Great to see you again. Lots of friends here.
And thank you for the invitation to speak. I can't tell you how much fun I've been having with this.
This is going to be a very elementary talk. It's a it's a colloquium talk, not a seminar talk.
The only thing is some of the letters seem to have gotten just a second copy.
Yeah. Oh, that's better. Okay. Okay.
Okay. So what I'm going to be talking about is something a little bit strange and a little bit weird.
I'm going to talk about extending quantum mechanics and classical mechanics into the complex plane.
As you know, mathematicians love to generalise things and extend things.
And it's very useful, as you know, to understand real variables.
The best way to understand real variables is to extend them into the complex plane and to extend them to complex variables.
And I'm going to try to do that with quantum mechanics.
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.
These are some examples. When I flew to the UK, actually, I took this plane over here.
It actually worked very well.
The only problem was that the people were sitting over here and the toilets were over there, so that that was so difficult.
Okay. So here's the point of the talk. The as you know, if you take an elementary course, introductory course in quantum mechanics,
what you learn is that the hermie that the Hamiltonian has to be her mission.
And what that means is that if you take the Hamiltonian, it has a symmetry.
If you take the Hamiltonian and you take its transpose and complex conjugate, you get back to the same Hamiltonian.
And this is a very nice symmetry because it guarantees that the energy is real.
And that's a good thing, because what you measure is the energy and that ought to be a real number.
Okay. And it also guarantees that probability is conserved.
But this axiom of quantum mechanics stands out from all the others as being different in some serious way.
It looks like it's been written by a mathematician. All the other axioms of quantum mechanics sound, physical.
You know, you would like to have a ground state of the system.
You'd like probability to be conserved. You would like the energy to be real.
You would like to have Lorentz invariance. You know, these these are very, very physical conditions.
Okay. But transposed and complex conjugate doesn't sound like it was written by a physicist.
It sounds like it was written by a mathematician. And I'm going to generalise that you're going to replace that.
Oh, turn down the lights. Okay. It's probably one of these, uh, uh, lights.
Here. Put. Do I push that button? No, no, that didn't work.
The Troubles are getting a reflection from your screen. Okay.
I don't think it's there. He. He pushed something over here. Lights, control panel.
Zoom lights. No, nothing up helps.
Cavalry's coming. Okay. I'm talking to his.
Work better. Is that better?
Okay. Now you can sleep without being noticed. Okay.
So the point is that this axiom of elementary quantum mechanics, namely Hermitage City,
stands out as being a little bit unusual in the sense that it sounds very mathematical.
And I'm going to simplify that axiom and generalise that axiom and replace it by something a little bit more physical sounding.
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.
And the condition is called P symmetry. And P, as you know, is parity stands for parity, reflection space reflection.
T is time reversal. So we're talking about theories that are invariant under space time reflection.
That is X goes to minus X and T goes to minus T.
Okay. So we're reflecting space, all of space.
And I'm going to argue that systems that have this symmetry are very interesting.
And this condition of symmetry can replace, in many cases, Dirac, Hermitage City, and simplify it.
And because it's a weaker condition, it allows us to study all kinds of new and very strange theories.
That's what we're going to talk about. Okay, so here's an example of this Hamiltonian.
I came across this Hamiltonian quite a while ago when I was visiting some clay and in near Paris.
And this Hamiltonian is sort of a quantum mechanical version of a conformal field theory that's associated with the leading edge singularity,
which I'm not going to talk about at all. But a number of people were working at it, some fancy mathematicians, mathematical physicists.
Daniel This is interesting.
A number of other people were working on it and they mentioned this Hamiltonian to me.
And when I saw this Hamiltonian, I said, This is ridiculous.
It's it's not permission. So this Hamiltonian really forget it.
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.
It was at my university and people were actually getting up and leaving.
Just unbelievably awful. This was much worse than the second worst colloquium I had ever heard.
And I wanted to get up and leave as well. But I couldn't because I had invited the seminars and the colloquium speaker.
So instead, I decided to escape by doodling.
And this problem came to me.
And I want to point out that this Hamiltonian, you know, h equals p squared plus I x cubed.
This is symmetric because under parity reflection exchanges sine and under time reversal.
I change this sign, I goes to minus II. Okay.
And just for just to emphasise that you might ask why does it change sine and the fundamental equation of of quantum mechanics.
The Heisenberg Algebra says that X commuted with P is eight times H.
Okay. And this ought to be invariant under.
You sign. So the equation remains invariant. P changes sign.
So the left side changes side. So the right side ought to change sign as well.
So time reversal involves complex conjugation.
Okay, another way to see that is to look at the Schrödinger equation.
Just look at the Schrödinger equation and you can see that on the right hand side there is a D by d,
t, and if you change the sign of T and you want to keep the equation invariant, okay.
You also have to change the sign of odd. Okay. So if you had this equation to solve.
And you're doodling in a colloquium, what might you do?
Well, you say, maybe I can solve this equation using perturbation theory.
