12th UK Foundations of Physics — Antimatter, QFT & Noether's Theorem

0:00 Thank you. This transformation, per se, doesn't have a localisation at all. The way we localise it is we require a tree to best like this. And then we localise it here. Well, since you have to know, this is before deo and algebra, you can represent these in any basis to get the name. So, there's an equally valid elementic representation. Why not give the elementic basis, So the empty bases or the X plus 130 squared bases, or they're not many bases united. Third problem, there's an artifact of the one particle situation, it's that the quantum wave function looks like a new space. It doesn't, it's an configuration space. It happens, configuration space is like a more physical space, when you've got one particle, only you've got one particle. So really, you'll be able to think about that. And then if you want to get into that, then presumably your key gauge in coordinates depends on x and y. If you do that, you don't have that much of the consistent. If you do that, you get up with g-sign in an a-field, which is a function of accumulation space. Everything's about that. The time, the only problem is it doesn't simulate it, which works in reality. Continuing, this now makes the sum of Poirot-Hogler's making. the phase symmetry in quantum planes has a large level space. It's completely universal. You don't need functional space, or a cubit has better than phase symmetry. It's generally taken in some sense as being unphysical. The phase rotation is generally taken to just take the state to a different script at the same stage. And we can make that direct because there are some extremely natural reformulations of quantum mechanics which is just eliminate, and we can replace the sign with the projection operator, for instance, in which case it takes, that's not what we might replace, which just vanishes. Or we can just say, well, we're not considering the quantum state, we're considering the ray of space.

2:30 Both of those are making action by the Cholkin. In both of those formulations, there's nothing to think of space symmetry. And the heuristic doesn't seem to fit the situation for something like Jan Mill's theory, something like colour symmetry would be the obvious example of this where we see the dating not this general quantum phase, but it's a much more specific quantum property system that you sort of require, I suppose, to have some freedom in terms of freedom, and you've got some red, green, blue, three dimensions of space, and what we get to these rotations like that. And there, obviously, not any random Pupid has a colour symmetry, between them and our symmetry is something that's special, and if we just gauge the general phase of the symmetry, there seems to be rather unpleasant personality between the electromagnetic E1 phase of symmetry on one hand and the other phase of the other hand. Okay, so those are some reasons why I think back here is the effort to be treated with extreme caution, not just in front of them all together. There's an alternative route to unsunggaged, of course, which I'm going to talk about in some detail, could be viewed as transforming the level of a classical field. So I'm going to just quickly review that and I'll suggest that that's the better starting point for understanding what's going on the Nullvallicistic quantum and sort of the gauge principle of the Nullvallicistic level as soon as coming out of that rather than running into it. So for the moment we're going to see a completely classical situation. We've got a classical field in physical space. So the classical field satisfies some field equations, it doesn't matter what, it's got some internal degree of freedom, so the A is indexing some internal space, colour space, so the B, thigh, actually, in spite of what type of tree, it satisfies some equation, we don't care what, some group acts on the thigh and its symmetry group, so it leaves the field invariant, and it's an internal group, so it doesn't sort of take 5x to 5y, it just re-imagines the indices of 5x point, and it's irreducible, which is just to say that we can't, that's an internal space of indices, we can't decompose it into subspaces such that any subspace is in their original dimension, okay.

5:00 Then you can imagine localizing that symmetry, the representation arc is, just to remember, the group in abstract, the representation is some way that's going to see it matrices to the group so that that matrix acts on the space and the algebra of the matrix satisfies the group algebra. So, it's often the law of the difference between the big factors we find in this case. So, here's the representation acting in the standard global case. If we localise, then we allow the group help to achieve both point and therefore the representation can achieve both point. Again, that doesn't generate the group relation invariant. Again, we make a transformation like taking the site of SMA class IA. That's the most ill form of it. a certain transformation relation or a pose that relation, then we feel the equation becomes a day and a day. The actual form of the relation is somewhat unattractive. I'm not going to use it too wide as written as well. Okay, so I think it's actually, that situation is in a much stronger position than the element of divistic case, because the internal space transformations, even if they're something like a complex transformation, are describing something which really is a property of the spatial field. There's no worry about it only accidentally being intubated because of being one particle. It's straightforward with classical field like electromagnetism. And the complex space transformation is now on a part with something like a colour transformation. So if we had, say, a colour field, it might have three intermobiles of freedom, and we might look at transformations like that. If it's complex, well a complex field is really just an ordinary pair of real fields, and the transformation is just a rotational kind of space. So that's all fine, the only problem is we don't want to deal with classical fields, we want to deal with quantum particles. So, however nice that is, it's not what we want to finish up. So when we do want to finish up, we need to see how a theory of this form behaves when you quantize it, and then look at the particles that emerge from that quantizing area. For obvious reasons of time, I don't have the time to do a very detailed description of how the field quantization and how the emergence of particles proceeds. So this is going to be some quantization light. I'm going to start by assuming the field equation is linear.

