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PROFESSOR: OK, let us start.
So maybe start by
reminding you what
we did in the last lecture.
So you take the
black hole metric
so we explain that
you can actually
extend-- so this is the other
black hole horizon-- so you can
actually extend the
region outside the horizon
to actually four total regions
for the four black hole
spacetime.
And then this [? tag, ?]
this is a singularity.
And of course, this
is only the rt plane,
and then you also have
[? s2. ?] So now [INAUDIBLE],
let me show that in
order for the metric
to be regular at the horizon
into this when you go
to Euclidean signature then
you want to identify Tau to be
periodic with this period.
So in essence, when you
go to Euclidean signature,
then this rt plane becomes,
essentially, a disc.
And this Tau is in the angular
direction, and then the rho,
this r is in the
radial direction.
And then the region of
this disc is the horizon.
The region of the
disc is the horizon.
So when you say, yeah, so
this Tau has to be periodic.
So that means if you fit any
theories in this spacetime,
then that theory must be
at a finite temperature
with inverse of this, which
gives you this [? H ?]
kappa divided by beta.
And the kappa is
the surface gravity.
So you can do a similar
thing with the [INAUDIBLE].
And the [INAUDIBLE] is just
1/4 of a Minkowski, which
[? forniated ?] in this way.
This is a constant
of the rho surface,
and this is a constant
of the eta surface.
And, again, [INAUDIBLE]
in eta to Euclidean.
And in order for the Euclidean
need [? congestion ?]
to be regular, the
region, then the theta
have to be periodical in 2pi.
And then this theta
essentially, again,
becomes an angular direction.
And then this implies
that the local observer
in the window of spacetime must
observe a temperature given
by this formula, OK?
Any questions regarding that?
Good, so today we
will talk about
the physical interpretation
of this temperature.
OK?
So let me just write down
what this temperature
means in words, so I will
use black hole as an example.
We can stay exactly
parallel [INAUDIBLE].
It says if you consider
a quantum fields
theory in the black
hole spacetime, then
the vacuum state-- so when
you put the quantum fields
theory, say, in the spacetime,
always ask, what is the vacuum?
Then the vacuum state
for this curve t
obtained via the analytic
continuation procedure
where the analytic continuation
procedure from Euclidean
signature is a
thermal equilibrium
state with the stated
temperature, OK?
So emphasize when you want
to talk about temperature,
you first have to specify
what is the time you use.
So this temperature refers to
this particular choice of time,
and so this is the temperature
corresponding to an observer
while using this time.
And then this is the time, as we
said, corresponding to observer
leaving at infinity.
And so this would
be the temperature
observed by the infinity.
So similarly, this
temperature is the temperature
for given [? with ?]
the observer
at some location of rho.
So this is for local observer.
If you just talk about the
eta, then the temperature
is just h divided by 2pi because
theta is periodical in 2pi.
And if you ask what is
the temperature associated
with eta, it's just
h pi divided by 2pi.
So any questions
regarding this statement?
So now let me
[INAUDIBLE] is going
to be abstract, so let me
elaborate a little bit,
and let me make some remarks.
So the first is that
the choice of vacuum
for QFT in the curved
spacetime is not unique.
So this is a standard feature.
This is a standard
feature, so if you
want to define a quantum fields
theory in the curved space time
because, in general, in
the curved space time,
there's no prefer
the choice of time.
So in order to quantize here,
you have to choose a time.
So depending on
your choice of time,
then you can quantize your
theory in a different way.
And then you have a
different possibility
of what is your vacuum.
So in other words, in
the curved spacetime,
in general, the vacuum
is observer dependent.
So of course, the black hole
is the curved spacetime, so
the particular
continuation procedure
we described corresponds
to a particular choice,
to a special choice,
of the vacuum.
In the case of a
black hole, this
has been called to be
the Hartle-Hawking vacuum
because the Hartle-Hawking
first defined it
from this Euclidean procedure.
And for the Rindler
spacetime, time
so we see a little bit
later in today's class
that this is actually--
the choice of vacuum
actually just goes one into the
standard of the Minkowski way
vacuum.
reduced to the Rindler patch.
So of course, you
can actually choose
the vacuum in some other way.
For example, say if
for a black hole,
instead of periodically
identify Tau with this period,
we can choose not to
identify Tau at all.
Suppose for a black hole,
in Euclidean signature
we take Tau to be uncompact.
So we don't identify.
So this, of course, runs into
a different Euclidean manifold.
Then that corresponds to a
different Euclidean manifold.
And, again, you can really
continue the Euclidean theory
on this manifold
back to [INAUDIBLE]
to define our vacuum.
And that gives you
a different vacuum
from the one in which
you identified Tau.
So this gives you a so-called
Schwarzschild vacuum.
Or sometimes people
call it Boulware vacuum
because Boulware had worked
on it in the early days
before other people.