So let's take a theory that I can solve, namely the Hamiltonian P squared plus X squared.
That's the harmonic oscillator. And I know how to solve that. And let's just.
I will only write a few more lines. So, okay, let's just take this Hamiltonian and introduce a perturbation from.
Salon. And treat Epsilon as a small parameter.
Okay. Now, if Epsilon is zero, I can solve this problem because it's the harmonic oscillator.
And at the end of the calculation, I would like to set epsilon equal one.
To recover the problem that I want to solve. Okay. So I can expand in powers of epsilon.
And that's what I did. I calculated a few terms in the perturbation expansion.
This is very powerful, by the way. This technique of putting a small parameter in the exponent,
you can use it to solve all kinds of nonlinear problems, like the effect of freeze equation.
You can use it to solve the Thomas Fermi equation, all sorts of interesting nonlinear problems and order by order and powers of epsilon.
To my astonishment, the eigenvalues of the Hamiltonian turned out to be real.
Okay. Which is hard to believe. Okay.
But you see, the reason for introducing Epsilon this way is you notice that this problem remains pretty symmetric for any epsilon,
so long as epsilon is real. And that's because, again, you send X into minus X and I into minus sign.
Okay. So, in fact, I ran out of this colloquium when it was finally over with.
Okay. And went to my office and calculated numerically the eigenvalues of this Hamiltonian and on the horizontal axis is just epsilon, okay?
And on the vertical axis I've plotted the eigenvalues and you can see that the eigenvalues are discrete and real and positive.
Okay? And so there's the plot of the eigenvalues.
It's really nice. You can see it, okay? And they're real.
And that was the beginning of p symmetric quantum mechanics.
This goes back to 1998. Okay. And I wanted to give you an outline.
That was the introduction. This is an outline of my talk today.
I think that's fairly accurate. Okay.
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,
real and positive. And this is an infinite class of symmetric theories which are not permission in the usual quantum mechanical sense.
Okay, but they have real discrete, positive energy levels.
Okay. And you might ask, wait a minute, what happens if Epsilon is equal to two?
Because if Epsilon is equal to two, you get the Hamiltonian for epsilon equals two.
Here epsilon equals two, you get the Hamiltonian P squared minus X to the four.
And that's an upside down potential.
And the picture I showed you before had positive real eigenvalues, a spectrum going off to infinity, and this is an upside down potential.
So what I'm telling you is really strange.
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.
Okay, so what I'm telling you is quite radical, but it is not just true.
It's rigorously true at a mathematical level. Okay, so that was the beginning.
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
nature physics that there is now a very big field of symmetric quantum mechanics.
People are working on it. These are the papers from 2008 to 2010, and there are lots more in 2011 and 12.
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.
These are the papers in 2014 and already in fancy journals there are four review articles so far.
There's a book under preparation right now and more review articles that are being worked on right now.
Since its beginning in 1998, there have been 20 international conferences in physics goes very fast.
Okay? There are now nearly 2000 published papers.
But the most exciting thing for me as a mathematical physicist is that in the last four years, there have been piles of.
Our mental results. And people have observed in the laboratory what we've been talking about,
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.
So there is a rigorous proof that what I've told you is true, that the eigenvalues are real.
I'm not going to talk about it at all, but you can read it.
It involves developing something called the ODI correspondence.
There's a deep a very deep and profound correspondence between ordinary differential equations,
namely the Schrodinger equation and integral role models.
Okay. And I want to go back now to something that I glossed over because I really meant to gloss over it.
You notice in this picture that when Epsilon is positive, the eigenvalues are strictly real positive and discrete.
But you notice that when Epsilon, which is the the power over here, if you can see it in the dark,
X squared times X to the epsilon, when epsilon goes negative, something interesting seems to happen.
You notice that eigenvalues are coming together and becoming degenerate and disappearing.
Well, they're not disappearing. But what is happening is that they're becoming complex, okay?
They become degenerate and then disappear into the complex plane.
So something happens at Epsilon equals zero.
There's a transition that occurs. And when epsilon is equal to zero, that's the harmonic oscillator.
So the harmonic oscillator lives at this transition. That's, in fact, where her mission hamiltonians live.
They live right at the transition between a region of unbroken symmetry and a region of broken symmetry,
which is where the eigenvalues, which can be complex, that's where they actually become complex.
So there's a region where they all the eigenvalues are real and positive and where they begin to go complex.
And that's called the boundary. Okay. And whenever you have a transition, that means there's a possibility of doing experiments.
And that's these experiments that I mentioned before. And one of the things that these experiments observe is this transition.
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.
Okay. Or if you go to Paris, Steve, you've seen this lots of times in Paris.
This is this is an example right there on the sidewalk in Paris of Broken Peaty.
Okay. So I want to emphasise that her mission hamiltonians are really boring and they're boring because nothing interesting happens.
The eigenvalues are always real and they don't undergo a transition.
Okay, but but p t symmetric hamiltonians are astonishing because there is this transition
between a region of of unbroken symmetry where the eigenvalues are real and broken.
