If you haven't had a course of general relativity, don't worry about that.
But it's not really a physical symmetry. It's more-- or we wouldn't really-- in modern parlance, people wouldn't usually call it a symmetry, or at least a non-trivial symmetry. It's really a redundancy of description.
To make the mathematics easier, we introduce some extra variables that have no physical meaning, and then we throw them away at the end of the day. And there are no experimental predictions that follow from the most general kind of diffeomorphism or general coordinate transformation. And so now, people would say that that's trivial.
And in fact, there's a formulation of general relativity given by Tullio Regge a long time ago that didn't have general covariants. So that's really a property of how the theory is described. So physical symmetries have associated conservation laws by Noether theorem, and those are the ones that were interested in.
So for example, most of the laws of physics we study are invariant under translations in space and in time. And these have, by Noether's theorem, associated conserved charges, which are energy and angular momentum. And also, from energy and momentum. And also from rotational invariance, we get our angular momentum conservation.
So this is easy to derive in special relativity. But what about general relativity? So in 1962, Bondi, van der Burg, Metzner, and Sachs asked a very important question, which is, how do Poincaré symmetries and the associated charges appear in general relativity. So in general, if you're in a highly curved spacetime near a black hole, there's no translation invariance.
What you mean by energy? It depends on your reference frame and so on. How do you get these conservation laws back? And they were expecting that if you considered a spacetime which at long distances was flat, and maybe had some black holes or crazy stuff going on in the middle, that you would get back the results that we know and love from special relativity. Not so.
So what did they do? So here's the key idea. Here's a picture, and in explaining their ideas, it's nice to draw these so-called Penrose diagrams. And in these Penrose diagrams, time is vertical, space as horizontal, and light rays go at 45 degrees.
And you sort of pull all of the spacetime in so that the place that light rays go and the place that light rays come from is at a finite distance and you can draw it on the blackboard. So the origin, r equals 0 would, be here or on the slide. And r equals infinity is out here. And so light rays come in from here or here, and they go out here or here.
Now, what is an asymptotic symmetry? Well, to define this problem, you have to impose some boundary conditions. And you impose some boundary conditions, and then you do some coordinate transformations.
And the non-trivial symmetries should be, so they thought-- so Bondi, Metzner, and Sachs thought-- should be those coordinate transformations which preserved the boundary conditions, a kind of quotient of all these coordinated transformations, by the ones that act trivially in the sense that they didn't change any observable physics out at infinity. And they thought this should be the Poincaré group. But it's not.
So what they found is they got a bigger group, which was boosts, rotations, translations. But the translations at infinity are allowed to-- so out at infinity here, there's a sphere. And one has allowed rigid translations, which independently move the different points on the sphere in infinity.
And in fact, the reason this happens is that in a curved space-- and general relativity is the theory of curved space. And it's very tricky to compare something that's going on here, like the direction that something is pointing or the time on a watch, with the direction or the time somewhere else. Because if you have 100 people, they'll choose 100 different procedures for how different things will be compared and they won't all be equivalent when the space is curved.
So these are symmetries which are on exactly the same footing as translations in space or translations in time. And they change the physical spacetime. So I could have-- these are called super translations. By the way, this word was used for something else again, in the context of supersymmetry, 15 years later. But this is the original use of this word.
So you could have a photon at 10 o'clock in Ithaca, one coming out in Boston, also at 10 o'clock. And then you do the super translation and the one in Boston comes out at midnight. So they're different. They change time at different places in a different way.
And it's very surprising and in deep contradiction with our intuition that there should be such a symmetry of nature, yet they showed that it was-- because of the difficulty of comparing things at different spacetime points, there was no consistent meaning, no invariant meaning to say that the photon comes out in Boston and Ithaca at the same time.
Now, there's a similar symmetry, a similar kind of action, past no infinity that affects incoming things rather than outgoing things. Now, what do we do with this? So BMS didn't really-- there was a lot of confusion about what the meaning of this BMS symmetry was and what the meaning of the super translations were.
Many people in the general relativity community didn't like it. They tried to eliminate it with some kind of boundary condition. It wasn't understand why it was a good thing or what we were supposed to do with it.
