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PROFESSOR: Yeah, so I'm Hong,
so you can just call me Hong.

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I'm a theoretical physicist.

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I work on high energy theory,
including string theory,

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quantum gravity, non equilibrium
statistical physics, et cetera,

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and many different topics.

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So now let me say a few words
regarding the QFT, the quantum

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field theory itself.

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So the goal of this
quantum field theory class

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is to develop concepts,
formalism, and techniques

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for quantum dynamics of fields.

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Say, I'm sure all of you have
studied Maxwell's equations,

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OK?

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So Maxwell's equations describe
fully classical dynamics

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of electric and magnetic fields.

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But we live in
the quantum world,

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and so we should also treat
electric and magnetic fields

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using quantum language, OK?

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And then you find that
once you do that, and then

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the concept of photon, once
you treat the electric field

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and the magnetic field
quantum mechanically,

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and then you get the
concept of the photon.

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And the photon now is
the fundamental particle,

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which mediates electromagnetic
interactions, OK?

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So you actually get a completely
different physical picture

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from what you get classically.

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And yeah, so one
goal of the class

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is for you to appreciate,
say, to understand the quantum

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electrodynamics, OK?

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And that's actually where
we will end this class,

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is we will discuss the
quantum electrodynamics.

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And the quantum
field theory is also

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very important for other
interactions in nature.

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So among the four fundamental
interactions in nature,

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three of them are
described by quantum field

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theory completely, OK?

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And you can also use
quantum field theory

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to describe gravity, to
describe quantum gravity.

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But at the moment,
quantum field theory

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does not offer a complete
description of quantum gravity.

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But still, you can use the
quantum field theory techniques

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for understanding certain
questions of quantum gravity.

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And also, quantum field
theory over the years,

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even though it's initially
developed for particle physics,

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but over the years
has been also found

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many applications in many,
many other branches of physics,

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in condensed matter,
statistical physics, et cetera.

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And it's fair to say,
nowadays, quantum field theory

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has become a universal language,
OK, for theoretical physics,

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essentially many in
all different fields.

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And so if you are
serious, certainly you

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need to master this language.

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And even if you are
only interested, say,

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in atomic physics or
statistical physics,

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the field theory concept
will be very important.

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And for experimentalists to know
basic concept of quantum field

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theory and to have some
basic understanding of it

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should also go a long way to
help you to appreciate, say,

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the most recent theoretical
developments and also for help

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you to communicate with
theorists, and yeah.

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And so this class is the first
of a three-semester sequence.

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And so this semester
will mostly develop

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the fundamental concepts,
and the quantum field theory

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2 and 3 will be more about
technical development.

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And say, if your
experimentalist,

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and then you can view this
quantum field theory 1

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as a stand alone class, which
is just enough for you to get

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the basic idea of quantum theory
of fields, which you don't--

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yeah, you may not--

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it depends on your need and
whether you have 2 or 3 or not.

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Yeah, may depend on your needs.

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OK, so under the
main topics, I plan

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to cover are listed
in the outline,

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this document of the
outline, which is already

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on the website.

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And so I should emphasize
that outline is rough,

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is only a rough
roadmap and may change.

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It depends on the pace.

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I may change things
along the way.

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And sometimes I
change my mind, say,

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half through the
course, somehow I feel--

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yeah, anyway.

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So don't treat it too literally.

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Any questions about this subject
of quantum field theory so far?

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OK, good.

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And also, let me say a few
words that the quantum field

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theory has a reputation of being
a very difficult subject, OK?

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Actually, indeed, myself
have suffered a lot

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when I learned it myself, OK?

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But with 20/20
hindsight and also

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from interacting with
large number of students

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through teaching various level
of quantum field theory classes

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in many years, I
can assure you now

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that actually quantum
field theory is actually

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not difficult at all, OK--

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[LAUGHTER]

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--if you learned
it the right way.

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And of course, the learning
thing-- yeah, of course,

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anything is not difficult if
you learned it in the right way.

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And so in a sense this is empty
words, but keep it in mind.

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Whenever you think it's too
hard and there might be--

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the reason might not be--

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the reason might be you have
to change your perspective, OK?

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You have to change
your perspective.

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And so quantum field
theory, one thing people

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complain about is that
quantum field theory often

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involve a lot of
calculations, and that's true.

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That's just a fact of life.

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You cannot avoid it.

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But that's not what it
makes it difficult, OK?

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Complicated calculations,
you can just

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go through them from one line to
the next line to the next line.

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If you're careful
enough, patient enough,

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you can go through.

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So the difficulty, I think for
most people, of quantum field

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theory, it's more at
a conceptual level.

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It's because this subject is
not a very intuitive subject.

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It's not something you can just
understand just by thinking,

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OK?

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So that's why I emphasized
earlier the exercise

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and really working through
it is very important.

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It's a little bit like
quantum mechanics.

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In quantum mechanics
your intuition

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was developed through examples.

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By working through
many examples,

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you slowly develop intuition
about the quantum mechanics.

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And if you learn all
the lessons, and then

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you get good feeling
about quantum mechanics.

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And quantum field theory
is the same thing.

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It said, you have to--

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some kind of intuition
has to be developed, OK?

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It's not something
you can easily--

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just like mechanics, which
you can in some other subject,

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maybe if you have
very good intuition,

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you can just imagine it, OK?

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Yeah, so to help you develop
good intuition about quantum

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field theory, I can offer you
three pieces of advice, OK?

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So first, yeah, so the
first piece of advice

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is that the quantum field
theory is essentially

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quantum mechanics but dealing
with infinite number of degrees

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of freedom, OK?

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So in your quantum
mechanics class,

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you always treat with finite
number of degrees of freedom.

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But quantum field theory, the
difference for quantum field

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theory is now you treat
an infinite number

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of degrees of freedom.

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It turns out that this treating
infinite number degrees

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freedom makes a difference.

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So more is different
and actually, sometimes

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make conceptual differences.

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And so that's why
sometimes the quantum field

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theory is unintuitive, OK?

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But that said, I
found, for many people,

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including myself when I learned
it, for many difficulties you

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encounter in learning
quantum field theory,

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it's not due to the difficulty
in quantum field theory itself.

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It's actually due to
your gap in understanding

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of quantum mechanics.

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So whenever you
encounter something

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you don't quite understand
in quantum field theory,

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try to step it
back, to say, can I

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formulate this difficulty in
terms of quantum mechanics

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with only a finite number
of degrees of freedom?

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And often, you find actually
your difficulty can already

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be formulated in
quantum mechanics.

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And then that way
then you should

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be able to just
settle it yourself,

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OK, because we are
supposed already

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to be a master of
quantum mechanics, OK?

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And certainly, when I
learned quantum field theory

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I was stuck at a certain
point for a long time.

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And then later I realized, just
because I didn't understand

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certain Heisenberg picture
of quantum mechanics

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very well, OK?

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Somehow I realized when I
understood the Heisenberg

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picture of quantum
mechanics well,

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and those difficulties
just went away.

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And have nothing to do with
quantum field theory itself.

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So that's why, in
your first Pset

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you will get familiar
with Heisenberg

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picture of quantum
mechanics, OK?

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That came from my
own experience.

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And also, the second point is
that quantum field theory deals

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with formalisms.

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And sometimes the subject
seems very formal, OK?

