1
00:00:02,520 --> 00:00:05,610
Great to see you again. Lots of friends here.
2
00:00:07,320 --> 00:00:14,530
And thank you for the invitation to speak. I can't tell you how much fun I've been having with this.
3
00:00:14,550 --> 00:00:21,720
This is going to be a very elementary talk. It's a it's a colloquium talk, not a seminar talk.
4
00:00:23,880 --> 00:00:27,990
The only thing is some of the letters seem to have gotten just a second copy.
5
00:00:28,680 --> 00:00:32,950
Yeah. Oh, that's better. Okay. Okay.
6
00:00:33,750 --> 00:00:40,980
Okay. So what I'm going to be talking about is something a little bit strange and a little bit weird.
7
00:00:41,280 --> 00:00:47,430
I'm going to talk about extending quantum mechanics and classical mechanics into the complex plane.
8
00:00:47,790 --> 00:00:52,649
As you know, mathematicians love to generalise things and extend things.
9
00:00:52,650 --> 00:00:56,960
And it's very useful, as you know, to understand real variables.
10
00:00:57,030 --> 00:01:03,810
The best way to understand real variables is to extend them into the complex plane and to extend them to complex variables.
11
00:01:04,320 --> 00:01:07,800
And I'm going to try to do that with quantum mechanics.
12
00:01:08,530 --> 00:01:15,150
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.
13
00:01:15,480 --> 00:01:23,730
These are some examples. When I flew to the UK, actually, I took this plane over here.
14
00:01:24,210 --> 00:01:25,890
It actually worked very well.
15
00:01:26,130 --> 00:01:33,690
The only problem was that the people were sitting over here and the toilets were over there, so that that was so difficult.
16
00:01:35,250 --> 00:01:46,590
Okay. So here's the point of the talk. The as you know, if you take an elementary course, introductory course in quantum mechanics,
17
00:01:47,040 --> 00:01:51,660
what you learn is that the hermie that the Hamiltonian has to be her mission.
18
00:01:52,350 --> 00:01:57,450
And what that means is that if you take the Hamiltonian, it has a symmetry.
19
00:01:57,720 --> 00:02:02,490
If you take the Hamiltonian and you take its transpose and complex conjugate, you get back to the same Hamiltonian.
20
00:02:03,090 --> 00:02:10,700
And this is a very nice symmetry because it guarantees that the energy is real.
21
00:02:11,040 --> 00:02:16,800
And that's a good thing, because what you measure is the energy and that ought to be a real number.
22
00:02:17,250 --> 00:02:21,030
Okay. And it also guarantees that probability is conserved.
23
00:02:21,570 --> 00:02:29,550
But this axiom of quantum mechanics stands out from all the others as being different in some serious way.
24
00:02:31,290 --> 00:02:37,620
It looks like it's been written by a mathematician. All the other axioms of quantum mechanics sound, physical.
25
00:02:38,910 --> 00:02:44,790
You know, you would like to have a ground state of the system.
26
00:02:44,790 --> 00:02:49,710
You'd like probability to be conserved. You would like the energy to be real.
27
00:02:49,890 --> 00:02:56,610
You would like to have Lorentz invariance. You know, these these are very, very physical conditions.
28
00:02:56,820 --> 00:03:02,280
Okay. But transposed and complex conjugate doesn't sound like it was written by a physicist.
29
00:03:02,280 --> 00:03:07,380
It sounds like it was written by a mathematician. And I'm going to generalise that you're going to replace that.
30
00:03:07,980 --> 00:03:15,720
Oh, turn down the lights. Okay. It's probably one of these, uh, uh, lights.
31
00:03:16,050 --> 00:03:22,770
Here. Put. Do I push that button? No, no, that didn't work.
32
00:03:26,850 --> 00:03:30,240
The Troubles are getting a reflection from your screen. Okay.
33
00:03:33,050 --> 00:03:39,470
I don't think it's there. He. He pushed something over here. Lights, control panel.
34
00:03:46,710 --> 00:03:51,200
Zoom lights. No, nothing up helps.
35
00:03:51,650 --> 00:03:57,940
Cavalry's coming. Okay. I'm talking to his.
36
00:04:03,360 --> 00:04:06,820
Work better. Is that better?
37
00:04:07,870 --> 00:04:14,050
Okay. Now you can sleep without being noticed. Okay.
38
00:04:14,230 --> 00:04:21,550
So the point is that this axiom of elementary quantum mechanics, namely Hermitage City,
39
00:04:21,790 --> 00:04:27,070
stands out as being a little bit unusual in the sense that it sounds very mathematical.
40
00:04:27,400 --> 00:04:35,680
And I'm going to simplify that axiom and generalise that axiom and replace it by something a little bit more physical sounding.
41
00:04:35,840 --> 00:04:48,430
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.
42
00:04:48,910 --> 00:04:56,890
And the condition is called P symmetry. And P, as you know, is parity stands for parity, reflection space reflection.
43
00:04:57,190 --> 00:05:04,390
T is time reversal. So we're talking about theories that are invariant under space time reflection.
44
00:05:04,780 --> 00:05:08,500
That is X goes to minus X and T goes to minus T.
45
00:05:09,040 --> 00:05:12,070
Okay. So we're reflecting space, all of space.
46
00:05:13,810 --> 00:05:19,629
And I'm going to argue that systems that have this symmetry are very interesting.