7:30 The reason for that is particles are an emerging phenomenon you get in weakly interacting circumstances. If you're going to see that they're strongly interacting, then you just got some mass of interacting stuff. It doesn't really have a clean part of the system. More accurately, you have a lot of different particle descriptions, different ones that tell you different things about what's going on with the stuff. Particles are emergent when you haven't got that situation, when the dust has settled. That's not the thing that controversial I came before, but it's not going to be time to do it now. like solid-state physics, where if we have, say, a crystal whose vibrations are linear or nearly so, then you can describe the system in terms of phonons, quantum vibrations that interact a little bit. As you turn up the non-linearity, you interact with it bigger, and when you think it's bigger, the phononol description is less and less helpful. It comes to a point where you might want to throw away the description and try to find some article in the description So, solid state physics is actually analogous to quantum field theory and it's the same sort of So I'm going to use the theories exactly, if you want to imagine there's a little perturbation term then the other head doesn't make a massive difference. So, I take classical field, I make some sort of analysis into modes. So, this should be any standard from just analyzing a classical field, you've got modes that you've got a mode here that means you can sum up the normal modes of the equation, different modes have different frequencies, and right through all of this I'm requiring that the fields are real, so I mean anything that comes to this is a mode in the first place. What this is after is the time you're in the field, just what degree of freedom, the modes are just the same ways. We quantise, and again this is not something I'm trying to justify. The quantisation algorithm that we use is just that we replace the coefficients of these modes with constant operators. We require them to satisfy connotation and relationships like that, or anti-quantation and relationships like that. Once you do that, this thing becomes a quantum field observable. I've suppressed the time for all of it because I want to work in a future perspective. So this thing is now an operator. It's not a description of a particular physical state. It's an abstract law-like observable

10:00 system, but the algebraic observables for the classical theory were generated by the of the products of the engulfurators, so the algebra of the reserve balls in the pocket case were generated similarly by things like this. So that's quantisation, we haven't talked about things yet, about particles yet. The particle story, I'm going to talk to you, so physically I'm going to specialise to a masonic case. The situation purposes of looking at the one is not valid for a femoral case, there's a little more complicated, and I don't think it has a fundamental film that comes to me. So, I've got my field observable, I've got a halftime in the quantum state built up for the field observable, I've got all sorts of problems with the normalization and infinities that we can completely ignore, and formally at least I've got a backlink state, which is just the lowest energy state of this whole mass system. So far I've learned about particles. If I want to start with the particles, well one quick way to see where the particle structure comes from is to look at how we might have the structure and then build markers up, I see the positions of those. There's quite a bit of slight of hand in that, because by no means there's a low-transportage state in particular, but it happens to work in the particular structure I'm using. And easy way to do that, you've got a field of circles. The field of circles needs to describe materials with the stripes, stuff around the point X. So act with the field of circles on a vacuum, and you expect to get a particle local, some of the things are localised around the point X. So you apply that. The phenol-beservable is something like phi equals some AK joint AK plus AK AK. And you can see if that thing acts on the vacuum, then this section is about to annihilate the vacuum. So at the end of the matter, it is here. So these low-flying states will be just generated by the action of a composter-free-kits part in the field of the state that has that. Suppose for now that those states really do count, as in some sense analogies to position-oriented states in numberistic quantum mechanics, so build-up, general particle states in the same way, just integrates over a class of these form, meaning there's a sum over A in place in there as well. So, if you do that, and you analyze them effectively, you can get a transformation to a momentum basis for the particle states.

