1
00:00:00,540 --> 00:00:08,130
PROFESSOR: I now have that new
n of t that we wrote there.

2
00:00:08,130 --> 00:00:10,800
I have to write
it as what it is.

3
00:00:10,800 --> 00:00:18,680
It's i psi n of r of t times--

4
00:00:18,680 --> 00:00:22,550
I will write it here
this way-- d dt--

5
00:00:22,550 --> 00:00:26,900
the dot will be
replaced by the d dt--

6
00:00:26,900 --> 00:00:29,371
psi n of r of t.

7
00:00:32,600 --> 00:00:36,650
And then, of course,
the gamma n of t

8
00:00:36,650 --> 00:00:40,280
will be just the
integral from 0 to t

9
00:00:40,280 --> 00:00:43,740
of new n of t prime bt prime.

10
00:00:43,740 --> 00:00:45,620
So that's the next step.

11
00:00:49,630 --> 00:00:53,370
Well, if you have
to differentiate

12
00:00:53,370 --> 00:01:01,080
a function that depends on
r of t, what do you have?

13
00:01:01,080 --> 00:01:09,680
Let me do it for a simpler
case, d dt of f of r of t.

14
00:01:09,680 --> 00:01:17,370
This means d dt of a
function of r1 of t

15
00:01:17,370 --> 00:01:19,830
are all the ones up to rn of t.

16
00:01:25,860 --> 00:01:27,660
And what must you do?

17
00:01:27,660 --> 00:01:37,920
Well, you should do
df dr1 times dr1 dt

18
00:01:37,920 --> 00:01:46,860
all the way up to
the df dr and drn dt.

19
00:01:46,860 --> 00:01:49,830
You want to find the time
dependence of a function that

20
00:01:49,830 --> 00:01:54,030
depends on a collection of
time-dependent coordinates.

21
00:01:54,030 --> 00:01:56,280
Well, the chain rule applies.

22
00:01:59,770 --> 00:02:05,980
But this can be written
in a funny language--

23
00:02:05,980 --> 00:02:07,690
maybe not so funny--

24
00:02:07,690 --> 00:02:21,280
as the gradient sub r vector
of f dotted dr vector dt.

25
00:02:24,262 --> 00:02:29,770
See, the gradient, in
general, is d dx1 d dx2 d dx3.

26
00:02:29,770 --> 00:02:31,720
It's a vector operator.

27
00:02:31,720 --> 00:02:37,150
The gradient sub r would
mean d dr1 d dr2 d dr3,

28
00:02:37,150 --> 00:02:42,970
just the gradient in
this Euclidean vector

29
00:02:42,970 --> 00:02:47,690
space times dr dt.

30
00:02:47,690 --> 00:02:52,520
So that's what I want to
use for this derivative.

31
00:02:52,520 --> 00:02:54,950
I have to differentiate
that state.

32
00:02:54,950 --> 00:02:57,470
And therefore, I'll
write it that way.

33
00:03:03,420 --> 00:03:11,580
So gamma n of t is equal
to i, from the top line,

34
00:03:11,580 --> 00:03:25,810
psi n of r of t times gradients
of r acting on the state psi

35
00:03:25,810 --> 00:03:36,310
n of r of t dotted with dr dt.

36
00:03:41,100 --> 00:03:44,450
This is dot product.

37
00:03:44,450 --> 00:03:48,520
So just to make sure
you understand here,

38
00:03:48,520 --> 00:03:55,340
you have one ket here, and
you have this gradient.

39
00:03:55,340 --> 00:04:00,420
So that gives you
capital N components,

40
00:04:00,420 --> 00:04:05,410
the derivative of the ket
with respect to r1 r2 r3 r4.

41
00:04:05,410 --> 00:04:10,090
Then with the inner product,
it gives your capital

42
00:04:10,090 --> 00:04:14,320
N numbers, which are the
components of a vector that is

43
00:04:14,320 --> 00:04:18,589
being dotted with this vector.

44
00:04:18,589 --> 00:04:20,750
It's all about
trying to figure out

45
00:04:20,750 --> 00:04:24,900
that this language makes sense.

46
00:04:24,900 --> 00:04:30,990
If this made sense to you,
this should make sense,

47
00:04:30,990 --> 00:04:35,380
a little more, maybe a
tiny bit more confusing.