And, in fact, this
Schwarzschild vacuum
is the most natural vacuum.
In fact, this is
the vacuum that one
would get by doing
canonical connotation
in the black hole in terms
of this Schwarzschild time t.
So let me just emphasize, so
this is a curbed spacetime.
You can just put
a quantum fields
theory in this spacetime,
and then there's
a time here which, actually,
the spacetime is time dependent.
So you have a time-independent
Hamiltonian [? so ?] respect
to this time.
And so, in principal,
you can just
do a straightforward
canonical connotation.
And then that connotation
will give you a vacuum.
And that vacuum would
corresponding to the same
as you do analytic
continuation to the Euclidean.
And don't compact Phi
Tau because there's
nothing special.
Just like when you do
a 0 temperature fields
theory in the standard
Minkowski spacetime.
But when you compactify
Tau, then you
get a different vacuum.
It is what would be now called
a Hartle-Hawking vacuum,
and it's what you get
a finite temperature.
Is this clear?
Yes?
AUDIENCE: So in what sense
are these different vacuums
physical or not?
PROFESSOR: Yeah, I will answer
this question in a little bit.
Right now, I want
you to be clear.
Right now we have two
different vacuum here
corresponding to
two different ways
of doing analytic continuation.
And one way you compactify
Tau, and one way
you don't compactify Tau.
And the way you
don't compactify Tau
is actually the most
straightforward way.
So if you just do a
straightforward canonical
connotation using
this time, then that
will be the vacuum you get.
So that's why this is called
a Schwarzschild vacuum, OK?
AUDIENCE: So do you not have
a similarity in writing?
PROFESSOR: I will
talk about that.
So similarly for
the Rindler, again,
if you do straightforward
canonical connotation,
in this spacetime with
eta has your time,
then when you go to
Euclidean signature,
you want compactify the
Euclidean version of eta.
Then you won't compactify theta.
So if you take the
theta to be uncompact,
then you get what is
so-called Rindler vacuum.
So it just, again,
can be obtained.
So then you ask them, why do we
bother to define those vacuums?
Why do we bother to make this
identification given that this
is the most
straightforward thing
to do, say, when we
consider this quantum fields
theory inside curved spacetimes.
So here is the
key, and so that's
where the geometric
consideration becomes
important.
So, again, let me just
first use the black hole
as an example case.
So in the Schwarzschild vacuum,
the corresponding Euclidean
manifold is singular
add to the horizon.
So we explained last time,
when we do this [INAUDIBLE],
the spacetime is
regular only for
this periodic identification.
For any other
possibilities, then you
will have a conical
singularity add to the horizon.
So this implies the Euclidean
manifold singular horizon.
Say, if you cut computer
correlation function
[INAUDIBLE], then you can have
single behavior at the horizon.
In particular, when
you analytically
continue to
[INAUDIBLE] signature,
then that means that
physical observables often
are singular after the horizon.
So we need to continue
to [INAUDIBLE] signature.
For example-- which actually
I should have maybe the
[INAUDIBLE] problem but
then forgot earlier.
For example, you
can check yourself
that just take a free
scalar fields theory.
You compute the stress tensor
from this free scalar fields
theory, and you find
the stress tensor
of this theory blows up after
the horizon in the Minkowski
vacuum.
And if this is not the
case, for the vacuum
we obtained by doing analytic
continuation this way--
so this is not the case for
a Hartle-Hawking vacuum.
So Hartle-Hawking vacuum
just is [? lame ?]
for the vacuum we obtained
by this analytic continuation
procedure for which all
observer was regular just,
essentially, by definition,
by construction.
So essentially by construction
because Euclidean manifold
is completely smooth, and if you
want to compute any correlation
function-- for example, you can
compute the Euclidean signature
and then I need you to continue
back to [INAUDIBLE] signature
because it's the smooth side
of the horizon of the Euclidean
signature.
I need the continuation back.
We also give you a regular
function at the horizon there.
So similar remarks apply
to the Rindler space,
and this Rindler vacuum will
be singular at this Rindler
horizon.
It will be singular
at Rindler horizon.
So since from the
general relativity,
the horizon is a smooth place.
The curvature there
can be very small
if you have a big
black hole [INAUDIBLE].
So physically, we believe that
these Hartle-Hawking vacuums
are more physical then, for
example, the Schwarzschild
vacuum.
But Schwarzschild is still
often used for certain purposes.
But if you want to consider
these physical observables,
then the Hartle-Hawking
vacuum should be the right one
to use if you don't want
to encounter singularities
at the horizon.
Any questions?
Yes?
AUDIENCE: [INAUDIBLE]?
PROFESSOR: No.
AUDIENCE: [INAUDIBLE].
PROFESSOR: Yeah.
Yeah, because, for example,
Euclidean Tau is uncompact.
AUDIENCE: So the singularity
of the physical variables
[INAUDIBLE] to become
singular to Hartle-Hawking?