12:30 This is the position representation, this is the momentum representation. And in the momentum representation, you can see that you've got something quite structurally analogous to the original classical way of what you started with. I can say the classical field you started with. Here you've just got almost choice of alpha k's, and here you've got almost choice of alpha k's and you've got the conjugate of the function included on the other side. So there's a sort of isomorphism between the classical field configurations and the quantum one-particle wave functions, which is what often gets called field-particle duality. But the reality is a bit subtle because it's not between just the configuration space, it's between the configuration space and the one-particle space. It's between the complexification of the configuration space and the one-particle space. I mean, that's not completely true, but you can see the reality condition you impose on this effectively cuts down the sort of notion span, these things, to just a real subspace of the space of complex solutions to my function. And this thing just would span the whole set of dots. First of all, what's important about it is it's roughly speaking political. In other words, if I've got this classical solution, and I've got this quantum state that's isomorphic to it under the isomorphism, And if this classical solution is localized right here, then this quantum particle is also localized right here. So, effectively we can treat the solutions of the complexified classical theory of gradient just being the particle space. And that sort of gets down in one step often in discussions that the words here as a certain . Let's do that. G, remember, just acts on the internal space of these classical field states. We have an isomorphism between the classical field states and the quantum quantum particle states. The space is the expectation of that, so if the internal space of the field, So, for instance, the color that appears in three dimensional space is B, then a classical

15:00 theory is basically an association according to B to every spatial point. So, for just the real trigorgon theory, it's just a number to every point. For some sort of color field, it's a point in color space to every point. So, the one part of the waveform can associate a point in the complexification of that internal space, with the test, and then the action of the group extends from the reason they speak to the classifications of the reality. So that seems to suggest the following picture, that in the one-particle case, where there was previously a real field with some internal space, there should now be a complex one-particle wave function with the same internal space. And that's just fine for something like that. That says that we start with a color field, the particles, and the particles in some internal space, and the particles in internal space. That's no problem. But that's not the stone we'd like to see for the complex field we started with. If we take a complex field like the electron and we quantize it, we don't see some internal space of the electron degree of freedom. We see electrons, and secondly we see positrons. There's two lots of things, but they're clearly distinguishable. It's not just a matter that could be the convention, what's the electron, what's the positron, we take each other one in terms of the other thing together. So, what's going on there? And to see what's going on, we could look at how this whole process works specifically for a complex field. So, here's the picture. We've got some field by its classical, it's the length of the linear, and a complex. Now, we quantize, we get this one line of complex. Now, and this is where we came in, I actually has two roles here. I instituted classical symmetry, and it's also acting as a quantum phase transformation. And I'm just going to sort of cut through the diffusion between those roles and distinguish the two sharpness. So I'm going to reformulate the classical theory. Where I had a complex field, I'm going to take it all in care of real fields. Where I act with I, I'm going to act with this thing here. That's mathematically completely equivalent. If you imagine that the two real fields are the real and the imaginary component, So you don't act with this thing on the real component, you can get the imaginary component and the imaginary component and you can get minus the real bit, I squarely minus one. So this is mathematically a completely equivalent structure, but now we distinguish i, which I'm just reserving for the complex structure you get in conversation,

17:30 from j, which is this specific thing acting on the classical fields. And the new one symmetry is then generated by j, so the one particle symmetry is generated by the complex action of j on the complex side in terms of the two real functions, with complex side we now have one part in terms of the space which has two complex functions and j acts on between the two complex functions. But, I specified that the representation of rubies we take was irreducible, the way they're not really talking about classical fields. The reason for that to be simple, if it was reducible, basically you'd have two fields. You'd be calling one of the same fields, but actually you'd have... They wouldn't be connected in any way, they're just fields which have never same mass. And we actually see this in practice in the, um, the flavor symmetry appears to be a case where we call a reducible representation of the pedagogic gaze groups, and, you know, white water, that's always there sort of stuff. So, we require it to be a reducible just to avoid redundancy. But, is it the case that a, an irreducible real representation is still irreducible when you collect it? And the answer is no, it's not. It's a very simple example, which is that one we'll look at. Look at U1. To get the feel for a minute, R2 takes the internal coordinates of this thing, but just if you think that is abstract, if you'd like. So U1 is represented in R2 just by ordinary rotation of R2, and that representation is in Tucson. That's pretty intuitive. If you reduceability means there's some subspace that's left to vary, and the only subspaces of R2 are dimensional, clearly there's no vector vector plane rotation. So that's, um, that's irreducible. And here's the denominator. Now we flexify, and if you flexify this matrix, as I'm in France, um, the other things are on positive outside. They weren't in the real space, obviously, but they are in the vector plane space. And that means that the vector vector excitation is irreducible because the vector b plus when you act on it with this transformation just goes to e to the i in theta b plus and b minus goes to e to the minus i think to the b minus. So when we complexify this irreducible representation, we actually get out a reducible representation, which is quite threatening.