Symmetry with the eigenvalues go complex. Okay.
So I want to give you an intuitive explanation of why there's a transition.
So this is really simple argument.
This is I'm giving you basically an undergraduate quantum mechanics argument.
Suppose you have a box I'm calling it box one.
And in this box you have some system and you have an antenna in the box which is taking stuff out of the box.
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.
It doesn't really matter. But stuff is being taken out of the box.
So there's a sink in that box. Okay.
Now, if you were to describe that box using a Schrödinger equation, there is the Schrödinger equation, okay, underneath the box.
And in general, the Hamiltonian is an operator, an infinite dimensional matrix or a finite dimensional matrix.
Here is just a one by one matrix. Just, just a number.
Okay. And that number is just what I'm calling e one over here.
It's just. Ah, E to the fader. Okay.
And if you solve this trivial first order linear differential equation,
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.
Okay. And that's all. So this is no surprise.
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.
This is radioactive decay. You could have another box with a sauce antenna in it.
And all you have to do is take the complex conjugate of the energy of the Hamiltonian in this box.
So you get the Hamiltonian is just r e to the minus theta.
And again, the solution is the same, and this is growing exponentially.
Okay. So these, these are not her emission hamiltonians, but that's what you get.
You can think of these as effective hamiltonians. Okay.
Now let's put these two boxes together and consider them as a single system,
even though they're not connected to one another and they're not interacting with one another.
So the Hamiltonian is diagonal,
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.
And the point here is that this Hamiltonian is symmetric and is p t symmetric where p
parity is the matrix that I've written down 0110 and T is just complex conjugation.
Okay, so we have a two by two symmetric Hamiltonian.
Okay. And now what I'm going to do is I'm going to couple the boxes together.
So you have to imagine that there's a pipe going from box number one to box number two.
Stuff is going into box number two and it's being taken out of box number one.
But stuff is flowing from box number two to box number one so that the system can be in equilibrium.
In principle, it might be in equilibrium if you couple the boxes strongly enough.
And indeed, if you calculate the eigenvalues of this Hamiltonian, which is not our mission but is pretty symmetric,
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,
and that's the value of the critical coupling. Okay, squared science squared theta.
Okay. So here is the full theory.
Consider a quantum mechanical theory governed by that Hamiltonian.
It has it's a three parameter Hamiltonian. It contains R, s and theta.
It is symmetric. And you can see that the energy levels become real.
If x squared is greater than R squared sine squared theta.
And interestingly, there's another matrix which I'm going to call which I'm calling C,
which I'll talk about in a minute, but that's the formula for this operator.
C So this is a really trivial example of a two by two pretty symmetric Hamiltonian that that exhibits this transition.
Okay. All right. So how do we understand this transition in a more sophisticated system?
Let's look at p t symmetric classical mechanics.
So take this Hamiltonian. I started out my talk with this p squared plus x cubed and you notice that the interaction term,
this this potential is proportional to EI and you notice that the sign of that potential,
the sign of eye changes as you go from X, which is negative to X, that is positive.
Okay. So in fact, this Hamiltonian is no longer a matrix simple matrix Hamiltonian.
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
sync antenna which is spread over the entire real axis as x goes to plus infinity.
And we now ask a very simple question How could this system possibly be in equilibrium?
How could the eigenvalues of this Hamiltonian be real if the antenna,
the source antenna and the zinc antenna are becoming infinitely strong as you run off to infinity?
And the only way that this system could be in equilibrium would be if the particles that are being created,
the stuff that's being created at minus infinity can get to plus infinity where it's destroyed fast enough.
So classical and classical mechanics, let's calculate the amount of time it takes for a particle to travel from minus infinity,
where it's created to plus infinity where it's destroyed.
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.
Okay, in the Hamiltonian it's the integral d x over the square root of v minus x cubed.
And that integral is finite. Okay. And that's because the power of x is bigger than two in the square root,
and therefore it only takes a finite amount of time for the particles that are created at minus infinity to get to,
plus infinity where they're destroyed. The particles can travel infinitely fast.
Evidently they can travel over an infinite distance in a finite time.
And that's how the system can be in equilibrium.
And that's why the eigenvalues of this Hamiltonian that I started out with can actually be, you know, the eigenvalues can be real.
The system is in equilibrium. Okay, so to give you another classical example, suppose you were taking a course in first gear,
undergraduate physics, and someone was explaining to you the classical harmonic oscillator.
So what do they tell you? They say the Hamiltonian is p squared plus x squared,
and now all you're told is that there are turning points and this is the edge of the classically allowed region.
The particle travels say to the right until it hits a turning point and it stops and turns around and goes back again.
And then it stops and turns around and goes back again. So that's simple harmonic motion,
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,
What happens if you put a particle initially to the right of the right hand turning point?
And of course, the lecturer says to the student, You idiot, you can't do that.
You don't say that. Okay, get fired.
If you do that, you then you say, interesting question.
Okay, but you can't do that because that's the classically forbidden region.