PROFESSOR: Yeah, that's right.
AUDIENCE: [INAUDIBLE]?
PROFESSOR: I would say this
is the physical reason.
If I think about it, I might
other not-so-essential reasons.
Yeah, but at the moment, I
could not think of any others.
Yeah?
AUDIENCE: So if I
understand correctly,
when you quantize a theory
in curved spacetime,
you have to choose a space-like
foliation of your spacetime,
then you quantize it on
that foliation basically.
So the problem is that I
still don't understand.
So the different
ground suits that you
get for different possible
quantizations based
on different foliations.
Like, to what degree are
they compatible versus
incompatible with each other?
Are they at odds
with each other?
Like is one more real
than another one,
or is it just sort of an
artifact of your coordinates,
and there are things
that we can do
which are independent of
which one we end up with.
PROFESSOR: Yeah, so
first, in general, they
are physically
inequivalent to each other.
For example, in this case,
the Schwarzschild vacuum
is physically inequivalent
to the Hartle-Hawking vacuum.
And actually, with a
little bit of work,
one can write down the
explicit relation between them.
And so from the perspective
of the Schwarzschild vacuum,
the Hartle-Hawking vacuum
is a highly excited
state and vice versa.
So vice versa.
And so, yeah, for local
observable, say in the local,
it depends.
I think the vacuum is
more like-- sometimes
certain vacuums are more
convenient for a certain
observer and a
certain vacuum is more
convenient for that observer.
Sometimes it depends on
your physical [INAUDIBLE],
et cetera.
And other questions?
AUDIENCE: So we cannot treat
these two vacuums as like some
transformation?
PROFESSOR: Yeah, there is some
transformation between them.
AUDIENCE: So they
stay just the same?
Like do [INAUDIBLE]?
PROFESSOR: No, they
are different states.
They are completely
different states.
You can relate to
them in a certain way,
but they're different states.
Yeah, they're different states.
AUDIENCE: If they are
different, than why can
we only say one is wrong?
Because, for example, if it's
singular, it's not physical,
why don't we just discard it?
PROFESSOR: I'm sorry?
AUDIENCE: Why we--
PROFESSOR: Yeah, physically
we do discard it.
Physically we don't think
the Schwarzschild vacuum is--
so here there's assumption.
So the assumption is that we
believe that the physics is now
single at the horizon.
So this is a basic assumption.
And if that's the
basic assumption,
then you should abandon
Schwarzschild vacuum
as the physical
choice of the vacuum.
Then you say, then if we really
do experiment in the black hole
geometry, then what
you will discover
is the property of the
Hartle-Hawking vacuum.
But if our physical
assumption is wrong
that actually a black
hole horizon is singular,
then maybe Schwarzschild horizon
will turn out to be right.
But in this case, we
actually don't know.
We cannot really do
experiments in the black hole,
so we cannot really check it.
AUDIENCE: But we can measure
black hole temperature
from infinitely far away,
and the Schwarzschild
vacuum it would be 0.
PROFESSOR: No, but
you cannot measure it.
We have not been
able to measure it.
AUDIENCE: And what
about the [INAUDIBLE]?
PROFESSOR: Yeah, first
produce the black hole first.
[LAUGHTER]
Yeah, we will worry
about it after we
have produced the black hole.
AUDIENCE: But if
they didn't radiate,
they would not decay at all.
PROFESSOR: Yeah, then
you may also not see it.
Anyway, so this can considered
as a physical interpretation
of the temperature inside
the regularity at the horizon
force us to be in this
Hartle-Hawking vacuum.
And then the
Hartle-Hawking vacuum
is like [? a similar ?] state.
OK, so now they may
explain why [? they ?] say,
they will behavior like
[? a similar ?] state, OK?
So let me go to the second
thing I will explain today.
It's the physical origin
of the temperature.
So, again, I can use either
black hole or the Rindler
example.
The mathematics are
almost identical.
But I will use the
Rindler example
because the mathematics
are slightly simpler.
So I will use the
window of spacetime.
So I will explain using
the Rindler example.
So I will explain at A
that this choice of theta
to periodically identify theta
plus theta plus 2pi [INAUDIBLE]
into the choice of
a Minkowski vacuum.
Then the second
thing I will do is
I will derive this temperature--
derive the temperature using
a different method.
So that's I will do today.
So here, when me say,
this temperature,
we just directly
read the temperature
from the period of the
Tau, OK but I will really
derive with the
thermal density matrix.
And then you will see that
this temperature is indeed
the temperature with appears
in the density matrix.
Any questions?
So these two
together also amounts
to the following statement.
So this is an
important statement,
so let me just write it
down, so this A and B also
amounts to be the
following-- also
is equivalent to the
following statement.
It says, the vacuum
inside the Minkowski--
the standard of the
Minkowski vacuum where
you do your quantum
fields theory,
the Minkowski vacuum appears
to be in a similar state.
