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Great to see you again. Lots of friends here.

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And thank you for the invitation to speak. I can't tell you how much fun I've been having with this.

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This is going to be a very elementary talk. It's a it's a colloquium talk, not a seminar talk.

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The only thing is some of the letters seem to have gotten just a second copy.

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Yeah. Oh, that's better. Okay. Okay.

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Okay. So what I'm going to be talking about is something a little bit strange and a little bit weird.

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I'm going to talk about extending quantum mechanics and classical mechanics into the complex plane.

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As you know, mathematicians love to generalise things and extend things.

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And it's very useful, as you know, to understand real variables.

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The best way to understand real variables is to extend them into the complex plane and to extend them to complex variables.

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And I'm going to try to do that with quantum mechanics.

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And you'll see what I'm getting at in a second, just to make sure you understand what I mean by the complex plane.

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These are some examples. When I flew to the UK, actually, I took this plane over here.

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It actually worked very well.

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The only problem was that the people were sitting over here and the toilets were over there, so that that was so difficult.

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Okay. So here's the point of the talk. The as you know, if you take an elementary course, introductory course in quantum mechanics,

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what you learn is that the hermie that the Hamiltonian has to be her mission.

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And what that means is that if you take the Hamiltonian, it has a symmetry.

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If you take the Hamiltonian and you take its transpose and complex conjugate, you get back to the same Hamiltonian.

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And this is a very nice symmetry because it guarantees that the energy is real.

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And that's a good thing, because what you measure is the energy and that ought to be a real number.

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Okay. And it also guarantees that probability is conserved.

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But this axiom of quantum mechanics stands out from all the others as being different in some serious way.

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It looks like it's been written by a mathematician. All the other axioms of quantum mechanics sound, physical.

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You know, you would like to have a ground state of the system.

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You'd like probability to be conserved. You would like the energy to be real.

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You would like to have Lorentz invariance. You know, these these are very, very physical conditions.

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Okay. But transposed and complex conjugate doesn't sound like it was written by a physicist.

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It sounds like it was written by a mathematician. And I'm going to generalise that you're going to replace that.

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Oh, turn down the lights. Okay. It's probably one of these, uh, uh, lights.

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Here. Put. Do I push that button? No, no, that didn't work.

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The Troubles are getting a reflection from your screen. Okay.

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I don't think it's there. He. He pushed something over here. Lights, control panel.

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Zoom lights. No, nothing up helps.

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Cavalry's coming. Okay. I'm talking to his.

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Work better. Is that better?

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Okay. Now you can sleep without being noticed. Okay.

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So the point is that this axiom of elementary quantum mechanics, namely Hermitage City,

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stands out as being a little bit unusual in the sense that it sounds very mathematical.

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And I'm going to simplify that axiom and generalise that axiom and replace it by something a little bit more physical sounding.

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Okay. So the point of this talk is that I'm going to replace Dirac, Hermitage City that is transpose and complex conjugate by a weaker condition.

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And the condition is called P symmetry. And P, as you know, is parity stands for parity, reflection space reflection.

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T is time reversal. So we're talking about theories that are invariant under space time reflection.

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That is X goes to minus X and T goes to minus T.

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Okay. So we're reflecting space, all of space.

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And I'm going to argue that systems that have this symmetry are very interesting.

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And this condition of symmetry can replace, in many cases, Dirac, Hermitage City, and simplify it.

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And because it's a weaker condition, it allows us to study all kinds of new and very strange theories.

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That's what we're going to talk about. Okay, so here's an example of this Hamiltonian.

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I came across this Hamiltonian quite a while ago when I was visiting some clay and in near Paris.

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And this Hamiltonian is sort of a quantum mechanical version of a conformal field theory that's associated with the leading edge singularity,

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which I'm not going to talk about at all. But a number of people were working at it, some fancy mathematicians, mathematical physicists.

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Daniel This is interesting.

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A number of other people were working on it and they mentioned this Hamiltonian to me.

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And when I saw this Hamiltonian, I said, This is ridiculous.

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It's it's not permission. So this Hamiltonian really forget it.

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This is a stupid Hamiltonian. About four or five years later, I was in a colloquium, and this was the worst colloquium I had ever heard in my life.

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It was at my university and people were actually getting up and leaving.

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Just unbelievably awful. This was much worse than the second worst colloquium I had ever heard.

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And I wanted to get up and leave as well. But I couldn't because I had invited the seminars and the colloquium speaker.

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So instead, I decided to escape by doodling.

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And this problem came to me.

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And I want to point out that this Hamiltonian, you know, h equals p squared plus I x cubed.

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This is symmetric because under parity reflection exchanges sine and under time reversal.

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I change this sign, I goes to minus II. Okay.

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And just for just to emphasise that you might ask why does it change sine and the fundamental equation of of quantum mechanics.

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The Heisenberg Algebra says that X commuted with P is eight times H.

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Okay. And this ought to be invariant under.

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You sign. So the equation remains invariant. P changes sign.

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So the left side changes side. So the right side ought to change sign as well.

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So time reversal involves complex conjugation.

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Okay, another way to see that is to look at the Schrödinger equation.

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Just look at the Schrödinger equation and you can see that on the right hand side there is a D by d,

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t, and if you change the sign of T and you want to keep the equation invariant, okay.

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You also have to change the sign of odd. Okay. So if you had this equation to solve.

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And you're doodling in a colloquium, what might you do?

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Well, you say, maybe I can solve this equation using perturbation theory.

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So let's take a theory that I can solve, namely the Hamiltonian P squared plus X squared.

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That's the harmonic oscillator. And I know how to solve that. And let's just.

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I will only write a few more lines. So, okay, let's just take this Hamiltonian and introduce a perturbation from.

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Salon. And treat Epsilon as a small parameter.

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Okay. Now, if Epsilon is zero, I can solve this problem because it's the harmonic oscillator.

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And at the end of the calculation, I would like to set epsilon equal one.

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To recover the problem that I want to solve. Okay. So I can expand in powers of epsilon.

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And that's what I did. I calculated a few terms in the perturbation expansion.

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This is very powerful, by the way. This technique of putting a small parameter in the exponent,

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you can use it to solve all kinds of nonlinear problems, like the effect of freeze equation.

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You can use it to solve the Thomas Fermi equation, all sorts of interesting nonlinear problems and order by order and powers of epsilon.

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To my astonishment, the eigenvalues of the Hamiltonian turned out to be real.

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Okay. Which is hard to believe. Okay.

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But you see, the reason for introducing Epsilon this way is you notice that this problem remains pretty symmetric for any epsilon,

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so long as epsilon is real. And that's because, again, you send X into minus X and I into minus sign.

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Okay. So, in fact, I ran out of this colloquium when it was finally over with.

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Okay. And went to my office and calculated numerically the eigenvalues of this Hamiltonian and on the horizontal axis is just epsilon, okay?

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And on the vertical axis I've plotted the eigenvalues and you can see that the eigenvalues are discrete and real and positive.

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Okay? And so there's the plot of the eigenvalues.

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It's really nice. You can see it, okay? And they're real.

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And that was the beginning of p symmetric quantum mechanics.

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This goes back to 1998. Okay. And I wanted to give you an outline.

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That was the introduction. This is an outline of my talk today.

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I think that's fairly accurate. Okay.

