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HONG LIU: OK, let us start.
Welcome, everybody.
So first, let me just say a
few general things regarding
the logistics of this class.
So you should have
these three handouts.
One, organization.
One, outline.
And then, one is for PSET 1.
So let me just go over
briefly this organization
and the outline.
So firstly in this class,
there's no textbooks.
However, there are
many reviews available.
But unfortunately, a
lot of those reviews,
none of the single review is
suitable for the whole course.
Yeah.
So I have listed
some of the reviews
at the end of this organization.
Yeah.
So along the way, I
will point it out,
some specific references
or some specific parts
of certain reviews if they
are directly relevant.
And also for this class,
there's no recitation.
Yeah, for advanced
courses like this,
department does not assign TA.
So we don't have recitation.
So I have my own office hour,
which is Monday 3:30 to 4:30.
Just one hour before
the lecture on Monday.
Yeah.
So if you have questions
or if you have-- yeah,
just feel free to come.
So if you have some
questions other time,
say, if this office hour
is not good for you,
I'm also happy to set up
some other time to meet you.
Or, you can stay after the class
to ask questions, et cetera.
Yeah.
So whichever way is
more convenient to you.
So today, we have this--
yeah, so form now on,
everything will be
just on the web.
You should download
the future PSETs.
Or if I put any lecture
notes or materials,
you should find them on the web.
And I assume you can all find
out the website of the course.
And it's also listed here
in the organization page.
In the organization page.
So any questions so far?
Good.
So this class there's no exam.
And the grades are solely based
on PSETs and the final project.
And the PSETs will be
due every two weeks.
So if you look at the calendar,
there are about like 5 PSETs.
Then, for the last
three weeks, instead
of, say, handing a PSET, you're
handing in the small paper.
Just whatever topics
you would like to choose
about the holographic duality.
Yeah, just choose one topic.
Write, say, 8 to 10-page paper.
Either review or
some calculation
you would like to do, et cetera.
Yeah.
So 5 PSET is 75%.
And this final
project would be 25%.
That's pretty much the
logistics regarding this class.
Do you have any
questions regarding
the grading, regarding the
project or PSET, et cetera?
Good.
So let me say a few
words about this outline.
So this outline is
way too ambitious.
It gives you--
yeah, it just serves
to give you a rough
contour what we will cover.
But really, don't take it too
literally because-- first, we
are not going to cover
all seven chapters.
That would be too much.
So last time when I taught it,
I covered only three chapters.
And so hopefully, this
time I will do four.
Hopefully, time I will do four.
And I may also just deviate
from some of the things
here, depending on the pace or
depend on people's interests,
et cetera.
So this is going to be flexible.
If there are certain things
you would really very much like
to hear, you can
also let me know.
I can think about it.
In particular, among
the five, six, seven.
And I can think about it, maybe
I'll discuss some of them.
Discuss some of them.
Yeah.
Also, another thing if you keep
in mind is that in this class,
this is a class which will touch
on many different subjects,
say gravity, a little
bit of string theory.
And also, quantum field
theories, et cetera.
So in such a class, it's
almost impossible-- I
think it's just impossible
to really derive everything
in the self-contained way.
And also, some of-- yeah, I
think some of your background
are also very different.
So the certain
part may not appear
self-contained to some of you.
So if there is anything
which is not familiar to you,
if I just mention it,
if I just code it,
if it's not familiar
with you, then
you make sure-- don't
hesitate to ask me,
either during the class
or after the class,
so that I can help you
to make up on that.
To make up on that.
And also, to keep in mind,
like other advanced graduate
classes, you actually mostly
learn outside the class.
Inside the class,
it's mostly try
to provide you some motivations,
guidance, et cetera,
and give you a rough contour.
And your own reading
outside the class
actually should be the main
routes you learn things.
The main route you learn things.
So any questions?
So during the class, please
feel free to ask questions.
Yeah.
Asking questions
are a great thing
because if something
you don't understand,
there's a very good chance quite
a few of your fellow students
also don't understand.
So if you ask the
question, you're
not only helping
yourself, you're
also helping other people.
And you also help
me because then I
will know certain aspect
is not clear to you.
Then, I will try to emphasize
it, or repeat it, et cetera.
So please, do ask questions.
And also, if you have any
feedback on the class,
whether it's too fast, too slow,
or problems are too hard or too
easy, the PSET is
too short for you,
just not challenging
enough, just let me know.
Just let me know.
Yeah.
Then, I will try to adjust.
And again, this is
a very important.
Again, this is very
important because the-- yeah.
This will help us to
improve the class.
To improve the class.
Any other questions?
Or any questions?
OK.
Then, let us start our lecture.
So first, we look at the hints
for the holographic duality.
OK.
So we start by doing a
multiple choice problem.
OK.
So what is your answer?
AUDIENCE: [INAUDIBLE]
HONG LIU: Good.
So why is gravity?
AUDIENCE: We don't have a good
quantum theory of gravity.
HONG LIU: Good.
So we don't have a good
quantum theory of gravity.