This temperature which
is pi divided by 2pi or T
equal to hbar divided
by 2pi in terms of eta.
OK, it depends on which time
you use to a Rindler observer
of constant acceleration rate.
So a says, actually, this choice
[INAUDIBLE] a periodic of 2pi,
actually we are choosing
the Minkowski vacuum.
And then the second
statement you said,
but the stand of the
Minkowski vacuum, which
appears to be 0 temperature
to ordinary Minkowski observer
than appears to be a similar
state to a Rindler observer.
OK, so that's the
physical content
of this things
which we will show.
Any questions on this?
Yes?
AUDIENCE: [INAUDIBLE]?
PROFESSOR: It's actually
a tricky question.
In some sense, they don't really
belong to the same Hilbert
space.
Yeah just when you talk
about quantum fields theory,
it's a little bit
tricky when you have
an infinite number [INAUDIBLE].
Yeah, but one can write down
the relation between them.
AUDIENCE: [INAUDIBLE].
PROFESSOR: Yeah, one can write
down a relation between them,
and then if you take the
[? modulus ?] of that vacuum.
Then you'll find that
it's actually infinite
because you have an infinite
number that [INAUDIBLE].
It's not possible to
normalize that state.
OK, so that's what
we are going to show.
Hopefully, we will reach
it by the end of this hour.
But before that, we need a
little bit of preparation
to remind you of a few things.
So once we go through
these preparation,
then final derivation only
takes less than 10 minutes.
So first, he said, they are
actually two descriptions
of a similar state.
So let me remind you.
They are two descriptions
of a similar state.
So we will use harmonic
oscillator as an example.
So now let's consider a
single harmonic oscillator
a finite temperature.
So the standard
way of doing it is
that if you want to
compute the expectation
value of some operator,
of some observable
at finite temperature,
you just do
the standard canonical average.
So the H would be,
say, the Hamiltonian
of this single
harmonic oscillator,
and the z is the
partition function.
So can also be written as trace
x and as a thermal density
matrix.
So the thermal density
matrix is 1/z minus beta h,
and the z is just the sum
of all possible states.
So this is a standard
way you would do, say,
the finite temperature physics.
So this, of course, applies
to any quantum systems
causal quantum fields
theories, et cetera.
But actually, this
alternative way
to do thermal physics-- so
this was realized by Umezawa
in the 1960s.
So he said, instead
of considering
this thermal density
matrix, let me just consider
two copies of the same.
Let's consider two copies
of harmonic oscillator.
Let's double the copy,
and then now we have H1.
Then the foreseeable
space for these two copies
will be the H1 of one
system tensor product of H2
and the Hilbert
space of the other.
And then you will have H1
H2, the Hamiltonian H1 H2
associated with each of them.
But, of course, these
two H's are the same.
Then the [INAUDIBLE]
system in a typical state
would be, say, of the form
sum mn, amn, m1 [? canceled ?]
with n and 2.
So the general state, say,
of this doubled system
will be like this.
So this is two copies of the
same system with no interaction
with each other.
So now he says, in order to
consider thermal physics,
let's consider
special state defined
by the following, 1 over
square root of z sum over n.
And he said, now let's
consider the following states.
So this is an entangled state
between the two systems.
So this is an entangled state.
So the key observation
is that if you
want to consider
the thermal physics,
it says, for any observable,
say this x-- so [INAUDIBLE]
1, which only acts in
one of the systems.
[INAUDIBLE] act on
the first system.
So let's consider
any such observable.
So now, if you to
take the expectation
value between the sides,
the x between the sides,
so you essentially
have a side squared.
Then you have 1/z.
Then also Tau will
become better En,
and then this just becomes 1/z
sum over n [INAUDIBLE] and x n.
So this is the thermal average.
This is sum average.
So if you understand
the why, you can also
consider because if we are
interested only in system 1,
then you can just
integrate our system 2.
We can trace our system 2.
Say, suppose we
trace our system 2
of this state, then, of
course, what do you find?
Which is the thermal density
matrix of the system 1.
So in other words, the thermal
density matrix in one system
can be considered as
[? entangled ?] pure states
of a doubled system.
And because we know nothing
about the other system,
once we trace the l to
the other system, then
you get a density matrix.
And this density matrix comes
from our insufficient knowledge
of the other system.
So this is another way to think
about this thermal behavior.
Do you have any
questions on this?
So the temperature arises due
to an ignorance of system 2.
So because if you now have full
knowledge of one of the two,
then this would be just the
[? size ?] of a pure state.
It's just a very
special pure state.
But if you don't know
anything about system 2
and if you consent of
only about system 1,
then when you trace
out system 2 then you
get the thermal
density matrix for 1.
So let me make some
additional remarks on this.
I'll make some further remarks.