20:00 And if you now go back to the quantized complex fields, well of course the symmetry group is just u1. So what we find is our irreducible description back on the classical level has now gone through a reducible description. subspaces of the one-particle space on each of those subspaces, the group acts irreducibly, and it acts differently in the two cases. So we don't have, as in the case of Tallahassee, a situation where there's this transform arbitrary between the inter-particle space. We have a sharp, independent division of the one-particle spaces to the spaces, and Christ And by surprise, one of those spaces is positive for charge, in the sense it's associated with the plus-like feature excitation. That's matter. The other space is negatively charged, the same thing with the minus-like feature excitation. That's matter matter. So here's a way of seeing what's going on now. Go back to the number of the case. View one group seems to be doing this. It just seems to be a straight phase transformation. because we have to be considering a space that's not only in the one-particle subspace, that's fine, but also in the matter subspace of the one-particle space. The full transformation of the whole one-particle space is not trivial. It's not generated as trivial, but it differs on the two subspaces. So this is now a bona fide transformation. You won't find it on a random qubit, on a given a random process. It's also relative to significance. It just happens that we were looking at states that are able to charge, so the transformation looks like an organ phase transformation. And more generally, if you don't restrict yourself to the one particle space, if you look at the general space of this theory, then of course the generator just rotates the charge n section by each of the i n theta. So that's very interesting because it shows that sometimes when we quantize we get, in terms of this, we get matter-antimatter. And we notice to see what circumstances that occurs. I mean, the heuristic we tend to use, which is to assume that we've got the real thing, we get matter. If we don't get antimatter, you've got a complex here, we get antimatter. But it's interesting to ask me, should there be some internal symmetry group that generates antimatter in the same way,

22:30 or maybe generates antimatter and meta-antimatter or something like that, and can things fall apart into multiple irreducible representations? Well, there's a general answer to that, and it's very boring, as they've got to discover meta-antimatter perhaps soon. The general theory looks like this. Arbitrary depth space V, this is your intelligence freedom of the field. field, R to G, R represents G irreducibly. Redexify, so RC is just the one you get on the complex kind of space if you extend it linearly. And there are exactly two possibilities. Firstly, the complexification is also irreducible, and secondly that it's reducible but it breaks down to two parts which are conjugate between the other. And the former case corresponds to situations And it turns out that the way of telling what you've got is, it breaks down, the move breaks down this way, or representation breaks down this way, even only if there's some way of extending the representation to include an action of view 1, which is essentially to say it's possible if it was really complex in the first place and we were just pretending it was complex and so it matters. So we actually get a fairly formal way of seeing why it is that, in a complex case, we're going to get the term degrees of freedom, which we know enough from things like the power situation. And I'll stop now. This might relate to the very last thing you said, but suppose that you were considering a theory with a family of realists, with different internal torture groups, and they both satisfied with the second possibility. Would you then, for the two different types of field, be able to make a map between the two, You said that these are the natural ones, at least the earthenatic of the sort of distinction between the natural and the natural ones, and the earthenatic of the sort of distinction between the natural and the natural ones, and the earthenatic of the natural ones. And my snap is that the action is equal to the natural ones, and not predict the world.

25:00 I mean, let's think about actual physical situations, and I don't think there's a fundamental reason that says the electron should write down the protocol rather than the anti-protocol. I think there are obvious, practical reasons why we decided to call third and then we actually matter. But I don't see any kind of natural evolution connected to the world. Maybe we should go and look at this. It's a very easy way to understand that. Can I ask for a minute? It's kind of, yeah. I'll say the wine glass is quite a minute. So, I don't think that's it. A couple of years ago in Belfast, I gave a talk on Noda's second theorem and touched on these issues of how the determination engage theories and so on, but particularly in relation to applications of general relativity. And today I want to look at Noda's first theorem, go back to Noda's first theorem, and just mention one or two subtleties, and in particular some very simple, very, very preliminary applications in quantum mechanics and electromagnetism. This work is based mostly on a paper I built by Peter Holland, but it also touches on some issues in a paper that Catherine Brady and I are writing on sort of a review paper on notice theorems. We've had some wonderful talks this morning, so I really will make an attempt to keep this very short. this is this is the heroine piece i think some of you have seen these this slide before let me just refer to a to a summary about a woman vile just made in a kind of conference dedicated to notice memory to such guy she was not clay pressed by the artistic hands of god