Any other reason?
AUDIENCE: Except string theory.
HONG LIU: Yeah,
string theory is still
not the-- I would say not yet
at the [INAUDIBLE] quantum
theory of gravity.
AUDIENCE: [INAUDIBLE]
HONG LIU: Yeah.
Yeah, this is the
similar to that.
We don't have a full theory
of gravity nonrealizable.
AUDIENCE: [INAUDIBLE]
HONG LIU: Good.
Anything else?
Or any other opinions?
Say you should choose any other
interaction other than gravity?
AUDIENCE: Strong is
the strongest one.
Strong is the strongest one.
HONG LIU: That's right.
You can also choose the gravity
because this is the weakest
one.
Yeah, so the gravity
is the weakest one.
OK.
So yeah.
So I think you have
said most of the things.
And the gravity is very
different from the others.
And in some sense, all
the others are understood.
So a to c are now
understood to be
described by gauge theories
in the fixed spacetime.
So the fixed spacetime.
In fact, mostly
we use Minkowski.
Minkowski.
And for example,
electromagnetism, [INAUDIBLE]
by QED.
And the QED [INAUDIBLE]
gauge field.
You want gauge theory
plus a Dirac fermion.
Dirac theory.
A Dirac theory.
And similarly, if you include
the weak interaction--
so there's something
called [INAUDIBLE] weak--
it's described by su2
times u1 gauge theory.
Described by su2
times u1 gauge theory.
And also, the strong
interaction you know now
is described by
su3 gauge theory.
So for all those cases, the
basic theoretical structure
is understood.
So the basic.
OK.
In principle, we can
formulate those theories
from first principle,
just use path integral.
Say, [INAUDIBLE]
where [INAUDIBLE]
group just based on
principle, relation group.
And then you can actually
formulate all the theory
in the [INAUDIBLE] way.
So in the sense, then any
calculation of theories
can be reduced to
certain algorithm.
OK.
Of course, this does
not imply we actually
know how to calculate
many quantities here.
It just say in principle,
they are calculable.
We often don't have
the technical tool
to calculate them.
For example, even in QCD.
[INAUDIBLE] joint action, we
don't have the technical tool
to calculate certain things.
But they are, in
principle, calculable.
And we can write down the
algorithm to calculate it.
But for gravity,
this is different.
So for gravity,
it's very different.
So for gravity.
So now we understand
that classical gravity,
it's just equal to spacetime.
It's just a theory of spacetime.
So this is the content over
the general relativity.
The general quantum activity.
But at a quantum
level, we really
don't have a precise idea
how to formulate the theory.
So there are many questions
still not understood.
Many, say, conceptual
questions still not understood.
For example, a spacetime
should fluctuate.
OK.
The spacetime should
fluctuate because
of the quantum
fluctuation [INAUDIBLE]
in quantum anything fluctuates.
And then, the natural question
is whether this is still--
so natural question is whether
spacetime is still continuous.
OK.
So this is still [INAUDIBLE].
You need to replace it
by something discrete.
And it's not even
clear right now
whether spacetime will remain
a fundamental notion when
you go to quantum gravity.
It could well be
that spacetime would
be replaced by something else.
And also, there are
many other questions.
So for example, quantum
nature of black holes.
Or what should be the beginning
of the universe, et cetera.
So all these questions
are not understood.
Also, gravity is the
weakest interaction.
It is actually
much, much weaker.
We will see, it's
much, much weaker
than all the other interactions.
And also, people have been
speculate-- maybe actually
this feature can also be
actually a fundamental feature
underlying in the special
role of the gravity.
For many years, it
seemed to people
that these questions and
the gravity questions,
they're completely unrelated.
They're completely unrelated.
They're a completely
different subject.
And those are well understood,
but somehow those we
need completely new ideas.
So it came out a great surprise.
So it came out that
great surprise in 1997
by [INAUDIBLE] that actually
they actually are equivalent.
So the proposal
of the [INAUDIBLE]
is that the-- chalk.
Is that quantum
gravity is actually
equal to field theories
on a fixed spacetime.
So on a fixed
spacetime just means
that there's no gravity
here on the right-hand side.
Because if there's gravity,
then you cannot have a fixed
spacetime.
And spacetime
should be dynamical.
So this is the equivalence
between quantum gravity
and just our ordinary
quantum field theories.
And this is ordinary
quantum mechanics.
And this is a very
nontrivial quantum mechanics,
which we don't yet understand.
OK.
This equality should
be really understood
as the different descriptions
of the same quantum systems.
So two sides are
different descriptions
of the same quantum system.
So it just depends on
how you want to see it.
So one way you see it,
you see a quantum gravity.
Then from some other
way you see it,
you don't see gravity at all.
You just see ordinary
quantum mechanics.
OK.
Yeah, so this is what this
equality is supposed to mean.
So this equality essentially--
let me call this equation 1,
which I will often use later.
So this equation 1 is
really a unification.
Unification, which
unifies quantum gravity
with other field theory in
a rather unconventional way.