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So I emphasise I am standing here telling you that the spectrum of the Hamiltonian P squared plus x squared times x to the epsilon is discrete,

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real and positive. And this is an infinite class of symmetric theories which are not permission in the usual quantum mechanical sense.

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Okay, but they have real discrete, positive energy levels.

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Okay. And you might ask, wait a minute, what happens if Epsilon is equal to two?

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Because if Epsilon is equal to two, you get the Hamiltonian for epsilon equals two.

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Here epsilon equals two, you get the Hamiltonian P squared minus X to the four.

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And that's an upside down potential.

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And the picture I showed you before had positive real eigenvalues, a spectrum going off to infinity, and this is an upside down potential.

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So what I'm telling you is really strange.

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What I'm telling you is that this potential here binds, has bound states, and the energy levels of these bound states are like this.

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Okay, so what I'm telling you is quite radical, but it is not just true.

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It's rigorously true at a mathematical level. Okay, so that was the beginning.

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And since then I've published lots of papers. These are there are seven proposals since then and a paper that has just come out in

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nature physics that there is now a very big field of symmetric quantum mechanics.

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People are working on it. These are the papers from 2008 to 2010, and there are lots more in 2011 and 12.

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And these are the papers. These are just the papers in fancy journals, you know, PRL and nature and science and and there are zillions of papers.

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These are the papers in 2014 and already in fancy journals there are four review articles so far.

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There's a book under preparation right now and more review articles that are being worked on right now.

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Since its beginning in 1998, there have been 20 international conferences in physics goes very fast.

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Okay? There are now nearly 2000 published papers.

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But the most exciting thing for me as a mathematical physicist is that in the last four years, there have been piles of.

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Our mental results. And people have observed in the laboratory what we've been talking about,

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and this is something that I never thought I would ever see in my life because of the problems that I choose to work on.

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So there is a rigorous proof that what I've told you is true, that the eigenvalues are real.

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I'm not going to talk about it at all, but you can read it.

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It involves developing something called the ODI correspondence.

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There's a deep a very deep and profound correspondence between ordinary differential equations,

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namely the Schrodinger equation and integral role models.

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Okay. And I want to go back now to something that I glossed over because I really meant to gloss over it.

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You notice in this picture that when Epsilon is positive, the eigenvalues are strictly real positive and discrete.

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But you notice that when Epsilon, which is the the power over here, if you can see it in the dark,

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X squared times X to the epsilon, when epsilon goes negative, something interesting seems to happen.

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You notice that eigenvalues are coming together and becoming degenerate and disappearing.

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Well, they're not disappearing. But what is happening is that they're becoming complex, okay?

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They become degenerate and then disappear into the complex plane.

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So something happens at Epsilon equals zero.

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There's a transition that occurs. And when epsilon is equal to zero, that's the harmonic oscillator.

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So the harmonic oscillator lives at this transition. That's, in fact, where her mission hamiltonians live.

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They live right at the transition between a region of unbroken symmetry and a region of broken symmetry,

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which is where the eigenvalues, which can be complex, that's where they actually become complex.

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So there's a region where they all the eigenvalues are real and positive and where they begin to go complex.

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And that's called the boundary. Okay. And whenever you have a transition, that means there's a possibility of doing experiments.

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And that's these experiments that I mentioned before. And one of the things that these experiments observe is this transition.

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Just to give you just to emphasise, you know, there are there are two cases you could have a case of unbroken Piti and a case of broken peaty.

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Okay. Or if you go to Paris, Steve, you've seen this lots of times in Paris.

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This is this is an example right there on the sidewalk in Paris of Broken Peaty.

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Okay. So I want to emphasise that her mission hamiltonians are really boring and they're boring because nothing interesting happens.

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The eigenvalues are always real and they don't undergo a transition.

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Okay, but but p t symmetric hamiltonians are astonishing because there is this transition

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between a region of of unbroken symmetry where the eigenvalues are real and broken.

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Symmetry with the eigenvalues go complex. Okay.

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So I want to give you an intuitive explanation of why there's a transition.

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So this is really simple argument.

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This is I'm giving you basically an undergraduate quantum mechanics argument.

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Suppose you have a box I'm calling it box one.

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And in this box you have some system and you have an antenna in the box which is taking stuff out of the box.

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And I'm not even telling you what the stuff is. The stuff could be probability, density could be could be energy, it could be particle number.

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It doesn't really matter. But stuff is being taken out of the box.

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So there's a sink in that box. Okay.

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Now, if you were to describe that box using a Schrödinger equation, there is the Schrödinger equation, okay, underneath the box.

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And in general, the Hamiltonian is an operator, an infinite dimensional matrix or a finite dimensional matrix.

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Here is just a one by one matrix. Just, just a number.

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Okay. And that number is just what I'm calling e one over here.

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It's just. Ah, E to the fader. Okay.

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And if you solve this trivial first order linear differential equation,

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the solution gives you psi of t is sine of zero times e to the IEEE t and this is if you are taking stuff out of the box is exponential lead decay.

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Okay. And that's all. So this is no surprise.

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You know, there's a negative antenna in the box. You have a complex energy, complex Hamiltonian and stuff is going out of the box and it's.

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This is radioactive decay. You could have another box with a sauce antenna in it.

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And all you have to do is take the complex conjugate of the energy of the Hamiltonian in this box.

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So you get the Hamiltonian is just r e to the minus theta.

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And again, the solution is the same, and this is growing exponentially.

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Okay. So these, these are not her emission hamiltonians, but that's what you get.

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You can think of these as effective hamiltonians. Okay.

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Now let's put these two boxes together and consider them as a single system,

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even though they're not connected to one another and they're not interacting with one another.

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So the Hamiltonian is diagonal,

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and the upper left in entry in the Hamiltonian describes the box with lost in the lower right hand corner of the Hamiltonian describes gain.

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And the point here is that this Hamiltonian is symmetric and is p t symmetric where p

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parity is the matrix that I've written down 0110 and T is just complex conjugation.

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Okay, so we have a two by two symmetric Hamiltonian.

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Okay. And now what I'm going to do is I'm going to couple the boxes together.

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So you have to imagine that there's a pipe going from box number one to box number two.

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Stuff is going into box number two and it's being taken out of box number one.

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But stuff is flowing from box number two to box number one so that the system can be in equilibrium.

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In principle, it might be in equilibrium if you couple the boxes strongly enough.

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And indeed, if you calculate the eigenvalues of this Hamiltonian, which is not our mission but is pretty symmetric,

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the eigenvalues suddenly become real if the coupling s is sufficiently large and there's a critical value of the coupling where this takes place,

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and that's the value of the critical coupling. Okay, squared science squared theta.

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Okay. So here is the full theory.

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Consider a quantum mechanical theory governed by that Hamiltonian.

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It has it's a three parameter Hamiltonian. It contains R, s and theta.

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It is symmetric. And you can see that the energy levels become real.

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If x squared is greater than R squared sine squared theta.

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And interestingly, there's another matrix which I'm going to call which I'm calling C,

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which I'll talk about in a minute, but that's the formula for this operator.

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C So this is a really trivial example of a two by two pretty symmetric Hamiltonian that that exhibits this transition.

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Okay. All right. So how do we understand this transition in a more sophisticated system?

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Let's look at p t symmetric classical mechanics.

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So take this Hamiltonian. I started out my talk with this p squared plus x cubed and you notice that the interaction term,

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this this potential is proportional to EI and you notice that the sign of that potential,

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the sign of eye changes as you go from X, which is negative to X, that is positive.