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You have a lot of
formalisms, OK?

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But just keep in mind,
any formalism in physics,

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no matter how abstract
it is, it was always

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designed to solve some
concrete physical problems

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and physical questions, very
concrete physical questions.

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And if you understand what kind
of concrete physical questions

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quantum field theory
was designed to solve,

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then that can give you
a very good perspective

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on those formalism
and why people

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do this why people do that,
why people do this trick, why

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people do that trick because
they were invented to solve

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certain concrete problems, OK?

00:11:23.160 --> 00:11:25.740 align:middle line:84%
And once you understand
the questions,

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understand the problems,
then the formalism

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becomes much easier
to understand.

00:11:31.080 --> 00:11:34.980 align:middle line:84%
And the third thing
we already said

00:11:34.980 --> 00:11:38.790 align:middle line:84%
is that in quantum field
theory, as in quantum mechanics,

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intuition was built
through experiences, OK,

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through examples.

00:11:45.540 --> 00:11:47.610 align:middle line:84%
So when you do
your Pset, when you

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look at the examples
in the class,

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you should always ask
yourself afterwards,

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say, after you have
done your Pset problems,

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always look back
at that problem.

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Say, what did I learn
from this problem, OK?

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And just think through it again.

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Think through what you learned
from that problem again.

00:12:07.350 --> 00:12:10.290 align:middle line:84%
And that is a very
good way to help

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you to learn from
your experiences

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and to help you
develop intuitions, OK?

00:12:17.780 --> 00:12:21.290 align:middle line:84%
And so yeah, so also
a very important thing

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you should keep in mind is
that in the graduate course

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like this most of
the things should

00:12:28.350 --> 00:12:30.300 align:middle line:90%
be learned outside the class.

00:12:30.300 --> 00:12:34.230 align:middle line:84%
So inside the class the
purpose is to give you a guide,

00:12:34.230 --> 00:12:37.080 align:middle line:84%
OK, is to emphasize
the conceptual picture

00:12:37.080 --> 00:12:39.820 align:middle line:84%
and the physical
intuition, et cetera.

00:12:39.820 --> 00:12:42.810 align:middle line:84%
And so sometimes I will
leave some details for you

00:12:42.810 --> 00:12:44.550 align:middle line:90%
to finish in the Pset.

00:12:44.550 --> 00:12:48.030 align:middle line:84%
And sometimes the Pset
will involve problems

00:12:48.030 --> 00:12:52.020 align:middle line:84%
which I did not, say,
fully discuss in lecture,

00:12:52.020 --> 00:12:54.780 align:middle line:84%
but I want you to
work out yourself, OK?

00:12:54.780 --> 00:12:58.800 align:middle line:84%
And so P set is important
part of the learning,

00:12:58.800 --> 00:13:02.190 align:middle line:84%
even new things, OK, not
just to practice something,

00:13:02.190 --> 00:13:08.085 align:middle line:84%
but it's also a very important
part for learning new things.

00:13:08.085 --> 00:13:11.430 align:middle line:90%


00:13:11.430 --> 00:13:12.850 align:middle line:90%
Right.

00:13:12.850 --> 00:13:13.630 align:middle line:90%
Good?

00:13:13.630 --> 00:13:20.830 align:middle line:84%
Also, finally, I would
like to make an apology.

00:13:20.830 --> 00:13:25.780 align:middle line:84%
So you will soon find that
notations rotations are

00:13:25.780 --> 00:13:29.560 align:middle line:84%
used in the lecture are
different from the notations

00:13:29.560 --> 00:13:33.430 align:middle line:84%
in the recommended
reading books, OK?

00:13:33.430 --> 00:13:36.640 align:middle line:84%
So I recommended reading
Peskin and Weinberg.

00:13:36.640 --> 00:13:39.550 align:middle line:84%
And you will find that
my notations are actually

00:13:39.550 --> 00:13:40.660 align:middle line:90%
different from them.

00:13:40.660 --> 00:13:44.740 align:middle line:84%
Also, the order of presentations
are also different from them,

00:13:44.740 --> 00:13:45.580 align:middle line:90%
OK?

00:13:45.580 --> 00:13:50.140 align:middle line:84%
I know this is very
annoying, but there's just

00:13:50.140 --> 00:13:53.710 align:middle line:90%
no perfect textbooks.

00:13:53.710 --> 00:13:57.850 align:middle line:84%
And there's no perfect set
of notations everybody use.

00:13:57.850 --> 00:14:01.510 align:middle line:84%
And we all use the
notations which we find

00:14:01.510 --> 00:14:03.820 align:middle line:90%
the most convenient to use, OK?

00:14:03.820 --> 00:14:08.260 align:middle line:84%
And so even though I
realize this problem,

00:14:08.260 --> 00:14:10.450 align:middle line:84%
but I don't have a
good resolution, OK?

00:14:10.450 --> 00:14:13.870 align:middle line:84%
So just keep in mind, the
notations in my lecture

00:14:13.870 --> 00:14:18.310 align:middle line:84%
can be different from the
notations in those textbooks,

00:14:18.310 --> 00:14:19.490 align:middle line:90%
OK?

00:14:19.490 --> 00:14:20.030 align:middle line:90%
Good?

00:14:20.030 --> 00:14:23.290 align:middle line:84%
So do you have any
other questions?

00:14:23.290 --> 00:14:29.490 align:middle line:90%


00:14:29.490 --> 00:14:33.040 align:middle line:84%
Good, OK, so if you don't
have any other questions,

00:14:33.040 --> 00:14:33.900 align:middle line:90%
so let's start.

00:14:33.900 --> 00:14:45.820 align:middle line:90%


00:14:45.820 --> 00:14:58.550 align:middle line:84%
So the Chapter 1 will be about
why we consider quantum field

00:14:58.550 --> 00:14:59.165 align:middle line:90%
theory, OK?

00:14:59.165 --> 00:15:02.960 align:middle line:90%


00:15:02.960 --> 00:15:08.720 align:middle line:84%
So first, we talk a little
bit about the classical field

00:15:08.720 --> 00:15:15.950 align:middle line:84%
theories to set the stage or
quantum field theories, OK?

00:15:15.950 --> 00:15:20.050 align:middle line:90%


00:15:20.050 --> 00:15:24.220 align:middle line:84%
And the first important
concept is called

00:15:24.220 --> 00:15:25.720 align:middle line:90%
the principle of locality.

00:15:25.720 --> 00:15:30.180 align:middle line:90%


00:15:30.180 --> 00:15:35.990 align:middle line:84%
So if you remember from your
high school days Newtonian

00:15:35.990 --> 00:15:47.240 align:middle line:84%
mechanics, so in
Newtonian mechanics

00:15:47.240 --> 00:15:57.450 align:middle line:84%
you have action
at a distance, OK?

00:15:57.450 --> 00:16:00.750 align:middle line:84%
For example, if you look
at gravity and the gravity

00:16:00.750 --> 00:16:03.690 align:middle line:84%
is exerted by the
sun on the Earth,

00:16:03.690 --> 00:16:09.780 align:middle line:84%
but they are very far
away, OK, and yeah.

00:16:09.780 --> 00:16:13.080 align:middle line:84%
And the same thing with
the Coulomb interactions

00:16:13.080 --> 00:16:14.640 align:middle line:90%
between the charged particles.