Not in a standard
sense of unification.
But that's bringing
them together.
OK.
So now, let me make some
philosophical remark.
Now, let me make some
philosophical remark.
So last time, I
have checked when
preparing this class this
[INAUDIBLE] original paper
has been cited more
than 10,000 times
in the [INAUDIBLE] Database.
And so there's a
huge amount of work
has been devoted to
understanding this relation.
But in my opinion, this
subject's still in its infancy.
And with many
elementary [INAUDIBLE]
we will see in the
course as course goes on,
there's still many
elementary issues
which are not understood.
And to us, this relation
is still like a magic.
And we don't really
have very good idea
where this relation comes home.
We have some rough idea,
but not very precise idea.
So the purpose of
physics, of course,
is to turn what looks like a
magic or art into some rules
so that it becomes
something trivial.
So at the beginning of
course of the gravity,
Newtonian gravity was magic.
But in the end--
and what Newton did
is to make the gravity to be
not different from the force we
see in ordinary life.
And similarly, for other--
yeah, for other interactions
we have seen before.
Anyway.
So personally, I believe when
we really fully understand
this relation,
this will really be
a huge landmark in the physics.
And comparable, say, maybe
to Newton's understanding
of gravity or
Maxwell-Boltzmann, et cetera.
So the goal of this
course is to help
you understand how we--
our current understanding
of this relation.
And also, to help you to
derive the related-- yeah,
so these are two very
different objects.
And in order to set
up an inequality,
you also need to
define a dictionary.
For example, this
is like Chinese.
This is like English.
So you need to define a
dictionary between the two
and what one said to the other.
So we will also work
out the dictionary.
And also, we will
develop with the tools
and how to use this relation.
And how to use the relation.
And also, discuss many
features of this relation
and the implications.
So the purpose is to
help you to-- really,
to bring to the forefront of
this very exciting subject.
So any questions?
Good.
No questions?
OK, so now let's move to--
so after this introduction.
So let's talk about this.
OK.
So if you look at this relation.
So looking from the
right-hand side.
So you just have some kind
of ordinary field theory--
ordinary quantum
mechanical system, which
we, in principle, know
very well conceptually,
without any gravity at all.
Somehow, if you view it in some
different way than the gravity
and the spacetime, and
the dynamical spacetime
should emerge out of this.
So in some sense, this 1 implies
that the quantum gravity-- so
let me just use
the quantum gravity
plus the dynamical spacetime.
OK, dynamical spacetime.
So it's important it's
a dynamical spacetime--
can really merge from
non-gravitational system.
It can emerge from a
non-gravitational system.
So the idea of emergence of
gravity from some other degrees
freedom is not a
really new idea.
So [INAUDIBLE] in
1967, Sakharov,
who was a Russian physicist.
Sakharov is a very common name,
a very common Russian name.
But this is the most famous one.
This is the most
famous Sakharov who
has invented the hydrogen bomb
for the former Soviet Union.
And he was also the guy
who also later, I think,
got Nobel Peace Prize for
some human rights stuff.
But he was also a very excellent
physicist, including this idea
of emergence of the gravity.
So what he observed
is that-- he found
actually certain [INAUDIBLE].
He would study, say, some
materials, et cetera.
Then, he finds that
certain [INAUDIBLE] systems
can actually have
mathematical description which
looks like a magic
connection, et cetera.
Yeah, just
mathematical, it looks
like the equation for the
magic and the equation
for the connection, et cetera.
And then he was speculating,
maybe the general relativity
can actually arise just from
some kind of electron systems,
just from the ordinary
[INAUDIBLE] systems,
and as effective description.
And in fact, in the 1950s,
even before the Sakharov
have [INAUDIBLE] idea,
people working in GR,
in General Relativity,
they already
found many striking parallel
between Einstein's equations
and also associated many
phenomena we hydrodynamics.
And so we know that
hydrodynamics is really
just an effective description.
And if you just look at
the river-- so in fact,
it's the discrete
water molecules.
But if you describe the
motion of the water molecules
at a microscopic level, then
you find the hydrodynamics.
But the hydrodynamics
does not apply
at more fundamental levels.
So it's more like some
kind of effective theory.
So in the 1950s,
people already found
that there are many features
of Einstein equation--
it's reminiscent.
Actually, just reminiscent
of certain features
of hydrodynamics.
So there's already
this suspicion
that maybe the gravity
or spacetime actually can
emerge out of something else.
Just like the hydrodynamics
is not a fundamental theory.
It actually emerges
from molecules,
dynamics of the molecules.
Anyway.
So from the field
theory perspective,
it's also natural to ask whether
massless spin-2 particles can
arise as bound state, say,
of lower spin particles.
Say, like spin-1.
Like photons, which are spin-1.
So photons.
Or gluons in QCD.
Electrons, or quarks, et cetera.
So these are the spin
[INAUDIBLE] objects.
And if yes-- so massless
spin-2 particle in some sense--
so when you learn
QR, you might have
learned that the massless spin-2
particle may be considered
as a hallmark form of
gravity because of the-- so
the basic propogation.