48
00:04:35,380 --> 00:04:37,980
But maybe you should
write it all out.

49
00:04:37,980 --> 00:04:39,360
What do you think it is?

50
00:04:39,360 --> 00:04:41,480
And that might help you.

51
00:04:41,480 --> 00:04:44,710
Or we could do that later.

52
00:04:44,710 --> 00:04:53,470
So if we have that, we can go
to gamma n, the geometric phase.

53
00:04:53,470 --> 00:05:01,510
So this is 0 2t, the integral
with respect to prime time,

54
00:05:01,510 --> 00:05:03,630
so new m.

55
00:05:03,630 --> 00:05:09,060
So it's i psi n r of t prime--

56
00:05:09,060 --> 00:05:14,560
there's lots of vectors
here, gradient r vector

57
00:05:14,560 --> 00:05:29,320
of psi n r of t prime
dotted dr dt prime dt prime.

58
00:05:29,320 --> 00:05:31,310
That's the last dt prime.

59
00:05:34,680 --> 00:05:39,730
And the good thing that
happened, the thing that

60
00:05:39,730 --> 00:05:42,430
really makes all the
difference, the thing that

61
00:05:42,430 --> 00:05:46,300
is responsible for
that conceptual thing

62
00:05:46,300 --> 00:05:51,430
is just this cancellation.

63
00:05:51,430 --> 00:05:56,340
This cancellation means that
you can think of the integral

64
00:05:56,340 --> 00:06:01,380
as happening just in
the configuration space.

65
00:06:01,380 --> 00:06:04,840
This is not really an
integral over time.

66
00:06:04,840 --> 00:06:09,860
This is an integral
in configuration space

67
00:06:09,860 --> 00:06:16,070
because now this integral is
nothing else than the integral

68
00:06:16,070 --> 00:06:20,090
over the path gamma.

69
00:06:20,090 --> 00:06:22,340
Because the path
gamma represents

70
00:06:22,340 --> 00:06:29,210
the evolution of the coordinate
capital R from 0 to time t.

71
00:06:29,210 --> 00:06:32,480
This is nothing else than
the integral over the path

72
00:06:32,480 --> 00:06:37,520
gamma of i psi n of r--

73
00:06:37,520 --> 00:06:41,750
I don't have to
write the t anymore--

74
00:06:41,750 --> 00:06:47,390
dr psi n of r--

75
00:06:47,390 --> 00:07:07,460
again no t-- dot dr. And this
is the geometric phase gamma n

76
00:07:07,460 --> 00:07:12,162
that depends on r on the path.

77
00:07:12,162 --> 00:07:16,090
I'll write it like that.

78
00:07:16,090 --> 00:07:18,980
You see, something very
important has happened here.

79
00:07:18,980 --> 00:07:24,970
It's a realization that
time plays no role anymore.

80
00:07:24,970 --> 00:07:26,210
This is the concept.

81
00:07:26,210 --> 00:07:30,580
This is what you have to
struggle to understand here.

82
00:07:30,580 --> 00:07:36,160
This integral says
take this path.

83
00:07:36,160 --> 00:07:46,520
Take a little dr dot it with
this gradient of this object,

84
00:07:46,520 --> 00:07:49,390
which is kind of the gradient
of this ket, which is

85
00:07:49,390 --> 00:07:51,480
a lot of kets with this thing.

86
00:07:51,480 --> 00:07:53,140
So it's a vector.

87
00:07:53,140 --> 00:07:55,920
Dot it with this and integrate.

88
00:07:55,920 --> 00:07:58,570
And time plays no role.

89
00:07:58,570 --> 00:08:02,120
You just follow the path.

90
00:08:02,120 --> 00:08:06,800
So whether this
thing took one minute

91
00:08:06,800 --> 00:08:11,930
to make the path
or a billion years,

92
00:08:11,930 --> 00:08:16,220
the geometric phase will
be exactly the same.

93
00:08:16,220 --> 00:08:21,990
It just depends on
the path it took.

94
00:08:21,990 --> 00:08:24,300
Time for some names
for these things.

95
00:08:27,760 --> 00:08:28,710
Let's see.

96
00:08:36,210 --> 00:08:53,330
So a first name is
that this whole object

97
00:08:53,330 --> 00:08:55,870
is going to be called
the Berry connection.