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,
that means that x squared is greater than E, and that means that p squared is negative.
And you can't have a negative value for P squared. And the student says That's okay.
It just means that P is not real. It's imaginary.
So instead of restricting the motion of this particle to the real axis, let's extended it to the complex plane.
And now, if you solve the equation F equals M A, what you find is that the particle,
if it starts to the right at the turning point, just goes around in the ellipse.
And no matter where you start the particle, it always goes around in an ellipse.
And interestingly, the turning points are the foci of the ellipse.
And this is complex classical mechanics where we've gone off the real axis into the complex plane.
It's interesting, by the way, that the back and forth motion that you talk about in an elementary physics class,
simple harmonic motion, is not back and forth. In fact, it's going around and around.
It's a degenerate ellipse. Okay.
It's really an ellipse. That's degenerate. Okay, so now what happens if we consider the Hamiltonian p squared plus x cubed?
Now you have two turning points and the particles are going around in these turning points, and these are just deformed ellipses.
Okay. But it's the same basic picture. And now if you have P squared minus,
remember I was going to answer the question of how you could have eigenvalues
with positive eigenvalues with the Hamiltonian P squared minus X to the four.
This is the picture. Now for a minus X to the four Hamiltonian,
there are four turning points and the classical paths go around and around these pairs of of turning points.
Okay. And if you'd like to quantised the system and ask why are the eigenvalues real?
You remember what you learn in elementary quantum mechanics.
You learn that a particle is a wave, and as you go around in a closed path and the complex explain,
you better have a finite number of you better have an integer number of wavelengths plus a half.
Because as this wave goes around and follows the path of the classical particle, it has to add up and phase.
And so this is W KB and if you integrate around a closed contour and set it equal to and plus a half pi,
that gives you the energy levels of the system. It's very simple.
Okay. So if you do that for this case, you'll find the eigenvalues that I talked about.
If you ask what happens when the symmetry is broken?
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.
You have open orbits. In fact, the transition that occurs is a classical transition occurs at the classical level.
And what happens when epsilon goes below zero is that the paths are no longer closed.
They're open and the paths go, the classical paths and the complex plane go off to infinity.
And you can no longer write down an equation like this,
because this equation has that that little circle on the integral sign that says you have
to integrate along a closed path in the complex plane and there aren't any closed paths.
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.
What's happening on the real axis? Well, if you push that, if you if you go closer and closer to the real axis,
the paths get bigger and bigger and bigger and they begin to look like this.
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.
It then zips around instantly through the complex plane back to minus infinity and runs back up the real axis again.
So the point is that it only takes a finite amount of time for a particle in
an upside down potential like minus X to the four to roll out to infinity.
And then where does it go? The answer is it goes all the way to minus infinity and comes back up to the origin.
And if you then say, Where are we most likely to find the particle?
The answer is you're most likely to find the particle at the origin,
because the classical probability of finding a particle is proportional to one over the velocity of the particle.
The faster it goes, the less likely it is to be there.
That's why if you're driving in a car and you drive through an intersection,
when you reach the intersection, you should drive as fast as possible so that you don't have an accident.
Okay, so in the car.
So this is what the classical probability looks like.
And this looks like a bound state, a classical.
Bound state, it is most likely to find the particle at the origin and that's the reason why you have a localised downstate.
Even though you have an upside down potential, it's very radical stuff.
This is not what you would expect. Maybe.
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.
It zips around in the complex plane, it comes back to the origin again.
So it spends most of its time at the origin. That's what the probability looks like.
Okay. Okay. So now I can't resist showing you a derivation, but this is a colloquium.
So this derivation is going to be the fastest derivation you ever saw.
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.
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,
the minus X to the four Hamiltonian has exactly the same energy levels as the plus Z to the four Hamiltonian.
Okay. But there's a difference between these two hamiltonians.
You notice there's a term in the middle. Okay.
And that is proportional to H. And that's an anomaly.
And that is actually a parody anomaly, which I'm I won't.
Quantum anomaly means something that vanishes in the limit as Planck's constant vanishes.
Okay. And. So that is a pure quantum effect.
So the the lower Hamiltonian, you know, that it has real positive eigenvalues because it's a confining potential.
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.
And that's the rigorous proof. And the proof goes like this you start with the top Hamiltonian.
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.
And now step one, I make a change of variables and this is a first year undergraduate mathematics change of variables,
and you end up with a disgusting differential equation. That's the eigenvalue problem.
Okay, trust me, the arithmetic is correct.
Step two you do a 48 transform of that ugly equation and you get another ugly equation.
And that's what's written on the bottom of this transparency. Okay.
It's pretty ugly, which is an oxymoron, I guess.
Pretty ugly. And then you make a change of dependent variable and you get this new equation on the bottom of this transparency.
And finally, you make a scale change.
And the resulting how the resulting differential equation is.
The differential equation is a simple differential equation for right side up potential.
That's it. I have never changed e in the process of this derivation and therefore the eigenvalues of the
upside down potential are exactly the same as the eigenvalues of a right side up potential.