47
00:05:19,630 --> 00:05:27,850
And this condition of symmetry can replace, in many cases, Dirac, Hermitage City, and simplify it.
48
00:05:28,180 --> 00:05:34,210
And because it's a weaker condition, it allows us to study all kinds of new and very strange theories.
49
00:05:34,480 --> 00:05:40,209
That's what we're going to talk about. Okay, so here's an example of this Hamiltonian.
50
00:05:40,210 --> 00:05:48,220
I came across this Hamiltonian quite a while ago when I was visiting some clay and in near Paris.
51
00:05:48,730 --> 00:06:00,910
And this Hamiltonian is sort of a quantum mechanical version of a conformal field theory that's associated with the leading edge singularity,
52
00:06:00,910 --> 00:06:10,840
which I'm not going to talk about at all. But a number of people were working at it, some fancy mathematicians, mathematical physicists.
53
00:06:11,110 --> 00:06:14,200
Daniel This is interesting.
54
00:06:14,410 --> 00:06:19,360
A number of other people were working on it and they mentioned this Hamiltonian to me.
55
00:06:19,570 --> 00:06:22,840
And when I saw this Hamiltonian, I said, This is ridiculous.
56
00:06:23,290 --> 00:06:27,750
It's it's not permission. So this Hamiltonian really forget it.
57
00:06:27,850 --> 00:06:41,170
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.
58
00:06:41,470 --> 00:06:45,940
It was at my university and people were actually getting up and leaving.
59
00:06:46,520 --> 00:06:53,740
Just unbelievably awful. This was much worse than the second worst colloquium I had ever heard.
60
00:06:54,220 --> 00:07:01,660
And I wanted to get up and leave as well. But I couldn't because I had invited the seminars and the colloquium speaker.
61
00:07:02,740 --> 00:07:06,970
So instead, I decided to escape by doodling.
62
00:07:07,300 --> 00:07:10,840
And this problem came to me.
63
00:07:13,270 --> 00:07:21,370
And I want to point out that this Hamiltonian, you know, h equals p squared plus I x cubed.
64
00:07:21,670 --> 00:07:29,980
This is symmetric because under parity reflection exchanges sine and under time reversal.
65
00:07:30,010 --> 00:07:33,280
I change this sign, I goes to minus II. Okay.
66
00:07:33,850 --> 00:07:45,190
And just for just to emphasise that you might ask why does it change sine and the fundamental equation of of quantum mechanics.
67
00:07:45,190 --> 00:07:50,560
The Heisenberg Algebra says that X commuted with P is eight times H.
68
00:07:50,890 --> 00:07:54,230
Okay. And this ought to be invariant under.
69
00:07:58,480 --> 00:08:05,080
You sign. So the equation remains invariant. P changes sign.
70
00:08:06,340 --> 00:08:10,750
So the left side changes side. So the right side ought to change sign as well.
71
00:08:10,990 --> 00:08:14,020
So time reversal involves complex conjugation.
72
00:08:14,280 --> 00:08:19,240
Okay, another way to see that is to look at the Schrödinger equation.
73
00:08:19,840 --> 00:08:28,120
Just look at the Schrödinger equation and you can see that on the right hand side there is a D by d,
74
00:08:28,120 --> 00:08:32,860
t, and if you change the sign of T and you want to keep the equation invariant, okay.
75
00:08:32,950 --> 00:08:39,220
You also have to change the sign of odd. Okay. So if you had this equation to solve.
76
00:08:40,550 --> 00:08:44,180
And you're doodling in a colloquium, what might you do?
77
00:08:44,210 --> 00:08:48,890
Well, you say, maybe I can solve this equation using perturbation theory.
78
00:08:49,310 --> 00:08:55,960
So let's take a theory that I can solve, namely the Hamiltonian P squared plus X squared.
79
00:08:55,970 --> 00:09:00,830
That's the harmonic oscillator. And I know how to solve that. And let's just.
80
00:09:02,900 --> 00:09:11,140
I will only write a few more lines. So, okay, let's just take this Hamiltonian and introduce a perturbation from.
81
00:09:15,560 --> 00:09:18,800
Salon. And treat Epsilon as a small parameter.
82
00:09:19,130 --> 00:09:24,440
Okay. Now, if Epsilon is zero, I can solve this problem because it's the harmonic oscillator.
83
00:09:24,950 --> 00:09:29,180
And at the end of the calculation, I would like to set epsilon equal one.
84
00:09:29,330 --> 00:09:36,650
To recover the problem that I want to solve. Okay. So I can expand in powers of epsilon.
85
00:09:37,250 --> 00:09:41,240
And that's what I did. I calculated a few terms in the perturbation expansion.
86
00:09:41,450 --> 00:09:46,669
This is very powerful, by the way. This technique of putting a small parameter in the exponent,
87
00:09:46,670 --> 00:09:51,860
you can use it to solve all kinds of nonlinear problems, like the effect of freeze equation.
88
00:09:52,220 --> 00:10:02,360
You can use it to solve the Thomas Fermi equation, all sorts of interesting nonlinear problems and order by order and powers of epsilon.
89
00:10:02,360 --> 00:10:08,390
To my astonishment, the eigenvalues of the Hamiltonian turned out to be real.
90
00:10:09,080 --> 00:10:10,800
Okay. Which is hard to believe. Okay.