The gravitational wave.
So the basic propogation
of the gravity
is the gravitational wave.
And that's propogated
as spin-2 particles.
And if you quantize the theory--
a propogation as spin-2 object.
And when you quantize
the gravitational waves,
then they become massless
spin-2 particles.
They become spin-2 particles.
So in some sense,
if your theory can
allow the massless
spin-2 particles
as-- so if your theory contains
the massless spin-2 particles,
then they must contain gravity.
OK.
So for example, even in strong
interaction, which are, say,
the theory describes
gluons and the quarks.
And the gluons and the
quarks, they can indeed form.
So these are the
gluons and the quarks.
In [INAUDIBLE], they're
indeed spin-2 bound states.
The gluons are spin-1,
quarks are spin [INAUDIBLE].
But you combine
them together, you
can make [INAUDIBLE] the
spin-2 bound state, even just
in strong interactions.
But of course in nature,
the spin-2 bound state,
they're all massive.
They're all massive.
And there are some very
unstable, massive particles.
When you create them,
immediate decay.
So they cannot be gravity.
For gravity, you need
massless spin-2 particles.
But if you look at this fact,
you cannot have the feeling.
Say, maybe I take the QCD.
Maybe I take a little bit.
Maybe I can make that
spin-2 massive particles
into some massless
spin-2 particles.
Then I would have generated the
gravity from a QCD like series.
And then, that would
be a revolution.
And then you will be immortal.
But unfortunately,
even though this
was a very promising
idea for many years,
and this hope was
actually completely dashed
by a powerful theorem
of Weinberg and Witten.
So there's a powerful
theorem from Weinberg-Witten
that say this can never happen.
So this is the--
they're not possible.
So this is 1980.
So the paper.
If you want, you can
take a look at the paper.
So this is volume
96 and the page 59.
So they proved two
theorems in that paper.
So let me just copy
the theorem down.
Copy the theorem down.
So they say a theory that
allows the construction
of a Lorentz-covariant
and the conserved current
for a vector-- current.
Say, j mu, cannot contain
massless particles.
Cannot contain massless
particles of spin greater than
1/2 with non-vanishing
value of-- OK,
so this is first theorem.
So the second theorem-- I think
I should have enough space.
So the second theorem.
Second theorem says
that a theory that
allows a covariant,
Lorentz-covariant,
and the conserved stress tensor.
So let me just call it T mu mu.
A stress tensor cannot contain
massless particles of spin j
greater than 1.
So the key-- so
let me just repeat
the theorem a little bit.
So the first theorem said if
you have a Lorentz-covariant
and the conserved current,
and in such a theory
there's no charged particles
can have spin more than 1/2.
So there's no charged particle
can have spin more than 1/2.
Of course, this cannot
contain graviton.
And the second theorem says
if you have a conserved stress
tensor, Lorentz-covariant and
a conserved stress tensor,
then this theory cannot contain
any particles with spin greater
than 1.
So of course, this cannot
also contain a graviton.
Graviton would be a spin-2.
So this theorem-- yes?
AUDIENCE: Is there anything
like RG flow of mass?
[INAUDIBLE]
HONG LIU: This is final mass.
Yeah.
Just the [INAUDIBLE] mass.
You don't talk about
even normalization here.
Yeah, this is just
[INAUDIBLE] statement.
So this theorem turned out to
be actually very easy to prove.
And in some sense, it's
rather instructive.
And just to give you the sense
of the power of this theorem,
we will prove it
in a little bit.
So before proving
it, I will first
make some remarks to
make you appreciate
what these two theorems means.
Do you have any questions
before I do that?
OK, good.
So let me just make some
remarks on this theorem.
So the first remark is that
the theorems really apply
to any particles, to both.
So normally we say the
fundamental particle
is the particle which
you put in your theory is
the particle which say--
yeah, let me say [INAUDIBLE].
Rather fundamental, let
me call it elementary.
So the particle which already
appear in your Lagrangian.
And composite particles.
Composite particles, say, would
be some kind of bound state
or some particles
which don't appear
in your original Lagrangian.
So this theorem does
not distinguish them.
As far as you have some
particles, [INAUDIBLE] region.
So now, let's try to see whether
this theorem is compatible
to [INAUDIBLE] these
things we already know.
So let's first try to
apply to, say, QED.
So the theorem is
compatible with QED.
So we know that QED is a Maxwell
theory plus Dirac's theory.
And then the Dirac-- and the
fermions in the Dirac theory
interact with the photons
in the Maxwell theory.
So this theory have a
massless photon of spin-1.
And this will have a conserved
charge because of, say,
electrons that do have a charge.
This theory has a
conserved current.
Has a conserved current.
But this theory does
allow a spin-1 particle.
It does allow the
spin-1 particle.
And this is not contradictory
with the theorem 1.
So theorem 1 said if you
have a Lorentz [INAUDIBLE]
and the conserved current,
then you cannot have massless
particle of spin 1/2 with
non-vanishing charge.