00:16:14.640 --> 00:16:18.250 align:middle line:90%


00:16:18.250 --> 00:16:22.460 align:middle line:84%
But then in the
19th century, they

00:16:22.460 --> 00:16:25.400 align:middle line:84%
came from this
principle locality.

00:16:25.400 --> 00:16:38.510 align:middle line:84%
And that's formulated by
Faraday around 1830, OK?

00:16:38.510 --> 00:16:47.940 align:middle line:84%
So the principle of
locality said all points--

00:16:47.940 --> 00:16:52.380 align:middle line:84%
you actually don't have
action at a distance.

00:16:52.380 --> 00:17:03.240 align:middle line:84%
He said all points
in space participate

00:17:03.240 --> 00:17:14.240 align:middle line:90%
in the physical process, OK?

00:17:14.240 --> 00:17:19.069 align:middle line:84%
And the effect, so if
you have interactions,

00:17:19.069 --> 00:17:30.205 align:middle line:84%
OK, so effect propagates from
points to a neighboring point,

00:17:30.205 --> 00:17:30.705 align:middle line:90%
OK?

00:17:30.705 --> 00:17:45.850 align:middle line:90%


00:17:45.850 --> 00:17:52.560 align:middle line:84%
OK, so in this
principle locality

00:17:52.560 --> 00:17:55.000 align:middle line:84%
you don't have
action at a distance.

00:17:55.000 --> 00:17:58.470 align:middle line:84%
So action at a distance
is always conveyed--

00:17:58.470 --> 00:18:02.310 align:middle line:84%
the action is always conveyed
from one point to another point

00:18:02.310 --> 00:18:07.160 align:middle line:84%
through the propagating
in the space, OK?

00:18:07.160 --> 00:18:14.900 align:middle line:84%
And the fields, the
concept of fields

00:18:14.900 --> 00:18:35.480 align:middle line:84%
is the mathematical
device or vehicle

00:18:35.480 --> 00:18:43.805 align:middle line:84%
that the principle of
locality is at work, OK?

00:18:43.805 --> 00:18:47.060 align:middle line:90%


00:18:47.060 --> 00:18:50.150 align:middle line:84%
So this is
essentially the device

00:18:50.150 --> 00:18:55.700 align:middle line:84%
we need to use to realize
this principle of locality.

00:18:55.700 --> 00:19:02.690 align:middle line:84%
And so the main
idea of the field

00:19:02.690 --> 00:19:16.990 align:middle line:84%
is that we associate each
point with each point in space

00:19:16.990 --> 00:19:30.670 align:middle line:84%
a dynamical variable, OK,
or dynamical variables, OK?

00:19:30.670 --> 00:19:43.310 align:middle line:84%
So for example, so for example,
so if you have electric field--

00:19:43.310 --> 00:19:49.820 align:middle line:90%


00:19:49.820 --> 00:19:57.960 align:middle line:84%
so the electric field is
defined for all space, OK?

00:19:57.960 --> 00:20:02.760 align:middle line:84%
So at each point x, we can
introduce an electric field,

00:20:02.760 --> 00:20:04.170 align:middle line:90%
OK?

00:20:04.170 --> 00:20:12.730 align:middle line:84%
And then this electric field can
also depend on time, and yeah.

00:20:12.730 --> 00:20:16.990 align:middle line:84%
So of course, normally
we write it this way.

00:20:16.990 --> 00:20:20.200 align:middle line:90%
E of x, t.

00:20:20.200 --> 00:20:23.690 align:middle line:84%
And the reason I
write it this way

00:20:23.690 --> 00:20:27.230 align:middle line:84%
is to emphasize that
in the definition

00:20:27.230 --> 00:20:31.300 align:middle line:84%
of the electric field, the
space and time actually

00:20:31.300 --> 00:20:33.700 align:middle line:90%
play a very different role.

00:20:33.700 --> 00:20:38.020 align:middle line:84%
The space plays the
role of a label.

00:20:38.020 --> 00:20:41.170 align:middle line:84%
So at each point we
have electric field, OK?

00:20:41.170 --> 00:20:43.730 align:middle line:84%
At each point x, we
have electric field.

00:20:43.730 --> 00:20:46.480 align:middle line:84%
And so x here is
just a label, OK?

00:20:46.480 --> 00:20:53.140 align:middle line:84%
And the t is used to describe
the evolution, the change

00:20:53.140 --> 00:20:54.470 align:middle line:90%
of the electric field.

00:20:54.470 --> 00:20:58.270 align:middle line:84%
So the x and t plays
a very different role.

00:20:58.270 --> 00:21:00.475 align:middle line:84%
And similarly, you can do
it for magnetic fields.

00:21:00.475 --> 00:21:05.000 align:middle line:90%


00:21:05.000 --> 00:21:12.310 align:middle line:90%
It's the same thing, OK?

00:21:12.310 --> 00:21:16.680 align:middle line:84%
And here you should always
view x as labels, OK?

00:21:16.680 --> 00:21:20.550 align:middle line:84%
So the spatial point, which
we denote as vector x,

00:21:20.550 --> 00:21:22.020 align:middle line:90%
is the labels.

00:21:22.020 --> 00:21:29.940 align:middle line:90%


00:21:29.940 --> 00:21:35.880 align:middle line:84%
And we know that the evolution
of electric and magnetic fields

00:21:35.880 --> 00:21:37.980 align:middle line:84%
are described by
Maxwell's equations.

00:21:37.980 --> 00:21:42.530 align:middle line:90%


00:21:42.530 --> 00:21:45.750 align:middle line:84%
So let me just write
them down to remind you.

00:21:45.750 --> 00:21:57.170 align:middle line:84%
So if you have dot v 0
dot E equal to 4 pi rho.

00:21:57.170 --> 00:22:08.400 align:middle line:90%


00:22:08.400 --> 00:22:10.440 align:middle line:84%
So this so-called
the differential form

00:22:10.440 --> 00:22:12.060 align:middle line:90%
of the Maxwell equations.

00:22:12.060 --> 00:22:16.410 align:middle line:90%


00:22:16.410 --> 00:22:21.710 align:middle line:84%
So the reason I'm taking
trouble to write them down

00:22:21.710 --> 00:22:24.840 align:middle line:84%
is to emphasize the
following point.

00:22:24.840 --> 00:22:32.300 align:middle line:84%
So this set of equations
exemplifies perfectly

00:22:32.300 --> 00:22:35.710 align:middle line:90%
the principle of locality.

00:22:35.710 --> 00:22:40.450 align:middle line:84%
It's because you see
those equations only

00:22:40.450 --> 00:22:44.590 align:middle line:84%
involve the value of electric
fields and the magnetic field

00:22:44.590 --> 00:22:47.430 align:middle line:90%
at a single point, OK?

00:22:47.430 --> 00:22:51.600 align:middle line:84%
So if here is at x point,
so here is also at x point.

00:22:51.600 --> 00:22:57.300 align:middle line:84%
You never say x here and
here is some other point y.

00:22:57.300 --> 00:23:00.880 align:middle line:84%
And so this is
reflected the principle

00:23:00.880 --> 00:23:04.110 align:middle line:90%
of locality that everything--

00:23:04.110 --> 00:23:08.160 align:middle line:84%
so the effect is propagate
from point to point, OK?

