WEBVTT
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PROFESSOR: I now have that new
n of t that we wrote there.
00:00:08.130 --> 00:00:10.800
I have to write
it as what it is.
00:00:10.800 --> 00:00:18.680
It's i psi n of r of t times--
00:00:18.680 --> 00:00:22.550
I will write it here
this way-- d dt--
00:00:22.550 --> 00:00:26.900
the dot will be
replaced by the d dt--
00:00:26.900 --> 00:00:29.371
psi n of r of t.
00:00:32.600 --> 00:00:36.650
And then, of course,
the gamma n of t
00:00:36.650 --> 00:00:40.280
will be just the
integral from 0 to t
00:00:40.280 --> 00:00:43.740
of new n of t prime bt prime.
00:00:43.740 --> 00:00:45.620
So that's the next step.
00:00:49.630 --> 00:00:53.370
Well, if you have
to differentiate
00:00:53.370 --> 00:01:01.080
a function that depends on
r of t, what do you have?
00:01:01.080 --> 00:01:09.680
Let me do it for a simpler
case, d dt of f of r of t.
00:01:09.680 --> 00:01:17.370
This means d dt of a
function of r1 of t
00:01:17.370 --> 00:01:19.830
are all the ones up to rn of t.
00:01:25.860 --> 00:01:27.660
And what must you do?
00:01:27.660 --> 00:01:37.920
Well, you should do
df dr1 times dr1 dt
00:01:37.920 --> 00:01:46.860
all the way up to
the df dr and drn dt.
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You want to find the time
dependence of a function that
00:01:49.830 --> 00:01:54.030
depends on a collection of
time-dependent coordinates.
00:01:54.030 --> 00:01:56.280
Well, the chain rule applies.
00:01:59.770 --> 00:02:05.980
But this can be written
in a funny language--
00:02:05.980 --> 00:02:07.690
maybe not so funny--
00:02:07.690 --> 00:02:21.280
as the gradient sub r vector
of f dotted dr vector dt.
00:02:24.262 --> 00:02:29.770
See, the gradient, in
general, is d dx1 d dx2 d dx3.
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It's a vector operator.
00:02:31.720 --> 00:02:37.150
The gradient sub r would
mean d dr1 d dr2 d dr3,
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just the gradient in
this Euclidean vector
00:02:42.970 --> 00:02:47.690
space times dr dt.
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So that's what I want to
use for this derivative.
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I have to differentiate
that state.
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And therefore, I'll
write it that way.
00:03:03.420 --> 00:03:11.580
So gamma n of t is equal
to i, from the top line,
00:03:11.580 --> 00:03:25.810
psi n of r of t times gradients
of r acting on the state psi
00:03:25.810 --> 00:03:36.310
n of r of t dotted with dr dt.
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This is dot product.
00:03:44.450 --> 00:03:48.520
So just to make sure
you understand here,
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you have one ket here, and
you have this gradient.
00:03:55.340 --> 00:04:00.420
So that gives you
capital N components,
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the derivative of the ket
with respect to r1 r2 r3 r4.
00:04:05.410 --> 00:04:10.090
Then with the inner product,
it gives your capital
00:04:10.090 --> 00:04:14.320
N numbers, which are the
components of a vector that is
00:04:14.320 --> 00:04:18.589
being dotted with this vector.
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It's all about
trying to figure out
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that this language makes sense.
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If this made sense to you,
this should make sense,
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a little more, maybe a
tiny bit more confusing.
00:04:35.380 --> 00:04:37.980
But maybe you should
write it all out.
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What do you think it is?
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And that might help you.
00:04:41.480 --> 00:04:44.710
Or we could do that later.
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So if we have that, we can go
to gamma n, the geometric phase.
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So this is 0 2t, the integral
with respect to prime time,
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so new m.
00:05:03.630 --> 00:05:09.060
So it's i psi n r of t prime--
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there's lots of vectors
here, gradient r vector
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of psi n r of t prime
dotted dr dt prime dt prime.
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That's the last dt prime.
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And the good thing that
happened, the thing that
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really makes all the
difference, the thing that
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is responsible for
that conceptual thing
00:05:46.300 --> 00:05:51.430
is just this cancellation.
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This cancellation means that
you can think of the integral
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as happening just in
the configuration space.
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This is not really an
integral over time.
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This is an integral
in configuration space
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because now this integral is
nothing else than the integral
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over the path gamma.
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Because the path
gamma represents
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the evolution of the coordinate
capital R from 0 to time t.
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This is nothing else than
the integral over the path
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gamma of i psi n of r--
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I don't have to
write the t anymore--
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dr psi n of r--
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again no t-- dot dr. And this
is the geometric phase gamma n
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that depends on r on the path.