In this case,
photon is massless.
But photon does
not carry charge.
Photon is neutral under
the electric charge.
So the existence
of the photon is
compatible with that theorem.
And now, let's look at
the [INAUDIBLE] theory
which also contain
massless spin-1 particle.
And the [INAUDIBLE] theory which
contain the spin-1 particle
is the Yang-Mill theory.
So Yang-Mill theory.
For example, let's consider
su2 Yang-Mill theory.
So the Yang-Mill theory have
three gauge field, A mu a.
And a is equal to 1, 2, 3.
Because 1 and 2 [INAUDIBLE]
of su2 gauge symmetry.
And from this three
gauge field, you
can combine them into
the following form.
I have A mu 3.
Then, I have A mu plus minus.
You could do 1 over square root
2 a mu 1 plus minus A mu 2.
OK.
So [INAUDIBLE].
So you can easily check yourself
that this a mu plus minus
is massless spin-1 particle.
Particles.
Can create massless
spin-1 particles
charged under the u1 subgroup
generated by sigma 3.
So sigma 3 is the generator
associated with A mu 3.
And so those two are charged
under this generator.
So now, [INAUDIBLE] we
have a contradiction
because those are massless
spin-1 particles charged
under this.
And the theorem 1 said you
have a conserved current,
then you cannot have
any particle recharged--
any charged particle with
spin greater than 1/2.
But those are the
spin-1 particles.
But actually, we're consistent.
Because as you may remember,
in Yang-Mills theory
there does not exist-- that
actually does not exist.
The way out [INAUDIBLE]
there does not
exist a conserved
Lorentz-covariant current
for actually-- for this u1.
For the u1 generated by
this sigma 3 divided by 2.
So this is actually compatible.
This is actually compatible
because such a current does not
exist.
OK.
So you will show this fact
yourself in your PSET.
But you may
remember, if you have
studied quantum
field theory 2, then
you may remember this
fact from the discussion
of [INAUDIBLE] gauge theories.
So in this course, we
will from time to time
use various facts of,
say, [INAUDIBLE] theories.
But they're not essential.
If you don't allow
[INAUDIBLE] theories,
you can still
understand this duality.
You can still understand
the relation one.
But of course, if you know gauge
field, it will help you a lot.
Any questions about this?
Good.
Let's continue.
So this theorem does not
forbid graviton from GR.
So if you try to quantize
Einstein's general relativity
around the flash
space for example,
then you will find the
massless spin-2 particle.
So this is what
we call graviton.
And this is not contradictory
with that theorem 2.
It's because you
may still remember
when you learn GR is that in
GR, General Relativity, actually
there is not.
There's no conserved.
So here, it's conserved.
Not covariantly
converged, just conserved.
Lorentz-covariant stress tensor.
OK.
So in GR, actually
there's no conserved
in the Lorentz-covariant
stress tensor.
So actually, the GR
[INAUDIBLE] graviton.
But this theorem 2, it's
nevertheless, very powerful.
It's nevertheless very powerful.
It said very powerful.
It said all [INAUDIBLE].
None of [INAUDIBLE] QFTs
in Minkowski spacetime,
which is the one we
normally work with,
can have emergent gravity.
OK.
So QCD like theories,
no matter how
you try to trick the theory,
the spin-2 massive bound state
can never become massless.
They can never become massless.
Because those series,
they do-- for example,
our QCD, our strong
interaction, they
do have a covariantly
conserved stress tensor.
A Lorentz-covariant
conserved stress tensor.
But this theorem
does not validate
this equation 1 I just erased.
Because this theorem, even
though it's very powerful,
it does have a
hidden assumption.
It does have a
hidden assumption.
This assumption is so
obvious and so self-evident,
that even though
Weinberg and Witten,
they're extremely
careful people,
they did not bother
to mention it.
In those times, nobody
with their right mind
would have mentioned
such a thing.
So the hidden assumption is
that the kind of particles
they are talking about-- is that
the particles are they talking
about, the particles, which
they tried to rule out,
live in the same spacetime
as original theory.
OK.
So if you write down,
say, QED in Minkowski
spacetime, and
then you ask about
whether you can have some spin-2
particles coming out from QED.
And then of course,
you ask about
whether you have
particles come out
QED living in the same
Minkowski spacetime.
Yeah.
So in some sense, this
assumption is self-evident.
But precisely,
this is assumption,
which taking advantage by
this relation, which first
envisioned by [INAUDIBLE],
is that precisely
in this relation, in this
equation 1 which I raised,
that the gravity
actually does not
live in the same spacetime
as the original theory.
And does not live in the same
spacetime as original theory.
So even in the early
days, when people
were trying to dream having
such kind of spin-2 particles
that come from
QCD like theories,
there's already an
immediate puzzle.
Suppose the QCD like theories
can have emergent graviton
in the same spacetime.
Then firstly, your QCD theory
is defined in a fix spacetime.
But now if it contains spin-2
particle in the spacetime
itself, it will be dynamical.