And in fact, they didn't understand how to relate what happened in the past to how you tied these super translations in the past with the super translations in the future. Now, the Einstein equations are really incredible equations with all kinds of hidden structure in them. And in the '80s and '90s, there was really spectacular progress by Christodoulou and Klainerman and other people understanding the structure of infinity. This is 30 years after BMS.
They didn't directly address this issue that I just talked about. But using that work of Christodoulou and Klainerman, it could be showed, weirdly, that if you do a super translation in which you redefine time on the sphere at future infinity by a function of theta phi-- outgoing time, retarded time-- and you redefine advanced time by the antipodal function-- so if you shift something-- if you shift the outgoing time at the North Pole, you have to shift the ingoing time at the South Pole. Sounds crazy.
But it's been proven that this gives you an exact symmetry of scattering processes with any collection of ingoing and outgoing particles. Now, we can phrase this in the language of the S-matrix. So the S-matrix is the matrix that in quantum field theory-- general relativistic, generally covariant quantum field theory-- in fact, this is true even in ordinary quantum field theories without gravity-- that there's an operator B, which commutes with the S-matrix.
An example of an operator which commutes with the S-matrix that you're familiar with is the energy. So if you act on an ingoing state, if the E were the energy, the energy of the outgoing state is equal the energy the ongoing state or the total electric charge. Those are simple examples of operators that commute with the S-matrix.
But it turns out that a careful analysis of this structure here shows that there are, in fact, infinitely many such generalizations of such conserved quantities like charge and energy. And we'll talk a little bit later about what experiments measure them. And so these are true symmetries of general relativity, but the story doesn't end there. That's not all of the symmetries. And I won't really talk about all of them, because I only have an hour.
OK, so what is this more general conservation law? So we have some generalization of time translation where we allow time to be translated differently at every point on the sphere at infinity. We know that if we translate it the same way at every point on the sphere at infinity that we get energy conservation by Noether's theorem. Energy is the conserved quantity that follows from the symmetry of the laws of physics under T goes to T plus constant.
What do we get now? Well, we get that the total incoming energy flux at any angle from the far past is equal to the outgoing energy flux in the far future. Well, that sounds crazy when you first hear it, because, OK, well everybody just take two things. I got one coming from there and one coming from there, let's say from the equator. And then they come in and they scatter and they go out like that.
So clearly I've got incoming energy from two angles and outgoing energy at two different angles. What on Earth am I talking about? That the ingoing energy always equals the outgoing energy.
And the important point is that in general relativity that the energy contains a crucial term, which is linear in the deviation of the metric, as defined by the Bondi mass aspect. It's linear in the deviation of the metric about the flat metric. And we can conclude that it all must-- and indeed, has been proven to be equal at every angle.
OK, so now I'm going to turn to what seems like a completely different topic. So I've just talked about the BMS, the symmetries here. Now I'm going to talk about something that seems completely different, which are these soft theorems.
So there are all kinds of soft theorems. And for every soft theorum. There's a story like the one I'm telling you now. And this soft theorem says, let's take any scattering amplitude-- here I've got n particles, I don't care what they are. And the n plus first particle-- sorry, n minus 1 particles-- I don't care what they are or how they're moving or whatever. And the nth particle is a graviton with polarization epsilon and momentum q.
And I consider this amplitude and the limit that I take that momentum to 0. Interestingly, that is equal to the scattering amplitude with this external line completely deleted. So an n minus 1 particle scattering amplitude times a function here, which is the sum over all incoming particles weighted by this weird combination of-- constructed from the graviton polarization tensor, the gravity momentum, and then the momenta of the n minus 1 particles.
So this isn't, in fact, an exact theorem. It's known to have-- not only it's a classically exact, but it's quantum mechanically an exact theorem. And there are similar exact theorems like that. They're more soft theorems of this type with various degrees of exactness. And there's certainly one in quantum electrodynamics, which we'll come to in a little bit.
OK, so what is the relationship between this? Well, I'm not going to go through with the gory mathematical details. But the statement is this. So we talked before-- we talked about these asymptotic symmetry generator of super translations. There's an operator B that commutes with the S-matrix.