But this new Hamiltonian has a term proportional to H four, which is very strange, and that's an anomaly.
And it's an anomaly that I call a parody anomaly.
And if you're interested, we can talk about it later.
But the result is that we have a pair of exactly ISO spectral hamiltonians and that means they have the same identity,
they have identical energy levels, identical eigenvalues.
And the first one is an upside down potential. Which is strange.
And the second one is a right side of potential. Okay.
So what I'm arguing is that we have a whole bunch of new hamiltonians that we can study with that look
strange that you would never have studied if you were just taking an elementary course in quantum mechanics.
And these new hamiltonians are in some sense intermediate between the two basic types of hamiltonians that you would study.
There is the usual Hamiltonian, which is a commission Hamiltonian, and that describes a closed system,
and there's the usual non-commissioned Hamiltonian that describes an open system.
So these are phenomenological hamiltonians that describe maybe some scattering process
in the that you would use to describe maybe a scattering process in nuclear physics.
And these new hamiltonians are midway between them.
These are symmetric hamiltonians, because if you are in a broken, symmetric region where the eigenvalues are complex,
they look as if they're coming from an on her mission Hamiltonian for an open system.
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.
They're not really closed because there are source antennas and sync antennas.
So they are coupled to the outside world. But they behave as if they're coming.
There are hamiltonians associated with a closed system.
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.
See, this is my I don't know if you can read it from where you are sitting.
If you read Portuguese, this says this says in Portuguese, if Piti is in your heart, I want to talk to you.
And since you're here, Piti must be in your heart.
And here I am talking to you. And so. Okay, good.
All right. So the T-shirts will be on sale in the lobby after this time.
So and I want to emphasise that at a mathematical level, that that was at the physical level.
But at a mathematical level, what we are talking about is extending conventional classical mechanics and her mission,
quantum mechanics into the complex plane. And when I say extending her mission, quantum mechanics into the complex plane,
the condition that the Hamiltonian be her mission is transpose and complex conjugate being a symmetry of the Hamiltonian.
Okay. And if you violate that symmetry, it is as if you're going off the real axis.
It's in that sense that I mean, extending quantum mechanics into the complex plane.
Okay. So the eigenvalues are real and positive, but that doesn't make this quantum mechanics.
The question is, do we really have quantum mechanics here? Do we have a probabilistic interpretation?
Do we have a Hilbert space with the positive metric? Do we have unitary time evolution?
And the answer to those questions is yes. That's why I'm here.
Okay. This is not a mathematics talk. This is a physics talk here.
Okay. And the way you show it is actually rather interesting and very simple.
It turns out that if you are in a region of unbroken symmetry where all the eigenvalues are real,
you can show that there exists a new symmetry of the Hamiltonian that you wouldn't have expected.
It's a secret symmetry, and it's represented by the symmetry operator.
C And I've already shown you that there's an operator. C Showed you a formula for one, some transparencies.
You go and you have to look for this symmetry. And C satisfies three simultaneous equations.
The first equation says that C commutes with p t so C itself the C operator, this linear operator is pretty symmetric.
The second condition is that C squared equals one. So C is like parity reflection OC or charge conjugation.
Okay. And the third symmetry is that C commutes with the Hamiltonian.
The third equation is that C computes for the Hamiltonian. So it's a symmetry of the Hamiltonian.
And now the trick is to replace dagger, which means transpose in complex conjugate by C, p, t.
And if you do that,
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.
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.
You have conservation of probability, unitary, everything works, and you now have a quantum mechanical theory.
Okay, so the way. So what I'm saying to.
Is that the Hamiltonian? You have to solve these three equations.
The Hamiltonian determines the Hilbert space in which it wants to live.
Okay, so how do. How do I explain this?
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,
let's postulate that the metric, you know, this tensor GMM you knew the four, four by four matrix is this.
And I said, let's postulate this matrix as this. And I wrote down a four by four matrix on the board.
You would laugh, you would say, you can postulate what the metric is.
You have to solve Einstein's equations and determine what it is.
But in fact, if we do conventional quantum mechanics, this is exactly what we do.
We assume before we even looked at that, before we even look at the Hamiltonian that describes the theory that we want to study,
we assume that it has a symmetry which is transpose and complex conjugate.
That is dagger h. Dagger equals H. Okay.
And what I'm saying is we have a whole bunch of new hamiltonians.
And if the spectrum of these hamiltonians, these symmetric hamiltonians,
it happens to be positive, we can then find out what the metric should be in the Hilbert space.
Okay. We don't assume dagger, but we calculate it, we find out what it is.
So this is the Hamilton this is the dagger that is appropriate for the Hamiltonian that you're studying.
Okay. So I emphasise with respect to CP Adjoint, the theory has unitary time evolution norms are strictly positive, probability is conserved.
So we have a generalisation of conventional quantum mechanics.
So this is an overview of my talk so far.
Okay. And I would like to just mention briefly that pretty symmetric systems are now being observed in the lab.