91
00:10:11,090 --> 00:10:19,280
But you see, the reason for introducing Epsilon this way is you notice that this problem remains pretty symmetric for any epsilon,
92
00:10:19,280 --> 00:10:24,910
so long as epsilon is real. And that's because, again, you send X into minus X and I into minus sign.
93
00:10:25,280 --> 00:10:32,690
Okay. So, in fact, I ran out of this colloquium when it was finally over with.
94
00:10:33,080 --> 00:10:45,420
Okay. And went to my office and calculated numerically the eigenvalues of this Hamiltonian and on the horizontal axis is just epsilon, okay?
95
00:10:45,800 --> 00:10:54,200
And on the vertical axis I've plotted the eigenvalues and you can see that the eigenvalues are discrete and real and positive.
96
00:10:54,830 --> 00:10:58,280
Okay? And so there's the plot of the eigenvalues.
97
00:10:58,280 --> 00:11:01,370
It's really nice. You can see it, okay? And they're real.
98
00:11:03,860 --> 00:11:07,429
And that was the beginning of p symmetric quantum mechanics.
99
00:11:07,430 --> 00:11:14,270
This goes back to 1998. Okay. And I wanted to give you an outline.
100
00:11:14,360 --> 00:11:18,950
That was the introduction. This is an outline of my talk today.
101
00:11:19,190 --> 00:11:22,610
I think that's fairly accurate. Okay.
102
00:11:25,610 --> 00:11:37,669
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,
103
00:11:37,670 --> 00:11:46,820
real and positive. And this is an infinite class of symmetric theories which are not permission in the usual quantum mechanical sense.
104
00:11:47,120 --> 00:11:54,230
Okay, but they have real discrete, positive energy levels.
105
00:11:54,710 --> 00:12:01,640
Okay. And you might ask, wait a minute, what happens if Epsilon is equal to two?
106
00:12:02,150 --> 00:12:09,290
Because if Epsilon is equal to two, you get the Hamiltonian for epsilon equals two.
107
00:12:09,320 --> 00:12:16,940
Here epsilon equals two, you get the Hamiltonian P squared minus X to the four.
108
00:12:17,600 --> 00:12:19,370
And that's an upside down potential.
109
00:12:19,670 --> 00:12:28,970
And the picture I showed you before had positive real eigenvalues, a spectrum going off to infinity, and this is an upside down potential.
110
00:12:29,210 --> 00:12:31,760
So what I'm telling you is really strange.
111
00:12:31,760 --> 00:12:41,630
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.
112
00:12:42,200 --> 00:12:46,429
Okay, so what I'm telling you is quite radical, but it is not just true.
113
00:12:46,430 --> 00:12:51,680
It's rigorously true at a mathematical level. Okay, so that was the beginning.
114
00:12:52,070 --> 00:13:02,990
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
115
00:13:02,990 --> 00:13:11,330
nature physics that there is now a very big field of symmetric quantum mechanics.
116
00:13:11,330 --> 00:13:23,180
People are working on it. These are the papers from 2008 to 2010, and there are lots more in 2011 and 12.
117
00:13:23,660 --> 00:13:33,920
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.
118
00:13:33,920 --> 00:13:46,280
These are the papers in 2014 and already in fancy journals there are four review articles so far.
119
00:13:46,340 --> 00:13:52,340
There's a book under preparation right now and more review articles that are being worked on right now.
120
00:13:53,840 --> 00:14:02,120
Since its beginning in 1998, there have been 20 international conferences in physics goes very fast.
121
00:14:02,330 --> 00:14:06,650
Okay? There are now nearly 2000 published papers.
122
00:14:06,980 --> 00:14:14,930
But the most exciting thing for me as a mathematical physicist is that in the last four years, there have been piles of.
123
00:14:15,200 --> 00:14:20,810
Our mental results. And people have observed in the laboratory what we've been talking about,
124
00:14:21,050 --> 00:14:27,050
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.
125
00:14:28,700 --> 00:14:34,820
So there is a rigorous proof that what I've told you is true, that the eigenvalues are real.
126
00:14:35,120 --> 00:14:40,100
I'm not going to talk about it at all, but you can read it.
127
00:14:41,030 --> 00:14:45,570
It involves developing something called the ODI correspondence.
128
00:14:45,590 --> 00:14:50,960
There's a deep a very deep and profound correspondence between ordinary differential equations,
129
00:14:51,290 --> 00:14:54,890
namely the Schrodinger equation and integral role models.
130
00:14:55,520 --> 00:15:05,210
Okay. And I want to go back now to something that I glossed over because I really meant to gloss over it.
131
00:15:05,480 --> 00:15:13,490
You notice in this picture that when Epsilon is positive, the eigenvalues are strictly real positive and discrete.
132
00:15:13,820 --> 00:15:19,670
But you notice that when Epsilon, which is the the power over here, if you can see it in the dark,
133
00:15:19,670 --> 00:15:27,250
X squared times X to the epsilon, when epsilon goes negative, something interesting seems to happen.
134
00:15:27,260 --> 00:15:32,060
You notice that eigenvalues are coming together and becoming degenerate and disappearing.
135
00:15:32,480 --> 00:15:38,220
Well, they're not disappearing. But what is happening is that they're becoming complex, okay?