And now we sandwich it between any two incoming states and any two outgoing states. And that gives us-- if out-S-in was an n minus 1 particle scattering amplitude, then the insertion of B here gives us more particles because it's an operator in the [INAUDIBLE] space. And so it relates-- this is an infinite number of relation between different scattering amplitudes in any-- and one of the important things about Weinberg's theorem is it applies to any quantum theory of gravity-- string theory, loop gravity, M-theory-- any quantum theory.
You just have to assume that the theory exists and that it has to obey Weinberg's soft gravity quantum theorum. So now, I'm not going to go through the details here, but the statement here is that the soft graviton theorem is the Ward identity associated with the supertranslation symmetry. So this relates, oddly, to papers which were written within three years of each other in the '60s.
Now, this has some interesting conceptual consequences. One is that the metric is not preserved by super translations. And indeed, you can change the angular momentum b a supertranslation.
And another way of saying that is that you can take a soft graviton, one of the things that Weinberg was considering, and it has zero energy in the limit where q goes to 0. You add it to the vacuum. A single graviton always has two units of angular momentum, but if it's completely soft, it has zero energy. So you're changing the angular momentum of the vacuum by 2.
So this implies that the vacuum-- and one can understand this both classically and quantum mechanically from this type of reasoning-- that the vacuum in general relativity is infinitely degenerate. There is not a unique vacuum. And supertranslation symmetry is spontaneously broken because the generators of it do not annihilate the vacuum, they change one vacuum into another vacuum.
OK, so it's fascinating that these two essentially equivalent discoveries were made a few years apart. And I asked Steve Weinberg about this once, if he thought this was a coincidence. And he said at that time, I'm not sure. But at that time, everybody was thinking about symmetries and what can we do with-- how far can we push them and how much can we get out of them.
OK now, I'm moving to the third corner of the triangle, the gravitational memory effect. What is the gravitational memory effect? Well, I've written down the equation describing it here, but I'm not going to go through this in detail. Let me just describe what it does.
So you have to inertial particles just floating out in space, and then a gravity wave comes by. And there's some distance apart. And after the gravity wave comes by, they differ by both angle gets twisted and they're at a different distance apart.
Now, we can understand this as-- in the next [INAUDIBLE] I'll explain how it's related to the soft theorum-- but we can understand this as being related to the vacuum degeneracy. So since there are these infinitely degenerate possibilities-- different vacuum in general relativity-- if you bang spacetime, it settles back down to a zero energy space-- say, by pushing a gravity wave through it-- it settles back down to a zero energy space, but it won't be the same one.
And so it'll differ by a supertranslation. And that's what the equations on the right hand side show, that the vacuum will differ by a supertranslation. And the positions of the particles in the supertranslated vacuo will be different.
And so when you see the particles moving after the passage of a gravitational wave into a different spot, what you're really seeing is you've sort of put a tag on the vacuum state. And you're tracking how it moves from initial vacuum to a final vacuum.
Now, so this gravitational memory effect has not been measured. My understanding is that LIGO is hoping to measure it in the next 5, 10 years. eLISA will also be a very good place to measure it. And there are other possibilities for measuring it the pulsar timing arrays, which see time delays very accurately.
And so this is a young audience, I think many of you will certainly live to see gravitational memory seen. And I think it's really interesting to see it. Now I'm going to close the triangle.
I described how gravitational memory was related to the symmetry transformations. But now I want to close the third side of the triangle. And I could have done this first. It's actually the easiest one.
The third side of the triangle gives the relationship between the soft theorem and the memory effect. And what I've done here is I've copied a paragraph on the left out of Weinberg's paper for the pre-factor in the soft theorem. And on the right, I've copied a paragraph out of a paper by Braginski and Thorne from 1987. And what is the main difference?
Well, Weinberg was a particle physicist. They always work in momentum space. And relativists always work in position space. And so these formulas, to relate from one to another, you have to do a Fourier transform.
And the other difference-- so this formula was written down in '65, this is '87. The fact that they were the same formula was understood less than 10 years ago. It's also amusing that Weinberg was scattering elementary particles and Braginski and Thorne we're scattering black holes and stars.
The fact that the same formula describes both of these things is a testimony to the universality of the symmetries, They apply to everything. They apply to little things, they apply to big things. The whole structure of this scattering is completely dictated by the symmetry whether you're studying black holes and stars or whether you're studying elementary particles.