Then, where does the original
QCD theory [INAUDIBLE]?
OK.
And then you will
just similarly go
into this kind of
self-contradiction,
like somehow-- how
can that-- yeah.
Anyway.
So in this relation, in
the holographic duality,
gravity does not live
in the same spacetime.
OK.
So this, in fact, live in
one dimension or higher,
or they can be several
dimension higher depending
on the examples.
So this void this
theorem in obvious way.
And also, void this
conundrum I said earlier.
Because in a
different spacetime.
So the field theory is still
in the fixed spacetime,
but then the gravity
then can be dynamical.
Then, the spacetime which
gravity lives can be dynamical.
So any questions
regarding this so far?
Yes.
AUDIENCE: [INAUDIBLE]
in GR, there
was no conserved
Lorentz-covariant stress
tensor.
Because the covariant
derivative of the stress
tensor [INAUDIBLE].
HONG LIU: No covariant
derivative is not conserved.
This covariant is not conserved.
Conserved means the
ordinary derivatives, not
the covariant derivatives.
A covariant conserved stress
tensor is not conserved.
If you write down a
conserved stress tensor,
then it's not covariant.
Then it's not covariant.
Yeah.
Similar thing with this
case for the gauge fields.
And in this case, gauge
invariant currents--
the gauge invariant and
the conserved currents
is not Lorentz-covariants.
And the Lorentz-covariants,
gauge invariant current
then is not conserved.
Any other question?
AUDIENCE: I have a question.
HONG LIU: Yes.
AUDIENCE: [INAUDIBLE]
HONG LIU: Sorry.
Say it again?
AUDIENCE: [INAUDIBLE] theory,
every particle is massless.
HONG LIU: Yes.
AUDIENCE: And the [INAUDIBLE]
spin-2 particle, which carries
[INAUDIBLE]
HONG LIU: Yeah.
For [INAUDIBLE] field theory,
indeed everything's massless.
You can construct
Lorentz-covariants
conserved stress tensor.
So that theory [INAUDIBLE] does
not allow spin-2 excitations
in the same spacetime.
AUDIENCE: [INAUDIBLE]
HONG LIU: There's no
massless spin-2 particle.
AUDIENCE: [INAUDIBLE]
HONG LIU: Yeah.
Yeah, but you don't
have massless spin-2.
You don't have massless
spin-2 particles.
AUDIENCE: [INAUDIBLE]
HONG LIU: No.
It depends.
Actually, talking about the
massless in the [INAUDIBLE]
field theory is a
little bit tricky.
It's a little bit tricky.
They come in all
kind of spectrum.
They come in the
continuous spectrum.
Yeah.
Yeah, but this theorem does
apply to ordinary [INAUDIBLE]
field theories.
So in those theories,
you won't have massless.
Yeah, when we prove
the theorem, then you
can see precisely
what do we mean
by massless spin-2 particles.
If you stay in the course,
then you will know.
Yeah.
Yeah, we will see it later on.
But not for a while.
Not for a couple of weeks.
And maybe in a couple of
weeks, we will see it.
In two weeks.
One or two weeks.
Any other questions?
OK.
So now, let me give you the
proof, which is pretty simple.
And use some elementary
facts, very elementary facts.
Because you see, this theorem
contains very little input.
That theorem really contained
very, very little input.
So if you can
prove such a thing,
it cannot be complicated.
If you can prove it, it
cannot be complicated.
Because the input is so little.
It does not require any details.
It does not require any
details of a theory.
So if you can prove
such a theorem,
it must be just from kinematics,
not come from any dynamics.
So indeed, that's
how the theorem goes.
So the proof.
So we will do proof
by contradiction
as you would naturally expect.
So let us suppose there exists
a massless spin-j particle.
Massless particles.
So let's assume
this theory contains
some theories [INAUDIBLE] exist
massless spin-2 particles.
Massless particles of spin-j.
And just from the
Lorentz symmetry,
you can immediately
write down-- so
from what we learned
in QFT 1 and 2.
So just by Lorentz symmetry--
it doesn't matter what
is the nature of those states.
Because the mass
formal representations
over the, say, Lorentz
group and [INAUDIBLE] group.
So [INAUDIBLE] immediately write
down the one particle state
of such particles.
And you normally can
write them as k and sigma.
So k just is the ordinary
spacetime momentum.
So for massless
particle, of course
the k0 squared should
be equal to k squared.
And the sigma is the helicity.
Helicity is a projection
of the angular momentum
in the direction
of your momentum.
And for a massless
particle of spin-j,
the sigma can only
be plus minus j.
So this is helicity.
So if you don't
remember this, you
can go back to review
Wemberg's book.
Wemberg's volume 1 QFT.
Volume 1, Section 2.5
discussed how you write down
the general one particle
states for, say,
any quantum-- in
any quantum theory
as far as you have
Lorentz symmetry.
OK.
And the one properties
of this sigma
is that let's now
consider rotation.
Now, let's consider rotation.
So let's consider
rotation operator.
