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00:00:00,360 --> 00:00:01,270
PROFESSOR: All right.

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00:00:01,270 --> 00:00:06,450
So that concludes the chapter,
a big chapter in this course.

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00:00:06,450 --> 00:00:11,410
It concludes time-dependent
perturbation theory.

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00:00:11,410 --> 00:00:15,520
The whole course is
largely organized

5
00:00:15,520 --> 00:00:17,730
with approximation methods.

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00:00:17,730 --> 00:00:21,070
We have done time-independent
perturbation theory.

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00:00:21,070 --> 00:00:24,710
We applied it to the
hydrogen atom in detail.

8
00:00:24,710 --> 00:00:31,430
Then we did WKB, which is
slowly varying things in space.

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00:00:31,430 --> 00:00:34,490
Then we did time-dependent
perturbation theory.

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00:00:34,490 --> 00:00:36,850
It had many parts to it.

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00:00:36,850 --> 00:00:39,920
It had Fermi's
golden rule, atoms.

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00:00:39,920 --> 00:00:44,630
It had just
transitions in general.

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00:00:44,630 --> 00:00:48,950
The next step would be
adiabatic approximation,

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00:00:48,950 --> 00:00:51,610
which is time-dependent
changes but slow.

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00:00:51,610 --> 00:00:56,730
This is the time analog of WKB.

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00:00:56,730 --> 00:00:58,580
We're going to take
a little break.

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00:00:58,580 --> 00:01:02,540
And for half a lecture
today and the full lecture

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00:01:02,540 --> 00:01:05,000
next time, we're
going to do particles

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00:01:05,000 --> 00:01:07,850
in electromagnetic fields.

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00:01:07,850 --> 00:01:11,040
That's an important subject.

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00:01:11,040 --> 00:01:13,640
It will involve Landau
levels and things

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00:01:13,640 --> 00:01:16,140
that you always hear about.

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00:01:16,140 --> 00:01:20,990
And it will be useful for
some of the things we've done.

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00:01:20,990 --> 00:01:25,530
We've had a little exposure
to it with the Pauli equation.

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00:01:25,530 --> 00:01:28,820
You may remember the Dirac
and Pauli equation that

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00:01:28,820 --> 00:01:30,660
had this electromagnetic field.

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00:01:30,660 --> 00:01:32,840
So now we're going
to take seriously

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00:01:32,840 --> 00:01:37,700
the electromagnetic fields
and study how they interact

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00:01:37,700 --> 00:01:38,810
with quantum mechanics.

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00:01:38,810 --> 00:01:43,920
And there's a lot of very
interesting and beautiful ideas

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00:01:43,920 --> 00:01:46,020
having to do with
gauge invariance

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00:01:46,020 --> 00:01:50,030
and why potentials
are important.

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00:01:50,030 --> 00:01:52,880
Charged particles in
electromagnetic fields.

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00:01:52,880 --> 00:02:03,236
So charged particles
in EM fields.

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00:02:10,630 --> 00:02:13,330
So what is our useful problem?

36
00:02:13,330 --> 00:02:18,550
You have a particle with
mass m, now charge q.

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00:02:18,550 --> 00:02:23,500
And finally, there's an electric
field and a magnetic field.

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00:02:23,500 --> 00:02:25,760
And there's going to be
interactions between them.

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00:02:30,030 --> 00:02:31,340
So there's E and B.

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00:02:31,340 --> 00:02:33,780
And you've gone through
this transition.

41
00:02:36,680 --> 00:02:40,340
Probably, you've been told
in electromagnetism something

42
00:02:40,340 --> 00:02:42,080
along these lines.

43
00:02:42,080 --> 00:02:44,910
There is this E and B fields.

44
00:02:44,910 --> 00:02:47,400
And those are the
physical fields.

45
00:02:47,400 --> 00:02:51,980
These are the things that do
stuff on charges and particles.

46
00:02:51,980 --> 00:02:56,070
And there is this mathematical
entities that can help you,

47
00:02:56,070 --> 00:02:58,790
which are potentials,
that allow you

48
00:02:58,790 --> 00:03:02,780
to rewrite these fields in
terms of these quantities,

49
00:03:02,780 --> 00:03:03,770
potentials.

50
00:03:03,770 --> 00:03:06,560
And they're practical
to solve the equation.