136
00:15:38,570 --> 00:15:41,750
They become degenerate and then disappear into the complex plane.
137
00:15:42,170 --> 00:15:46,100
So something happens at Epsilon equals zero.
138
00:15:46,490 --> 00:15:51,920
There's a transition that occurs. And when epsilon is equal to zero, that's the harmonic oscillator.
139
00:15:52,130 --> 00:15:58,810
So the harmonic oscillator lives at this transition. That's, in fact, where her mission hamiltonians live.
140
00:15:58,820 --> 00:16:05,480
They live right at the transition between a region of unbroken symmetry and a region of broken symmetry,
141
00:16:05,780 --> 00:16:11,720
which is where the eigenvalues, which can be complex, that's where they actually become complex.
142
00:16:12,080 --> 00:16:19,520
So there's a region where they all the eigenvalues are real and positive and where they begin to go complex.
143
00:16:19,520 --> 00:16:27,100
And that's called the boundary. Okay. And whenever you have a transition, that means there's a possibility of doing experiments.
144
00:16:27,110 --> 00:16:34,940
And that's these experiments that I mentioned before. And one of the things that these experiments observe is this transition.
145
00:16:37,370 --> 00:16:46,910
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.
146
00:16:47,060 --> 00:16:53,470
Okay. Or if you go to Paris, Steve, you've seen this lots of times in Paris.
147
00:16:53,480 --> 00:16:59,180
This is this is an example right there on the sidewalk in Paris of Broken Peaty.
148
00:17:00,620 --> 00:17:08,710
Okay. So I want to emphasise that her mission hamiltonians are really boring and they're boring because nothing interesting happens.
149
00:17:08,720 --> 00:17:12,860
The eigenvalues are always real and they don't undergo a transition.
150
00:17:13,110 --> 00:17:22,460
Okay, but but p t symmetric hamiltonians are astonishing because there is this transition
151
00:17:22,730 --> 00:17:28,880
between a region of of unbroken symmetry where the eigenvalues are real and broken.
152
00:17:29,180 --> 00:17:32,930
Symmetry with the eigenvalues go complex. Okay.
153
00:17:33,860 --> 00:17:38,030
So I want to give you an intuitive explanation of why there's a transition.
154
00:17:39,620 --> 00:17:43,099
So this is really simple argument.
155
00:17:43,100 --> 00:17:48,080
This is I'm giving you basically an undergraduate quantum mechanics argument.
156
00:17:49,280 --> 00:17:53,420
Suppose you have a box I'm calling it box one.
157
00:17:53,840 --> 00:18:02,510
And in this box you have some system and you have an antenna in the box which is taking stuff out of the box.
158
00:18:02,510 --> 00:18:11,030
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.
159
00:18:11,030 --> 00:18:14,870
It doesn't really matter. But stuff is being taken out of the box.
160
00:18:14,870 --> 00:18:17,940
So there's a sink in that box. Okay.
161
00:18:18,320 --> 00:18:26,780
Now, if you were to describe that box using a Schrödinger equation, there is the Schrödinger equation, okay, underneath the box.
162
00:18:27,050 --> 00:18:32,570
And in general, the Hamiltonian is an operator, an infinite dimensional matrix or a finite dimensional matrix.
163
00:18:32,570 --> 00:18:36,740
Here is just a one by one matrix. Just, just a number.
164
00:18:37,070 --> 00:18:41,209
Okay. And that number is just what I'm calling e one over here.
165
00:18:41,210 --> 00:18:44,220
It's just. Ah, E to the fader. Okay.
166
00:18:45,380 --> 00:18:49,970
And if you solve this trivial first order linear differential equation,
167
00:18:50,240 --> 00:19:01,010
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.
168
00:19:01,550 --> 00:19:05,840
Okay. And that's all. So this is no surprise.
169
00:19:05,840 --> 00:19:13,400
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.
170
00:19:14,480 --> 00:19:21,880
This is radioactive decay. You could have another box with a sauce antenna in it.
171
00:19:22,280 --> 00:19:28,220
And all you have to do is take the complex conjugate of the energy of the Hamiltonian in this box.
172
00:19:28,230 --> 00:19:32,360
So you get the Hamiltonian is just r e to the minus theta.
173
00:19:32,660 --> 00:19:36,410
And again, the solution is the same, and this is growing exponentially.
174
00:19:36,770 --> 00:19:42,780
Okay. So these, these are not her emission hamiltonians, but that's what you get.
175
00:19:42,800 --> 00:19:46,340
You can think of these as effective hamiltonians. Okay.
176
00:19:46,580 --> 00:19:51,379
Now let's put these two boxes together and consider them as a single system,
177
00:19:51,380 --> 00:19:55,220
even though they're not connected to one another and they're not interacting with one another.
178
00:19:55,550 --> 00:19:57,890
So the Hamiltonian is diagonal,
179
00:19:59,000 --> 00:20:07,550
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.
180
00:20:08,000 --> 00:20:15,770
And the point here is that this Hamiltonian is symmetric and is p t symmetric where p
181
00:20:15,770 --> 00:20:22,190
parity is the matrix that I've written down 0110 and T is just complex conjugation.
182
00:20:22,630 --> 00:20:27,080
Okay, so we have a two by two symmetric Hamiltonian.
183
00:20:27,600 --> 00:20:34,930
Okay. And now what I'm going to do is I'm going to couple the boxes together.