And so the nice thing about having this third corner is it shows that the whole discussion is not just about mathematical formalism. There's measurements that we can make that will test these ideas and test this very beautiful symmetry structure underlying the physical world around us. Now, this is not the only triangle.
There's soft theorems in gauge theories, there's soft theorums in QED. There's also subleading soft theorum. The first subleading soft theorem was discovered by Francis Low in 1958. And there's sub-subleading soft theorems.
And then, in fact, now we understand their whole infinite powers of soft theorems. And these both constrain the world around us. They give a lot of constraints on the world around us. They make experimental predictions.
So for example, there's a subleading soft graviton theorem. And it implies another kind of memory effect that you see with orbiting objects. And Nicholls has made a proposal for measuring that in LIGO. Maybe it's a little optimistic that it will be measured soon, but it's-- and I'll talk about some other experimental consequences.
But I think also understanding these symmetries, I'm hopeful that it will help us with other puzzles that we have in trying to put together general relativity and quantum mechanics and black holes and so on. OK, so now I'm going to-- for the next example, I'm going to turn to QED, which would seem to be-- you would think would be just simpler, but in a way, it's more subtle.
And I'm going to-- there are many ways to describe this. And I'm going to describe the whole structure in a complementary way in the hope that you-- so everything I'm going to say-- I was going to talk about gauge theory, but I think I'm going a little too slow. I was going to talk about QCD but I think I'm going a little too slow to get to that.
And I think everything I'm going to say for the rest of the talk basically could be understood with Jackson electrodynamics, who doesn't mention that there are infinite number of conserved charge in electromagnetic systems. But it follows from the equations in that book.
So let's start here with just the 1 over r squared potential Frt, electric field, following for a point particle. And now, let's use special relativity and we'll boost it. Now, there's something very strange about this formula for the boosted electric field. And that is that if you go to-- we want to ask what it goes to at infinity. But we have to say what we mean by going to infinity.
So the two different things-- if we're talking about things we observe in the future, something that comes out of accelerator as opposed to what we send in from the past, what we want to do is hold the retarded time fixed and send r to infinity. Conversely, if we want to talk about what we had in the past, we need to hold the advanced time fixed and take r to infinity. So we need to understand-- so what is F doing when you take r to infinity? It depends what we hold fixed.
OK, so here's the crazy thing. If we now boost the particle, here's what the world line looks like. And if we hold advanced time-- so just look at these formulas here, if you substitute u equals t minus r and then take r to infinity or you take v equals t plus r and take r to infinity, that corresponds to going up to here and then down to here, which is spatial infinity. Or starting here, going down to there, and then going up to there.
You don't get the same thing. You get a different sign down here. So the electric field of a single boosted particle is not single valued at spatial infinity. That seems really weird, but it does obey a different-- if you think about it longer, it's not weird anymore when I first understood it. I'm not the first one to understand this, obviously.
But what it does obey is it's antipodally matched. The electric field back here at the past of the future is equal to the electric field at the antipodal point at the future of the past. So there's an antipodal matching.
And here I've just written it for one charge, but it's actually true for any-- it's a property of Green's function, you can prove it it's true for any-- it's any collection of charges with any use, with any Green function. OK. OK, so now, I just told you that there were infinitely many things that were equal. The electric field at the-- OK, so I'm now going to find this function, an arbitrary function epsilon on the sphere, with the property that it's antipodally matched from the past to the future.
So I can write now a conservation law. I take the electric field at the past of the future, and that equals the antipodally-weighted with any function, and it equals the antipodally-weighted electric field at the future of the past. Now, that's an exact conservation law for any process.
And for the special epsilon equals 1 using Gauss's law, this equates all the incoming charges to all the outgoing charges. So what on Earth is it for a general function epsilon? So we're discussing here some infinite dimensional generalization of global charge conservation.
Well, what do I do now? I take all the incoming charges and I weight them by an arbitrary function of the angles. And I add to that a very strange term coming from the electric field. And that equals all the outgoing charges plus a related outgoing function.