This [INAUDIBLE] by angle theta
around the direction of the k.
Around the direction of the k,
which is the direction of the k
here.
So let's consider
the action of these
under state, this one particle--
this massless particle state.
Then by definition,
this just give you
i sigma theta k, sigma.
So this is essentially come from
the definition of the helicity.
And the k does not change
because this is the rotation
along the direction of the k.
The k itself does not change.
But this eigenstate of
this rotation [INAUDIBLE].
OK.
Again, if you don't remember
this, go back to Wemberg's book
and you can easily find it.
So now if you have
a conserved current,
if you have a covariant
conserved current,
J mu, then you can
always construct
the charge, the
conserved charge,
which is the integral over J0.
And similarly, if you have
T mu mu, then from T mu mu,
you can construct a momentum
operator, which is the T mu.
So this is just the
standard momentum operator
in your field theory.
So this is just still the setup.
Of course just by definition,
the p mu acting on k sigma,
you just get back k mu.
OK, because this is the
momentum eigenstate.
And if this particle
is also charged
under this conserved charge,
then-- so if charged under k,
under this conserved charge,
then Q act on this state
will also give you the charge q.
So this is operator.
This is operator.
These are all operator.
Operator.
And then, this should
give you charge q.
So now we want to show
this 2 theorem implies--
so with this set up,
this theorem can--
so we want to show two things.
1, if q is nonzero, that means
if the particle is charged
under a conserved charge, then
j must be smaller than 1/2.
It cannot be greater than 1/2.
So we assume it's a
conserved stress tensor.
Otherwise, j should be always
smaller or equal to 1 at most.
OK.
So that's what
that theorem means
translated in this language.
OK.
Any questions?
So now, let's prove it.
Let me just do it here.
So now, let's prove it.
I think we just have enough
time, maybe to prove it.
So to prove it, let's just first
list some elementary facts.
First, just do some
elementary fact.
First, from Lorentz
symmetry-- we
do have to assume
Lorentz symmetry.
OK.
You can show-- so this I will
just write down the answer
and leave yourself to show it.
OK?
In your PSET.
So if you look at the
matrix [INAUDIBLE]
of this conserved current
between two such states
of different k, in the limit
which k goes to k prime,
you have q k mu k0.
And similarly, in
this limit-- so this
we will do it in your PSET.
So let me call this
equation 2, equation 3.
So this is based on the
following normalization.
Based on the following
normalization between the two
states.
OK.
So this relation is actually
very easy to see intuitively.
Yeah, you can
prove it by-- yeah,
let me give me just a quick
hint for [INAUDIBLE] regarding
2, is that you can first
show that k sigma J0
sandwiched between the
two, in the limit when
this k become the
same is just given
by q divided by 2 pi cubed.
So this is, again,
somewhat intuitive
because k is eigenstate
of Q. Because the k is
an eigenstate of this Q hat.
And Q hat is just
the integral of J0.
It is integral of J0.
And so you can imagine
on the right-hand side,
there must be a Q because these
are essentially the eigenstate
of the integration with J0.
And then when you take into
account all these [INAUDIBLE]
functions here, and the
integral-- and the relation
between the integral,
then you will naturally
find the right-hand side
is just q divided by this.
So once you know this,
then you can immediately
see the first [INAUDIBLE]
equation 2 there.
Because at first,
there's a j mu there.
Then the right-hand
side must have index mu.
So the only thing can
carry index mu is the k mu.
There's nothing else.
And also, you know
when the mu equal to 0,
you should just get the q.
Then, downstairs we might
have a k0 to cancel them.
So this relation,
in principle you
can just-- just
by intuition, you
can just-- by dimensional
analysis, [INAUDIBLE] just
directly write it down.
And similar in the
second relation--
in the second relation,
the analog of q.
So q is the charge
associated with Q hat.
The analog of the q for the
stress tensor is just the k mu.
So you just replace
this by k mu.
It just relates
to the q by k mu.
Then you get the second line.
Because k by k mu [INAUDIBLE].
OK.
So I will leave yourself
to prove this rigorously.
So this is the
second observation.
It said for massless
particles, which
have k squared equal to 0, and
k prime squared equal to 0,
because [INAUDIBLE] particles.
So k squared and k prime
squared should be equal to 0.
And that means for any two
particles, a massless momentum
like this, you see immediately
just by writing this down
this must be smaller than 0.
So this should come from
your high school physics.
The k dot k prime should
be smaller than 0.
And that means that k plus
k prime must be time-like.
OK, must be time-like.
Because this square is
essentially given by this guy
because k square and
k prime equal to 0.
So this guy must be time-like.
OK.
And so that means
you can always choose
a frame that the special
part of this is 0.
We can choose a frame k
plus k prime equal to 0.
So such a frame is
just k mu, say E, 0.
Say we put all of these
in the z-direction,
in the 3-direction.
Then you can choose k
prime k mu equal to-- OK?
And this just follows.
These two are two
[INAUDIBLE] particles.
OK, good.