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Okay. So in fact, this Hamiltonian is no longer a matrix simple matrix Hamiltonian.

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It's a differential equation, the Schrödinger equation. But we have a source that becomes infinitely strong as X goes to minus infinity and a

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sync antenna which is spread over the entire real axis as x goes to plus infinity.

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And we now ask a very simple question How could this system possibly be in equilibrium?

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How could the eigenvalues of this Hamiltonian be real if the antenna,

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the source antenna and the zinc antenna are becoming infinitely strong as you run off to infinity?

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And the only way that this system could be in equilibrium would be if the particles that are being created,

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the stuff that's being created at minus infinity can get to plus infinity where it's destroyed fast enough.

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So classical and classical mechanics, let's calculate the amount of time it takes for a particle to travel from minus infinity,

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where it's created to plus infinity where it's destroyed.

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And the time t is the interval of d t and by the chain rule that's d x over p, and if you replace p by E minus x cubed.

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Okay, in the Hamiltonian it's the integral d x over the square root of v minus x cubed.

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And that integral is finite. Okay. And that's because the power of x is bigger than two in the square root,

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and therefore it only takes a finite amount of time for the particles that are created at minus infinity to get to,

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plus infinity where they're destroyed. The particles can travel infinitely fast.

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Evidently they can travel over an infinite distance in a finite time.

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And that's how the system can be in equilibrium.

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And that's why the eigenvalues of this Hamiltonian that I started out with can actually be, you know, the eigenvalues can be real.

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The system is in equilibrium. Okay, so to give you another classical example, suppose you were taking a course in first gear,

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undergraduate physics, and someone was explaining to you the classical harmonic oscillator.

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So what do they tell you? They say the Hamiltonian is p squared plus x squared,

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and now all you're told is that there are turning points and this is the edge of the classically allowed region.

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The particle travels say to the right until it hits a turning point and it stops and turns around and goes back again.

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And then it stops and turns around and goes back again. So that's simple harmonic motion,

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and that's all you can say about the harmonic oscillator until a very bright student comes up to the lecture after the lecture and says,

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What happens if you put a particle initially to the right of the right hand turning point?

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And of course, the lecturer says to the student, You idiot, you can't do that.

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You don't say that. Okay, get fired.

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If you do that, you then you say, interesting question.

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Okay, but you can't do that because that's the classically forbidden region.

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If X is greater than the turning point, which is the square root of V, if X is greater than the square root of V,

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that means that x squared is greater than E, and that means that p squared is negative.

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And you can't have a negative value for P squared. And the student says That's okay.

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It just means that P is not real. It's imaginary.

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So instead of restricting the motion of this particle to the real axis, let's extended it to the complex plane.

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And now, if you solve the equation F equals M A, what you find is that the particle,

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if it starts to the right at the turning point, just goes around in the ellipse.

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And no matter where you start the particle, it always goes around in an ellipse.

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And interestingly, the turning points are the foci of the ellipse.

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And this is complex classical mechanics where we've gone off the real axis into the complex plane.

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It's interesting, by the way, that the back and forth motion that you talk about in an elementary physics class,

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simple harmonic motion, is not back and forth. In fact, it's going around and around.

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It's a degenerate ellipse. Okay.

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It's really an ellipse. That's degenerate. Okay, so now what happens if we consider the Hamiltonian p squared plus x cubed?

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Now you have two turning points and the particles are going around in these turning points, and these are just deformed ellipses.

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Okay. But it's the same basic picture. And now if you have P squared minus,

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remember I was going to answer the question of how you could have eigenvalues

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with positive eigenvalues with the Hamiltonian P squared minus X to the four.

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This is the picture. Now for a minus X to the four Hamiltonian,

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there are four turning points and the classical paths go around and around these pairs of of turning points.

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Okay. And if you'd like to quantised the system and ask why are the eigenvalues real?

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You remember what you learn in elementary quantum mechanics.

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You learn that a particle is a wave, and as you go around in a closed path and the complex explain,

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you better have a finite number of you better have an integer number of wavelengths plus a half.

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Because as this wave goes around and follows the path of the classical particle, it has to add up and phase.

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And so this is W KB and if you integrate around a closed contour and set it equal to and plus a half pi,

260
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that gives you the energy levels of the system. It's very simple.

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Okay. So if you do that for this case, you'll find the eigenvalues that I talked about.

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If you ask what happens when the symmetry is broken?

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When we first saw this, we were we were astonished. What you find is that if Epsilon goes negative, you do not have closed orbits.

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You have open orbits. In fact, the transition that occurs is a classical transition occurs at the classical level.

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And what happens when epsilon goes below zero is that the paths are no longer closed.

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They're open and the paths go, the classical paths and the complex plane go off to infinity.

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And you can no longer write down an equation like this,

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because this equation has that that little circle on the integral sign that says you have

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to integrate along a closed path in the complex plane and there aren't any closed paths.

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So that's the transition that occurs. And if you ask, you know, if you say to me, yeah, yeah, but these are closed paths in the complex plane.

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What's happening on the real axis? Well, if you push that, if you if you go closer and closer to the real axis,

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the paths get bigger and bigger and bigger and they begin to look like this.

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Okay. And so what as you approach the real axis, what is actually happening is that the path goes all the way out to infinity.

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It then zips around instantly through the complex plane back to minus infinity and runs back up the real axis again.

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So the point is that it only takes a finite amount of time for a particle in

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an upside down potential like minus X to the four to roll out to infinity.

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And then where does it go? The answer is it goes all the way to minus infinity and comes back up to the origin.

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And if you then say, Where are we most likely to find the particle?

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The answer is you're most likely to find the particle at the origin,

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because the classical probability of finding a particle is proportional to one over the velocity of the particle.

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The faster it goes, the less likely it is to be there.

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That's why if you're driving in a car and you drive through an intersection,

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when you reach the intersection, you should drive as fast as possible so that you don't have an accident.

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Okay, so in the car.

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So this is what the classical probability looks like.

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And this looks like a bound state, a classical.

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Bound state, it is most likely to find the particle at the origin and that's the reason why you have a localised downstate.

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Even though you have an upside down potential, it's very radical stuff.

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This is not what you would expect. Maybe.

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And if you look at what is actually happening in the complex plane, you know, the particle is very likely to be at the origin.

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It zips around in the complex plane, it comes back to the origin again.

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So it spends most of its time at the origin. That's what the probability looks like.

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Okay. Okay. So now I can't resist showing you a derivation, but this is a colloquium.

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So this derivation is going to be the fastest derivation you ever saw.

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I'm going to prove rigorously that the energy levels in quantum mechanics for a minus X to the four potential are strictly real and positive.

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Okay. And this is a rigorous proof. So what I'm going to show you in 30 seconds is that the Hamiltonian, the first the top Hamiltonian,

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the minus X to the four Hamiltonian has exactly the same energy levels as the plus Z to the four Hamiltonian.

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Okay. But there's a difference between these two hamiltonians.

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You notice there's a term in the middle. Okay.

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And that is proportional to H. And that's an anomaly.

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And that is actually a parody anomaly, which I'm I won't.

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Quantum anomaly means something that vanishes in the limit as Planck's constant vanishes.

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Okay. And. So that is a pure quantum effect.

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So the the lower Hamiltonian, you know, that it has real positive eigenvalues because it's a confining potential.