What is this strange thing? Well, it involves the integral over all the past retarded in time, the integral over the sphere-- sorry, all the past advanced time, the integral over the sphere of the derivative of this arbitrary function times some component of the electromagnetic field, which is actually transverse to the sphere at infinity. If there were an e to the i omega v here, we would call this the wave function for a photon.
So this is a conservation law just like any other. If you're an experimentalist and you take all the incoming charges-- you weight them to any function, all the incoming charges. You form this sum and then you look at the electromagnetic field and you do this integral. You have a prediction about an infinite number of predictions about what will come out at infinity.
OK, so at the quantum level, we say this in terms of an S-matrix-- an outgoing state, which is the S-matrix times an incoming state-- and there's a charge which commutes with the S-matrix. And it tells us this charge has a term, which is linear in the electric field. It's at 0 momentum. We can think of it as the limit as the energy goes to 0 of the operator, which creates a photon of energy omega.
So it's precisely the zero energy limit of out-S-in with an extra photon added. And now, you can write this all out in detail and see how everything works. And indeed, this is exactly the soft photon theorem, which Weinberg wrote in the form we like to usually refer to it in. But really, it was discovered much earlier by Francis Low or some sense even much earlier by Bloch and Nordsieck.
Conversely, you can reverse the logic. The soft photon theorem, which is often thought of as some kind of technical relation among diagrams involving photons in quantum field theory, can be used to imply that there's an infinite number of these conserved charges, symmetries in nature. OK, now whenever you have conservation laws, you have symmetries. I worked it the other way before.
We have this operator. We can see what it does to the electromagnetic field at infinity. And what it does is it shifts it by the derivative of this function epsilon. So the associated symmetry is a gauge transformation, which acts on the electromagnetic field at infinity.
So usually, when we talk about quantum electrodynamics, we don't consider such gauge transformations. But here, we demand that the gauge transformations go to 0 at infinity or at most, a constant. But here, we see that they play a natural role in the theory.
They are these large gauge transformations. They do act non-trivially on physical states. And they can be measured in an electromagnetic memory effect.
So in gravity, the passage of a gravity wave acted with a diffeomorphism on the particles that it passed by. In electromagnetism, what happens is in the quantum theory, the quantum state associated with a particle of charge q, gets a phase, according to a gauge transformation. So what happens here is if some energy pulse electromagnetic wave passes by and we have a pair of charged particles, they get a phase. They will get a relative phase instead of a relative displacement.
And relative phases are trickier to measure. It's kind of like that's what you do in the Aharonov-Bohm effect. But we'd like to measure these. Yeah, OK, so this is what I was just saying.
So we have some stuff coming by. We have a pair of electrons. And after the stuff has gone by, it may cause them to move around. But after the stuff has gone by, they get a relative phase, which again, is measuring the vacuum transition in quantum electrodynamics.
So those are the three corners in the case of quantum electrodynamics. Now, what about measuring the electromagnetic memory effect? Well, it hasn't been measured. It should be measured.
Now, there's a similar question that-- you've all heard of the on Aharonov-Bohm effect. There's some proposals to measure it have been made. There's been a lot of discussion of measuring it, but I think these proposals that were published were just to show that it could be measured in principle. They weren't really aimed at an easy-to-build experiment.
But actually, there is a related endeavor in the literature out there, which is there's something called electric Aharonov-Bohm effect. Where instead of having a loop in space as you do and the usual Aharonov-Bohm effect, you have a loop in time. And actually, the electric Aharonov-Bohm effect has not yet been measured.
Nobody's wondering whether it's true or not, but it hasn't been measured yet. And so it's akin to measuring that. In fact, electric Aharonov-Bohm is kind of a subset of this more general electromagnetic memory effect.
There's a proposal to-- there's also something called color memory in gauge theory. But let me also say, just to give you some idea of the different directions that these ideas are taken in, that this has some information for-- has some implications for the infamous black hole information paradox. So one statement of the black hole information paradox is-- you what is it?
You have a black hole that's sitting there. It radiates quanta and slowly evaporates. And Hawking actually calculated this. And in Hawking's calculation, the quantum states of these particles, they're just in a thermal ensemble. And they're almost, in his calculation, completely uncorrelated with the quantum state of the black hole.