98
00:09:00,114 --> 00:09:12,340
i psi n of r
gradient r psi n of r

99
00:09:12,340 --> 00:09:22,090
is called the Berry
connection a n vector of r.

100
00:09:22,090 --> 00:09:24,780
Berry connection.

101
00:09:29,560 --> 00:09:35,300
OK, a few things to notice,
the Berry connection

102
00:09:35,300 --> 00:09:41,000
is like a vector in the
configuration space.

103
00:09:41,000 --> 00:09:46,520
It has capital N components
because this is a gradient.

104
00:09:46,520 --> 00:09:51,860
And therefore, it produces
of this ket n kets

105
00:09:51,860 --> 00:09:55,280
and, therefore, n numbers
because of the bra.

106
00:09:55,280 --> 00:09:59,160
So this is a thing with
capital N components.

107
00:10:04,930 --> 00:10:08,250
So it's a vector in RN.

108
00:10:08,250 --> 00:10:12,800
But people like the
name connection.

109
00:10:12,800 --> 00:10:14,350
Why Connection?

110
00:10:14,350 --> 00:10:17,650
Because it's a little
more subtle than a vector.

111
00:10:21,060 --> 00:10:24,090
It transforms under
Gage transformation,

112
00:10:24,090 --> 00:10:27,720
your favorite things.

113
00:10:27,720 --> 00:10:30,605
And it makes it interesting
because it transforms

114
00:10:30,605 --> 00:10:31,780
under Gage transformation.

115
00:10:31,780 --> 00:10:33,790
We'll see it in a second.

116
00:10:33,790 --> 00:10:36,340
So it's a connection
because of that.

117
00:10:36,340 --> 00:10:41,220
And there's one Berry
connection for every eigenstate

118
00:10:41,220 --> 00:10:42,760
of your system.

119
00:10:42,760 --> 00:10:46,780
Because we fix some n,
and we got the connection.

120
00:10:46,780 --> 00:10:52,170
And we're going to get different
connections for different n's.

121
00:10:52,170 --> 00:11:02,140
So n components,
one per eigenstate,

122
00:11:02,140 --> 00:11:06,640
and they live all over
the configuration space.

123
00:11:06,640 --> 00:11:09,310
You can ask, what is
the value of the Berry

124
00:11:09,310 --> 00:11:11,330
connection at this point?

125
00:11:11,330 --> 00:11:12,820
And there is an answer.

126
00:11:12,820 --> 00:11:15,340
At every point, this
connection exists.

127
00:11:18,280 --> 00:11:26,630
Now, let's figure out the issue
of gauge transformations here.

128
00:11:26,630 --> 00:11:35,140
And it's important because
this subject somehow--

129
00:11:35,140 --> 00:11:37,570
these formulas, I
think in many ways,

130
00:11:37,570 --> 00:11:42,340
were known to everybody
for a long time.

131
00:11:42,340 --> 00:11:47,800
But Berry probably clarified
this issue of the time

132
00:11:47,800 --> 00:11:51,940
independence and
emphasized that this could

133
00:11:51,940 --> 00:11:54,520
be interesting in some cases.

134
00:11:54,520 --> 00:12:00,040
But in fact, in most
cases, you could say

135
00:12:00,040 --> 00:12:02,290
they're not all that relevant.

136
00:12:02,290 --> 00:12:03,910
You can change them.

137
00:12:03,910 --> 00:12:06,205
So here is one thing
that can happen.

138
00:12:09,310 --> 00:12:12,010
You have your
energy eigenstates,

139
00:12:12,010 --> 00:12:14,620
your instantaneous eigenstates.

140
00:12:14,620 --> 00:12:17,350
You solve them,
and you box them.

141
00:12:17,350 --> 00:12:20,590
You're very happy with them.

142
00:12:20,590 --> 00:12:23,680
But in fact, they're
far from unique.

143
00:12:23,680 --> 00:12:27,280
Your energy eigenstates,
your instantaneous energy

144
00:12:27,280 --> 00:12:29,830
eigenstates can be changed.

145
00:12:29,830 --> 00:12:37,180
If you have an energy
eigenstate psi n of r--

146
00:12:37,180 --> 00:12:39,460
that's what it is--

147
00:12:39,460 --> 00:12:43,180
well, you could decide
to find another one.