51
00:03:06,560 --> 00:03:10,640
But the potentials are less
physical than the fields,

52
00:03:10,640 --> 00:03:13,160
because the potentials
are ambiguous.

53
00:03:13,160 --> 00:03:15,740
You can change the
value of the potential

54
00:03:15,740 --> 00:03:17,880
without changing
the electric field.

55
00:03:17,880 --> 00:03:19,220
You can do things like that.

56
00:03:21,920 --> 00:03:29,750
Well, that is somewhat
misleading, put charitably.

57
00:03:29,750 --> 00:03:34,550
It's a pretty wrong
viewpoint, actually.

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00:03:34,550 --> 00:03:38,090
As we will discover
and understand here,

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00:03:38,090 --> 00:03:43,700
the potentials are really more
important than the fields.

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00:03:43,700 --> 00:03:47,850
Quantum mechanics couples
to the potentials.

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00:03:47,850 --> 00:03:51,710
And as an afterthought,
if couples to the fields,

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00:03:51,710 --> 00:03:54,290
because it couples
to the potentials.

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00:03:54,290 --> 00:03:59,480
But even the definition of
an electromagnetic field,

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00:03:59,480 --> 00:04:01,280
at the end of the
day-- nowadays,

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00:04:01,280 --> 00:04:06,770
we really understand it as
a definition of potentials.

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00:04:06,770 --> 00:04:09,320
And I'll try to explain that.

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00:04:09,320 --> 00:04:14,210
This is pretty important if
you try to describe anything

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00:04:14,210 --> 00:04:16,880
that has topological content.

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00:04:16,880 --> 00:04:20,300
Most of the times, if
you're considering open

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00:04:20,300 --> 00:04:25,100
space, open Minkowsky space,
it doesn't make much difference

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00:04:25,100 --> 00:04:27,500
whether you take the electric
and magnetic field to be

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00:04:27,500 --> 00:04:29,510
fundamental or the potentials.

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00:04:29,510 --> 00:04:33,710
But as soon as you consider
anything topological,

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00:04:33,710 --> 00:04:38,000
a sample that lifts on a
toroidal surface, that's

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00:04:38,000 --> 00:04:41,030
a topological
space, then, if you

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00:04:41,030 --> 00:04:42,770
don't think about
potentials, you

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00:04:42,770 --> 00:04:45,920
could be absolutely
wrong, completely wrong.

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00:04:45,920 --> 00:04:49,310
Or if you have an
extra dimension, and in

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00:04:49,310 --> 00:04:52,700
many theories of physics, you
could have extra dimensions--

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00:04:52,700 --> 00:04:55,430
and you think about the
electric and magnetic fields

81
00:04:55,430 --> 00:04:57,830
then, if you don't
think of potentials,

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00:04:57,830 --> 00:04:59,850
you can be quite wrong.

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00:04:59,850 --> 00:05:02,480
So let's explain
what's happening

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00:05:02,480 --> 00:05:04,280
with these potentials.

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00:05:04,280 --> 00:05:07,250
Basically, it all
starts from the idea

86
00:05:07,250 --> 00:05:14,500
that you have a Maxwell's
equation del dot B equal 0.

87
00:05:14,500 --> 00:05:17,500
And this is solved,
it is said, by setting

88
00:05:17,500 --> 00:05:24,400
B equal to the curl
of A. And indeed,

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00:05:24,400 --> 00:05:30,550
if B is equal to the curl of
A, this equation is satisfied.

90
00:05:30,550 --> 00:05:37,150
The other equation that is
relevant is this Faraday's

91
00:05:37,150 --> 00:05:55,060
law, which if B is already curl
of A, you have that curl of E

92
00:05:55,060 --> 00:06:05,200
plus 1 over C da/dt
is equal to 0.

93
00:06:05,200 --> 00:06:10,790
You see, I substitute
B here equal curl of A.

94
00:06:10,790 --> 00:06:16,330
I commute the order of
derivatives, the dt the curl.

95
00:06:16,330 --> 00:06:20,640
And this gives me this equation.

96
00:06:20,640 --> 00:06:24,660
From this equation, we say
that anything that has 0 curl

97
00:06:24,660 --> 00:06:27,090
is the gradient of something.

