WEBVTT

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PROFESSOR: I now have that new
n of t that we wrote there.

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I have to write
it as what it is.

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It's i psi n of r of t times--

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I will write it here
this way-- d dt--

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the dot will be
replaced by the d dt--

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psi n of r of t.

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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.

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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.

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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.

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And what must you do?

00:01:27.660 --> 00:01:37.920
Well, you should do
df dr1 times dr1 dt

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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.

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Well, the chain rule applies.

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But this can be written
in a funny language--

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maybe not so funny--

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as the gradient sub r vector
of f dotted dr vector dt.

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See, the gradient, in
general, is d dx1 d dx2 d dx3.

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It's a vector operator.

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The gradient sub r would
mean d dr1 d dr2 d dr3,

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just the gradient in
this Euclidean vector

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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.

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So gamma n of t is equal
to i, from the top line,

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psi n of r of t times gradients
of r acting on the state psi

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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.

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So that gives you
capital N components,

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the derivative of the ket
with respect to r1 r2 r3 r4.

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Then with the inner product,
it gives your capital

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N numbers, which are the
components of a vector that is

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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.

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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.

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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.

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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

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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.

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But Berry probably clarified
this issue of the time

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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

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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.

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Psi prime of r is going to
be equal to e to the minus

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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

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are as good as
your original psi n

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because this equation also
holds for the psi n primes.

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If you add with the Hamiltonian,
the Hamiltonian in here

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just goes through this and
hits here, produces the energy,

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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

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would care about this factor.

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The r's are just parameters.

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Yes?

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AUDIENCE: [INAUDIBLE]

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PROFESSOR: No, they're
still normalized.

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I should put a phase here--
thank you very much--

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minus i.

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Thank you.

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Yes, I want the states
to be normalized,

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and I want them
to be orthonormal.

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And all that is not changed
if I put them phase.

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So this is the funny thing
about quantum mechanics.

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It's all about phases
and complex numbers.

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But you can, to a large degree,
change those phases at will.

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And whatever survives is some
sort of very subtle effects

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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

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connection associated with
this new state a n prime of r.

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So I must do that operation
that we have up there

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with the news state.

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So I would have i psi n of r
times e to the i beta of r.

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That's The bra.

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Then I have dr and now the
ket, e to the minus i beta of r

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psi n of r.

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So this is, by definition,
the new Berry connection

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associated to your new,
redefined eigenstates.

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Now this nabla is acting
on everything to the right.

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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.

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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.

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Here is a 1, 2, 3, 4 5.

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OK, so this is the
first one, and then you

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have the term for this gradient
acts on this exponential.

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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.

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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.

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Maybe I'll put the r of the r.

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And then these cancel, and you
have the state with itself,

00:17:05.280 --> 00:17:06.619
which gives you 1.

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So that's all it is, all that
the second term gives you.

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So here we get a n of r plus
gradient r of beta of r.

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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.

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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.