148
00:12:43,180 --> 00:12:51,070
Psi prime of r is going to
be equal to e to the minus

149
00:12:51,070 --> 00:12:57,005
some function, arbitrary
function, of r times this.

150
00:13:00,030 --> 00:13:05,340
And these new states
are energy eigenstates,

151
00:13:05,340 --> 00:13:07,830
instantaneous energy
eigenstates that

152
00:13:07,830 --> 00:13:11,460
are as good as
your original psi n

153
00:13:11,460 --> 00:13:16,410
because this equation also
holds for the psi n primes.

154
00:13:16,410 --> 00:13:21,180
If you add with the Hamiltonian,
the Hamiltonian in here

155
00:13:21,180 --> 00:13:27,810
just goes through this and
hits here, produces the energy,

156
00:13:27,810 --> 00:13:30,830
and then the state
is just the same.

157
00:13:35,010 --> 00:13:38,360
The r of t's are parameters
of the Hamiltonian.

158
00:13:38,360 --> 00:13:40,080
They're not operators.

159
00:13:40,080 --> 00:13:43,650
So there's no reason
why the Hamiltonian

160
00:13:43,650 --> 00:13:46,860
would care about this factor.

161
00:13:46,860 --> 00:13:49,090
The r's are just parameters.

162
00:13:49,090 --> 00:13:49,987
Yes?

163
00:13:49,987 --> 00:13:53,730
AUDIENCE: [INAUDIBLE]

164
00:13:53,730 --> 00:13:56,010
PROFESSOR: No, they're
still normalized.

165
00:13:56,010 --> 00:13:59,740
I should put a phase here--
thank you very much--

166
00:13:59,740 --> 00:14:01,410
minus i.

167
00:14:01,410 --> 00:14:02,880
Thank you.

168
00:14:02,880 --> 00:14:05,910
Yes, I want the states
to be normalized,

169
00:14:05,910 --> 00:14:08,460
and I want them
to be orthonormal.

170
00:14:08,460 --> 00:14:12,780
And all that is not changed
if I put them phase.

171
00:14:12,780 --> 00:14:17,700
So this is the funny thing
about quantum mechanics.

172
00:14:17,700 --> 00:14:20,510
It's all about phases
and complex numbers.

173
00:14:20,510 --> 00:14:25,410
But you can, to a large degree,
change those phases at will.

174
00:14:25,410 --> 00:14:30,240
And whatever survives is some
sort of very subtle effects

175
00:14:30,240 --> 00:14:32,170
between the phases.

176
00:14:32,170 --> 00:14:37,310
So here I put the i
and beta of r is real.

177
00:14:42,500 --> 00:14:46,830
PROFESSOR: So you can say
let's compute the new Berry

178
00:14:46,830 --> 00:14:55,645
connection associated with
this new state a n prime of r.

179
00:14:55,645 --> 00:15:01,400
So I must do that operation
that we have up there

180
00:15:01,400 --> 00:15:02,600
with the news state.

181
00:15:02,600 --> 00:15:11,030
So I would have i psi n of r
times e to the i beta of r.

182
00:15:11,030 --> 00:15:13,460
That's The bra.

183
00:15:13,460 --> 00:15:22,700
Then I have dr and now the
ket, e to the minus i beta of r

184
00:15:22,700 --> 00:15:24,830
psi n of r.

185
00:15:24,830 --> 00:15:29,970
So this is, by definition,
the new Berry connection

186
00:15:29,970 --> 00:15:36,520
associated to your new,
redefined eigenstates.

187
00:15:39,730 --> 00:15:45,200
Now this nabla is acting
on everything to the right.

188
00:15:45,200 --> 00:15:50,590
Suppose it acts on the state
and then the two exponentials

189
00:15:50,590 --> 00:15:55,960
will cancel, and then you
get the old connection.

190
00:15:55,960 --> 00:16:02,170
So there is one term here,
which is just the old a n of r.

191
00:16:04,800 --> 00:16:07,320
There's all these arrows there.

192
00:16:07,320 --> 00:16:11,090
There's probably five arrows
at least I miss on every board.

193
00:16:11,090 --> 00:16:17,135
Here is a 1, 2, 3, 4 5.

194
00:16:20,310 --> 00:16:25,310
OK, so this is the
first one, and then you

195
00:16:25,310 --> 00:16:32,400
have the term for this gradient
acts on this exponential.