00:23:08.160 --> 00:23:13.050 align:middle line:84%
You just have the derivative
of the same point, OK?

00:23:13.050 --> 00:23:16.020 align:middle line:84%
You don't involve
separated points, OK?

00:23:16.020 --> 00:23:19.230 align:middle line:90%


00:23:19.230 --> 00:23:22.300 align:middle line:90%
So these are local equations.

00:23:22.300 --> 00:23:24.750 align:middle line:84%
So these are what we
call the local equations.

00:23:24.750 --> 00:23:29.110 align:middle line:90%


00:23:29.110 --> 00:23:43.210 align:middle line:84%
And they contain only E
and B at the same point

00:23:43.210 --> 00:23:45.850 align:middle line:84%
and also the charge
density and current density

00:23:45.850 --> 00:23:50.557 align:middle line:84%
at the same point, with
finite number of derivatives.

00:23:50.557 --> 00:24:00.100 align:middle line:90%


00:24:00.100 --> 00:24:06.300 align:middle line:84%
So the derivatives are the ones
to help you to propagate, OK,

00:24:06.300 --> 00:24:11.220 align:middle line:84%
because it relates the point
to the neighboring point, OK?

00:24:11.220 --> 00:24:14.280 align:middle line:90%
And so that's how it propagates.

00:24:14.280 --> 00:24:15.930 align:middle line:84%
It's through the
derivatives, OK?

00:24:15.930 --> 00:24:17.735 align:middle line:90%
So the derivatives are key, OK?

00:24:17.735 --> 00:24:18.735 align:middle line:90%
The derivatives are key.

00:24:18.735 --> 00:24:21.330 align:middle line:90%


00:24:21.330 --> 00:24:25.930 align:middle line:84%
And another example with
examples of locality,

00:24:25.930 --> 00:24:31.390 align:middle line:84%
which I will not go into here,
which some of you may know,

00:24:31.390 --> 00:24:35.170 align:middle line:84%
is the Einstein gravity, so
Einstein's general relativity.

00:24:35.170 --> 00:24:41.810 align:middle line:90%


00:24:41.810 --> 00:24:45.280 align:middle line:84%
So in Einstein gravity
the dynamical variables

00:24:45.280 --> 00:24:48.200 align:middle line:90%
are so-called spacetime metric.

00:24:48.200 --> 00:24:54.320 align:middle line:84%
So they are object with two
indices, two spacetime indices,

00:24:54.320 --> 00:24:54.870 align:middle line:90%
OK?

00:24:54.870 --> 00:24:56.850 align:middle line:84%
And then the Einstein
equations are

00:24:56.850 --> 00:24:58.700 align:middle line:84%
equations for this
kind of object.

00:24:58.700 --> 00:25:01.040 align:middle line:84%
And again, the equations
are local equations

00:25:01.040 --> 00:25:02.870 align:middle line:84%
in the sense that
they only depend

00:25:02.870 --> 00:25:07.820 align:middle line:84%
on the G evaluated at the same
point with a finite number

00:25:07.820 --> 00:25:09.440 align:middle line:90%
of derivatives, OK?

00:25:09.440 --> 00:25:12.740 align:middle line:84%
So in Einstein's
gravity you no longer

00:25:12.740 --> 00:25:14.720 align:middle line:90%
have action at a distance, OK?

00:25:14.720 --> 00:25:20.330 align:middle line:84%
So the effect of the gravity
is propagated through the space

00:25:20.330 --> 00:25:22.340 align:middle line:90%
time, through space time.

00:25:22.340 --> 00:25:25.740 align:middle line:90%


00:25:25.740 --> 00:25:32.077 align:middle line:84%
So in fact, so here we say
these two examples exemplifies

00:25:32.077 --> 00:25:33.160 align:middle line:90%
the principle of locality.

00:25:33.160 --> 00:25:35.590 align:middle line:84%
In fact, the
principle of locality

00:25:35.590 --> 00:25:39.670 align:middle line:84%
plays a very important role
in formulating those equations

00:25:39.670 --> 00:25:42.580 align:middle line:84%
because, as we
will very soon see,

00:25:42.580 --> 00:25:44.500 align:middle line:84%
when you have
principle of locality,

00:25:44.500 --> 00:25:49.280 align:middle line:84%
you can significantly constrain
the theory you can write down.

00:25:49.280 --> 00:25:51.910 align:middle line:84%
So that's a very, very
powerful principle.

00:25:51.910 --> 00:25:52.930 align:middle line:90%
Yes.

00:25:52.930 --> 00:25:54.430 align:middle line:84%
AUDIENCE: Are there
popular examples

00:25:54.430 --> 00:25:57.070 align:middle line:84%
you think we might have
seen of nonlocal equations?

00:25:57.070 --> 00:26:00.850 align:middle line:84%
PROFESSOR: Yeah, you can
have nonlocal equations,

00:26:00.850 --> 00:26:03.070 align:middle line:84%
but it's believed that
the fundamental equations

00:26:03.070 --> 00:26:07.310 align:middle line:84%
in nature, they are
all local, yeah.

00:26:07.310 --> 00:26:11.780 align:middle line:84%
Yeah, and so far,
the equations govern

00:26:11.780 --> 00:26:13.130 align:middle line:90%
all fundamental interactions.

00:26:13.130 --> 00:26:15.465 align:middle line:84%
All four different interactions
in nature, the equations

00:26:15.465 --> 00:26:15.965 align:middle line:90%
are local.

00:26:15.965 --> 00:26:18.653 align:middle line:90%


00:26:18.653 --> 00:26:19.320 align:middle line:90%
Other questions?

00:26:19.320 --> 00:26:19.575 align:middle line:90%
Yes.

00:26:19.575 --> 00:26:21.600 align:middle line:84%
AUDIENCE: When you wrote down
the Einstein's gravity thing,

00:26:21.600 --> 00:26:23.990 align:middle line:84%
why did you pick out a
particular time component t?

00:26:23.990 --> 00:26:24.740 align:middle line:90%
PROFESSOR: Sorry?

00:26:24.740 --> 00:26:26.448 align:middle line:84%
AUDIENCE: When you
wrote down the G mu nu

00:26:26.448 --> 00:26:28.280 align:middle line:84%
as a function of x
and t, why did you

00:26:28.280 --> 00:26:32.330 align:middle line:84%
pick at a particular
time component?

00:26:32.330 --> 00:26:33.330 align:middle line:90%
PROFESSOR: Say it again.

00:26:33.330 --> 00:26:35.220 align:middle line:84%
I don't quite understand
the question, yeah.

00:26:35.220 --> 00:26:36.360 align:middle line:90%
AUDIENCE: Sure.

00:26:36.360 --> 00:26:38.977 align:middle line:84%
You kind of treated the
x and t asymmetrically

00:26:38.977 --> 00:26:41.310 align:middle line:84%
when you write down the
equation for Einstein's gravity.

00:26:41.310 --> 00:26:41.670 align:middle line:90%
PROFESSOR: Yeah.

00:26:41.670 --> 00:26:42.450 align:middle line:84%
AUDIENCE: Is there
a particular reason

00:26:42.450 --> 00:26:43.670 align:middle line:90%
why you wrote it in that way?