27:30 into a harmonious form but rather a chunk of human primary rock into which he had blown his creative breath of life and the story goes that einstein was at the conference immediately piped up and said to him, well Herman, you're not exactly a picture postcard either, or something along those lines. All of you will know that, I mean, Nerka had several important results In fact, in Carl's talk he gave a beautiful example, a very nice example of what might be called Newton's third theory. That's a phrase that was made by John Erman. It's the claim that if you combine the first and the second theory, that's to say you start with dynamics that have a local symmetry, but you choose the subgroup in that local symmetry, global symmetries, Then you apply Newton's first theorem, which relates to global symmetry for conservation. Then you find that the conservation principle that's coming out of the subgroup of the local group turns out to be trivial. That's Newton's third theorem. I'll give a nice example of that. But of course the first theorem has to do with the connection between global symmetries. transformations that rely on constant parameters, not space-time dependence, and the existence of continuity equations and, in some cases, conservationism. Now, in 1954, Wichler warned against what he called the facile identification of symmetry And this is simply based on the remark that not every interesting piece of dynamics in physics is Lagrangian. And if you have some physics that's not Lagrangian, then you can't appeal towards machinery, and indeed there are cases where you might have dynamics that have asymmetry but no corresponding conservation, something like that. But let's consider the case where the physics is Lebrun.

30:00 Then a number of sort of health-borne equations. The symmetry just may not lead to a continuity equation. And this happens because, well we'll see in a moment why this is, But this can come about when, for example, not all of the dynamical, not all of the variables that appear in the world, in fact, are solutions of oil-legrind equations. Or if you like, not all of the fields are dynamic, but the absolute structure in the problem, and that means that the root of the continuity equation is broken. The continuity equation may not actually lead to a conserved charge. that if you have a continuity equation, which is what you get from Nervous Theorem, putting aside this particular problem, I mean, when this doesn't problem, you get a continuity equation. Well, a continuity equation doesn't automatically lead to this sort of charge. It depends on boundary conditions. And again, we'll see a little bit about that later. But that's widely recognized. But then, there are cases where the continuity equation may be trivial In fact, they may even coincide with mutual equations of motion. There are situations where the charge, the nerve of charge, the conserved charge, need not be real valued. And the nerve of symmetry itself may not take the stakes of the system and the stakes of the system. And it's these three cases here that I want to comment on today, by way of these examples of quantum mechanics and electromagnetism. start by mentioning another slight subtlety, which is, I have the idea that, you know, in a typical case you have a definite symmetry, and at least it's associated with a definite conservative quantity. So, normally we would say, you know, the variance under spatial translations is associated with conservation of linear romantic. But there are cases where which symmetry is associated with a given conservation, independent of the choice of Lagrangian. So let me just give you an example of this, which I came to just recently. A very nice review paper of the inverse problem in the calculus of variations that involves the two-dimensional harmonic oscillator. So you have the two degrees of freedom, Q1 and Q2.

32:30 This is the standard Lagrangian for the two-dimensional harmonic oscillator, and it's invariant under, Here, of course, this is an isotropic system. It's a variant under two-dimensional locations. Here. Here. Ah, this one here. That's good. That should be a plus. Now, it's well known in the literature that there's a kind of a gauge frame that it is the Lagrangian. That's to say, you can always add, well here we're talking about, we're talking about classical particle mechanics, you can always add some total time derivative to the Lagrangian, you get exactly the same equations of motion. In the general case, the relativistic case, it's a total divergence. You can always add up to Lagrangian. It's widely known that Lagrangians are only defined, I mean, they're only unique up to... Once you've got a Lagrangian, you can always create another Lagrangian that gives rise to exactly the same role of Lagrangian equations if you add up, well, in this case, it's a total time derivative. But there are cases where you can introduce what's sometimes called an alternative Lagrangian, which is not related to the first Lagrangian by a total time derivative or, in general, a total divergence. This is the case here. This gives rise to exactly the same point of Lagrangian equation. But it's not equivalent to this one, that's to say the same up to a total time derivative. Now, this Lagrangian here is invariant under what might be called squeezes, sort of a scaling transformation, where you just multiply these Q1 by this perspective, each to the eta, eta is a constant parameter, Q2 transforms to Q2 prime, which is equal minus eta. Now, in both cases, the conserved charge is angular momentum. So, if you say, for example, what is the symmetry that's associated with conservation of annual momentum, it depends on Lagrangian. But the other thing I want you to notice, by the way, is that if you take, for example, this Lagrangian here,