98
00:06:27,090 --> 00:06:34,950
And therefore, we say that
E plus the 1 over c da/dt

99
00:06:34,950 --> 00:06:41,520
is the gradient
of phi, from where

100
00:06:41,520 --> 00:06:43,590
we find the second
equation, which

101
00:06:43,590 --> 00:06:49,890
says that E can be obtained
as minus the gradient of phi

102
00:06:49,890 --> 00:06:55,010
minus 1 over c da/dt.

103
00:06:55,010 --> 00:06:56,190
I hope I have it right.

104
00:06:56,190 --> 00:06:56,690
Yes.

105
00:07:01,920 --> 00:07:04,320
So this is the origin
of the potentials.

106
00:07:04,320 --> 00:07:05,760
You've seen that.

107
00:07:05,760 --> 00:07:07,530
That's how they
explain, probably,

108
00:07:07,530 --> 00:07:11,130
to you why you have potentials.

109
00:07:11,130 --> 00:07:13,785
And indeed, this
is so far so good.

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00:07:17,050 --> 00:07:20,640
But now, there is a freedom
with this potentials which are

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00:07:20,640 --> 00:07:23,180
called gauge transformations.

112
00:07:23,180 --> 00:07:26,850
So what are the gauge
transformations?

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00:07:26,850 --> 00:07:33,210
It's the possibility of changing
the potentials without changing

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00:07:33,210 --> 00:07:36,130
the electromagnetic fields.

115
00:07:36,130 --> 00:07:44,370
So for example, since the curl
of the grad of anything is 0--

116
00:07:44,370 --> 00:07:47,820
it's almost like
curl cross curl is 0.

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00:07:47,820 --> 00:07:52,740
This is the curl of the
grad of anything is 0.

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00:07:52,740 --> 00:07:58,760
You can change A
into a different A

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00:07:58,760 --> 00:08:05,210
that we'll call A prime,
given by A plus gradient

120
00:08:05,210 --> 00:08:08,640
of a function lambda.

121
00:08:08,640 --> 00:08:13,640
And this will not change the
value of the magnetic field B,

122
00:08:13,640 --> 00:08:18,620
because if you calculate
the magnetic field B,

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00:08:18,620 --> 00:08:23,390
the new magnetic field
associated to the new vector

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00:08:23,390 --> 00:08:31,220
potential is equal
to curl of A prime.

125
00:08:31,220 --> 00:08:35,179
But that's equal to
curl of A. And that's

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00:08:35,179 --> 00:08:41,990
equal to the old B due to the
old magnetic vector potential.

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00:08:41,990 --> 00:08:49,770
So the vector potential has
changed, but B the not change.

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00:08:49,770 --> 00:08:54,560
On the other hand, if you
change the vector potential

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00:08:54,560 --> 00:08:59,690
by something like this,
now it can affect the E.

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00:08:59,690 --> 00:09:07,940
But if simultaneously you change
phi, E will be left unchanged.

131
00:09:07,940 --> 00:09:12,020
So what should you do to phi?

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00:09:12,020 --> 00:09:14,930
So this is the first one for A.

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00:09:14,930 --> 00:09:17,360
If you want to keep
E unchanged, you

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00:09:17,360 --> 00:09:24,740
define phi prime equal to phi
minus 1 over c d lambda/dt.

135
00:09:30,780 --> 00:09:33,815
And then when you
compute the new E,

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00:09:33,815 --> 00:09:37,440
you would do the gradient
of the new phi, which

137
00:09:37,440 --> 00:09:40,290
will be the gradient
of the old phi,

138
00:09:40,290 --> 00:09:45,540
but an extra term, which
would be plus 1 over C

139
00:09:45,540 --> 00:09:50,720
d/dt of gradient of lambda.

140
00:09:50,720 --> 00:09:53,860
And it will cancel
with the minus 1

141
00:09:53,860 --> 00:09:59,020
over C d/dt of the
gradient of lambda.

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00:09:59,020 --> 00:10:08,110
So by changing A and phi in
this way, B and E are unchanged.

143
00:10:08,110 --> 00:10:11,470
So I can summarize this
gauge invariance statement

144
00:10:11,470 --> 00:10:18,490
by saying that the new electric
field due to due to phi

145
00:10:18,490 --> 00:10:25,810
prime and A prime is equal to
the electric field due to phi

146
00:10:25,810 --> 00:10:33,910
and A. And the magnetic
field new due to phi prime

147
00:10:33,910 --> 00:10:38,620
and A prime is equal
to the magnetic field

148
00:10:38,620 --> 00:10:47,780
due to phi and A.