184
00:20:35,140 --> 00:20:40,030
So you have to imagine that there's a pipe going from box number one to box number two.
185
00:20:40,450 --> 00:20:46,240
Stuff is going into box number two and it's being taken out of box number one.
186
00:20:46,480 --> 00:20:52,390
But stuff is flowing from box number two to box number one so that the system can be in equilibrium.
187
00:20:53,050 --> 00:20:57,910
In principle, it might be in equilibrium if you couple the boxes strongly enough.
188
00:20:58,120 --> 00:21:04,599
And indeed, if you calculate the eigenvalues of this Hamiltonian, which is not our mission but is pretty symmetric,
189
00:21:04,600 --> 00:21:14,500
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,
190
00:21:14,740 --> 00:21:18,730
and that's the value of the critical coupling. Okay, squared science squared theta.
191
00:21:19,330 --> 00:21:24,310
Okay. So here is the full theory.
192
00:21:24,340 --> 00:21:28,990
Consider a quantum mechanical theory governed by that Hamiltonian.
193
00:21:29,860 --> 00:21:34,000
It has it's a three parameter Hamiltonian. It contains R, s and theta.
194
00:21:34,630 --> 00:21:40,780
It is symmetric. And you can see that the energy levels become real.
195
00:21:41,080 --> 00:21:44,080
If x squared is greater than R squared sine squared theta.
196
00:21:44,800 --> 00:21:51,160
And interestingly, there's another matrix which I'm going to call which I'm calling C,
197
00:21:51,160 --> 00:21:55,180
which I'll talk about in a minute, but that's the formula for this operator.
198
00:21:55,180 --> 00:22:02,740
C So this is a really trivial example of a two by two pretty symmetric Hamiltonian that that exhibits this transition.
199
00:22:03,100 --> 00:22:12,940
Okay. All right. So how do we understand this transition in a more sophisticated system?
200
00:22:14,410 --> 00:22:18,220
Let's look at p t symmetric classical mechanics.
201
00:22:18,580 --> 00:22:26,680
So take this Hamiltonian. I started out my talk with this p squared plus x cubed and you notice that the interaction term,
202
00:22:27,130 --> 00:22:34,570
this this potential is proportional to EI and you notice that the sign of that potential,
203
00:22:34,570 --> 00:22:42,010
the sign of eye changes as you go from X, which is negative to X, that is positive.
204
00:22:42,430 --> 00:22:48,880
Okay. So in fact, this Hamiltonian is no longer a matrix simple matrix Hamiltonian.
205
00:22:49,090 --> 00:22:58,180
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
206
00:22:58,180 --> 00:23:04,720
sync antenna which is spread over the entire real axis as x goes to plus infinity.
207
00:23:05,140 --> 00:23:12,130
And we now ask a very simple question How could this system possibly be in equilibrium?
208
00:23:12,610 --> 00:23:17,559
How could the eigenvalues of this Hamiltonian be real if the antenna,
209
00:23:17,560 --> 00:23:22,690
the source antenna and the zinc antenna are becoming infinitely strong as you run off to infinity?
210
00:23:23,780 --> 00:23:29,570
And the only way that this system could be in equilibrium would be if the particles that are being created,
211
00:23:30,170 --> 00:23:36,890
the stuff that's being created at minus infinity can get to plus infinity where it's destroyed fast enough.
212
00:23:37,700 --> 00:23:45,589
So classical and classical mechanics, let's calculate the amount of time it takes for a particle to travel from minus infinity,
213
00:23:45,590 --> 00:23:48,620
where it's created to plus infinity where it's destroyed.
214
00:23:48,830 --> 00:23:59,720
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.
215
00:24:00,260 --> 00:24:05,659
Okay, in the Hamiltonian it's the integral d x over the square root of v minus x cubed.
216
00:24:05,660 --> 00:24:12,350
And that integral is finite. Okay. And that's because the power of x is bigger than two in the square root,
217
00:24:12,650 --> 00:24:20,330
and therefore it only takes a finite amount of time for the particles that are created at minus infinity to get to,
218
00:24:20,330 --> 00:24:25,940
plus infinity where they're destroyed. The particles can travel infinitely fast.
219
00:24:26,180 --> 00:24:30,050
Evidently they can travel over an infinite distance in a finite time.
220
00:24:30,530 --> 00:24:32,930
And that's how the system can be in equilibrium.
221
00:24:33,110 --> 00:24:40,519
And that's why the eigenvalues of this Hamiltonian that I started out with can actually be, you know, the eigenvalues can be real.
222
00:24:40,520 --> 00:24:51,919
The system is in equilibrium. Okay, so to give you another classical example, suppose you were taking a course in first gear,
223
00:24:51,920 --> 00:24:57,560
undergraduate physics, and someone was explaining to you the classical harmonic oscillator.
224
00:24:57,980 --> 00:25:02,090
So what do they tell you? They say the Hamiltonian is p squared plus x squared,
225
00:25:03,620 --> 00:25:10,250
and now all you're told is that there are turning points and this is the edge of the classically allowed region.
226
00:25:10,550 --> 00:25:17,990
The particle travels say to the right until it hits a turning point and it stops and turns around and goes back again.