35:00 The equations of motion are invariant, form invariant. Under this, under a squeeze, that's to say the squeezes are symmetries of the equations, but the Lagrangian is not invariant, even up to a total vibration. Just bear that example in mind. And conversely here, this Lagrangian here is not invariant under an O2 rotation. Let's go back to Nertha. We ask ourselves, what was Nertha really doing in the original 1918 paper? Well, she was considering essentially a system of fields, independent fields, psi-i. And for the sake of argument, I'm restricting the dependency and the Lagrange of the first-order derivatives, but Nertha was, didn't restrict herself to that. But I mean, there's nothing, you don't, there's no real loss of generality. Now, we consider, Nerda considered simultaneously transformations in the independent variables and the dependent variables, so it's unlike Hamilton's principle where one has sort of variations in the fields on the dynamical variables. these transformations need not vanish on the boundary of the reason of integration that's another difference between Hamilton's variational problem and the question is well, we can write down the variation in the action which is defined by this that's to say it's just this integral minus the original action where we replace to replace the variables by their transformed versions. And to first order, this variation in the action will take this form, where we end up with a linear combination of Euler expressions with the left-hand sides of the Euler-Lagrange equations. And then we also have a total divergence term. Okay, and Noertes asked the following question, what conditions must hold if the first order

37:30 variation in the action, as a result of these variations, vanishes? Nowadays, there's a generalized Noertes variational problem, and Noertes herself, by the way, suggested later after her 1980 paper that her variational problem should be generalized. you allow for the first order variation in the action to vanish up to a surface term, that's just out to an integral of a divergence. And this just follows, the generalization simply follows from the fact that I mentioned earlier there's a gauge freedom in the Lagrange essentially. So for the same world of Lagrange equations, you can take that are generated by a Lagrange using Calvin's principle, you can a divergence term. You get the same oil and gravity. So this really just reflects this gauge freedom in the dynamics. Okay? Now, you're beginning to see where the conservation principle comes from because, well, in Nerda's case, in the case of the first theorem, we're dealing with transformations that involve constant parameters. I mean, I haven't written it down explicitly, but not functions that rely on space and time, which makes this term here take a somewhat simplified form. But the thing is that if we apply Hamilton's principle to all of the fields, all of the psi i, then this linear combination will vanish because that's just another way each of these e i's is just the left-hand side of the Euler-Lugrange equation. We're going to assume that they're valid. All the fields are dynamical as solutions of Lagrange equations, which means that term vanishes, and you end up with just a surface term, which we get to pull it off with divergence. This has to equal another surface term, so we're going to get essentially a surface term equals zero. And because this has to hold independently of the choice of the region of integration, we're going to have to have but a divergence equals zero, that's a continuity of divergence. Now, the interesting question is, at this stage, what is the connection between Nernitz's condition and symmetry? Well, it's a common place in the literature, well known,

40:00 that if you have, by symmetry here, I mean a transformation of the dependent-independent variables is to take solutions of the Euler-Lagrange equations into solutions. Or, if you prefer, it preserves the form of the Euler-Lagrange equations. That's what we normally mean by symmetry. So the question is, if you have a symmetry of the equations, does that imply there is generalized condition? And it's widely known in the literature. This is not the case. It's just not enough. And we've already seen an example, a counter-example, in the case of the harmonic oscillator. the standard Lagrangian, and we saw that it's not invariant, it sort of squeezes our symmetries of the equations of motion, well we didn't see it but I claimed it. But the Lagrangian is not invariant, so the generalized motor condition doesn't hold, not even up to a service term. And this has been widely known, for example, particularly since Hill's review paper on the variational principle in 1951, Troutman's work, Boya's work, Doty's work, I should mention Joe Rosen's also working on this, and there's a certain amount of disagreement in the literature on this question as to where the divergence term, if any, should come in. I don't want to go into the details, but but it's sort of an interesting side issue. It occurred to me recently that, in fact, in a sense, we're looking in the wrong place. What we really need to be asking ourselves is not under what conditions or what else do you have to add to the symmetry to get the generalized neurotic condition. It's does the neurotic condition itself imply symmetry? Now, one of the things that surprises me about the literature on the nervous universe as long as nothing, you can very, very rarely come across an analysis of this particular question. The only one I've come across, and in fact this was looking at the last few days, is Oliver's wonderful work on the application of weak roots to differential equations, which has a very large discussion of Newton's first and second theorems. all the claims that it is in fact the case that it is a theorem that if you have a generalized nerdy condition