196
00:16:32,400 --> 00:16:34,550
When the gradient acts
on the exponential,

197
00:16:34,550 --> 00:16:36,810
it gives the same
exponential times

198
00:16:36,810 --> 00:16:39,900
the gradient of the exponent.

199
00:16:39,900 --> 00:16:43,110
The exponentials then cancel.

200
00:16:43,110 --> 00:16:48,030
The gradient of the exponent
would give me plus i times

201
00:16:48,030 --> 00:16:51,475
minus i gradient of beta.

202
00:16:56,040 --> 00:17:00,420
Maybe I'll put the r of the r.

203
00:17:00,420 --> 00:17:05,280
And then these cancel, and you
have the state with itself,

204
00:17:05,280 --> 00:17:06,619
which gives you 1.

205
00:17:06,619 --> 00:17:12,109
So that's all it is, all that
the second term gives you.

206
00:17:12,109 --> 00:17:22,490
So here we get a n of r plus
gradient r of beta of r.

207
00:17:31,720 --> 00:17:33,880
So this is the gauge
transformation.

208
00:17:33,880 --> 00:17:39,280
And you say, wow,
I can see now why

209
00:17:39,280 --> 00:17:40,900
this is called a connection.

210
00:17:40,900 --> 00:17:44,080
Because just like
the vector potential

211
00:17:44,080 --> 00:17:47,380
under a gauge
transformation, it transforms

212
00:17:47,380 --> 00:17:50,450
with a gradient of a function.

213
00:17:50,450 --> 00:17:56,530
So it really transforms
as a vector potential, all

214
00:17:56,530 --> 00:18:00,400
in this space called
the configuration space,

215
00:18:00,400 --> 00:18:03,100
not in real space.

216
00:18:03,100 --> 00:18:08,930
In the configuration space it
acts like a vector potential.

217
00:18:08,930 --> 00:18:12,280
And that's why it's
called a connection.

218
00:18:12,280 --> 00:18:13,350
But let's see.

219
00:18:13,350 --> 00:18:17,470
We have now what happens
to the connection.

220
00:18:17,470 --> 00:18:23,830
Let's see what happens to the
Berry's phase if you do this.

221
00:18:23,830 --> 00:18:28,885
So the Berry's phase over
there is this integral.

222
00:18:33,960 --> 00:18:35,670
So the Berry's phase can change.

223
00:18:46,890 --> 00:18:52,690
And let's see what happens
to the Berry's phase.

224
00:18:52,690 --> 00:19:02,200
So what is the geometric
phase gamma n of gamma?

225
00:19:02,200 --> 00:19:05,890
In plain language, it is
the integral over gamma--

226
00:19:05,890 --> 00:19:08,860
from here, I'm just
copying the formula--

227
00:19:08,860 --> 00:19:20,120
of a n of r, the Berry
connection, times dr.

228
00:19:20,120 --> 00:19:28,730
So what is the new Berry phase
for your new instantaneous

229
00:19:28,730 --> 00:19:30,770
energy eigenstates?

230
00:19:30,770 --> 00:19:33,140
Now you would say,
if the Berry phase

231
00:19:33,140 --> 00:19:36,950
is something that is
observable, it better not

232
00:19:36,950 --> 00:19:40,100
depend just on your
convention to choose

233
00:19:40,100 --> 00:19:42,740
the instantaneous
energy eigenstates.

234
00:19:42,740 --> 00:19:45,400
And this is just
your convention.

235
00:19:45,400 --> 00:19:49,310
Because if a problem
is sufficiently messy,

236
00:19:49,310 --> 00:19:55,700
I bet you guys would all
come up with different energy

237
00:19:55,700 --> 00:19:59,960
eigenstates because the phases
are chosen in different ways.

238
00:19:59,960 --> 00:20:04,380
So it better not change
if the Berry phase

239
00:20:04,380 --> 00:20:06,810
is to be significant.

240
00:20:06,810 --> 00:20:08,780
So what is the prime thing?

241
00:20:08,780 --> 00:20:12,560
Well, we still integrate
over the same path, but now

242
00:20:12,560 --> 00:20:14,270
the prime connection--

243
00:20:19,540 --> 00:20:28,330
but that is the old
connection a n of rd r,

244
00:20:28,330 --> 00:20:33,640
the old Berry's phase, plus
the integral over gamma,

245
00:20:33,640 --> 00:20:37,380
or I will write it from
initial the final r.