227
00:25:17,990 --> 00:25:23,059
And then it stops and turns around and goes back again. So that's simple harmonic motion,
228
00:25:23,060 --> 00:25:29,930
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,
229
00:25:30,170 --> 00:25:37,130
What happens if you put a particle initially to the right of the right hand turning point?
230
00:25:38,180 --> 00:25:44,000
And of course, the lecturer says to the student, You idiot, you can't do that.
231
00:25:44,300 --> 00:25:47,780
You don't say that. Okay, get fired.
232
00:25:47,780 --> 00:25:51,950
If you do that, you then you say, interesting question.
233
00:25:53,090 --> 00:25:57,740
Okay, but you can't do that because that's the classically forbidden region.
234
00:25:58,100 --> 00:26:05,420
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,
235
00:26:05,750 --> 00:26:10,730
that means that x squared is greater than E, and that means that p squared is negative.
236
00:26:13,110 --> 00:26:18,780
And you can't have a negative value for P squared. And the student says That's okay.
237
00:26:18,930 --> 00:26:22,710
It just means that P is not real. It's imaginary.
238
00:26:23,190 --> 00:26:30,330
So instead of restricting the motion of this particle to the real axis, let's extended it to the complex plane.
239
00:26:30,630 --> 00:26:35,490
And now, if you solve the equation F equals M A, what you find is that the particle,
240
00:26:35,490 --> 00:26:39,540
if it starts to the right at the turning point, just goes around in the ellipse.
241
00:26:40,530 --> 00:26:44,760
And no matter where you start the particle, it always goes around in an ellipse.
242
00:26:44,970 --> 00:26:48,930
And interestingly, the turning points are the foci of the ellipse.
243
00:26:49,590 --> 00:26:55,410
And this is complex classical mechanics where we've gone off the real axis into the complex plane.
244
00:26:55,770 --> 00:27:01,259
It's interesting, by the way, that the back and forth motion that you talk about in an elementary physics class,
245
00:27:01,260 --> 00:27:06,050
simple harmonic motion, is not back and forth. In fact, it's going around and around.
246
00:27:06,060 --> 00:27:09,270
It's a degenerate ellipse. Okay.
247
00:27:09,270 --> 00:27:18,600
It's really an ellipse. That's degenerate. Okay, so now what happens if we consider the Hamiltonian p squared plus x cubed?
248
00:27:18,930 --> 00:27:26,880
Now you have two turning points and the particles are going around in these turning points, and these are just deformed ellipses.
249
00:27:28,080 --> 00:27:33,950
Okay. But it's the same basic picture. And now if you have P squared minus,
250
00:27:33,980 --> 00:27:39,150
remember I was going to answer the question of how you could have eigenvalues
251
00:27:39,150 --> 00:27:44,100
with positive eigenvalues with the Hamiltonian P squared minus X to the four.
252
00:27:44,940 --> 00:27:48,989
This is the picture. Now for a minus X to the four Hamiltonian,
253
00:27:48,990 --> 00:27:56,880
there are four turning points and the classical paths go around and around these pairs of of turning points.
254
00:27:57,450 --> 00:28:05,100
Okay. And if you'd like to quantised the system and ask why are the eigenvalues real?
255
00:28:05,580 --> 00:28:08,940
You remember what you learn in elementary quantum mechanics.
256
00:28:09,270 --> 00:28:16,890
You learn that a particle is a wave, and as you go around in a closed path and the complex explain,
257
00:28:17,160 --> 00:28:23,160
you better have a finite number of you better have an integer number of wavelengths plus a half.
258
00:28:23,670 --> 00:28:30,270
Because as this wave goes around and follows the path of the classical particle, it has to add up and phase.
259
00:28:30,750 --> 00:28:37,890
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
00:28:38,070 --> 00:28:41,160
that gives you the energy levels of the system. It's very simple.
261
00:28:41,820 --> 00:28:48,270
Okay. So if you do that for this case, you'll find the eigenvalues that I talked about.
262
00:28:51,360 --> 00:28:54,630
If you ask what happens when the symmetry is broken?
263
00:28:55,020 --> 00:29:03,300
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.
264
00:29:03,480 --> 00:29:11,550
You have open orbits. In fact, the transition that occurs is a classical transition occurs at the classical level.
265
00:29:11,790 --> 00:29:16,739
And what happens when epsilon goes below zero is that the paths are no longer closed.
266
00:29:16,740 --> 00:29:22,710
They're open and the paths go, the classical paths and the complex plane go off to infinity.
267
00:29:22,860 --> 00:29:26,100
And you can no longer write down an equation like this,
268
00:29:26,520 --> 00:29:30,569
because this equation has that that little circle on the integral sign that says you have
269
00:29:30,570 --> 00:29:35,760
to integrate along a closed path in the complex plane and there aren't any closed paths.
270
00:29:36,570 --> 00:29:46,380
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.
271
00:29:46,680 --> 00:29:53,430
What's happening on the real axis? Well, if you push that, if you if you go closer and closer to the real axis,
272
00:29:53,790 --> 00:29:59,580
the paths get bigger and bigger and bigger and they begin to look like this.
273
00:30:00,960 --> 00:30:09,150
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.
274
00:30:09,600 --> 00:30:17,850
It then zips around instantly through the complex plane back to minus infinity and runs back up the real axis again.