42:30 that's to say you have some transformations as a dependent and dependent variables the first variation in the action vanishes up to a divergence up to a surface term which will have a divergence that implies that the transformations are symmetric I've got a little timid question right there because it's straightforward, perhaps as all it says, or at least there may be some some implicit assumptions in all this proof. For example, ask yourself the following question. If you have in space-time a Lagrangian density, which is a scalar density, that involves a number of different fields, or we know about G and U, which is Lagrangian density, how do you know for sure that the Is it obvious? Remember, the oil-to-grange equations in films contain ordinary derivatives, not co-grind derivatives. How can you be absolutely sure that the equations would be generally co-grind? Well, it's easy to construct counter-examples. But they involve cases where not all of the fields are dynamic, in other words, not all of the fields satisfy oil-to-grange equations. So I'm saying there is perhaps a little subtlety here. want to do myself is to go back and study all this proof. All the claims of the proof is not trivial. So one of the things I want to do shortly is to study all this proof in more detail. But of course, if in ordinary circumstances, the generalized condition implies a symmetry, and the generalized non-condition, in the case of global symmetries, implies conservation principles, then you see the connection. Because just looking at it from this point of view, it's not so clear what the connection is, because it's well known that the symmetry itself does not imply, but just by itself, a generalized neural condition. So the tightness of the link between symmetries and conservation principles becomes weak. Let's just consider the free-shorting equation. This is the typical Lagrangian, standard Lagrangian, written for a Lagrangian density, three-dimensional density, written for the Schrodinger equations. Free Schrodinger equation, that's to say the Schrodinger particle with no external potential. And let's just consider, for the sake of argument,

45:00 the following slightly perverse paradigm transformations, where we just add a constant, complex number to the wave function. Similarly, the conjugate complex number we add to the complex conjugate of the wave function. What happens to the Lagrangian? Well, the Lagrangian is invariant up to a total time derivative. So we're going to satisfy the NERTI condition. So another question is, what is the concert charge? Well, the NERTI results, since one of these fields are dynamical, we are going to get And if we use the standard machinery of murder's theorem, we can calculate what the density is and what the current is, and lo and behold we get back the original Schrodinger equation. This shouldn't altogether surprise us because the original Schrodinger equation is itself a continuity. Because it takes the form of d rho by dt, and then conducts the sin of the 3 vector, where the 3 vector itself is just proportionally degraded weight. So it is completely degraded. So, we push ahead a little bit. So again we have the same symmetry transformation here, we're seeing that we use the and we end up getting back the Schrodinger equation but then we ask ourselves what is the conserved charge well we disintegrate over space and by Gauss's theorem this will become the surface term so if the gradient of the wave function is rather I mean if it falls off the system of the class of the system we end up getting the integral of the wave function so that's the So it's going to be the symmetry. Now, of course, the symmetry is not taking, in general, it's not taking a state, that's to say, where it's going to be waveforms, where it's going to be waveforms, where it's going to be waveforms, but there's nothing in those terms that you have to transform to get to the state.

47:30 Of course, this is not like psi-square, this is a gate. So it's not something you consider going to be as fun. It's not something observable. But nonetheless, this is the charge associated with this particular symmetry. Now, something entirely analogous happened to me with Maxwellian on this planet. Because Maxwell's equations are false and very new in the standard Lagrangian is just, again, a continuity equation. So if we ask ourselves, well, if we had a continuity equation inside theory, it just happens to be the equation so what's the corresponding and the answer is well it's just like the translation to the space of the fourth vector so what we're doing here we're just adding a four vector to the vector potential and of course this is going to be this will be a symmetry this will satisfy the murder conditions, and again one finds that the equation of continuity that one gets at using the machine or it is just the original action of it. And again here, the charge, the sort of charge associated with the symmetry, is just usable as much as long as the magnetic field falls off the spaceship in the past. And this is not, I'm finding it surprising when you remember the comment, Maxwell's Maxwell's equation, just with the derivative, the side derivative of the electric field is associated with the problem of the magnetic field Finally, just another, um, I don't know what to say.