00:26:43.670 --> 00:26:46.550 align:middle line:84%
PROFESSOR: Oh, no,
no, no, here, I

00:26:46.550 --> 00:26:50.030 align:middle line:84%
just want to emphasize
again that, of course,

00:26:50.030 --> 00:26:54.950 align:middle line:84%
in Einstein's theory these
two are treated the same way.

00:26:54.950 --> 00:26:57.565 align:middle line:84%
But if you think in
terms of the fields,

00:26:57.565 --> 00:26:58.940 align:middle line:84%
they actually have
very different

00:26:58.940 --> 00:27:00.870 align:middle line:90%
physical interpretation.

00:27:00.870 --> 00:27:03.990 align:middle line:90%
So that's why I write this way.

00:27:03.990 --> 00:27:08.160 align:middle line:84%
Yeah, it's the same way I write
it like here, yeah, yeah, yeah,

00:27:08.160 --> 00:27:10.140 align:middle line:84%
same reason, just here
to emphasize for this--

00:27:10.140 --> 00:27:12.720 align:middle line:90%


00:27:12.720 --> 00:27:16.760 align:middle line:84%
to emphasize the different
role played by x and t.

00:27:16.760 --> 00:27:20.370 align:middle line:84%
Yeah, but I think you're
asking a very good question.

00:27:20.370 --> 00:27:25.160 align:middle line:84%
So I'm going to mention later
in the relativistic series

00:27:25.160 --> 00:27:28.670 align:middle line:84%
then, of course, then
these two become the same,

00:27:28.670 --> 00:27:32.040 align:middle line:90%
play the very equal role.

00:27:32.040 --> 00:27:33.440 align:middle line:90%
Does that answer your question?

00:27:33.440 --> 00:27:34.065 align:middle line:90%
AUDIENCE: Yeah.

00:27:34.065 --> 00:27:35.620 align:middle line:90%
PROFESSOR: OK, good.

00:27:35.620 --> 00:27:43.570 align:middle line:84%
Good, also yeah, so let me
also very quickly mention

00:27:43.570 --> 00:27:45.145 align:middle line:90%
the different types of fields.

00:27:45.145 --> 00:27:49.290 align:middle line:90%


00:27:49.290 --> 00:27:54.600 align:middle line:84%
OK, so you can have what's
called a scalar field, scalar.

00:27:54.600 --> 00:28:01.600 align:middle line:84%
So these are quantities,
which at a given point

00:28:01.600 --> 00:28:06.230 align:middle line:84%
there's only one value, OK, say,
for example, the temperature.

00:28:06.230 --> 00:28:15.680 align:middle line:84%
So it's just a single
quantity defined at the point,

00:28:15.680 --> 00:28:18.780 align:middle line:84%
for example, the temperature,
OK, and et cetera,

00:28:18.780 --> 00:28:20.440 align:middle line:90%
and maybe other quantities.

00:28:20.440 --> 00:28:24.400 align:middle line:84%
And then you also
have a vector field.

00:28:24.400 --> 00:28:27.010 align:middle line:84%
And E and B, they're
examples of a vector field

00:28:27.010 --> 00:28:33.560 align:middle line:84%
because at each point
you have a vector, OK?

00:28:33.560 --> 00:28:37.220 align:middle line:84%
So here I'm using the
three-dimensional notations.

00:28:37.220 --> 00:28:39.440 align:middle line:84%
But in the relativistic
series, You.

00:28:39.440 --> 00:28:42.760 align:middle line:84%
Can also use the four
vector notations.

00:28:42.760 --> 00:28:45.880 align:middle line:84%
Say, for example,
the vector potential

00:28:45.880 --> 00:28:47.590 align:middle line:90%
will have the following form.

00:28:47.590 --> 00:28:51.340 align:middle line:84%
Say, A mu will be a four-vector,
and then the space time I

00:28:51.340 --> 00:28:54.070 align:middle line:84%
combine them together
into a four vector, OK?

00:28:54.070 --> 00:28:58.990 align:middle line:84%
And so at each point, and then
you have a four vector, OK?

00:28:58.990 --> 00:29:02.060 align:middle line:90%
And so yeah.

00:29:02.060 --> 00:29:04.070 align:middle line:90%
And also you have tensor field.

00:29:04.070 --> 00:29:07.880 align:middle line:84%
So this metric is an
example of a tensor field.

00:29:07.880 --> 00:29:11.870 align:middle line:84%
Here you have two components
you have two indices, OK?

00:29:11.870 --> 00:29:15.740 align:middle line:84%
So you have many, many different
components depending on the--

00:29:15.740 --> 00:29:18.560 align:middle line:84%
so again, this is a
relativistic notation.

00:29:18.560 --> 00:29:20.750 align:middle line:84%
I can write it as a
relativistic notation, OK?

00:29:20.750 --> 00:29:25.080 align:middle line:84%
So at each point, and
now you have some object

00:29:25.080 --> 00:29:27.380 align:middle line:90%
with two indices, OK?

00:29:27.380 --> 00:29:29.030 align:middle line:90%
And you can also have--

00:29:29.030 --> 00:29:35.680 align:middle line:84%
later we will see you also have
something called spinor fields,

00:29:35.680 --> 00:29:36.370 align:middle line:90%
alpha.

00:29:36.370 --> 00:29:38.650 align:middle line:84%
And this alpha is some
other indices, which we

00:29:38.650 --> 00:29:40.060 align:middle line:90%
we'll define later, OK?

00:29:40.060 --> 00:29:42.980 align:middle line:90%


00:29:42.980 --> 00:29:45.920 align:middle line:84%
OK, so you can also have
so-called spinor and alpha,

00:29:45.920 --> 00:29:49.040 align:middle line:90%
some other indices, OK?

00:29:49.040 --> 00:29:59.880 align:middle line:84%
And as our convention, is that
mu is always from 0 to 3, OK?

00:29:59.880 --> 00:30:01.410 align:middle line:84%
Then when I sometimes
I just write

00:30:01.410 --> 00:30:07.590 align:middle line:84%
x, when I write x just means
a four vector, means x mu.

00:30:07.590 --> 00:30:09.450 align:middle line:90%
You could do Ct and x.

00:30:09.450 --> 00:30:15.360 align:middle line:84%
So x vector always denotes
a spatial vector, OK?

00:30:15.360 --> 00:30:23.690 align:middle line:84%
And yeah, and then this
is the same as Ct xi, OK?

00:30:23.690 --> 00:30:28.730 align:middle line:84%
So the spatial index is
always denoted by i, and yeah.

00:30:28.730 --> 00:30:35.480 align:middle line:84%
And so partial mu will be the
same as 1 over C partial t,

00:30:35.480 --> 00:30:39.140 align:middle line:84%
and then the derivative,
the gradient,

00:30:39.140 --> 00:30:41.360 align:middle line:90%
on spatial directions.

00:30:41.360 --> 00:30:43.730 align:middle line:84%
And say, if you
have a four-vector,

00:30:43.730 --> 00:30:48.830 align:middle line:84%
then again, we have the
convention that a 0.

00:30:48.830 --> 00:30:54.450 align:middle line:84%
Then the A vector, it's
the same also as A0 and Ai.

00:30:54.450 --> 00:30:58.790 align:middle line:84%
I use the i to denote
the spatial components.