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I'll write it like that.
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You see, something very
important has happened here.
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It's a realization that
time plays no role anymore.
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This is the concept.
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This is what you have to
struggle to understand here.
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This integral says
take this path.
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Take a little dr dot it with
this gradient of this object,
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which is kind of the gradient
of this ket, which is
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a lot of kets with this thing.
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So it's a vector.
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Dot it with this and integrate.
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And time plays no role.
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You just follow the path.
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So whether this
thing took one minute
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to make the path
or a billion years,
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the geometric phase will
be exactly the same.
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It just depends on
the path it took.
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Time for some names
for these things.
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Let's see.
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So a first name is
that this whole object
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is going to be called
the Berry connection.
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i psi n of r
gradient r psi n of r
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is called the Berry
connection a n vector of r.
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Berry connection.
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OK, a few things to notice,
the Berry connection
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is like a vector in the
configuration space.
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It has capital N components
because this is a gradient.
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And therefore, it produces
of this ket n kets
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and, therefore, n numbers
because of the bra.
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So this is a thing with
capital N components.
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So it's a vector in RN.
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But people like the
name connection.
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Why Connection?
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Because it's a little
more subtle than a vector.
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It transforms under
Gage transformation,
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your favorite things.
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And it makes it interesting
because it transforms
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under Gage transformation.
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We'll see it in a second.
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So it's a connection
because of that.
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And there's one Berry
connection for every eigenstate
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of your system.
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Because we fix some n,
and we got the connection.
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And we're going to get different
connections for different n's.
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So n components,
one per eigenstate,
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and they live all over
the configuration space.
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You can ask, what is
the value of the Berry
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connection at this point?
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And there is an answer.
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At every point, this
connection exists.
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Now, let's figure out the issue
of gauge transformations here.
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And it's important because
this subject somehow--
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these formulas, I
think in many ways,
00:11:37.570 --> 00:11:42.340
were known to everybody
for a long time.
00:11:42.340 --> 00:11:47.800
But Berry probably clarified
this issue of the time
00:11:47.800 --> 00:11:51.940
independence and
emphasized that this could
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be interesting in some cases.
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But in fact, in most
cases, you could say
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they're not all that relevant.
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You can change them.
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So here is one thing
that can happen.
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You have your
energy eigenstates,
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your instantaneous eigenstates.
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You solve them,
and you box them.
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You're very happy with them.
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But in fact, they're
far from unique.
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Your energy eigenstates,
your instantaneous energy
00:12:27.280 --> 00:12:29.830
eigenstates can be changed.
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If you have an energy
eigenstate psi n of r--
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that's what it is--
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well, you could decide
to find another one.
00:12:43.180 --> 00:12:51.070
Psi prime of r is going to
be equal to e to the minus
00:12:51.070 --> 00:12:57.005
some function, arbitrary
function, of r times this.
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And these new states
are energy eigenstates,
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instantaneous energy
eigenstates that
00:13:07.830 --> 00:13:11.460
are as good as
your original psi n
00:13:11.460 --> 00:13:16.410
because this equation also
holds for the psi n primes.
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If you add with the Hamiltonian,
the Hamiltonian in here
00:13:21.180 --> 00:13:27.810
just goes through this and
hits here, produces the energy,
00:13:27.810 --> 00:13:30.830
and then the state
is just the same.
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The r of t's are parameters
of the Hamiltonian.
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They're not operators.
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So there's no reason
why the Hamiltonian
00:13:43.650 --> 00:13:46.860
would care about this factor.
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The r's are just parameters.
00:13:49.090 --> 00:13:49.987
Yes?
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AUDIENCE: [INAUDIBLE]
00:13:53.730 --> 00:13:56.010
PROFESSOR: No, they're
still normalized.
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I should put a phase here--
thank you very much--
00:13:59.740 --> 00:14:01.410
minus i.
00:14:01.410 --> 00:14:02.880
Thank you.
00:14:02.880 --> 00:14:05.910
Yes, I want the states
to be normalized,
00:14:05.910 --> 00:14:08.460
and I want them
to be orthonormal.
00:14:08.460 --> 00:14:12.780
And all that is not changed
if I put them phase.
00:14:12.780 --> 00:14:17.700
So this is the funny thing
about quantum mechanics.
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It's all about phases
and complex numbers.
00:14:20.510 --> 00:14:25.410
But you can, to a large degree,
change those phases at will.
00:14:25.410 --> 00:14:30.240
And whatever survives is some
sort of very subtle effects
00:14:30.240 --> 00:14:32.170
between the phases.
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So here I put the i
and beta of r is real.