50:00 You can play the same game by introducing the following transformation, where instead of just adding a complex number, you add a number that depends on just the inner product of that vector with the x vector, and similarly with the complex conjugate. And again, these leave the Lagrangian quasi-invariant, so they satisfy the neural condition, and lo and behold you end up getting, as your equation of continuity, just the ordinary Schrodinger equation multiplied by an xi on both sides. So the xi's cancel, you get back the original Schrodinger equation. And here, if the wave function itself falls off sufficiently quickly with Sorry, that can't be right. It must be the gradient of the wave function pulls up with distance sufficiently fast. You get that the conserved charge is the wave function multiplied by the displacement. And again, there's an exact, well almost an exact analog in the case of electromagnetism, where either you now add an arbitrary constant anti-symmetric second-rank tensor to FmU, the electromagnetic tensor, or equivalently you add this four-vector to the vector potential where, again, this second-rank tensor here is constant. And again, you get a continuity equation. incidentally in this particular case the continuity equation does not coincide with maximum equations. Here's the conserved four-currents, but here it's not so easy to see what, I mean I don't think it's by any means straightforward to say that it's a conserved charge. So just to finish off, coming back to the original issue of subtleties

52:30 The continuity equation may be truly in the light of the equation of motion, and the examples we've looked at are cases where the continuity equation just is, just one size of the equation. You might not want to say this is a weakness, I mean, this is somehow a weakness of notice theorem. It's just that what we've really been looking at here are cases where equations of motion take the form of continuity equations, at least have the associated notice theorem. Now, Stephen, thank you very much for the invitation to speak. By now you will have realized that this has nothing to do with recent trends, engaged theories in physics. So before you kick me out of the session, I'm going to stop. Thank you very much. How if in the last example you could also have the... if you had made a change where the wave function goes to psi prime because psi plus a constant c times psi, then you seem to get the norm of the wave function as a control function. Is that why? That would be where we'd have to have normalized wave functions. Well, if you're adding a wave function to a wave function. Yeah. Yeah. And the wave function that you're adding is in other words, So suppose you go, well that's a wave function, so it will depend on space, right? So that's not going to come into the sway of this first thing, but let's just ask ourselves if we can play with this. Suppose that's a wave function, it's not arbitrary.

55:00 Suppose it's a solution itself of the Schrodinger equation. So I start with a solution of the Schrodinger equation. I add to it a solution of the Schrodinger equation. Is that a solution of the Schrodinger equation? Of course, the similar position principle. So the similar position principle itself is a sort of symmetry. okay it's a sort of symmetry but it's going to it wouldn't it doesn't have nothing to do with Newton's first theorem because this thing here depends on pi is so it involves functions but you might think well does the superposition symmetry have something to do with the second theorem, which actually does allow for functions rather than constants. The answer is no, because, once again, this does not satisfy this transformation. It does not satisfy the nodal condition. It's a symmetry of the equations of motion, but it doesn't satisfy the nodal condition. In other words, the Lagrangian is not invariant of two of daguerrions. So this symmetry has no role to play. I don't know whether that was true. Can I tell you a doubt on that? You can always put the A for that 5, that 5 just happens to be an auxiliary field that we've chosen to be a solution to show the information. So the A is the parameters that you enter here symmetry and respect and calculate the conserved charges respect to it. I'm conjecturing that the conserved charges respect to it is the inner products have been signed by if only started with the math. So you're saying if you have some action like that, doesn't it? Yeah, now you're allowed to calculate the conserved charges once you get to that. that value size, value size, value size, and I might be wrong on that line. I don't think that's going to be a nice solution. I mean, if...

57:30 So I can thank all the speakers for this morning, it was really excellent as well. Thank you. And then leave it with the red button, that's all you need to do. You need to press nothing else, you need to adjust nothing else, everything will be set up. And then just leave it, and it's an automatic reverse, so it will reverse automatically. half speed it should record after three hours so you won't have to do anything but just before the first speaker just press the red button but that'll be okay