First, this, of course,
applies to any--
even though I'm saying I'm using
the harmonic oscillator, this,
of course, applies to any
quantum systems including
the QFT's.
So the second remark is
that this side is actually
a very special state.
This size is invariant under
H1 minus H2, H1 minus H2.
So you can see it very
clearly from here.
It says, if you add H1 and
H2 on this state, H1 minus H2
because both n's have the same
energy, so they just cancel.
So this is invariant
under this-- yeah,
I should say, more precisely,
it's [? annihilated ?]
by H1 minus H2.
And it's invariant
under any translation
created by H1 minus H2.
AUDIENCE: [INAUDIBLE]?
PROFESSOR: Yeah, just
double the system.
It's two copies of
the same system.
So the third remark now relies
on the harmonic oscillator.
For the harmonic
oscillator, we can write
this psi in the following form.
So this you can check yourself.
I only write down the answer.
You can easily convince
yourself this is true.
Some of you might be able
to see it immediately just
on the blackboard.
As you actually
write this in the--
if you can write this as some
explanation of some-- so a1 a2
are creation operators
respectfully for two systems.
And then the 0 1 and 0 2
are the vacuum of the two
harmonic oscillator systems.
And you can easily
see yourself, when
you expand this exponential,
then you essentially
just take the power of n.
And then that will give you n.
Yeah, when you act on 0 0,
then that will n a times n, OK?
So this is normally
called a squeezed state.
So this tells you
that this psi is
related to the
vacuum of the system
by some kind of squeezed--
so this is a squeezed state
in terms of the vacuum.
So this form is useful
for the following reason.
It's that now, based
on these three,
one can show it's
possible to construct
two oscillators, which are
b1 and b2, which [INAUDIBLE]
the psi.
So the b1 and b2 are
constructed by the following.
So, again, this you
should check yourself.
It's easy to do a
little bit of algebra.
So cos theta is
equal to 1 over 1
minus the exponential
minus beta omega.
So the omega is the frequency
for this harmonic oscillator
system and the [INAUDIBLE].
So this b1 b2 are
related to a1 a2
by some linear transformation.
So what this shows is that
[? while ?] 0 1 and the 0 2
is a vacuum for a1
and a2, and this side
is a vacuum for b1 b2.
So then maybe you
can see that this
goes [? into ?] a
different choice of vacuum.
So as we'll see
later, the relation
between the so-called
Hartle-Hawking
vacuum and the Schwarzschild
vacuum is precisely like this.
They just, of course, run
into a different choice
of alternators.
And, in particular,
their relation
is precise over this
form because, say,
if you consider some
[? three ?] series
because the [? three ?] series
essentially just reduces
to harmonic oscillators.
OK, so this is the
first preparation.
And those things are
very easy check yourself
because this is just a
single harmonic oscillator.
Any questions on this?
Oh, by the way, this
kind of transformation
is often called
Bogoliubov transformation.
So the [INAUDIBLE] you
assume by this transformation
is that in the
expression for b1 b2,
a [? dagger ?] appears here.
If there is only a that appears
here, then b1 b2, a1, a2,
they will have the same
vacuum because they will
[INAUDIBLE] the same states.
So the [? nontrivial ?] thing
is because of the [? dagger ?]
appearing here, so now the state
which they are [INAUDIBLE] are
completely different and
[? graded ?] by this kind
of squeezed-state relation.
Yes?
AUDIENCE: So it's related
to the H2 minus H2?
PROFESSOR: I'm sorry?
AUDIENCE: The
operators like b1 b2
are somewhat similar
to the H1 minus H2?
PROFESSOR: Not really.
You can write down H1 H2
in terms of b1 and b2.
You can certainly do that.
Yeah, but H1 H2 is
also very simply.
You write in terms of a1 and a2.
AUDIENCE: If a letter
transformation [INAUDIBLE]
we get-- never mind.
PROFESSOR: Yeah, you can
find the b1 [INAUDIBLE].
Just take the conjugate.
Good, so this is the
first preparation,
which is just something about
the harmonic oscillator, which
actually can be generalized
also to general quantum systems.
So the second is I need
to remind you a little bit
the Schrodinger
representation of QFT's.
So normally when we talk
about quantum field theory,
you always use the
Heisenberg picture.
We don't talk about
wave function.
But this equivalent formulation
of, of course, quantum
fields theory, which you can
just talk about wave functional
and talk about it in terms
of the Schrodinger picture.
So for example, let me just
consider a scalar field's
theory.
Then the Hilbert space,
the configuration space,
of this theory of this
system is just phi x.
You validate it at
some given time.
Say, you validate it at t
equals to 0, for example.
So let me just write it as 5x.
The configuration space
is just all possibles
of phi defining
the spatial slice.
OK, so this your configuration
space for the quantum
fields theory.
And the Hilbert
space of the system
just given by all
possible y functionals
of this configuration
space variables.