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It's right side up potential, but the eigenvalues of that Hamiltonian are exactly the same as the eigenvalues of the Hamiltonian on the top.

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And that's the rigorous proof. And the proof goes like this you start with the top Hamiltonian.

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You integrate along a path that enters stokes wedges in the complex plane you so you see on the second line, there's the Schrodinger equation.

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And now step one, I make a change of variables and this is a first year undergraduate mathematics change of variables,

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and you end up with a disgusting differential equation. That's the eigenvalue problem.

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Okay, trust me, the arithmetic is correct.

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Step two you do a 48 transform of that ugly equation and you get another ugly equation.

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And that's what's written on the bottom of this transparency. Okay.

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It's pretty ugly, which is an oxymoron, I guess.

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Pretty ugly. And then you make a change of dependent variable and you get this new equation on the bottom of this transparency.

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And finally, you make a scale change.

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And the resulting how the resulting differential equation is.

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The differential equation is a simple differential equation for right side up potential.

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That's it. I have never changed e in the process of this derivation and therefore the eigenvalues of the

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upside down potential are exactly the same as the eigenvalues of a right side up potential.

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But this new Hamiltonian has a term proportional to H four, which is very strange, and that's an anomaly.

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And it's an anomaly that I call a parody anomaly.

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And if you're interested, we can talk about it later.

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But the result is that we have a pair of exactly ISO spectral hamiltonians and that means they have the same identity,

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they have identical energy levels, identical eigenvalues.

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And the first one is an upside down potential. Which is strange.

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And the second one is a right side of potential. Okay.

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So what I'm arguing is that we have a whole bunch of new hamiltonians that we can study with that look

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strange that you would never have studied if you were just taking an elementary course in quantum mechanics.

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And these new hamiltonians are in some sense intermediate between the two basic types of hamiltonians that you would study.

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There is the usual Hamiltonian, which is a commission Hamiltonian, and that describes a closed system,

331
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and there's the usual non-commissioned Hamiltonian that describes an open system.

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So these are phenomenological hamiltonians that describe maybe some scattering process

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in the that you would use to describe maybe a scattering process in nuclear physics.

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And these new hamiltonians are midway between them.

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These are symmetric hamiltonians, because if you are in a broken, symmetric region where the eigenvalues are complex,

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they look as if they're coming from an on her mission Hamiltonian for an open system.

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Okay. On the other hand, if you're in the unbroken region where the eigenvalues are real, these hamiltonians look as if they come from closed systems.

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They're not really closed because there are source antennas and sync antennas.

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So they are coupled to the outside world. But they behave as if they're coming.

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There are hamiltonians associated with a closed system.

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Okay, so I really like the guy who waving the flag in the middle and I should say I have a happy flag as well.

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See, this is my I don't know if you can read it from where you are sitting.

343
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If you read Portuguese, this says this says in Portuguese, if Piti is in your heart, I want to talk to you.

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And since you're here, Piti must be in your heart.

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And here I am talking to you. And so. Okay, good.

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All right. So the T-shirts will be on sale in the lobby after this time.

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So and I want to emphasise that at a mathematical level, that that was at the physical level.

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But at a mathematical level, what we are talking about is extending conventional classical mechanics and her mission,

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quantum mechanics into the complex plane. And when I say extending her mission, quantum mechanics into the complex plane,

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the condition that the Hamiltonian be her mission is transpose and complex conjugate being a symmetry of the Hamiltonian.

351
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Okay. And if you violate that symmetry, it is as if you're going off the real axis.

352
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It's in that sense that I mean, extending quantum mechanics into the complex plane.

353
00:37:55,680 --> 00:38:00,730
Okay. So the eigenvalues are real and positive, but that doesn't make this quantum mechanics.

354
00:38:00,750 --> 00:38:06,180
The question is, do we really have quantum mechanics here? Do we have a probabilistic interpretation?

355
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Do we have a Hilbert space with the positive metric? Do we have unitary time evolution?

356
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And the answer to those questions is yes. That's why I'm here.

357
00:38:15,670 --> 00:38:20,130
Okay. This is not a mathematics talk. This is a physics talk here.

358
00:38:20,460 --> 00:38:26,700
Okay. And the way you show it is actually rather interesting and very simple.

359
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It turns out that if you are in a region of unbroken symmetry where all the eigenvalues are real,

360
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you can show that there exists a new symmetry of the Hamiltonian that you wouldn't have expected.

361
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It's a secret symmetry, and it's represented by the symmetry operator.

362
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C And I've already shown you that there's an operator. C Showed you a formula for one, some transparencies.

363
00:38:50,860 --> 00:38:58,020
You go and you have to look for this symmetry. And C satisfies three simultaneous equations.

364
00:38:58,470 --> 00:39:06,150
The first equation says that C commutes with p t so C itself the C operator, this linear operator is pretty symmetric.

365
00:39:06,450 --> 00:39:15,390
The second condition is that C squared equals one. So C is like parity reflection OC or charge conjugation.

366
00:39:15,900 --> 00:39:20,700
Okay. And the third symmetry is that C commutes with the Hamiltonian.

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The third equation is that C computes for the Hamiltonian. So it's a symmetry of the Hamiltonian.

368
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And now the trick is to replace dagger, which means transpose in complex conjugate by C, p, t.

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And if you do that,

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the theory is now now has a quantum mechanical interpretation where the key thing that we have found now is what the metric is in the Hilbert space.

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That is the thing that replaces Dagger. And if you use CP, you can you you have a positive inner product in your Hilbert space.

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You have conservation of probability, unitary, everything works, and you now have a quantum mechanical theory.

373
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Okay, so the way. So what I'm saying to.

374
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Is that the Hamiltonian? You have to solve these three equations.

375
00:40:16,950 --> 00:40:21,390
The Hamiltonian determines the Hilbert space in which it wants to live.

376
00:40:22,260 --> 00:40:25,620
Okay, so how do. How do I explain this?

377
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If I were. If you invited me to give a talk and I said I was giving you a talk on G, R, for example, and I walked in and I said,

378
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let's postulate that the metric, you know, this tensor GMM you knew the four, four by four matrix is this.

379
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And I said, let's postulate this matrix as this. And I wrote down a four by four matrix on the board.

380
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You would laugh, you would say, you can postulate what the metric is.

381
00:40:51,330 --> 00:40:55,210
You have to solve Einstein's equations and determine what it is.

382
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But in fact, if we do conventional quantum mechanics, this is exactly what we do.

383
00:41:00,930 --> 00:41:07,650
We assume before we even looked at that, before we even look at the Hamiltonian that describes the theory that we want to study,

384
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we assume that it has a symmetry which is transpose and complex conjugate.

385
00:41:14,370 --> 00:41:17,620
That is dagger h. Dagger equals H. Okay.

386
00:41:18,000 --> 00:41:21,120
And what I'm saying is we have a whole bunch of new hamiltonians.

387
00:41:21,540 --> 00:41:26,280
And if the spectrum of these hamiltonians, these symmetric hamiltonians,

388
00:41:26,940 --> 00:41:34,350
it happens to be positive, we can then find out what the metric should be in the Hilbert space.

389
00:41:34,780 --> 00:41:39,450
Okay. We don't assume dagger, but we calculate it, we find out what it is.

390
00:41:39,660 --> 00:41:44,880
So this is the Hamilton this is the dagger that is appropriate for the Hamiltonian that you're studying.