This is one of the early experiments, and it was a rather rudimentary experiment.
It appeared in PRL. It was a beautiful experiment in which they observed the transition using optical waveguides.
But it was followed shortly thereafter by a spectacular experiment which was published in Nature Physics,
where they observed the phase transition with tremendous precision.
Absolutely beautiful. I mean, you cannot see any difference between the theoretical predictions and the experimental measurements.
Absolutely spectacular experiment. This is an experiment that was done in Shanghai, a collaboration between Shanghai and Rutgers,
where they saw the phase transition in PTC, symmetric in the symmetric diffusion equation using rubidium atoms.
This is an experiment at Caltech where they used photonic silicon photonics circuits.
This is an experiment in Indiana using symmetric superconducting wires.
And you notice this is that graph I copied directly out of their paper.
There you see that the the energy levels, the eigenvalues coming together, becoming degenerate and disappearing into the complex plane.
You see that two distinct real eigenvalues become degenerate and then become complex.
And what is plotted here is only the real part of the eigenvalue.
So it goes off into the complex plane. A few more experiments.
This is a spectacular experiment. Absolutely beautiful experiment done in Germany.
It is an experiment using microwave cavities.
The microwave cavity was broken into two pairs, a single cavity.
There's a diaphragm that cuts the cavity in half and with a hole in between so that you can
have stuff in one cavity flowing into the one side of the cavity flowing into the other side.
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.
Beautiful experiment.
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.
There is there are studies of symmetric, photonic graphene done in Israel.
These are more studies of symmetric lasers done in Vienna, Princeton, Yale and Zurich.
There are studies of nonlinear asymmetric systems.
These are multiple symmetric waveguides.
This is a collaboration between Germany and Florida in in it just recently published in Nature.
Another experiment on superconducting wire is done at Argonne.
Symmetric NMR done in Beijing. And this is a beautiful experiment.
Very, very simple experiment. How do you have a system with balanced loss and gain?
Well, all you need to do is to take two LRC circuits.
Couple them inductively, put energy into one circuit, take energy out of the other circuit.
Change the coupling between the circuits.
And as you change the coupling constant between the two LRC circuits, you can see the phase transition.
And so we looked at I looked at this experiment and I said, Wait a minute, I can do this.
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.
So I said, Look, let's just take two pendulums and put a couple them together.
You see, there's an epsilon y and an Epsilon X.
So these are two coupled linear harmonic oscillators with loss and gain.
The first pendulum, the x pendulum has lost the way pendulum has gained.
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.
So we have we can have balance, loss and gain. So this is my experiment that we did at the Keiko in London.
And you can see there are two pendulum, one here, one pendulum here, one there hanging from a clothesline.
So they're coupled. You see that, right? And all we did was to take an electromagnet.
That's the that you see that circular thing at the top? That's an electromagnet.
And we taped. You see that white thing at the top of the pendulum cord?
That's a nail taped to the string with a piece of white tape.
And we the only fancy part of the experiment is that we used infrared beams.
So that is the pendulum with swinging one pendulum swinging toward the electromagnet,
the electromagnetic fired, that is, it pushed the swing, put a little energy into the pendulum.
But in the other pendulum, when it was swinging away from the electromagnet,
the electromagnetic fired took a little bit of energy out of the pendulum.
So one pendulum has energy going in.
The other pendulum has energy going out there, coupled. This is a symmetric system where under parity, you interchange the two peninsula.
And it's a time reversal. You have loss. Go into gain and gain.
Go into loss. So it's pretty symmetric. Okay. And I have to say, we stared at the pendulum for a long time.
This is the reason why science teachers should not be given playground duty, because they they really like to steer a pendulum.
Anyway, this is what we saw. If you turn off the magnets so you don't couple that pendulum, you don't.
You don't you don't have loss and gain. Rather, you see that what you have is Rabi oscillations.
The top picture is the theory. The bottom picture is the experiment.
Okay. Of course, the experiment shows a little bit of loss after many swings because the system has friction.
But you can see the Rabi oscillations very clearly in the X and the Y pendulum.
And then you turn on the magnets weakly and you still have Rabi oscillations.
So the system is still in the unbroken symmetric phase.
It's in equilibrium. You have Rabi oscillations, you turn up the magnets a little bit more and bang you go out of equilibrium.
The Rabi oscillations disappear and and the experiment and the theory are in wonderful agreement.
Okay, so that's a rudimentary experiment, the kind of thing that you might expect the theorists to do.
And in fact, what is going on here, what is really going on here is that these two pendulums are actually a Hamiltonian system.
Energy is actually conserved. And that is the Hamiltonian that describes the system.
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.
So it seems to lose a finite amount of energy. But the other pendulum blows up to infinity.
So you might think how could the system possibly be Hamiltonian?
How could energy be conserved? Well, it is conserved, but the form of the Hamiltonian is not exactly what you think it is.
It's not just p squared plus X squared, plus Q squared, plus y squared.
It's a little bit different. And this is the Hamiltonian for that system.