So if you feel this is a
little bit too abstract to you,
then just think of
space is discrete,
and then you can just write
this as some discrete set
of variables that
them become identical
to the ordinary quantum system.
So two things to remind
you, two more things.
First, just as in
quantum mechanics,
if you ask the value
of, say, a time t2,
you are in the [INAUDIBLE]
state of this phi,
this [INAUDIBLE]
value phi 2 and the
overlap we said t1 in the
[INAUDIBLE] state of phi,
this [INAUDIBLE] value phi 1.
This is given by a path
integral [? d ?] phi.
You integrate with the
following boundary condition.
And actually, just
[INAUDIBLE] for this theory.
In particular, by taking
a linear of this formula,
one can write down a path
integral representation
of the vacuum wave function.
So the vacuum wave
function, in this case,
will be very functional.
[INAUDIBLE], so if you
have a vacuum state,
then you just
consider the overlap
with the vacuum to your
configuration space variable.
So I will denote
it by phi 0 phi x.
This is the vacuum.
And this has a path
integral [INAUDIBLE]
is that you
compactify your time.
OK, so it's where this
is your real time,
and this is your imaginary
time, which I call t e.
You compactify your time.
And then integrate the
path integral or back
to imaginary time.
So this can be written from the
path integral as t phi t e x.
So you go to Euclidean, the but
integrate all t's more than 0
with the boundary
condition te equal to 0x.
You can do phi x, and
your Euclidean action.
OK, I hope you're
familiar with this.
This is where we obtain the
vacuum [INAUDIBLE] function.
So if this is not for me
to [INAUDIBLE] say, well,
how do you get to
the wave function
of a harmonic oscillator in
the vacuum from path integral?
Yes?
AUDIENCE: [INAUDIBLE]?
PROFESSOR: You will
see it in a few minutes
because I'm going to
use those formulas.
Yes?
AUDIENCE: So one
question is that
what is the analog of the
wave equation for the wave
functional?
Like what is the
[INAUDIBLE] equation
for the wave functional?
PROFESSOR: The same thing.
i [? partial ?] t
phi equal to H phi.
AUDIENCE: OK.
PROFESSOR: Yeah, it just has
more [INAUDIBLE] freedom.
The quantum mechanics
works the same.
AUDIENCE: OK.
PROFESSOR: So is
this familiar to you?
If not familiar
to you, I urge you
to think about the case
of a harmonic oscillator.
For the harmonic oscillator,
that will be the way
you obtain the ground state wave
function from the path integral
is that you first
need to continue
the system to the Euclidean.
And they integrate the path
integral all the way from minus
Euclidean time equal to infinity
to Euclidean time equal to 0.
Let me just write down the--
so the standard of quantum
mechanics if you want to right
down to the ground state wave
function, so that's what you do.
Again, you go to
the path integral.
You go to go to
Euclidean, and you really
integrate of all
tE smaller than 0.
And with the boundary condition
that x [? evaluated ?]
at tE equal to tE
equal to 0 equal to x.
So that gives you
the wave function,
gives you the wave function.
OK, now with this
preparation, that's
just some ordinary
quantum mechanics.
And this is a
generalization of that
to quantum fields theory
just replace that x by 5.
Good, so now let's
go back to prove
the statement we claim you're
going to prove, this one, OK?
So now let's come back.
So we finished our preparation.
Now come back to Rindler space.
So it is considered scalar
field theory, for example.
So let me just remind you,
again, this Rindler-- so
write down some
key formulas here--
so Rindler is the right quadrant
of the Minkowski time, which is
separated by this right column.
Suppose this is X. This is
T. Then Minkowski [INAUDIBLE]
is minus dT squared
plus dX squared.
And then the Rindler path
is minus rho squared to eta
squared plus the rho squared.
And so this is the
rho equal to constant.
And this is the eta
equal to constant.
And here is eta equal
to minus infinity.
So here is eta equal
to plus infinity.
So spacetime foliates
like this, OK?
And when we go to Euclidean
signature for the Minkowski,
of course, it's just
T goes to minus iTE.
So I call this Euclidean time.
For the Rindler, I
will do minus i theta.
This is my notation before.
So in the Euclidean signature,
the standard Minkowski
just becomes TE squared
plus the dX squared.
And then this Rindler
is just rho squared
[? to ?] theta squared
plus the rho squared.
In particular, this
theta identified
to be plus period 2pi.
The Euclidean, under
the continuation
of Minkowski and Rindler,
are actually identical.
Both of them are the full
two-dimensional Euclidean
space.
So if you do the standard
of the Minkowski,
just replace X by TE.
But for the Rindler, you
just replace it by rho theta.
So they just go one into
a different foliation,
and this goes one
into the [INAUDIBLE].
And this one is bound for
the Cartesian coordinate.
So the remarkable
thing is that even
in the Lorentzian signature,
the Rindler is only part
of the Minkowski spacetime.