391
00:41:45,390 --> 00:41:57,090
Okay. So I emphasise with respect to CP Adjoint, the theory has unitary time evolution norms are strictly positive, probability is conserved.

392
00:41:57,330 --> 00:42:01,020
So we have a generalisation of conventional quantum mechanics.

393
00:42:02,820 --> 00:42:07,860
So this is an overview of my talk so far.

394
00:42:08,610 --> 00:42:20,460
Okay. And I would like to just mention briefly that pretty symmetric systems are now being observed in the lab.

395
00:42:23,610 --> 00:42:29,070
This is one of the early experiments, and it was a rather rudimentary experiment.

396
00:42:29,220 --> 00:42:38,250
It appeared in PRL. It was a beautiful experiment in which they observed the transition using optical waveguides.

397
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But it was followed shortly thereafter by a spectacular experiment which was published in Nature Physics,

398
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where they observed the phase transition with tremendous precision.

399
00:42:53,370 --> 00:43:01,320
Absolutely beautiful. I mean, you cannot see any difference between the theoretical predictions and the experimental measurements.

400
00:43:01,830 --> 00:43:12,360
Absolutely spectacular experiment. This is an experiment that was done in Shanghai, a collaboration between Shanghai and Rutgers,

401
00:43:12,720 --> 00:43:21,300
where they saw the phase transition in PTC, symmetric in the symmetric diffusion equation using rubidium atoms.

402
00:43:22,140 --> 00:43:29,490
This is an experiment at Caltech where they used photonic silicon photonics circuits.

403
00:43:31,290 --> 00:43:37,260
This is an experiment in Indiana using symmetric superconducting wires.

404
00:43:37,710 --> 00:43:42,180
And you notice this is that graph I copied directly out of their paper.

405
00:43:42,450 --> 00:43:51,839
There you see that the the energy levels, the eigenvalues coming together, becoming degenerate and disappearing into the complex plane.

406
00:43:51,840 --> 00:43:57,360
You see that two distinct real eigenvalues become degenerate and then become complex.

407
00:43:57,360 --> 00:44:00,659
And what is plotted here is only the real part of the eigenvalue.

408
00:44:00,660 --> 00:44:05,040
So it goes off into the complex plane. A few more experiments.

409
00:44:05,250 --> 00:44:10,710
This is a spectacular experiment. Absolutely beautiful experiment done in Germany.

410
00:44:12,780 --> 00:44:15,870
It is an experiment using microwave cavities.

411
00:44:16,170 --> 00:44:20,490
The microwave cavity was broken into two pairs, a single cavity.

412
00:44:20,790 --> 00:44:26,909
There's a diaphragm that cuts the cavity in half and with a hole in between so that you can

413
00:44:26,910 --> 00:44:32,160
have stuff in one cavity flowing into the one side of the cavity flowing into the other side.

414
00:44:32,370 --> 00:44:42,360
You put a source antenna on one side, a sink antenna on the other side, and bingo, you can observe the phase transition with absolute precision.

415
00:44:42,420 --> 00:44:43,650
Beautiful experiment.

416
00:44:44,910 --> 00:44:53,790
There are symmetric lasers that are being studied at Yale and there have been a whole bunch of papers and endless stream of papers coming out of Yale.

417
00:44:55,560 --> 00:45:02,250
There is there are studies of symmetric, photonic graphene done in Israel.

418
00:45:03,750 --> 00:45:10,830
These are more studies of symmetric lasers done in Vienna, Princeton, Yale and Zurich.

419
00:45:12,990 --> 00:45:17,490
There are studies of nonlinear asymmetric systems.

420
00:45:19,740 --> 00:45:23,460
These are multiple symmetric waveguides.

421
00:45:23,490 --> 00:45:31,680
This is a collaboration between Germany and Florida in in it just recently published in Nature.

422
00:45:33,210 --> 00:45:36,720
Another experiment on superconducting wire is done at Argonne.

423
00:45:38,860 --> 00:45:45,630
Symmetric NMR done in Beijing. And this is a beautiful experiment.

424
00:45:45,780 --> 00:45:50,400
Very, very simple experiment. How do you have a system with balanced loss and gain?

425
00:45:50,520 --> 00:45:54,630
Well, all you need to do is to take two LRC circuits.

426
00:45:55,140 --> 00:46:00,600
Couple them inductively, put energy into one circuit, take energy out of the other circuit.

427
00:46:01,230 --> 00:46:04,980
Change the coupling between the circuits.

428
00:46:05,250 --> 00:46:11,370
And as you change the coupling constant between the two LRC circuits, you can see the phase transition.

429
00:46:12,060 --> 00:46:17,520
And so we looked at I looked at this experiment and I said, Wait a minute, I can do this.

430
00:46:17,880 --> 00:46:27,450
I can do it even simpler. So I said, Look, let's I may not look like an experimentalist and I may not behave like one, but I couldn't resist.

431
00:46:27,840 --> 00:46:35,670
So I said, Look, let's just take two pendulums and put a couple them together.

432
00:46:35,700 --> 00:46:38,700
You see, there's an epsilon y and an Epsilon X.

433
00:46:39,030 --> 00:46:44,820
So these are two coupled linear harmonic oscillators with loss and gain.

434
00:46:44,940 --> 00:46:49,920
The first pendulum, the x pendulum has lost the way pendulum has gained.

435
00:46:50,190 --> 00:46:57,270
And all we need to do is remove some of the energy from the X pendulum and put it into the Y pendulum and let's see what happens.

436
00:46:57,810 --> 00:47:06,030
So we have we can have balance, loss and gain. So this is my experiment that we did at the Keiko in London.

437
00:47:06,660 --> 00:47:14,190
And you can see there are two pendulum, one here, one pendulum here, one there hanging from a clothesline.

438
00:47:14,550 --> 00:47:21,270
So they're coupled. You see that, right? And all we did was to take an electromagnet.

439
00:47:21,270 --> 00:47:25,080
That's the that you see that circular thing at the top? That's an electromagnet.

440
00:47:25,410 --> 00:47:30,570
And we taped. You see that white thing at the top of the pendulum cord?

441
00:47:30,900 --> 00:47:35,490
That's a nail taped to the string with a piece of white tape.

442
00:47:36,750 --> 00:47:42,810
And we the only fancy part of the experiment is that we used infrared beams.

443
00:47:43,020 --> 00:47:47,310
So that is the pendulum with swinging one pendulum swinging toward the electromagnet,

444
00:47:47,730 --> 00:47:53,580
the electromagnetic fired, that is, it pushed the swing, put a little energy into the pendulum.

445
00:47:53,970 --> 00:47:58,379
But in the other pendulum, when it was swinging away from the electromagnet,

446
00:47:58,380 --> 00:48:02,220
the electromagnetic fired took a little bit of energy out of the pendulum.

447
00:48:02,700 --> 00:48:06,570
So one pendulum has energy going in.

448
00:48:06,660 --> 00:48:14,370
The other pendulum has energy going out there, coupled. This is a symmetric system where under parity, you interchange the two peninsula.

449
00:48:14,580 --> 00:48:18,540
And it's a time reversal. You have loss. Go into gain and gain.

450
00:48:18,540 --> 00:48:26,010
Go into loss. So it's pretty symmetric. Okay. And I have to say, we stared at the pendulum for a long time.

451
00:48:26,250 --> 00:48:36,090
This is the reason why science teachers should not be given playground duty, because they they really like to steer a pendulum.