149
00:10:47,780 --> 00:10:53,130
This is lifted to a principal.

150
00:10:56,570 --> 00:11:00,530
So we started describing
electromagnetic fields

151
00:11:00,530 --> 00:11:06,130
with E and B. We can
describe them with A and phi.

152
00:11:06,130 --> 00:11:09,220
And let's take that
seriously, because we'll

153
00:11:09,220 --> 00:11:11,850
need it for quantum mechanics.

154
00:11:11,850 --> 00:11:17,880
And we'll add now the extra
important physical assumption,

155
00:11:17,880 --> 00:11:25,420
which is that phi
prime A prime is

156
00:11:25,420 --> 00:11:48,160
really equivalent, physically
equivalent to phi A.

157
00:11:48,160 --> 00:11:53,660
And an electromagnetic
field configuration,

158
00:11:53,660 --> 00:11:58,100
as a mathematician will say,
is the equivalent classes

159
00:11:58,100 --> 00:12:00,090
of potentials.

160
00:12:00,090 --> 00:12:04,640
So you want to describe
an electromagnetic field?

161
00:12:04,640 --> 00:12:05,900
I ask one of you.

162
00:12:05,900 --> 00:12:09,860
And you tell me what is phi and
what is A. I ask somebody else,

163
00:12:09,860 --> 00:12:12,050
they give me another
phi and another A.

164
00:12:12,050 --> 00:12:14,150
And you say, they
look different.

165
00:12:14,150 --> 00:12:15,650
But they may be the same.

166
00:12:15,650 --> 00:12:21,990
They are the same if they
are related by some lambda.

167
00:12:24,695 --> 00:12:27,680
I'll write it.

168
00:12:27,680 --> 00:12:41,360
If there exists a lambda
such that they are gauge

169
00:12:41,360 --> 00:12:45,880
transforms of each other.

170
00:12:52,110 --> 00:12:58,970
So before you say, you tell me
what the E is and what B is,

171
00:12:58,970 --> 00:13:00,970
and they have to agree.

172
00:13:00,970 --> 00:13:04,190
And you say, that's preferable,
because if E and B are

173
00:13:04,190 --> 00:13:05,390
the same, they agree.

174
00:13:05,390 --> 00:13:07,280
That's it.

175
00:13:07,280 --> 00:13:11,420
Why do we go through A and phi?

176
00:13:11,420 --> 00:13:12,800
They don't have to agree.

177
00:13:12,800 --> 00:13:14,750
And they can be the
same, because if they're

178
00:13:14,750 --> 00:13:19,340
related with some lambda,
they are the same.

179
00:13:19,340 --> 00:13:23,060
We seem to be going backwards
into a lot more complicated

180
00:13:23,060 --> 00:13:24,300
situation.

181
00:13:24,300 --> 00:13:25,580
But that's what we need.

182
00:13:25,580 --> 00:13:31,040
And that's what the physics
tells you is really going on.

183
00:13:31,040 --> 00:13:36,170
So some particular things that
are curious can happen now.

184
00:13:36,170 --> 00:13:37,730
And they're very curious.

185
00:13:37,730 --> 00:13:39,620
And there are examples.

186
00:13:39,620 --> 00:13:41,285
And we'll see some
of those examples.

187
00:13:44,540 --> 00:13:45,740
So curious effects.

188
00:13:50,840 --> 00:14:06,300
1, suppose phi and A and
phi prime and A prime

189
00:14:06,300 --> 00:14:19,360
give the same E and B. So
you and your peer editor

190
00:14:19,360 --> 00:14:26,260
came up with some phi and A
and phi prime and A prime.

191
00:14:26,260 --> 00:14:32,080
And they give the
same E and B. Are you

192
00:14:32,080 --> 00:14:34,990
and your peer
editor on the hands

193
00:14:34,990 --> 00:14:38,560
of the same electromagnetic
field configuration?

194
00:14:43,950 --> 00:14:45,920
Maybe but not obvious.

195
00:14:45,920 --> 00:14:48,170
In classical physics,
you would say, yes.