So now, we can
almost envision this.
Then, there's still a third
elementary observation
we need to put here.
So the third observation
which uses this formula.
So let's now take
sigma just to be j.
So now, under a rotation of
theta around 3-direction--
around this direction, which we
have the [INAUDIBLE] momentum.
So we choose a momentum
to be in this direction.
Then you should
have-- so let me call
that operator to be R
theta is the rotation
in the z-direction,
in the 3-direction.
So this operate on k, j.
So let's take sigma equal to j.
Take sigma equal to j.
So this should be just equal
to exponential i, j theta k, j.
And if this acting on k prime j.
So k prime have opposite
orientation as k
because this is minus e.
And the opposite orientation
means your helicity should also
change sign.
So that means here should be
minus i j theta k prime, j.
OK.
Yeah, because this
is the rotation
along the positive 3-direction.
OK.
So now we are in business.
So now finally, we will try
to-- from those facts together,
we will now see a contradiction.
Now see a contradiction.
Now you can see
that this object.
Consider k, j-- k prime, j.
Say this R minus 1 theta
acting on J mu R theta k, j.
So now, let's
consider this object.
So this object, I can
evaluate it by two ways.
First, I can just act
R theta on both states.
OK.
So R theta on this one
just gives me this.
R minus 1 acting on that just
give me opposite of this.
So all together,
if I do that way,
then I get exponential 2 i j
theta k prime, j J mu k, j.
But I can also act this
operator on this current.
So this current should
transform as a Lorentz vector.
So if I do this way, then I
should get lambda mu new k
prime, j J new k, j.
So these acting on that should
just effect a Lorentzian
summation on j.
And so this is in the
Lorentzian summation matrix.
So lambda mu mu just is the
standard rotation matrix
acting on the vector.
So lambda mu mu just the
standard rotation matrix.
For example, just
given by 1, 0, 0, 0.
Say, cosine theta
minus sine theta.
So just a rotation
along the 3-direction.
So this just the
rotation acting on the j.
So now similarly, if you
examine this quantity.
Yeah, similarly if I just
replace here by T mu mu, OK?
by T mu mu-- we are
out of time, so let me
be a little bit faster now.
Similarly, if we
replace that by T mu mu,
then we will get the same
thing 2 i j theta k prime,
j T mu mu k, j, which obtained
by acting directly on k, j.
By the way, that R
acting on T mu mu, then
T mu mu transform as a 2 tensor.
Then we should get
lambda mu rho mu lambda.
Lambda mu rho k prime,
j T lambda rho k, j.
So now, here's the key.
Now, here's the key.
So do you already see
the contradiction?
So let's look at this equation.
This is just like a vector.
The whole thing
is like a vector.
The same vector.
So this is like an
eigenvalue equation.
So this lambda mu mu
acting on this vector
get eigenvalue 2 i j theta.
So this is like an
eigenvalue equation.
But if you look at
this matrix, this
can only have
eigenvalue exponential
plus minus i theta and the 1.
So this is thing we
are familiar with.
So [INAUDIBLE] minus i theta, 1.
So similarly with this equation,
just you have 2 lambda, OK?
So now, that means
if this quantity
is nonzero, which we know
it is nonzero from 2 and 3.
It is nonzero.
That these 2 i j theta
can only take a value
plus minus i theta and the 1.
So that means that
j-- from here,
let's say equation 4 and 5.
From equation 4, that j
must be smaller than 1/2.
Because there's a 2j there.
Similarly, from equation 5,
means j must be smaller than 1.
So this is what
we want to prove.
So this is what
we want to prove.
Yes?
AUDIENCE: Aren't k and
k prime [INAUDIBLE]?
I mean, 2 and 3, the limit k
goes to k prime and k plus k
prime is 0.
HONG LIU: Yeah.
So I can take them-- if
j is greater than 1/2.
For example in this relation,
if j is greater than 1/2,
then the only possibility
is for this to be 0.
But we know when you take k
equal to k prime, at least
in that limit this is nonzero.
And so this cannot be 0.
So this quantity cannot
be identical to 0.
OK.
So this gives you the proof.
So if you find the
loophole in this proof,
other than they have live in
the same dimension-- yeah,
that would be great.
AUDIENCE: What is the
whole matrix element is 0?
What if the whole
matrix element is 0?
HONG LIU: No, the point is that
the whole matrix element cannot
be 0.
AUDIENCE: Yes.
HONG LIU: Because of this.
AUDIENCE: Isn't the
limit k goes to k prime?
And there, you assume
that k plus k prime is 0.
So it means both k and
k prime should be 0, no?
HONG LIU: I'm sorry?
No.
k plus k prime is not--
AUDIENCE: [INAUDIBLE]
HONG LIU: No, this
is a spatial vector.
AUDIENCE: [INAUDIBLE]
HONG LIU: No, this
is spatial vector.
No, this is spatial vector.
Yeah.
Yeah.
This is spatial vector.
OK.
Sorry, I'm a little bit late.
Yeah.
That's all for today.