452
00:48:38,520 --> 00:48:46,080
Anyway, this is what we saw. If you turn off the magnets so you don't couple that pendulum, you don't.

453
00:48:46,320 --> 00:48:54,180
You don't you don't have loss and gain. Rather, you see that what you have is Rabi oscillations.

454
00:48:54,510 --> 00:48:58,350
The top picture is the theory. The bottom picture is the experiment.

455
00:48:58,920 --> 00:49:05,130
Okay. Of course, the experiment shows a little bit of loss after many swings because the system has friction.

456
00:49:05,940 --> 00:49:10,650
But you can see the Rabi oscillations very clearly in the X and the Y pendulum.

457
00:49:11,130 --> 00:49:15,720
And then you turn on the magnets weakly and you still have Rabi oscillations.

458
00:49:15,730 --> 00:49:20,700
So the system is still in the unbroken symmetric phase.

459
00:49:21,660 --> 00:49:29,490
It's in equilibrium. You have Rabi oscillations, you turn up the magnets a little bit more and bang you go out of equilibrium.

460
00:49:29,880 --> 00:49:36,270
The Rabi oscillations disappear and and the experiment and the theory are in wonderful agreement.

461
00:49:36,900 --> 00:49:43,740
Okay, so that's a rudimentary experiment, the kind of thing that you might expect the theorists to do.

462
00:49:45,630 --> 00:49:54,959
And in fact, what is going on here, what is really going on here is that these two pendulums are actually a Hamiltonian system.

463
00:49:54,960 --> 00:49:59,820
Energy is actually conserved. And that is the Hamiltonian that describes the system.

464
00:50:00,090 --> 00:50:07,950
You might think it's not a Hamiltonian system because one pendulum, if they're not coupled, for example, one pendulum just dies out to zero.

465
00:50:08,040 --> 00:50:12,780
So it seems to lose a finite amount of energy. But the other pendulum blows up to infinity.

466
00:50:13,910 --> 00:50:17,260
So you might think how could the system possibly be Hamiltonian?

467
00:50:17,270 --> 00:50:24,670
How could energy be conserved? Well, it is conserved, but the form of the Hamiltonian is not exactly what you think it is.

468
00:50:24,680 --> 00:50:28,490
It's not just p squared plus X squared, plus Q squared, plus y squared.

469
00:50:28,880 --> 00:50:32,090
It's a little bit different. And this is the Hamiltonian for that system.

470
00:50:33,430 --> 00:50:37,670
Okay. I won't go into detail.

471
00:50:37,690 --> 00:50:42,790
But in fact, a system like this has to phase transitions, not one.

472
00:50:43,720 --> 00:50:52,300
Okay. And. One of the phase transitions has yet to be seen experimentally.

473
00:50:52,870 --> 00:50:56,220
So this is this is a challenge for experimentalists.

474
00:50:56,680 --> 00:51:06,550
I want to say, this is this is something I want to boast about. This is a paper that just came out just essentially this week in nature physics.

475
00:51:07,540 --> 00:51:13,570
And this is a very fancy system that we did at my university, Washington University.

476
00:51:13,870 --> 00:51:18,909
These are two coupled optical resonators.

477
00:51:18,910 --> 00:51:25,300
These are two coupled optical whispering galleries were really talked about whispering galleries.

478
00:51:26,260 --> 00:51:28,659
These are solid state devices.

479
00:51:28,660 --> 00:51:36,250
If you start a light beam going around, it will go around and around these whispering galleries a million times before it's absorbed.

480
00:51:36,850 --> 00:51:44,350
And then we can dope one of these whispering galleries with terbium and shine a laser light on it to pump it up.

481
00:51:44,680 --> 00:51:49,900
So that's a source. And the other one, the other whispering gallery has a natural loss.

482
00:51:50,410 --> 00:51:55,059
Okay. And then we can couple them together and we can observe the phase transition.

483
00:51:55,060 --> 00:52:01,510
And here it is. That's the phase transition and it is clean as a whistle.

484
00:52:01,960 --> 00:52:05,110
Okay, it is really beautiful. Comes out very, very well.

485
00:52:05,860 --> 00:52:16,450
Okay. And this is I won't discuss it, but if you read the last chapter of Jackson, you can read about something called the electromagnetic cell force.

486
00:52:16,870 --> 00:52:21,100
And instead of an x dot term, you can have an x triple dot.

487
00:52:21,430 --> 00:52:25,600
And if you couple two such systems together, they become symmetric.

488
00:52:25,840 --> 00:52:33,580
And this is the way to understand I claim to understand the self force, the electromagnetic self force.

489
00:52:34,960 --> 00:52:38,350
Okay, well, the point is pretty symmetric.

490
00:52:38,350 --> 00:52:44,200
Quantum mechanics is fun. And I just just concluded a minute or two,

491
00:52:44,200 --> 00:52:54,730
but I just wanted to remark that you can revisit things that people have already done where there were problems in quantum mechanics,

492
00:52:55,270 --> 00:53:01,510
and you can find that peachy symmetry pops up all over the place.

493
00:53:01,960 --> 00:53:15,160
Okay. And I'm just going to mention in passing three examples, there's a very famous model that was proposed back in the 1950s by TD Vee.

494
00:53:15,550 --> 00:53:22,270
The model is surprisingly called the Lee model, and the Lee model was proved to be wrong.

495
00:53:24,190 --> 00:53:28,360
Okay. It was just wrong. It violated fundamental physics.

496
00:53:29,350 --> 00:53:36,579
Violated units already. There is the pace uhlenbeck model, which is also an interesting model.

497
00:53:36,580 --> 00:53:46,540
It was a bad model. It was a model of a higher order field equation in quantum field theory, and it had ghosts.

498
00:53:46,900 --> 00:53:54,280
So it's no good. And in fact, the double scaling limit in quantum field theory is another such example where

499
00:53:54,790 --> 00:54:01,390
there's a problem of a essentially a ghost in on her mission Hamiltonian.

500
00:54:01,810 --> 00:54:07,270
And what happens with the lead model is that there's a fundamental problem with it.

501
00:54:07,270 --> 00:54:14,390
The lead model is a is a tri linear coupling and one essentially one year 90.

502
00:54:14,800 --> 00:54:24,810
We published this paper in 1954. One year later, in 1955, Pauli and Sheldon published a paper saying It's no good because it violates unitary.

503
00:54:25,360 --> 00:54:31,180
In fact, it's not no good. You can read about it in Barton's book.

504
00:54:31,600 --> 00:54:35,079
This is this is the epitaph in Barton's book on Quantum Field Theory.

505
00:54:35,080 --> 00:54:41,950
It says, A non-American Hamiltonian is unacceptable, partly because it may lead to complex energy eigenvalues,

506
00:54:41,950 --> 00:54:49,780
but chiefly because it implies a non unitary ESP matrix which fails to conserve probability and makes a hash of the physical interpretation.

507
00:54:50,230 --> 00:54:53,560
In fact, there's nothing wrong with the lead model. It's a wonderful model.

508
00:54:54,820 --> 00:54:59,980
But the reason that it appears to be wrong is that it isn't her mission.

509
00:55:00,250 --> 00:55:07,450
It is symmetric. And if you calculate the C operator and redefine your Hilbert space, it works perfectly.

510
00:55:07,780 --> 00:55:12,220
Nothing at all wrong with the model. The same is true with the space uhlenbeck model.