196
00:14:48,170 --> 00:14:50,770
If they give you the same E
and B, these are equivalent.

197
00:14:50,770 --> 00:14:55,730
No, now, you would
have to show that there

198
00:14:55,730 --> 00:15:01,610
is some lambda so that they are
gauge equivalent to each other.

199
00:15:01,610 --> 00:15:05,390
You see, the fact that they
are gauge equivalent guarantees

200
00:15:05,390 --> 00:15:10,520
they give the same B and E. But
if they give the same B and E,

201
00:15:10,520 --> 00:15:14,120
it may happen that you still
cannot find the lambda that

202
00:15:14,120 --> 00:15:19,220
relates them, because what
you have to do is try to take

203
00:15:19,220 --> 00:15:22,280
your friend's A and
your friend's phi,

204
00:15:22,280 --> 00:15:24,950
then this is yours and yours,
and now you have to find

205
00:15:24,950 --> 00:15:25,990
the lambda.

206
00:15:25,990 --> 00:15:28,650
And you have to solve a
differential equation.

207
00:15:28,650 --> 00:15:31,880
What if you can't
find the lambda?

208
00:15:31,880 --> 00:15:34,790
They give the same E
and B, but they're not

209
00:15:34,790 --> 00:15:38,050
gauge equivalent to each other.

210
00:15:38,050 --> 00:15:42,630
That means that you have
different field configurations.

211
00:15:42,630 --> 00:15:44,370
The fact that E
and B are the same

212
00:15:44,370 --> 00:15:49,570
doesn't mean they are the
same electromagnetic fields.

213
00:15:49,570 --> 00:15:56,550
And this can happen if you
have a little bit of topology.

214
00:15:56,550 --> 00:16:01,790
So suppose A and this
give the same B field

215
00:16:01,790 --> 00:16:14,650
but are not related by
a gauge transformation.

216
00:16:23,190 --> 00:16:29,700
Then these are
inequivalent EM fields.

217
00:16:35,310 --> 00:16:38,020
They could have different
quantum mechanics.

218
00:16:38,020 --> 00:16:39,460
And we will see that.

219
00:16:39,460 --> 00:16:44,340
So one example is what
people call Wilson loops.

220
00:16:44,340 --> 00:16:49,170
Wilson loops correspond to some
closed curve in space that you

221
00:16:49,170 --> 00:16:51,090
cannot contract.

222
00:16:51,090 --> 00:16:53,410
Of course, in Minkowski
space, in this room,

223
00:16:53,410 --> 00:16:55,170
I can contract anything.

224
00:16:55,170 --> 00:16:57,360
But sometimes, if
you live on a torus

225
00:16:57,360 --> 00:17:01,320
or your sample is on
a sphere or something,

226
00:17:01,320 --> 00:17:04,230
you may not be able
to contract the curve.

227
00:17:04,230 --> 00:17:09,990
Along that curve, you can
put a vector potential

228
00:17:09,990 --> 00:17:13,560
that is pointing tangent
to the curve all the time.

229
00:17:13,560 --> 00:17:18,990
And it's a constant, a
constant vector potential.

230
00:17:18,990 --> 00:17:25,380
That constant vector potential
has 0 B and 0 electric field.

231
00:17:25,380 --> 00:17:29,460
But if that constant is small
or that constant is big,

232
00:17:29,460 --> 00:17:30,660
it's inequivalent.

233
00:17:30,660 --> 00:17:32,880
You won't find the
gauge transformation

234
00:17:32,880 --> 00:17:35,340
that transforms each other.

235
00:17:35,340 --> 00:17:39,630
And therefore, those
would be configurations

236
00:17:39,630 --> 00:17:44,730
of electromagnetic fields
that give you the same E and B

237
00:17:44,730 --> 00:17:47,470
but are inequivalent.

238
00:17:47,470 --> 00:17:49,140
So that's the first
curious effect.

239
00:17:49,140 --> 00:17:51,795
There is another curious effect.

240
00:17:58,800 --> 00:18:02,140
And quantum mechanics is
forcing you to do that.

241
00:18:02,140 --> 00:18:06,030
That's the main thing.