Okay. I won't go into detail.
But in fact, a system like this has to phase transitions, not one.
Okay. And. One of the phase transitions has yet to be seen experimentally.
So this is this is a challenge for experimentalists.
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.
And this is a very fancy system that we did at my university, Washington University.
These are two coupled optical resonators.
These are two coupled optical whispering galleries were really talked about whispering galleries.
These are solid state devices.
If you start a light beam going around, it will go around and around these whispering galleries a million times before it's absorbed.
And then we can dope one of these whispering galleries with terbium and shine a laser light on it to pump it up.
So that's a source. And the other one, the other whispering gallery has a natural loss.
Okay. And then we can couple them together and we can observe the phase transition.
And here it is. That's the phase transition and it is clean as a whistle.
Okay, it is really beautiful. Comes out very, very well.
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.
And instead of an x dot term, you can have an x triple dot.
And if you couple two such systems together, they become symmetric.
And this is the way to understand I claim to understand the self force, the electromagnetic self force.
Okay, well, the point is pretty symmetric.
Quantum mechanics is fun. And I just just concluded a minute or two,
but I just wanted to remark that you can revisit things that people have already done where there were problems in quantum mechanics,
and you can find that peachy symmetry pops up all over the place.
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.
The model is surprisingly called the Lee model, and the Lee model was proved to be wrong.
Okay. It was just wrong. It violated fundamental physics.
Violated units already. There is the pace uhlenbeck model, which is also an interesting model.
It was a bad model. It was a model of a higher order field equation in quantum field theory, and it had ghosts.
So it's no good. And in fact, the double scaling limit in quantum field theory is another such example where
there's a problem of a essentially a ghost in on her mission Hamiltonian.
And what happens with the lead model is that there's a fundamental problem with it.
The lead model is a is a tri linear coupling and one essentially one year 90.
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.
In fact, it's not no good. You can read about it in Barton's book.
This is this is the epitaph in Barton's book on Quantum Field Theory.
It says, A non-American Hamiltonian is unacceptable, partly because it may lead to complex energy eigenvalues,
but chiefly because it implies a non unitary ESP matrix which fails to conserve probability and makes a hash of the physical interpretation.
In fact, there's nothing wrong with the lead model. It's a wonderful model.
But the reason that it appears to be wrong is that it isn't her mission.
It is symmetric. And if you calculate the C operator and redefine your Hilbert space, it works perfectly.
Nothing at all wrong with the model. The same is true with the space uhlenbeck model.
It seems to have a ghost. This is a higher order field equation and it seems to have a ghost.
But there isn't a ghost.
The reason why there seems to be a ghost is because you're assuming a priori that the Hamiltonian is invariant under dagger h.
Dagger equals equals h, but in fact p t equals h, and you can calculate the C operator and there's no ghost.
Okay. And finally, if you look at the double scaling limit in quantum field theory, which I'm not going to talk about here,
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.
And people thought this is a disaster because we have an upside down potential.
But as I talked about earlier, there's nothing wrong with an upside down potential.
You just have to treat it as a symmetric theory and calculate the C operator and it all works.
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.
Last slide. Here it is. Okay.
So there are many possible future applications.
There are so many problems to work on. It is just there's just a myriad of problems.
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.
For example, you can study a symmetric Higgs model in the Higgs sector instead of talking about a fight of the fourth theory.
You could talk about a minus fight of the fourth theory.
And such a theory is really interesting because that theory is asymptotically free and stable and conformal,
invariant, and the expectation value of fire is nonzero because parity symmetry is broken.
And you may not see. It may not appear that fight of the Four has a broken parity symmetry.
But that theory is not invariant under PHI goes to minus PHI.
That's because the boundary conditions are not invariant under PHI goes to minus PHI.
The boundary conditions on the functional integral. That's the reason for this parity anomaly I was talking about.
And the parity anomaly I think is associated with the Higgs mass.
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.
This has been known for a very long time,
and the running coupling constant allows the the coefficient of the fight of the fourth term in the Higgs sector to go negative.
And that's that may be a serious problem, but I don't think so.
I think pity symmetry solves it. You can construct a model of symmetric electrodynamics.
And it's very interesting because it looks a little bit like a theory of magnetic charge,
and it's an asymptotically free theory, which is very interesting.
You can construct a symmetric theory of gravity, and this theory has a repulsive force, which is very interesting.
Maybe there's some connection between that and dark energy.
You can study the symmetric Dirac equation, just just to look at neutrinos.
And what you find is that the symmetric Dirac equation allows for massless neutrinos to undergo oscillations.
So it is not necessarily true that if you observe neutrino oscillations, that this implies that neutrinos have mass.
It may be still true that neutrinos are massless, and this could have interesting astrophysical implications.
And I could go on and on and on and on. There are so many interesting problems to look at.
I hope you look at some of them. Thank you for listening and I hope you go away and think about symmetric quantum mechanics.
All right. We're running a little bit late, but I'll take one question.
Okay. Yes.
Yeah, I'll ask pretty much the question you might expect me to ask.