00:30:58.790 --> 00:31:02.070 align:middle line:84%
And this is the last time I
will write speed of light.

00:31:02.070 --> 00:31:04.010 align:middle line:84%
So the c I will
always take to be 1,

00:31:04.010 --> 00:31:09.640 align:middle line:84%
and h bar will taken
to be 1, OK, just

00:31:09.640 --> 00:31:14.130 align:middle line:90%
for notational convenience.

00:31:14.130 --> 00:31:15.570 align:middle line:90%
You have questions?

00:31:15.570 --> 00:31:20.470 align:middle line:90%


00:31:20.470 --> 00:31:26.100 align:middle line:84%
OK, so now, so with
this preparation then

00:31:26.100 --> 00:31:37.050 align:middle line:84%
we can talk about action
principle for classical fields,

00:31:37.050 --> 00:31:38.310 align:middle line:90%
OK?

00:31:38.310 --> 00:31:49.900 align:middle line:84%
So first, recall in your
classical mechanics,

00:31:49.900 --> 00:31:53.500 align:middle line:84%
so we introduce
the action, which

00:31:53.500 --> 00:31:56.540 align:middle line:84%
is the integral of
your Lagrangian.

00:31:56.540 --> 00:32:01.160 align:middle line:84%
And Lagrangian is a function
of x, this is your variable,

00:32:01.160 --> 00:32:04.550 align:middle line:90%
and x dot and the time, OK?

00:32:04.550 --> 00:32:07.430 align:middle line:84%
And so this is a
one-dimensional-- yeah,

00:32:07.430 --> 00:32:11.300 align:middle line:84%
this is the 8.01
one-dimensional particle motion.

00:32:11.300 --> 00:32:14.160 align:middle line:84%
And you can also introduce
momentum, canonical momentum,

00:32:14.160 --> 00:32:18.920 align:middle line:84%
which is defined by
partial L, partial x dot.

00:32:18.920 --> 00:32:21.620 align:middle line:84%
And then you can also
define a Hamiltonian.

00:32:21.620 --> 00:32:23.840 align:middle line:84%
And Hamiltonian is
related to the Lagrangian

00:32:23.840 --> 00:32:29.990 align:middle line:84%
by p x-dot minus L, from
a Legendre transform.

00:32:29.990 --> 00:32:41.660 align:middle line:84%
And the equation of motion is
obtained by extremize S, OK?

00:32:41.660 --> 00:32:46.820 align:middle line:84%
So S is considered to be a
functional of your trajectory.

00:32:46.820 --> 00:32:50.630 align:middle line:84%
So whatever trajectory you
have, and you extremize this S,

00:32:50.630 --> 00:32:52.640 align:middle line:84%
and then you get the
equation of motion, OK?

00:32:52.640 --> 00:32:55.310 align:middle line:90%


00:32:55.310 --> 00:33:01.310 align:middle line:84%
So now we can generalize
to field theory, OK?

00:33:01.310 --> 00:33:05.930 align:middle line:84%
So now, from
principle of locality,

00:33:05.930 --> 00:33:15.560 align:middle line:84%
so for field theory from
principle locality, so

00:33:15.560 --> 00:33:20.750 align:middle line:84%
the form of the Lagrangian, so
the form of the Lagrangian L--

00:33:20.750 --> 00:33:23.530 align:middle line:90%


00:33:23.530 --> 00:33:25.180 align:middle line:84%
so again, we here,
for field theory,

00:33:25.180 --> 00:33:28.330 align:middle line:84%
we can again define S as a
time integral of a Lagrangian.

00:33:28.330 --> 00:33:31.940 align:middle line:90%


00:33:31.940 --> 00:33:35.380 align:middle line:84%
And the form of L is
significantly constrained, OK?

00:33:35.380 --> 00:33:48.490 align:middle line:90%


00:33:48.490 --> 00:34:03.110 align:middle line:84%
To have the following form,
so you must have the form L,

00:34:03.110 --> 00:34:05.700 align:middle line:84%
you could choose a
spatial integral.

00:34:05.700 --> 00:34:08.840 align:middle line:84%
So this is an integration over
all spatial directions, OK,

00:34:08.840 --> 00:34:14.239 align:middle line:90%
so this d^3 x and some thing.

00:34:14.239 --> 00:34:17.510 align:middle line:84%
So let me first write
down the notation,

00:34:17.510 --> 00:34:19.205 align:middle line:84%
and then I will
explain the notation.

00:34:19.205 --> 00:34:24.340 align:middle line:90%


00:34:24.340 --> 00:34:32.699 align:middle line:84%
OK, yeah, maybe just
write partial i, OK?

00:34:32.699 --> 00:34:36.699 align:middle line:84%
So let me just explain
notation a little bit.

00:34:36.699 --> 00:34:42.810 align:middle line:84%
Well, first, so here I
use a shorthand notation

00:34:42.810 --> 00:34:44.940 align:middle line:90%
to denote the fields.

00:34:44.940 --> 00:34:49.860 align:middle line:84%
So phi a so will be a
function of spatial direction

00:34:49.860 --> 00:35:05.400 align:middle line:84%
and time is general field, and
a label different fields, OK?

00:35:05.400 --> 00:35:12.420 align:middle line:84%
So this index, a, labels
different fields, OK?

00:35:12.420 --> 00:35:16.420 align:middle line:84%
So for example, a can label
different scalar fields.

00:35:16.420 --> 00:35:19.290 align:middle line:84%
If you have multiple scalar
fields, they can label them.

00:35:19.290 --> 00:35:23.100 align:middle line:84%
And a can also refers to
indices, space time indices,

00:35:23.100 --> 00:35:27.180 align:middle line:84%
like A mu, and also,
yeah, et cetera, OK?

00:35:27.180 --> 00:35:31.440 align:middle line:84%
So a just label whatever
fields you have, OK?

00:35:31.440 --> 00:35:37.500 align:middle line:90%


00:35:37.500 --> 00:35:45.010 align:middle line:84%
And then the second
point, you said this L,

00:35:45.010 --> 00:35:51.990 align:middle line:90%
this script L is a function--

00:35:51.990 --> 00:36:11.000 align:middle line:84%
I emphasize here-- is a function
of phi a and its derivatives,

00:36:11.000 --> 00:36:12.530 align:middle line:90%
OK, and its derivatives.

00:36:12.530 --> 00:36:15.440 align:middle line:90%


00:36:15.440 --> 00:36:17.410 align:middle line:90%
So in other words, the L--

00:36:17.410 --> 00:36:20.230 align:middle line:90%
so this is a key point, OK?

00:36:20.230 --> 00:36:35.550 align:middle line:84%
So that this L only
depends on the value

00:36:35.550 --> 00:36:56.430 align:middle line:84%
of phi a and its derivatives
at a single point, OK, say x.

00:36:56.430 --> 00:36:58.545 align:middle line:84%
And then you integrate
over all x, OK?

00:36:58.545 --> 00:37:02.020 align:middle line:90%


00:37:02.020 --> 00:37:04.765 align:middle line:84%
So L is called the
Lagrangian density.

00:37:04.765 --> 00:37:08.580 align:middle line:90%


00:37:08.580 --> 00:37:10.510 align:middle line:84%
AUDIENCE: So these
are all still local?