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PROFESSOR: So you can say
let's compute the new Berry
00:14:46.830 --> 00:14:55.645
connection associated with
this new state a n prime of r.
00:14:55.645 --> 00:15:01.400
So I must do that operation
that we have up there
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with the news state.
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.
00:15:11.030 --> 00:15:13.460
That's The bra.
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Then I have dr and now the
ket, e to the minus i beta of r
00:15:22.700 --> 00:15:24.830
psi n of r.
00:15:24.830 --> 00:15:29.970
So this is, by definition,
the new Berry connection
00:15:29.970 --> 00:15:36.520
associated to your new,
redefined eigenstates.
00:15:39.730 --> 00:15:45.200
Now this nabla is acting
on everything to the right.
00:15:45.200 --> 00:15:50.590
Suppose it acts on the state
and then the two exponentials
00:15:50.590 --> 00:15:55.960
will cancel, and then you
get the old connection.
00:15:55.960 --> 00:16:02.170
So there is one term here,
which is just the old a n of r.
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There's all these arrows there.
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There's probably five arrows
at least I miss on every board.
00:16:11.090 --> 00:16:17.135
Here is a 1, 2, 3, 4 5.
00:16:20.310 --> 00:16:25.310
OK, so this is the
first one, and then you
00:16:25.310 --> 00:16:32.400
have the term for this gradient
acts on this exponential.
00:16:32.400 --> 00:16:34.550
When the gradient acts
on the exponential,
00:16:34.550 --> 00:16:36.810
it gives the same
exponential times
00:16:36.810 --> 00:16:39.900
the gradient of the exponent.
00:16:39.900 --> 00:16:43.110
The exponentials then cancel.
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The gradient of the exponent
would give me plus i times
00:16:48.030 --> 00:16:51.475
minus i gradient of beta.
00:16:56.040 --> 00:17:00.420
Maybe I'll put the r of the r.
00:17:00.420 --> 00:17:05.280
And then these cancel, and you
have the state with itself,
00:17:05.280 --> 00:17:06.619
which gives you 1.
00:17:06.619 --> 00:17:12.109
So that's all it is, all that
the second term gives you.
00:17:12.109 --> 00:17:22.490
So here we get a n of r plus
gradient r of beta of r.
00:17:31.720 --> 00:17:33.880
So this is the gauge
transformation.
00:17:33.880 --> 00:17:39.280
And you say, wow,
I can see now why
00:17:39.280 --> 00:17:40.900
this is called a connection.
00:17:40.900 --> 00:17:44.080
Because just like
the vector potential
00:17:44.080 --> 00:17:47.380
under a gauge
transformation, it transforms
00:17:47.380 --> 00:17:50.450
with a gradient of a function.
00:17:50.450 --> 00:17:56.530
So it really transforms
as a vector potential, all
00:17:56.530 --> 00:18:00.400
in this space called
the configuration space,
00:18:00.400 --> 00:18:03.100
not in real space.
00:18:03.100 --> 00:18:08.930
In the configuration space it
acts like a vector potential.
00:18:08.930 --> 00:18:12.280
And that's why it's
called a connection.
00:18:12.280 --> 00:18:13.350
But let's see.
00:18:13.350 --> 00:18:17.470
We have now what happens
to the connection.
00:18:17.470 --> 00:18:23.830
Let's see what happens to the
Berry's phase if you do this.
00:18:23.830 --> 00:18:28.885
So the Berry's phase over
there is this integral.
00:18:33.960 --> 00:18:35.670
So the Berry's phase can change.
00:18:46.890 --> 00:18:52.690
And let's see what happens
to the Berry's phase.
00:18:52.690 --> 00:19:02.200
So what is the geometric
phase gamma n of gamma?
00:19:02.200 --> 00:19:05.890
In plain language, it is
the integral over gamma--
00:19:05.890 --> 00:19:08.860
from here, I'm just
copying the formula--
00:19:08.860 --> 00:19:20.120
of a n of r, the Berry
connection, times dr.
00:19:20.120 --> 00:19:28.730
So what is the new Berry phase
for your new instantaneous
00:19:28.730 --> 00:19:30.770
energy eigenstates?
00:19:30.770 --> 00:19:33.140
Now you would say,
if the Berry phase
00:19:33.140 --> 00:19:36.950
is something that is
observable, it better not
00:19:36.950 --> 00:19:40.100
depend just on your
convention to choose
00:19:40.100 --> 00:19:42.740
the instantaneous
energy eigenstates.
00:19:42.740 --> 00:19:45.400
And this is just
your convention.
00:19:45.400 --> 00:19:49.310
Because if a problem
is sufficiently messy,
00:19:49.310 --> 00:19:55.700
I bet you guys would all
come up with different energy
00:19:55.700 --> 00:19:59.960
eigenstates because the phases
are chosen in different ways.