246
00:20:41,400 --> 00:20:45,120
Maybe I should have ir
and i f in the picture.

247
00:20:45,120 --> 00:20:51,140
If you want to, you can put
this r of time equals 0 as ri

248
00:20:51,140 --> 00:21:01,370
and r of time equal tf
is rf the extra term,

249
00:21:01,370 --> 00:21:16,750
the gradient of beta dot dr.
So this is the old Berry phase.

250
00:21:16,750 --> 00:21:24,150
So the new Berry phase
is the old Berry phase.

251
00:21:24,150 --> 00:21:26,600
And how about the last integral?

252
00:21:26,600 --> 00:21:27,930
Does it vanish?

253
00:21:27,930 --> 00:21:32,450
No, it doesn't vanish.

254
00:21:32,450 --> 00:21:34,110
It gifts you.

255
00:21:34,110 --> 00:21:36,600
But in fact, it can be done.

256
00:21:36,600 --> 00:21:39,500
This is like derivative
times this thing,

257
00:21:39,500 --> 00:21:42,542
so it's one of those
simple integrals.

258
00:21:42,542 --> 00:21:47,690
The gradient times the
d represents the change

259
00:21:47,690 --> 00:21:51,530
in the function as
you move a little dr.

260
00:21:51,530 --> 00:21:57,530
So when you go from ri to rf,
the integral of the gradient

261
00:21:57,530 --> 00:22:00,440
is equal to the
function beta at rf

262
00:22:00,440 --> 00:22:03,410
minus the function beta on ri.

263
00:22:03,410 --> 00:22:07,520
This is like when you
integrate the electric field

264
00:22:07,520 --> 00:22:09,980
along a line, and
the electric field

265
00:22:09,980 --> 00:22:11,930
is the gradient
of the potential.

266
00:22:11,930 --> 00:22:14,900
The integral of the electric
field through a line

267
00:22:14,900 --> 00:22:18,060
is the potential here
minus the potential there.

268
00:22:18,060 --> 00:22:28,710
So here this is plus beta
or rf minus beta of ri.

269
00:22:31,650 --> 00:22:35,700
So it's not gauge invariant
in the Berry phase.

270
00:22:39,320 --> 00:22:46,050
And therefore, it will mean that
most of the times it cannot be

271
00:22:46,050 --> 00:22:48,090
observed.

272
00:22:48,090 --> 00:22:49,410
It's not gauge invariant.

273
00:22:49,410 --> 00:22:53,100
Whatever is not gauge
invariant cannot be observed.

274
00:22:53,100 --> 00:22:56,460
You cannot say you make a
measurement and the answer is

275
00:22:56,460 --> 00:23:00,120
gauge-dependent because
everybody is going to get

276
00:23:00,120 --> 00:23:01,005
a different answer.

277
00:23:01,005 --> 00:23:03,870
And whose answer is right?

278
00:23:03,870 --> 00:23:05,620
That's not possible.

279
00:23:05,620 --> 00:23:12,330
So if this Barry phase seems to
have failed a very basic thing,

280
00:23:12,330 --> 00:23:14,790
then it's not gauge-invariant.

281
00:23:14,790 --> 00:23:20,160
But there is one way in
which this gets fixed.

282
00:23:20,160 --> 00:23:26,370
If your motion in the
configuration space

283
00:23:26,370 --> 00:23:36,030
begins and ends in the same
place, these two will cancel.

284
00:23:36,030 --> 00:23:38,230
And then it will
be gauge-invariant.

285
00:23:38,230 --> 00:23:43,730
So the observable Berry's
phase is a geometric phase

286
00:23:43,730 --> 00:23:50,000
accumulated by the system in a
motion in a configuration space

287
00:23:50,000 --> 00:23:53,990
where it begins and
ends in the same point.

288
00:23:53,990 --> 00:23:55,910
Otherwise, it's not observable.

289
00:23:55,910 --> 00:23:58,190
You can eliminate it.

290
00:23:58,190 --> 00:24:01,610
And so this is an
important result

291
00:24:01,610 --> 00:24:39,420
that the geometric Berry
phase for a closed path

292
00:24:39,420 --> 00:24:54,045
in the configuration
space is gauge-invariant.