00:37:10.510 --> 00:37:13.210 align:middle line:90%


00:37:13.210 --> 00:37:13.930 align:middle line:90%
PROFESSOR: Sorry?

00:37:13.930 --> 00:37:16.230 align:middle line:84%
AUDIENCE: These are
all still local?

00:37:16.230 --> 00:37:17.098 align:middle line:90%
PROFESSOR: Sorry?

00:37:17.098 --> 00:37:17.640 align:middle line:90%
Say it again.

00:37:17.640 --> 00:37:18.557 align:middle line:90%
AUDIENCE: Still local?

00:37:18.557 --> 00:37:20.125 align:middle line:84%
PROFESSOR: Yeah,
yeah, yeah, yeah.

00:37:20.125 --> 00:37:20.625 align:middle line:90%
Yes.

00:37:20.625 --> 00:37:23.150 align:middle line:84%
AUDIENCE: Can you even
higher order derivatives?

00:37:23.150 --> 00:37:24.330 align:middle line:84%
PROFESSOR: Yeah, yeah,
we'll mention that.

00:37:24.330 --> 00:37:25.372 align:middle line:90%
Yeah, we'll mention that.

00:37:25.372 --> 00:37:29.340 align:middle line:90%


00:37:29.340 --> 00:37:35.520 align:middle line:84%
So here I just explained
the notation, OK?

00:37:35.520 --> 00:37:39.240 align:middle line:84%
So now let me just
make some remarks

00:37:39.240 --> 00:37:45.990 align:middle line:84%
on why the Lagrangian
must have this form.

00:37:45.990 --> 00:37:48.610 align:middle line:90%


00:37:48.610 --> 00:37:59.940 align:middle line:84%
So first, the
principle of locality

00:37:59.940 --> 00:38:04.140 align:middle line:84%
implies here must only
involve a single integral, OK,

00:38:04.140 --> 00:38:14.990 align:middle line:84%
because the locality does not
allow something like this,

00:38:14.990 --> 00:38:30.960 align:middle line:84%
say, for example, does not allow
a term like this, OK, does not

00:38:30.960 --> 00:38:36.100 align:middle line:84%
allow a term like this,
which involving the phi

00:38:36.100 --> 00:38:39.410 align:middle line:90%
at a different point, OK?

00:38:39.410 --> 00:38:40.250 align:middle line:90%
Why?

00:38:40.250 --> 00:38:45.260 align:middle line:84%
It's because if you have terms
like this in your Lagrangian,

00:38:45.260 --> 00:38:52.730 align:middle line:84%
OK, so later you will--
oh, sorry, L, not L script.

00:38:52.730 --> 00:38:55.820 align:middle line:84%
So if you Lagrangian
have this kind of terms,

00:38:55.820 --> 00:39:00.550 align:middle line:84%
and then, so as we will describe
the equation of motion later,

00:39:00.550 --> 00:39:02.770 align:middle line:84%
you will see from the
equation of motion,

00:39:02.770 --> 00:39:06.230 align:middle line:84%
then your equation of motion
will not be local, OK?

00:39:06.230 --> 00:39:09.200 align:middle line:84%
So equation of notion
will involve the behavior

00:39:09.200 --> 00:39:14.570 align:middle line:84%
of your field at one point
and then influenced by point

00:39:14.570 --> 00:39:17.120 align:middle line:90%
at some point far away, OK?

00:39:17.120 --> 00:39:20.180 align:middle line:84%
And then you will
not have local.

00:39:20.180 --> 00:39:23.180 align:middle line:84%
So the locality
significantly constraint

00:39:23.180 --> 00:39:26.258 align:middle line:84%
because if you
give away locality,

00:39:26.258 --> 00:39:27.800 align:middle line:84%
and then in principle
your Lagrangian

00:39:27.800 --> 00:39:30.060 align:middle line:90%
can be arbitrarily complicated.

00:39:30.060 --> 00:39:33.180 align:middle line:84%
You can have many
integrals as you want, OK?

00:39:33.180 --> 00:39:35.390 align:middle line:84%
But because of
locality, you're only

00:39:35.390 --> 00:39:37.460 align:middle line:84%
allowed to have such
simple integral, one

00:39:37.460 --> 00:39:40.250 align:middle line:90%
integral of a function, OK?

00:39:40.250 --> 00:39:43.760 align:middle line:90%
So this is the key.

00:39:43.760 --> 00:39:46.610 align:middle line:84%
OK, so does not allow,
so here is a key point.

00:39:46.610 --> 00:39:49.290 align:middle line:90%


00:39:49.290 --> 00:40:03.720 align:middle line:84%
So the second point is that
we only allow first derivative

00:40:03.720 --> 00:40:07.170 align:middle line:90%
in time, OK?

00:40:07.170 --> 00:40:08.880 align:middle line:84%
We only allow first
derivative in time.

00:40:08.880 --> 00:40:11.620 align:middle line:84%
We don't allow the second
derivative in time.

00:40:11.620 --> 00:40:15.180 align:middle line:84%
So the reason is
that, again, as you

00:40:15.180 --> 00:40:18.570 align:middle line:84%
will see equation of
motion, if you involve

00:40:18.570 --> 00:40:22.620 align:middle line:84%
the second derivative
in time in your action

00:40:22.620 --> 00:40:25.020 align:middle line:84%
or in your Lagrangian,
and then when

00:40:25.020 --> 00:40:26.760 align:middle line:84%
you get the equation
of motion, you

00:40:26.760 --> 00:40:30.270 align:middle line:84%
will get equation of
motions involving more

00:40:30.270 --> 00:40:33.070 align:middle line:90%
than two derivatives in time.

00:40:33.070 --> 00:40:35.090 align:middle line:90%
So this will lead to--

00:40:35.090 --> 00:40:37.490 align:middle line:84%
so this implies the
equation of motion

00:40:37.490 --> 00:40:49.600 align:middle line:84%
only contain two derivatives,
two time derivatives, OK?

00:40:49.600 --> 00:40:54.770 align:middle line:84%
So these constraints come
from our experiences.

00:40:54.770 --> 00:40:56.380 align:middle line:84%
It's the same
reason here we only

00:40:56.380 --> 00:41:00.550 align:middle line:84%
include the first
derivative in time, OK?

00:41:00.550 --> 00:41:04.870 align:middle line:84%
It's because in real life all
the experiment is determined

00:41:04.870 --> 00:41:06.532 align:middle line:90%
by the initial condition.

00:41:06.532 --> 00:41:07.990 align:middle line:84%
The initial condition
you only need

00:41:07.990 --> 00:41:10.150 align:middle line:84%
to specify the location
and the velocity, OK?

00:41:10.150 --> 00:41:11.980 align:middle line:90%
You don't need to specify more.

00:41:11.980 --> 00:41:14.470 align:middle line:84%
If the equation of motion
involving more than two

00:41:14.470 --> 00:41:17.770 align:middle line:84%
derivatives, then you
need to specify more

00:41:17.770 --> 00:41:19.640 align:middle line:90%
general initial conditions.

00:41:19.640 --> 00:41:22.720 align:middle line:84%
And so yeah, so here
it's the same thing.

00:41:22.720 --> 00:41:24.970 align:middle line:84%
We only allow the first
derivative in time.