275
00:30:18,630 --> 00:30:22,950
So the point is that it only takes a finite amount of time for a particle in
276
00:30:22,950 --> 00:30:27,600
an upside down potential like minus X to the four to roll out to infinity.
277
00:30:27,960 --> 00:30:34,080
And then where does it go? The answer is it goes all the way to minus infinity and comes back up to the origin.
278
00:30:34,800 --> 00:30:38,490
And if you then say, Where are we most likely to find the particle?
279
00:30:38,490 --> 00:30:41,670
The answer is you're most likely to find the particle at the origin,
280
00:30:41,970 --> 00:30:48,990
because the classical probability of finding a particle is proportional to one over the velocity of the particle.
281
00:30:49,260 --> 00:30:52,350
The faster it goes, the less likely it is to be there.
282
00:30:52,590 --> 00:30:55,560
That's why if you're driving in a car and you drive through an intersection,
283
00:30:55,980 --> 00:31:00,420
when you reach the intersection, you should drive as fast as possible so that you don't have an accident.
284
00:31:00,570 --> 00:31:05,630
Okay, so in the car.
285
00:31:05,850 --> 00:31:08,940
So this is what the classical probability looks like.
286
00:31:09,120 --> 00:31:12,120
And this looks like a bound state, a classical.
287
00:31:12,250 --> 00:31:20,170
Bound state, it is most likely to find the particle at the origin and that's the reason why you have a localised downstate.
288
00:31:20,410 --> 00:31:24,970
Even though you have an upside down potential, it's very radical stuff.
289
00:31:25,150 --> 00:31:27,670
This is not what you would expect. Maybe.
290
00:31:28,690 --> 00:31:35,260
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.
291
00:31:35,500 --> 00:31:39,160
It zips around in the complex plane, it comes back to the origin again.
292
00:31:39,520 --> 00:31:44,500
So it spends most of its time at the origin. That's what the probability looks like.
293
00:31:45,250 --> 00:31:52,930
Okay. Okay. So now I can't resist showing you a derivation, but this is a colloquium.
294
00:31:52,930 --> 00:31:56,320
So this derivation is going to be the fastest derivation you ever saw.
295
00:31:57,250 --> 00:32:07,030
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.
296
00:32:07,190 --> 00:32:19,179
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,
297
00:32:19,180 --> 00:32:26,770
the minus X to the four Hamiltonian has exactly the same energy levels as the plus Z to the four Hamiltonian.
298
00:32:28,000 --> 00:32:31,030
Okay. But there's a difference between these two hamiltonians.
299
00:32:31,030 --> 00:32:34,710
You notice there's a term in the middle. Okay.
300
00:32:34,860 --> 00:32:38,100
And that is proportional to H. And that's an anomaly.
301
00:32:38,250 --> 00:32:42,030
And that is actually a parody anomaly, which I'm I won't.
302
00:32:43,590 --> 00:32:48,420
Quantum anomaly means something that vanishes in the limit as Planck's constant vanishes.
303
00:32:48,540 --> 00:32:53,610
Okay. And. So that is a pure quantum effect.
304
00:32:54,000 --> 00:33:02,040
So the the lower Hamiltonian, you know, that it has real positive eigenvalues because it's a confining potential.
305
00:33:02,040 --> 00:33:10,589
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.
306
00:33:10,590 --> 00:33:15,690
And that's the rigorous proof. And the proof goes like this you start with the top Hamiltonian.
307
00:33:16,290 --> 00:33:26,910
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.
308
00:33:27,240 --> 00:33:35,400
And now step one, I make a change of variables and this is a first year undergraduate mathematics change of variables,
309
00:33:35,670 --> 00:33:39,690
and you end up with a disgusting differential equation. That's the eigenvalue problem.
310
00:33:40,290 --> 00:33:43,290
Okay, trust me, the arithmetic is correct.
311
00:33:45,030 --> 00:33:50,400
Step two you do a 48 transform of that ugly equation and you get another ugly equation.
312
00:33:50,700 --> 00:33:54,230
And that's what's written on the bottom of this transparency. Okay.
313
00:33:54,270 --> 00:33:58,049
It's pretty ugly, which is an oxymoron, I guess.
314
00:33:58,050 --> 00:34:09,540
Pretty ugly. And then you make a change of dependent variable and you get this new equation on the bottom of this transparency.
315
00:34:10,470 --> 00:34:14,790
And finally, you make a scale change.
316
00:34:15,750 --> 00:34:20,489
And the resulting how the resulting differential equation is.
317
00:34:20,490 --> 00:34:24,960
The differential equation is a simple differential equation for right side up potential.
318
00:34:25,170 --> 00:34:32,760
That's it. I have never changed e in the process of this derivation and therefore the eigenvalues of the
319
00:34:32,760 --> 00:34:37,800
upside down potential are exactly the same as the eigenvalues of a right side up potential.
320
00:34:38,070 --> 00:34:44,340
But this new Hamiltonian has a term proportional to H four, which is very strange, and that's an anomaly.
321
00:34:44,640 --> 00:34:47,730
And it's an anomaly that I call a parody anomaly.
322
00:34:47,730 --> 00:34:50,580
And if you're interested, we can talk about it later.
323
00:34:52,170 --> 00:34:59,909
But the result is that we have a pair of exactly ISO spectral hamiltonians and that means they have the same identity,
324
00:34:59,910 --> 00:35:03,600
they have identical energy levels, identical eigenvalues.