So usually, when there's some process in which some particles, with [INAUDIBLE] particles, there will be quantum correlations. Like in the famous EPR experiment, angular momentum says that the spins have to be correlated of two photons produced from a spinless object. Now, why did I say almost uncorrelated?
Well, even in his original paper, though it didn't exactly follow from the calculation, Hawking presumed that this process would conserve energy exactly at the quantum mechanical level and momentum. And that is saying that that's an exact-- not just on the average, but exactly-- and saying that that's an exact quantum statement forces quantum correlations between these. And now, we see that there are an infinite number, and it tells you that however much charge these Hawking quanta take out, there must be that much less left on the black hole.
But now we have an infinite number of conservation laws, which are numerous enough. If you have an infinite number of conservation laws, that's enough conservation laws to force a quantum coherence between the Hawking radiation and the black hole. Which is exactly what-- that's the Holy Grail of this subject is try to understand how that happens.
Now, it hasn't been shown that it does happen, but it's certainly on the table and under discussion. I should also add that there's a very rich and exciting problem. And your own Tom Hartman has done some really beautiful work recently on this problem. And moreover, I'm going to be talking more about black holes-- though I'll be giving a much more general talk, it's not just physicists-- on Wednesday.
And then tomorrow, at the theory seminar, I'll be giving a more specialized talk on recent progress. I said that there are many more of these symmetries. What are all the symmetries? The question I started, what are all the symmetries of that nature?
It's our homework for the theoretical physicist to find all of them. We haven't done that, and tomorrow, I'll be giving a theory seminar on some recent progress in that direction. And so in conclusion, many theoretical physicists have been working on understanding the laws that we don't understand beyond the standard model, quantum gravity, string theory, and so on.
But there are really many things that we don't understand about the laws that we already have measured and verified experimentally. And a clean understanding of this is relevant both for upcoming experiments and for further progress on fundamental issues and black holes quantum field theory and quantum gravity. And finally, thank you for listening.
[APPLAUSE]
TOM HARTMAN: So a couple of questions. Before we get started, I want to remind the graduate students and postdocs that there will be an opportunity to meet Andy after we're done here. That'll be in room 403 of the Physical Sciences building. OK, so the questions.
People can raise hands on Zoom or we can start here in the room. Yeah, go ahead.
SPEAKER 1: So how important is the [INAUDIBLE] Lauritsen signature here? [INAUDIBLE]
ANDREW STROMINGER: Yeah, that's a great question. His question is, how important is Euclidean signature here? And I think it's-- sorry, how important is it to be working in Lauritsen signature? And can we also understand these things in Euclidean signature?
And I would say it's really of fundamental importance. Now in principle, you can understand-- if you understand things in one signature, you could translate it to another signature. But all of this has to do-- in fact, maybe I take that back. I'm not even sure it's possible, in principle.
All this has to do with is the nature of-- infinity and Minkowski space is a really weird place. There are five kinds of infinity in Minkowski space. There's the place that light rays go to, there's the place that light rays come from, there's the place that you'll go to if you live forever. You'll never be able to catch up with light rays.
And there's a place that you came from if you had always been alive. And then there's the place that, at a moment of time if you could travel way faster than the speed of light, that you would get to. So there's five places. Those places are all different.
There's a very complicated group, this BMS group, that you might think that infinity is a simple [INAUDIBLE] place, which it actually is in Euclidean space. There's not really any structure at infinity in Euclidean space. But in Minkowski space, infinity is like a crazy complicated place with all kinds of interesting structure, which really limits the kinds of laws of nature that can be consistent.
So I think it's-- Euclidean methods have been incredibly important in quantum field theory for certain kinds of questions. But there are certain kinds of questions that it is really bad for. And I think everything that I've discussed today in a way-- and there is a tendency among quantum field theorists-- less so among general relativists, but among quantum field theorists-- to think of everything as though-- that's also changing recently in the last 10 years, I would say-- but to think of everything as the old axiomatic field theory program and so on-- to think of everything as essentially being defined in Euclidean space.
And what we're seeing now is that they're really central aspects of both quantum field theory and quantum gravity that we really certainly have missed using those point of view. Whether we could somehow reconstruct it from that point of view now that we know the answer is a different question. But it's not the simple way to think about it.