511
00:55:13,000 --> 00:55:18,610
It seems to have a ghost. This is a higher order field equation and it seems to have a ghost.

512
00:55:19,270 --> 00:55:20,440
But there isn't a ghost.

513
00:55:20,740 --> 00:55:28,569
The reason why there seems to be a ghost is because you're assuming a priori that the Hamiltonian is invariant under dagger h.

514
00:55:28,570 --> 00:55:37,600
Dagger equals equals h, but in fact p t equals h, and you can calculate the C operator and there's no ghost.

515
00:55:38,020 --> 00:55:47,440
Okay. And finally, if you look at the double scaling limit in quantum field theory, which I'm not going to talk about here,

516
00:55:47,710 --> 00:55:55,060
but if you look at the double scaling limit, in that limit, a fight of the fourth theory becomes a minus fight of the fourth theory.

517
00:55:55,270 --> 00:55:59,800
And people thought this is a disaster because we have an upside down potential.

518
00:56:00,130 --> 00:56:04,210
But as I talked about earlier, there's nothing wrong with an upside down potential.

519
00:56:04,450 --> 00:56:09,790
You just have to treat it as a symmetric theory and calculate the C operator and it all works.

520
00:56:10,000 --> 00:56:18,790
It's no problem at all. So I'm going to whiz now to the end of the talk and I'm going to show you one.

521
00:56:20,670 --> 00:56:24,690
Last slide. Here it is. Okay.

522
00:56:25,410 --> 00:56:30,930
So there are many possible future applications.

523
00:56:32,580 --> 00:56:38,060
There are so many problems to work on. It is just there's just a myriad of problems.

524
00:56:38,070 --> 00:56:46,710
There is this is a very, very rich field. I am not describing a calculation, but rather a context in which you can do calculations.

525
00:56:47,490 --> 00:56:53,820
For example, you can study a symmetric Higgs model in the Higgs sector instead of talking about a fight of the fourth theory.

526
00:56:54,030 --> 00:56:56,370
You could talk about a minus fight of the fourth theory.

527
00:56:57,300 --> 00:57:03,209
And such a theory is really interesting because that theory is asymptotically free and stable and conformal,

528
00:57:03,210 --> 00:57:09,330
invariant, and the expectation value of fire is nonzero because parity symmetry is broken.

529
00:57:10,050 --> 00:57:15,810
And you may not see. It may not appear that fight of the Four has a broken parity symmetry.

530
00:57:16,050 --> 00:57:20,340
But that theory is not invariant under PHI goes to minus PHI.

531
00:57:21,450 --> 00:57:26,040
That's because the boundary conditions are not invariant under PHI goes to minus PHI.

532
00:57:26,310 --> 00:57:31,980
The boundary conditions on the functional integral. That's the reason for this parity anomaly I was talking about.

533
00:57:32,250 --> 00:57:35,340
And the parity anomaly I think is associated with the Higgs mass.

534
00:57:35,640 --> 00:57:43,200
And as you know or as you may know, there's a fundamental problem with the standard model, and that is that there is a running coupling constant.

535
00:57:43,470 --> 00:57:45,330
This has been known for a very long time,

536
00:57:45,660 --> 00:57:54,120
and the running coupling constant allows the the coefficient of the fight of the fourth term in the Higgs sector to go negative.

537
00:57:54,840 --> 00:57:57,860
And that's that may be a serious problem, but I don't think so.

538
00:57:57,870 --> 00:58:03,899
I think pity symmetry solves it. You can construct a model of symmetric electrodynamics.

539
00:58:03,900 --> 00:58:08,430
And it's very interesting because it looks a little bit like a theory of magnetic charge,

540
00:58:09,060 --> 00:58:12,930
and it's an asymptotically free theory, which is very interesting.

541
00:58:13,230 --> 00:58:20,550
You can construct a symmetric theory of gravity, and this theory has a repulsive force, which is very interesting.

542
00:58:20,580 --> 00:58:23,760
Maybe there's some connection between that and dark energy.

543
00:58:24,930 --> 00:58:33,780
You can study the symmetric Dirac equation, just just to look at neutrinos.

544
00:58:33,780 --> 00:58:41,790
And what you find is that the symmetric Dirac equation allows for massless neutrinos to undergo oscillations.

545
00:58:42,240 --> 00:58:49,740
So it is not necessarily true that if you observe neutrino oscillations, that this implies that neutrinos have mass.

546
00:58:50,040 --> 00:58:57,180
It may be still true that neutrinos are massless, and this could have interesting astrophysical implications.

547
00:58:57,390 --> 00:59:02,490
And I could go on and on and on and on. There are so many interesting problems to look at.

548
00:59:02,850 --> 00:59:09,300
I hope you look at some of them. Thank you for listening and I hope you go away and think about symmetric quantum mechanics.

549
00:59:18,390 --> 00:59:23,160
All right. We're running a little bit late, but I'll take one question.

550
00:59:23,550 --> 00:59:27,290
Okay. Yes.

551
00:59:27,570 --> 00:59:31,230
Yeah, I'll ask pretty much the question you might expect me to ask.

552
00:59:33,120 --> 00:59:36,990
If you look at, say, you take your favourite potential, the ice cube or something.

553
00:59:37,200 --> 00:59:41,970
Yep. And look, not at the first two or three eigenvalues, but the 10th of the 20th.

554
00:59:42,240 --> 00:59:48,630
Mm hmm. I believe you'll find those are exponentially sensitive to perturbations of the problem.

555
00:59:48,780 --> 00:59:53,190
Yes. And show up in a year. Yeah. So this is very interesting.

556
00:59:53,240 --> 01:00:03,600
So the the the unstated part of of Nick's question is the question of the pseudo spectrum.

557
01:00:04,410 --> 01:00:08,010
And this is a very interesting question. Okay.

558
01:00:09,300 --> 01:00:13,950
Let me give you a very brief answer, but we can talk about it afterward in detail.

559
01:00:15,090 --> 01:00:23,459
When you talk about the pseudo spectrum. So for those of you who don't know what it means, a pseudo spectrum, very roughly speaking,

560
01:00:23,460 --> 01:00:32,670
means this If you perturb the Hamiltonian remission, Hamiltonian by a small amount, the spectrum will be perturbed.

561
01:00:32,970 --> 01:00:44,370
But a small perturbation of the Hamiltonian will will create a bounded perturbation, especially in the higher lying energy levels.

562
01:00:44,730 --> 01:00:53,010
But if you have a non her mission Hamiltonian, it may be that a very,

563
01:00:53,010 --> 01:01:01,170
very small perturbation in the Hamiltonian may produce an arbitrarily large change in the high lying energy level.

564
01:01:01,190 --> 01:01:15,180
So this is a kind of instability. Okay. The answer to the question is that you have to ask how do you measure a perturbation of the Hamiltonian?

565
01:01:15,570 --> 01:01:19,380
And this is a measure this is something that you have to measure in the Hilbert space itself.

566
01:01:20,160 --> 01:01:24,510
So if you use a conventional measure, that is an L2 measure.

567
01:01:25,470 --> 01:01:29,790
Okay. So you're assuming dagger as a measure in your Hilbert space.

568
01:01:30,000 --> 01:01:36,060
You can indeed get arbitrarily large changes in the high lying energy levels.

569
01:01:36,270 --> 01:01:40,440
But you have to remember that we have redefined the Hilbert space.