00:19:59.960 --> 00:20:04.380
So it better not change
if the Berry phase
00:20:04.380 --> 00:20:06.810
is to be significant.
00:20:06.810 --> 00:20:08.780
So what is the prime thing?
00:20:08.780 --> 00:20:12.560
Well, we still integrate
over the same path, but now
00:20:12.560 --> 00:20:14.270
the prime connection--
00:20:19.540 --> 00:20:28.330
but that is the old
connection a n of rd r,
00:20:28.330 --> 00:20:33.640
the old Berry's phase, plus
the integral over gamma,
00:20:33.640 --> 00:20:37.380
or I will write it from
initial the final r.
00:20:41.400 --> 00:20:45.120
Maybe I should have ir
and i f in the picture.
00:20:45.120 --> 00:20:51.140
If you want to, you can put
this r of time equals 0 as ri
00:20:51.140 --> 00:21:01.370
and r of time equal tf
is rf the extra term,
00:21:01.370 --> 00:21:16.750
the gradient of beta dot dr.
So this is the old Berry phase.
00:21:16.750 --> 00:21:24.150
So the new Berry phase
is the old Berry phase.
00:21:24.150 --> 00:21:26.600
And how about the last integral?
00:21:26.600 --> 00:21:27.930
Does it vanish?
00:21:27.930 --> 00:21:32.450
No, it doesn't vanish.
00:21:32.450 --> 00:21:34.110
It gifts you.
00:21:34.110 --> 00:21:36.600
But in fact, it can be done.
00:21:36.600 --> 00:21:39.500
This is like derivative
times this thing,
00:21:39.500 --> 00:21:42.542
so it's one of those
simple integrals.
00:21:42.542 --> 00:21:47.690
The gradient times the
d represents the change
00:21:47.690 --> 00:21:51.530
in the function as
you move a little dr.
00:21:51.530 --> 00:21:57.530
So when you go from ri to rf,
the integral of the gradient
00:21:57.530 --> 00:22:00.440
is equal to the
function beta at rf
00:22:00.440 --> 00:22:03.410
minus the function beta on ri.
00:22:03.410 --> 00:22:07.520
This is like when you
integrate the electric field
00:22:07.520 --> 00:22:09.980
along a line, and
the electric field
00:22:09.980 --> 00:22:11.930
is the gradient
of the potential.
00:22:11.930 --> 00:22:14.900
The integral of the electric
field through a line
00:22:14.900 --> 00:22:18.060
is the potential here
minus the potential there.
00:22:18.060 --> 00:22:28.710
So here this is plus beta
or rf minus beta of ri.
00:22:31.650 --> 00:22:35.700
So it's not gauge invariant
in the Berry phase.
00:22:39.320 --> 00:22:46.050
And therefore, it will mean that
most of the times it cannot be
00:22:46.050 --> 00:22:48.090
observed.
00:22:48.090 --> 00:22:49.410
It's not gauge invariant.
00:22:49.410 --> 00:22:53.100
Whatever is not gauge
invariant cannot be observed.
00:22:53.100 --> 00:22:56.460
You cannot say you make a
measurement and the answer is
00:22:56.460 --> 00:23:00.120
gauge-dependent because
everybody is going to get
00:23:00.120 --> 00:23:01.005
a different answer.
00:23:01.005 --> 00:23:03.870
And whose answer is right?
00:23:03.870 --> 00:23:05.620
That's not possible.
00:23:05.620 --> 00:23:12.330
So if this Barry phase seems to
have failed a very basic thing,
00:23:12.330 --> 00:23:14.790
then it's not gauge-invariant.
00:23:14.790 --> 00:23:20.160
But there is one way in
which this gets fixed.
00:23:20.160 --> 00:23:26.370
If your motion in the
configuration space
00:23:26.370 --> 00:23:36.030
begins and ends in the same
place, these two will cancel.
00:23:36.030 --> 00:23:38.230
And then it will
be gauge-invariant.
00:23:38.230 --> 00:23:43.730
So the observable Berry's
phase is a geometric phase
00:23:43.730 --> 00:23:50.000
accumulated by the system in a
motion in a configuration space
00:23:50.000 --> 00:23:53.990
where it begins and
ends in the same point.
00:23:53.990 --> 00:23:55.910
Otherwise, it's not observable.
00:23:55.910 --> 00:23:58.190
You can eliminate it.
00:23:58.190 --> 00:24:01.610
And so this is an
important result
00:24:01.610 --> 00:24:39.420
that the geometric Berry
phase for a closed path
00:24:39.420 --> 00:24:54.045
in the configuration
space is gauge-invariant.