PAUL GINSPARG: Can you hear me?
TOM HARTMAN: Oh, he's up there.
ANDREW STROMINGER: Where is he? Oh. I [INAUDIBLE].
PAUL GINSPARG: [INAUDIBLE] so you can hear me. Probably nobody else can hear me, but I can say how proud I am of my academic nephew.
ANDREW STROMINGER: They could all hear you, Paul, be careful what you say.
PAUL GINSPARG: Oh. As well as all of my other academic nephews and nieces. So one thing that perplexes me is the ability to make the measurements of this gravitational memory / if I understood correctly, you didn't have an experiment to do that or you couldn't suggest an experiment to see similar effects, even in the case of QED. So--
ANDREW STROMINGER: Whoa, whoa, whoa, whoa, whoa. You have to listen to your nephews a little more carefully, Paul. OK, so I mean, you talk to Saul about how they're going to measure it. I mean, they're planning to measure it.
Now, so this is a picture here. And what happens is the distance between the mirrors comes in and oscillates like this. And then according to general relativity, it doesn't quite go back.
Now, the memory is this height here. Now, one of the things that would be very confusing if experimentalists had not already discussed how they were going to measure it-- if one was just in pure theory land-- that is, how can you measure something that is infinite wavelength or zero energy? How do you measure that? Is that what was bothering you?
PAUL GINSPARG: Yeah, you didn't--
ANDREW STROMINGER: Well, we can see the answer in this picture. And what happens is, in fact, at LIGO, the mirrors are not inertial. They're on pendulum. And the pendula swing back.
And so what I've drawn here is the timescale for the pendulum to swing back. So what has to happen is you need a separation of scales. And the scale for the event has to be small compared to the time scale for the pendulum to swing back.
And in fact, it seems to me-- and if that's true, you don't literally need to see this go on forever in order to be convinced that there's a memory effect. You just need some separation of scales between this and the pendulum swinging back. Now, if you talk to LIGO people, it turns out that that separation seems to be something like a factor of 3. And it sounds incredibly hard, but these guys are geniuses.
And so they seem to think that they're going to get this sooner or later. Now, this, of course, is an old prediction made in 1974. And by the way, for the same reason you can see-- I didn't go through this part-- but for the same reason we wrote a paper with this guy at Brookhaven at the electron ion collider in QCD-- of course, there the color is confined so how could you see color memory?
Well, same thing. You have to see it. You have these quarks, the valence quarks, the hard valence quarks because it's in deep [? inelastic ?] scattering on the Regge limit. There's a regime in which it looks like classical QCD, or so they say. And therefore, there's a hope of measuring also color memory
I'm sorry, can you talk louder? The mask--
SPEAKER 2: [INAUDIBLE]
ANDREW STROMINGER: OK, so if you add a small cosmological constant, it is an entirely new ball game. And you want to properly take it into account. If you have a small cosmological constant, it's not small at large distances, right? We're going to very, very large distances.
And instead of five kinds of infinity, you then only have one kind of infinity, or maybe two kinds of infinity depending on exactly what you're thinking of doing. So that is an important open problem, just understanding, what are the symmetries of a universe which is the [INAUDIBLE] in the far future? And it's-- which we believe our own universe is.
It's an argument, which is conceptually fascinating for many reasons. But it's also-- it's a, sorry, question which is conceptually fascinating for many reasons. But it's also less connected to experiment because very few experiments we do are affected by that cosmological constant.
TOM HARTMAN: OK, I think we're out of time. Let's thank Andy again.
[APPLAUSE]
Professor Andrew Strominger, Gwill E. York Professor of Physics, Harvard University, presents "Memory, Soft Theorems and the Symmetries of Nature"
Beginning in the 1960s a variety of exact “soft theorems” were discovered in QED and general relativity which universally govern the scattering of photons and gravitons at very low energies. Around the same period, general relativists discovered a mysterious infinite number of “asymptotic symmetries” generalizing boosts, rotations and translations. A decade later it was found that the passage of gravitational radiation produces a “memory effect” in which inertial objects undergo a relative displacement. In the last five years, it has been shown that these three seemingly unrelated phenomena are in fact mathematically equivalent, and moreover have ubiquitous extensions.