242
00:18:06,030 --> 00:18:12,120
So the second curious
effect is, given

243
00:18:12,120 --> 00:18:29,336
an E and B that satisfy
Maxwell's equation,

244
00:18:29,336 --> 00:18:40,430
are these allowed fields?

245
00:18:40,430 --> 00:18:45,160
So somebody gives you an
E field and a B field.

246
00:18:45,160 --> 00:18:47,500
And you say, OK,
I'm going to check

247
00:18:47,500 --> 00:18:52,180
if these are possible physical
electric and magnetic fields.

248
00:18:52,180 --> 00:18:56,620
And then you go and check
Maxwell's equations.

249
00:18:56,620 --> 00:18:59,290
And Maxwell's
equations work out.

250
00:18:59,290 --> 00:19:01,270
They're all satisfied.

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00:19:01,270 --> 00:19:03,790
You say, OK, this is good.

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00:19:03,790 --> 00:19:07,840
Those electric and
magnetic fields are fine.

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00:19:07,840 --> 00:19:10,810
Not so quick.

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00:19:10,810 --> 00:19:15,010
To be sure those electric
and magnetic fields are fine,

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00:19:15,010 --> 00:19:19,600
you should give me a phi
and an A that gives rise

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00:19:19,600 --> 00:19:24,580
to them, because fundamentally,
the electromagnetic fields are

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00:19:24,580 --> 00:19:25,470
A and phi.

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00:19:25,470 --> 00:19:29,840
And those you need anyway to
do quantum mechanics with it.

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00:19:29,840 --> 00:19:31,960
So if you have an
electromagnetic field and you

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00:19:31,960 --> 00:19:35,860
cannot do quantum mechanics with
it, you would be suspicious.

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00:19:35,860 --> 00:19:39,670
So you need to find phi and A.

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00:19:39,670 --> 00:19:41,890
Are these allowed?

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00:19:41,890 --> 00:19:57,910
Only if you can find a phi and
A. And sometimes that can fail.

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00:19:57,910 --> 00:19:59,390
On a torus again--

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00:20:05,170 --> 00:20:07,115
we've described in
this course as circle

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00:20:07,115 --> 00:20:14,810
as a line x identified with
x plus L. So here is L.

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00:20:14,810 --> 00:20:16,000
And you identify.

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00:20:19,320 --> 00:20:24,030
A torus can be done by
taking a piece of the plane

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00:20:24,030 --> 00:20:27,570
and identifying this line
with that and this line

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00:20:27,570 --> 00:20:29,190
with this line.

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00:20:29,190 --> 00:20:31,680
So you glue this,
form a cylinder,

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00:20:31,680 --> 00:20:35,480
and then you glue
the other ends.

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00:20:35,480 --> 00:20:39,530
On a torus, you can put a
constant magnetic field.

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00:20:39,530 --> 00:20:44,370
A constant magnetic fields
satisfies Maxwell's equations.

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00:20:44,370 --> 00:20:47,690
So you say, oh, so you can put
any constant magnetic field

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00:20:47,690 --> 00:20:49,160
in a torus.

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00:20:49,160 --> 00:20:50,430
Wrong.

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00:20:50,430 --> 00:20:54,680
The vector potential
has trouble existing.

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00:20:54,680 --> 00:20:59,630
And only for particular
values of the magnetic field,

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00:20:59,630 --> 00:21:02,270
there's a consistent
vector potential.

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00:21:02,270 --> 00:21:07,020
You get the quantization of
the flux of the magnetic field.

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00:21:07,020 --> 00:21:09,635
So that's another
example, an E and a B.

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00:21:09,635 --> 00:21:12,770
A constant magnetic
field on a torus

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00:21:12,770 --> 00:21:16,400
satisfies every Maxwell
equation you may want to do.

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00:21:16,400 --> 00:21:21,970
But still, it's not a valid
electromagnetic field,

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00:21:21,970 --> 00:21:27,700
because there's no A and
phi that give rise to it.

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00:21:27,700 --> 00:21:37,170
So those are our lessons in why
we need the vector potentials

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00:21:37,170 --> 00:21:40,990
to describe physics
in quantum mechanics,

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00:21:40,990 --> 00:21:43,150
electromagnetic physics.

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00:21:43,150 --> 00:21:46,840
And then let's write,
therefore, the Schrodinger

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00:21:46,840 --> 00:21:51,340
equation with the
electromagnetic fields.