But once you go to Euclidean,
if you do this identification
theta cos theta plus 2pi, they
have exactly the same Euclidean
manifold, just identical.
So this immediately
means one thing.
It's that all the
Euclidean observables just
with the trivial
coordinate transformation
from the Cartesian to
the polar coordinates,
all the Euclidean observables
are identical in two theories.
AUDIENCE: What is that down
there, "are identical?"
What's R sub E squared?
PROFESSOR: Yeah, this just means
the Euclidean two-dimensional
spacetime.
AUDIENCE: Oh, I see.
PROFESSOR: Yeah, let
me just right it.
This is R subscript E squared 2.
So this is just a
two-dimensional Euclidean
space.
So from here, we
can immediately lead
to the conclusion we said
earlier because for Minkowski--
say, if you compute the
Euclidean correlation
functions, and then [INAUDIBLE]
to the Lorentzian signature,
what you get is that
you get the correlation
function in the standard
Minkowski vacuum.
So for Minkowski and
then back to Lorentzian,
so the typical observables--
so Euclidean, say,
correlation functions just
goes to correlation functions
in the standard
Minkowski vacuum, OK?
Yeah, this is just trivial
QFT in your high school.
[LAUGHTER]
But with this top statement,
we reach a very [INAUDIBLE]
conclusion is that for Rindler,
when you go back to Lorentzian
signature, then that
tells you for the Rindler
when you do that
[INAUDIBLE] continuation
and back to the Lorentzian
signature, what do you get?
It's that you get a
correlation function
in the standard
Minkowski vacuum.
But for observables,
restricted to Rindler
because it's the same thing.
It's just the same function
when we do a [? continuation, ?]
just you do it differently.
It's just on the
continuation procedure,
it's a little bit different.
And a way to go into
Rindler, you just
get observable restriction
to the Rindler path.
So the Euclidean things
are exactly the same.
So that tells you that the
corresponding Lorentzian
Rindler correlation function
is the same as the correlation
function in the standard
Minkowski vacuum,
but you just restrict
to the Rindler patch.
OK, now is our final step.
So now let's talk a bit
more about the structure
of the Hilbert space.
So now I have to
derive the temperature,
but somehow in the Minkowski,
there's no temperature.
Here there's no temperature.
It's T equal to 0
from your high school.
And here, we must
see a temperature.
So where does this
temperature come from?
So now let's look a little bit
at the structure of the Hilbert
space.
So using that picture-- so let
me write here-- so the Hilbert
space of the Rindler-- so this
is all not in the Lorentzian
picture.
So the Hilbert
space in the Rindler
is essentially all possible
square integrable wave
functional of the phi, which
defines in the right patch.
So when we define
this wave functional,
we verify at the
single time slice.
So that's evaluated at
a slice which is eta
equal to 0, which is just here.
So with phi R is essentially
just phi for x greater than 0
and the T equal to 0.
This is the right
half of the real axis.
So this is a Hilbert
space of the Rindler.
And, of course, we can also
write down the Hamiltonian
for the Rindler respect to eta.
So this is called the
Rindler Hamiltonian.
And as I said before, you
can quantize this here unit,
this Hamiltonian, then
you will get-- say,
you can quantize
this Hamiltonian
to construct all the excited
states, so to me just label
them by n.
So I say, this complete
set of eigenstate for HR,
so the second
value will say, En.
OK, you can just
quantize it, then
you can find your full state.
In particular, the ground
state, when you do that,
is what we call
the Rindler vacuum.
We said before that if you
just to straightforward
canonical connotation, you
get the Rindler vacuum, which
is different from the vacuum
which we identify theta by 2pi.
And this we just straightforward
quantize respect to eta.
So now let's go back.
So this is a structure
of the Hilbert space
in some sense for this Rindler.
So now let's look at the
Hilbert space for the Minkowski.
So Minkowski is defined--
by Minkowski, I mean,
the theory defined for
the whole Minkowski
spacetime is this T. Quantize
with respect to this T.
So this would be just
standard psi phi x.
Now this x can be
anything, and then
we also have a Minkowski
Hamiltonian defined
with respect to T,
to capital T. OK,
so this is the standard of
the Minkowski Hamiltonian.
And then the vacuum, which
I denote to be the Minkowski
vacuum is M. And then vacuum
functional phi x is just phi
x 0 M.
So now the key
observation here is
at phi x, which this phi x
is for the full real axis,
for the full horizontal axis.
So this is in some
[? contained ?]
phi Lx and the phi Rx.
This goes one into
the variable of phi.
You [? variate ?] it
to the right patch,
and the value you've
added to the left patch.
So the space of phi x,
it's the combination
of space of phi L x for the
left part and space for the phi
right part.
In particular, this tells us
the Minkowski Hilbert space,
which is defined as a
functional of this full phi x,
should be the tensor
product of the Rindler
to the right tensor product
of the Rindler to the left.