325
00:35:04,050 --> 00:35:08,970
And the first one is an upside down potential. Which is strange.
326
00:35:09,210 --> 00:35:12,130
And the second one is a right side of potential. Okay.
327
00:35:13,920 --> 00:35:26,069
So what I'm arguing is that we have a whole bunch of new hamiltonians that we can study with that look
328
00:35:26,070 --> 00:35:31,250
strange that you would never have studied if you were just taking an elementary course in quantum mechanics.
329
00:35:31,800 --> 00:35:39,930
And these new hamiltonians are in some sense intermediate between the two basic types of hamiltonians that you would study.
330
00:35:40,650 --> 00:35:47,100
There is the usual Hamiltonian, which is a commission Hamiltonian, and that describes a closed system,
331
00:35:47,700 --> 00:35:51,960
and there's the usual non-commissioned Hamiltonian that describes an open system.
332
00:35:52,170 --> 00:35:57,120
So these are phenomenological hamiltonians that describe maybe some scattering process
333
00:35:57,510 --> 00:36:02,920
in the that you would use to describe maybe a scattering process in nuclear physics.
334
00:36:03,390 --> 00:36:06,750
And these new hamiltonians are midway between them.
335
00:36:07,290 --> 00:36:14,790
These are symmetric hamiltonians, because if you are in a broken, symmetric region where the eigenvalues are complex,
336
00:36:15,000 --> 00:36:20,310
they look as if they're coming from an on her mission Hamiltonian for an open system.
337
00:36:20,760 --> 00:36:29,640
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.
338
00:36:29,970 --> 00:36:34,950
They're not really closed because there are source antennas and sync antennas.
339
00:36:35,130 --> 00:36:39,740
So they are coupled to the outside world. But they behave as if they're coming.
340
00:36:39,750 --> 00:36:42,870
There are hamiltonians associated with a closed system.
341
00:36:43,910 --> 00:36:52,260
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.
342
00:36:52,530 --> 00:36:57,000
See, this is my I don't know if you can read it from where you are sitting.
343
00:36:57,210 --> 00:37:05,720
If you read Portuguese, this says this says in Portuguese, if Piti is in your heart, I want to talk to you.
344
00:37:05,730 --> 00:37:08,969
And since you're here, Piti must be in your heart.
345
00:37:08,970 --> 00:37:12,510
And here I am talking to you. And so. Okay, good.
346
00:37:13,230 --> 00:37:16,960
All right. So the T-shirts will be on sale in the lobby after this time.
347
00:37:18,870 --> 00:37:23,790
So and I want to emphasise that at a mathematical level, that that was at the physical level.
348
00:37:23,790 --> 00:37:31,020
But at a mathematical level, what we are talking about is extending conventional classical mechanics and her mission,
349
00:37:31,050 --> 00:37:37,290
quantum mechanics into the complex plane. And when I say extending her mission, quantum mechanics into the complex plane,
350
00:37:37,620 --> 00:37:45,480
the condition that the Hamiltonian be her mission is transpose and complex conjugate being a symmetry of the Hamiltonian.
351
00:37:45,620 --> 00:37:49,980
Okay. And if you violate that symmetry, it is as if you're going off the real axis.
352
00:37:50,010 --> 00:37:55,350
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
00:38:06,180 --> 00:38:11,100
Do we have a Hilbert space with the positive metric? Do we have unitary time evolution?
356
00:38:11,340 --> 00:38:15,600
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
00:38:27,420 --> 00:38:33,750
It turns out that if you are in a region of unbroken symmetry where all the eigenvalues are real,
360
00:38:34,020 --> 00:38:39,560
you can show that there exists a new symmetry of the Hamiltonian that you wouldn't have expected.
361
00:38:39,570 --> 00:38:43,469
It's a secret symmetry, and it's represented by the symmetry operator.
362
00:38:43,470 --> 00:38:50,860
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.
367
00:39:21,060 --> 00:39:26,310
The third equation is that C computes for the Hamiltonian. So it's a symmetry of the Hamiltonian.
368
00:39:26,910 --> 00:39:34,200
And now the trick is to replace dagger, which means transpose in complex conjugate by C, p, t.
369
00:39:34,560 --> 00:39:35,670
And if you do that,
370
00:39:36,300 --> 00:39:50,190
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.
371
00:39:50,460 --> 00:40:01,170
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.
372
00:40:02,220 --> 00:40:08,460
You have conservation of probability, unitary, everything works, and you now have a quantum mechanical theory.
373
00:40:09,070 --> 00:40:12,970
Okay, so the way. So what I'm saying to.
374
00:40:13,080 --> 00:40:16,710
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
00:40:25,920 --> 00:40:33,570
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
00:40:33,840 --> 00:40:41,210
let's postulate that the metric, you know, this tensor GMM you knew the four, four by four matrix is this.
379
00:40:41,220 --> 00:40:47,850
And I said, let's postulate this matrix as this. And I wrote down a four by four matrix on the board.
380
00:40:48,060 --> 00:40:51,330
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
00:40:55,980 --> 00:41:00,840
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
00:41:07,830 --> 00:41:14,070
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
00:42:39,030 --> 00:42:45,360
But it was followed shortly thereafter by a spectacular experiment which was published in Nature Physics,
398
00:42:46,800 --> 00:42:52,920
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?