If you look at, say, you take your favourite potential, the ice cube or something.
Yep. And look, not at the first two or three eigenvalues, but the 10th of the 20th.
Mm hmm. I believe you'll find those are exponentially sensitive to perturbations of the problem.
Yes. And show up in a year. Yeah. So this is very interesting.
So the the the unstated part of of Nick's question is the question of the pseudo spectrum.
And this is a very interesting question. Okay.
Let me give you a very brief answer, but we can talk about it afterward in detail.
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,
means this If you perturb the Hamiltonian remission, Hamiltonian by a small amount, the spectrum will be perturbed.
But a small perturbation of the Hamiltonian will will create a bounded perturbation, especially in the higher lying energy levels.
But if you have a non her mission Hamiltonian, it may be that a very,
very small perturbation in the Hamiltonian may produce an arbitrarily large change in the high lying energy level.
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?
And this is a measure this is something that you have to measure in the Hilbert space itself.
So if you use a conventional measure, that is an L2 measure.
Okay. So you're assuming dagger as a measure in your Hilbert space.
You can indeed get arbitrarily large changes in the high lying energy levels.
But you have to remember that we have redefined the Hilbert space.
That is not the way to measure a variation in the Hamiltonian.
The way you measure a variation is with respect to the CP T metric and with respect to the CP T metric,
you don't have an unbounded variation in the High Line energy levels.
Okay, so this has to do with the misuse of a metric.
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.
But it doesn't happen if you use the CP t metric.
It is bounded. That's, that's the that's the way to weasel out of that problem.
But but there's these that this is a very rich area and quite a few papers have now been written investigating this issue.
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.
I'm saying that these PD symmetric hamiltonians behave like ordinary formation.
Hamiltonians. But. But that's good that they they actually are stable.
The high lying energy levels are stable. Any other quick.
Yes, Steve. Yes. He talks a lot about the. Of Pakistan.
Yes. What about say it's the states? Oh, yes.
Oh, of course. Oh, yes. Oh, absolutely.
The Afghan states are or are corresponding to different energy levels are orthogonal.
That's actually very easy to prove directly from the Schrödinger equation.
And so you have all of the usual stuff in conventional quantum mechanics.
Okay. So we you don't have to worry.
You don't have to worry about those issues. Okay. And you also have completeness.
Okay. And so so that so the usual things that we expect for a removal problem, an unbounded removal problem, are still true.
You don't none of none of that machinery is lost.
Okay. And the problems were your actions were different.
Well, it's, it's reinterpreting.
Yeah. So yeah.
So the question is, I mean, here's what I would love.
I would love to have a smoking gun saying that this quantum system absolutely requires a p t symmetric Hamiltonian mean,
that's of course, you know, if I find that, then I retire, right.
And I'm it at this point, the kinds of symmetric systems that have been studied, you might say, are are synthetic symmetric systems.
So so we can do in the laboratory, if you build the p t symmetric system, it has remarkable properties.
Okay. What I mean by remarkable properties are, for example, you can see you need directional invisibility.
Okay. You can see something when the light goes one way and when it goes the other way, it's invisible.
You see remarkable things. You can see all sorts of strange and unusual properties.
But but these are synthetic symmetric systems.
And they're interesting because they offer the possibility in an optical system,
for example, of possibly building if you have a new control over light.
Okay. It does strange things as you introduce loss into the system, the power coming out of the system goes up.
Contrary to what you might expect, you would think that if you introduce loss into an optical system, the power is lost.
But in fact, it is to. Yeah, but.
But it goes the power that comes out of the system.
Power is going up. So this strange. So you have interesting control over light.
You might be able to build symmetric new kinds of symmetric metamaterials, symmetric computers and so on.
But what happens if we find a crystal on the ground and this crystal is has a symmetric spectrum that is uniquely different.
Okay. So I can give you one example. Okay.
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.
Okay. And so this is a band of allowed energies and then a gap and a band and a gap.
And at one edge of the conduction band, the wave function is two pi periodic.
At the other end, the wave function is for Pi periodic.
So at one end it looks like a boson, and the other end it looks like a fermion in a p t symmetric crystal.
So and a potential like ice sign of X instead of sign of x.
That's a symmetric crystal.
The bands and gaps are real because it is symmetric, but there's a remarkable difference at both edges of the conduction band.
The eigen functions are to Pi periodic, not to Pi Periodic for pi periodic, but to Pi Periodic.
It is as if you had two fermions coming together and making a boson.
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
and you measure the periodicity of the wave function at the edge of the conduction band,
and you see that that would be a smoking gun for a pretty symmetric Hamiltonian that you can't do with her mission.
Hamiltonian a conventional formation. HAMILTON That's what I would love to see.
No one has found such a crystal yet or such a clean example.
So right now, so far, the experiments that have been performed are on what is called P2 synthetic materials or synthetic devices,
where you can build the system and have it act like it, but not have it be, you know, crucially needed in nature.
Okay. Thank you. Yeah, really?