570
01:01:40,800 --> 01:01:44,940
That is not the way to measure a variation in the Hamiltonian.

571
01:01:44,940 --> 01:01:51,690
The way you measure a variation is with respect to the CP T metric and with respect to the CP T metric,

572
01:01:51,870 --> 01:01:56,280
you don't have an unbounded variation in the High Line energy levels.

573
01:01:56,410 --> 01:02:00,360
Okay, so this has to do with the misuse of a metric.

574
01:02:00,570 --> 01:02:11,190
So just a straightforward calculation would show that using an L2 norm that you can have an unbounded variation in the in this in the energy spectrum.

575
01:02:11,400 --> 01:02:15,630
But it doesn't happen if you use the CP t metric.

576
01:02:15,720 --> 01:02:22,410
It is bounded. That's, that's the that's the way to weasel out of that problem.

577
01:02:22,410 --> 01:02:29,730
But but there's these that this is a very rich area and quite a few papers have now been written investigating this issue.

578
01:02:30,000 --> 01:02:38,790
And I think that's the way to to resolve the issue. It's not a happy resolution in the sense that I'm saying it's just an ordinary.

579
01:02:39,280 --> 01:02:43,409
I'm saying that these PD symmetric hamiltonians behave like ordinary formation.

580
01:02:43,410 --> 01:02:50,490
Hamiltonians. But. But that's good that they they actually are stable.

581
01:02:50,500 --> 01:02:56,570
The high lying energy levels are stable. Any other quick.

582
01:02:58,300 --> 01:03:02,140
Yes, Steve. Yes. He talks a lot about the. Of Pakistan.

583
01:03:02,560 --> 01:03:07,140
Yes. What about say it's the states? Oh, yes.

584
01:03:07,160 --> 01:03:11,230
Oh, of course. Oh, yes. Oh, absolutely.

585
01:03:11,470 --> 01:03:18,550
The Afghan states are or are corresponding to different energy levels are orthogonal.

586
01:03:18,790 --> 01:03:22,150
That's actually very easy to prove directly from the Schrödinger equation.

587
01:03:23,410 --> 01:03:28,190
And so you have all of the usual stuff in conventional quantum mechanics.

588
01:03:28,210 --> 01:03:33,250
Okay. So we you don't have to worry.

589
01:03:33,270 --> 01:03:37,810
You don't have to worry about those issues. Okay. And you also have completeness.

590
01:03:38,440 --> 01:03:48,880
Okay. And so so that so the usual things that we expect for a removal problem, an unbounded removal problem, are still true.

591
01:03:49,180 --> 01:03:52,930
You don't none of none of that machinery is lost.

592
01:03:54,140 --> 01:03:58,040
Okay. And the problems were your actions were different.

593
01:04:00,540 --> 01:04:03,840
Well, it's, it's reinterpreting.

594
01:04:04,330 --> 01:04:09,000
Yeah. So yeah.

595
01:04:09,000 --> 01:04:12,960
So the question is, I mean, here's what I would love.

596
01:04:13,680 --> 01:04:26,450
I would love to have a smoking gun saying that this quantum system absolutely requires a p t symmetric Hamiltonian mean,

597
01:04:26,550 --> 01:04:30,720
that's of course, you know, if I find that, then I retire, right.

598
01:04:30,730 --> 01:04:45,059
And I'm it at this point, the kinds of symmetric systems that have been studied, you might say, are are synthetic symmetric systems.

599
01:04:45,060 --> 01:04:52,110
So so we can do in the laboratory, if you build the p t symmetric system, it has remarkable properties.

600
01:04:53,330 --> 01:05:02,060
Okay. What I mean by remarkable properties are, for example, you can see you need directional invisibility.

601
01:05:03,440 --> 01:05:07,820
Okay. You can see something when the light goes one way and when it goes the other way, it's invisible.

602
01:05:08,570 --> 01:05:14,660
You see remarkable things. You can see all sorts of strange and unusual properties.

603
01:05:15,840 --> 01:05:20,690
But but these are synthetic symmetric systems.

604
01:05:20,690 --> 01:05:24,589
And they're interesting because they offer the possibility in an optical system,

605
01:05:24,590 --> 01:05:29,540
for example, of possibly building if you have a new control over light.

606
01:05:30,110 --> 01:05:39,620
Okay. It does strange things as you introduce loss into the system, the power coming out of the system goes up.

607
01:05:41,270 --> 01:05:47,810
Contrary to what you might expect, you would think that if you introduce loss into an optical system, the power is lost.

608
01:05:48,110 --> 01:05:51,230
But in fact, it is to. Yeah, but.

609
01:05:51,380 --> 01:05:54,530
But it goes the power that comes out of the system.

610
01:05:54,920 --> 01:05:59,600
Power is going up. So this strange. So you have interesting control over light.

611
01:05:59,600 --> 01:06:06,770
You might be able to build symmetric new kinds of symmetric metamaterials, symmetric computers and so on.

612
01:06:07,220 --> 01:06:18,410
But what happens if we find a crystal on the ground and this crystal is has a symmetric spectrum that is uniquely different.

613
01:06:18,710 --> 01:06:21,170
Okay. So I can give you one example. Okay.

614
01:06:21,440 --> 01:06:30,650
If you have a bent if you have a crystal or you know it, say an insulator, it has bands and gaps, as we know.

615
01:06:31,040 --> 01:06:36,020
Okay. And so this is a band of allowed energies and then a gap and a band and a gap.

616
01:06:36,440 --> 01:06:42,590
And at one edge of the conduction band, the wave function is two pi periodic.

617
01:06:43,520 --> 01:06:46,730
At the other end, the wave function is for Pi periodic.

618
01:06:46,740 --> 01:06:53,930
So at one end it looks like a boson, and the other end it looks like a fermion in a p t symmetric crystal.

619
01:06:54,110 --> 01:06:58,820
So and a potential like ice sign of X instead of sign of x.

620
01:06:59,150 --> 01:07:01,340
That's a symmetric crystal.

621
01:07:02,180 --> 01:07:11,000
The bands and gaps are real because it is symmetric, but there's a remarkable difference at both edges of the conduction band.

622
01:07:11,660 --> 01:07:19,730
The eigen functions are to Pi periodic, not to Pi Periodic for pi periodic, but to Pi Periodic.

623
01:07:19,940 --> 01:07:25,340
It is as if you had two fermions coming together and making a boson.

624
01:07:26,480 --> 01:07:33,980
And it's it's uniquely different. If you pick up a crystal and you bring it to the laboratory and you do some neutron scattering

625
01:07:34,190 --> 01:07:38,839
and you measure the periodicity of the wave function at the edge of the conduction band,

626
01:07:38,840 --> 01:07:49,280
and you see that that would be a smoking gun for a pretty symmetric Hamiltonian that you can't do with her mission.

627
01:07:49,280 --> 01:07:53,210
Hamiltonian a conventional formation. HAMILTON That's what I would love to see.

628
01:07:53,600 --> 01:07:57,710
No one has found such a crystal yet or such a clean example.

629
01:07:58,880 --> 01:08:07,790
So right now, so far, the experiments that have been performed are on what is called P2 synthetic materials or synthetic devices,

630
01:08:07,790 --> 01:08:15,890
where you can build the system and have it act like it, but not have it be, you know, crucially needed in nature.

631
01:08:15,920 --> 01:08:19,760
Okay. Thank you. Yeah, really?