Notice that we talk
about this right Rindler,
but there's a similar
Rindler to the left,
but the structure of the
Minkowski space or Hilbert
space, it's the tensor
product of the two.
In particular, this ground
state wave functional,
you can write it as
phi Lx and phi Rx.
So this ground
state way functional
should also be understood
as the functional for phi L.
OK, so now here is my last key.
And here is last key.
So remember, the ground
state wave function
will be obtained by doing path
integral on the Euclidean half
plane.
So T is smaller than 0, OK?
So let me just draw the
Euclidean space again, X, TE.
So the wave function
can be written
as define [? tE ?] x [INAUDIBLE]
minus SE Euclidean action.
And the lower half plane
of the Euclidean space
with the boundary
condition-- OK,
remember from path
integral, that's
how you obtain the-- so that
means you integrate over
all this region, path integral
over all of this region.
Then with boundary
condition fixed at here,
we do this path integral, you
get the Minkowski vacuum wave
function.
Now here is the key.
So this half space, when
I write in this form,
you treat this T as
the time, but now let's
consider a different foliation.
I don't have colored chalk here.
I can consider a
different foliaton.
It's that foliation
in terms of the theta.
Then you just have
this foliation,
so for each value of
theta, you have a rho.
So integration for
minus theta to theta.
So if you think about
from this foliaion-- so
think about from this foliation,
then this path integral
can be written as a
following, D phi theta
rho, which you fix phi
theta equal to minus pi rho.
Rho here when theta equal
to minus pi becomes x,
you could do phi Lx.
And the phi theta
equal to pi-- theta
equal to 0rho equal to phi
Rx and exponential minus S,
so these two should be the same.
I just chose the
different foliation
to do my path integral.
But now if you think
from this point of view,
this a precisely like that
because theta 0 and theta
minus pi are just two
different times in terms
of this Euclidean Rindler time.
So this you can write it as phi,
so let me write one more step.
So this can written
as phi2 [INAUDIBLE]
minus iH t2 minus t1 phi1.
So this last step
can be now written
as I can think in
terms of the time
within the Rindler
of time, this can
be written as phi R
exponential minus i minus i
pi because we have a Euclidean
time H Rindler Hamiltonian phi
L, OK?
So this tells me that
Minkowski wave functional
can be written as
the following-- can
be written as phi R
exponential pi HR phi L.
So now let me expand this in the
complete set of states of HR.
Then this is sum of
n, so if you expand
in terms of a complete
[INAUDIBLE] to HR,
then this is, [? say, ?]
wave function.
So with this chi n equal
to chi n phi equal to phi.
So remember, this n is defined
from this Rindler Hamiltonian.
Now let me erase this.
So I can slightly rewrite
this because I don't
like this complex conjugate.
I can slightly rewrite this.
It's thi0 phi x is equal to sum
over n [? and then ?] minus pi
En chi n phi R and
the chi and tilde
phi L. So chi and tilde define
to be phi and star phi L.
So this chi and tilde
can be considered--
just like in ordinary
quantum mechanics,
this can be considered to belong
to a slightly different Rindler
Hilbert space whose time
duration is opposite.
So when you switch
the direction of time,
you put a complex conjugate.
So this complex
conjugate can also
be thought of just as a wave
function in the Hilbert space
with the opposite
time direction,
with opposite to the
Rindler we started with,
to the right patch Rindler.
So now this form is exactly
the form we have seen before.
In this [INAUDIBLE] way to
think about thermal state.
So this is exactly that
because this tells you
that the Cartesian in
the Minkowski vacuum
is the sum of
expression minus pi
En with n Rindler of 1 patch
[? tensor ?] and Rindler
of the other one.
So this or the
other Rindler should
be considered
Rindler of the left
except we should quantize the
theory with the opposite time
direction, which gives you this
Rindler tilde Rindler left.
So this is just precisely
the structure we saw before.
In particular, if
you are ignorant
about the left
Rindler, you can just
trace it out, trace out the left
part of this ground state wave
function.
Then, of course, you
just get rho Rindler,
the thermal density
matrix in the Rindler.
And this is equal to, of course,
1 over Rindler exponential
minus 2pi HR.
And we see that the
beta is equal to 2pi,
but this is the beta
associated with eta.
As we said, when you change
to a different observer,
a different rho, then you have
to go to the local proper time.
But this is precisely
what we found before.
So now we understand that the
thermal nature of this Rindler
observer is just precisely that
because this observer cannot
access the physics
to the left patch.
But the vacuum state of
the Minkowski spacetime
is an entangled state between
the left and the right.
And when you trace
out the left part,
then you get a similar
state on the right part,
so that's how the
temperature rises.
And the same thing can be
said about black hole, which
we'll say a little
bit more next lecture.
And let's stop here.