00:41:24.970 --> 00:41:30.340 align:middle line:84%
But you can, in principle, allow
arbitrary number of derivative

00:41:30.340 --> 00:41:34.460 align:middle line:90%
in spatial direction, OK?

00:41:34.460 --> 00:41:39.180 align:middle line:84%
But for simplicity, for the most
of the time, as we will see,

00:41:39.180 --> 00:41:42.660 align:middle line:84%
we will restrict to
quantum field theory

00:41:42.660 --> 00:41:44.740 align:middle line:90%
in special relativity, OK?

00:41:44.740 --> 00:41:47.460 align:middle line:84%
Means that there will be
a relativistic invariant,

00:41:47.460 --> 00:41:48.930 align:middle line:90%
will be Lorentz invariant.

00:41:48.930 --> 00:41:52.590 align:middle line:84%
And in Lorentz invariant
theory, space and time,

00:41:52.590 --> 00:41:55.180 align:middle line:84%
they can transform to each
other, play equal role.

00:41:55.180 --> 00:41:57.570 align:middle line:84%
So if you only have
single derivative in time,

00:41:57.570 --> 00:42:00.760 align:middle line:84%
you only have single time
in spatial derivatives.

00:42:00.760 --> 00:42:04.500 align:middle line:84%
So the example we
will see they will all

00:42:04.500 --> 00:42:07.500 align:middle line:84%
have only single derivative
in spatial directions, OK?

00:42:07.500 --> 00:42:10.020 align:middle line:84%
But certainly
nonrelativistic systems

00:42:10.020 --> 00:42:15.140 align:middle line:84%
you can have a higher number
of spatial derivatives, OK?

00:42:15.140 --> 00:42:15.640 align:middle line:90%
Good?

00:42:15.640 --> 00:42:17.010 align:middle line:90%
Any questions on this?

00:42:17.010 --> 00:42:25.550 align:middle line:90%


00:42:25.550 --> 00:42:26.320 align:middle line:90%
OK, good.

00:42:26.320 --> 00:42:29.640 align:middle line:90%


00:42:29.640 --> 00:42:32.990 align:middle line:84%
So now, as in
classical mechanics,

00:42:32.990 --> 00:42:35.960 align:middle line:84%
and now we can introduce
the canonical momentum,

00:42:35.960 --> 00:42:37.610 align:middle line:90%
Hamiltonian, et cetera, OK?

00:42:37.610 --> 00:42:40.550 align:middle line:90%


00:42:40.550 --> 00:42:43.430 align:middle line:84%
So here we can introduce
so-called the canonical

00:42:43.430 --> 00:42:44.210 align:middle line:90%
momentum density.

00:42:44.210 --> 00:42:52.010 align:middle line:90%


00:42:52.010 --> 00:42:53.915 align:middle line:84%
The reason we call it
density will be clear.

00:42:53.915 --> 00:42:56.930 align:middle line:90%


00:42:56.930 --> 00:43:00.800 align:middle line:90%
So remember, for this phi--

00:43:00.800 --> 00:43:04.520 align:middle line:90%
so x is just a label, OK?

00:43:04.520 --> 00:43:07.150 align:middle line:84%
So you can just
view this theory,

00:43:07.150 --> 00:43:11.080 align:middle line:84%
essentially it has an infinite
number of these such kind of.

00:43:11.080 --> 00:43:13.030 align:middle line:84%
Yeah, it's an unfortunate
notation here,

00:43:13.030 --> 00:43:15.370 align:middle line:84%
we use x as a
dynamical variable, OK?

00:43:15.370 --> 00:43:17.650 align:middle line:84%
But here the x is
only a label, OK?

00:43:17.650 --> 00:43:19.183 align:middle line:90%
Here x is a label.

00:43:19.183 --> 00:43:21.850 align:middle line:84%
So here you can just imagine you
have infinite number of degrees

00:43:21.850 --> 00:43:22.350 align:middle line:90%
of freedom.

00:43:22.350 --> 00:43:26.920 align:middle line:84%
Just each one is
labeled by x, OK?

00:43:26.920 --> 00:43:29.330 align:middle line:84%
So now just imagine here
you have many, many x.

00:43:29.330 --> 00:43:32.590 align:middle line:84%
Just you have some
labels for it.

00:43:32.590 --> 00:43:38.160 align:middle line:84%
And so for each such one we
can introduce its momentum, OK?

00:43:38.160 --> 00:43:47.200 align:middle line:84%
So each phi a we can introduce
its canonical momentum, OK,

00:43:47.200 --> 00:43:50.370 align:middle line:90%
phi_a dot, derivative, OK?

00:43:50.370 --> 00:43:53.910 align:middle line:90%
Yeah, yeah, let me just--

00:43:53.910 --> 00:43:58.020 align:middle line:84%
so defined as the derivative
of the Lagrangian density

00:43:58.020 --> 00:44:01.590 align:middle line:84%
divided by time
derivative of phi_a, OK?

00:44:01.590 --> 00:44:05.620 align:middle line:84%
So this is just the direct
generalization of here.

00:44:05.620 --> 00:44:08.790 align:middle line:84%
So remember, so this is
evaluated at each point,

00:44:08.790 --> 00:44:10.380 align:middle line:90%
OK, in spatial direction.

00:44:10.380 --> 00:44:13.230 align:middle line:90%


00:44:13.230 --> 00:44:17.803 align:middle line:84%
So this thing will also
depend on x and t, OK?

00:44:17.803 --> 00:44:19.845 align:middle line:84%
AUDIENCE: Sorry, what
symbol is that you're using

00:44:19.845 --> 00:44:21.350 align:middle line:90%
to represent the momentum?

00:44:21.350 --> 00:44:24.980 align:middle line:84%
PROFESSOR: Oh, this is just
the capital Pi, capital Pi.

00:44:24.980 --> 00:44:28.100 align:middle line:90%


00:44:28.100 --> 00:44:35.630 align:middle line:84%
OK, and this phi a dot just the
time derivative of phi a, OK?

00:44:35.630 --> 00:44:37.440 align:middle line:84%
And remember, x
are the labels, OK?

00:44:37.440 --> 00:44:39.520 align:middle line:90%
So x does not do anything here.

00:44:39.520 --> 00:44:43.050 align:middle line:84%
So the reason we call this the
Lagrangian density is because--

00:44:43.050 --> 00:44:45.540 align:middle line:84%
call it the momentum density
is because L is a Lagrangian

00:44:45.540 --> 00:44:49.800 align:middle line:90%
density, OK, and yeah.

00:44:49.800 --> 00:44:52.590 align:middle line:84%
And then we can also define
the Hamiltonian density

00:44:52.590 --> 00:44:58.855 align:middle line:84%
the same way, associated
with each degree of freedom.

00:44:58.855 --> 00:45:02.800 align:middle line:90%


00:45:02.800 --> 00:45:11.810 align:middle line:84%
So we have to find the script
H, which is defined by phi_a dot

00:45:11.810 --> 00:45:12.980 align:middle line:90%
Pi_a .

00:45:12.980 --> 00:45:16.010 align:middle line:84%
Again, this is all defined
on the same spatial point--

00:45:16.010 --> 00:45:20.200 align:middle line:84%
minus L. So this is
Langrangian density.

00:45:20.200 --> 00:45:28.460 align:middle line:84%
And then the Hamiltonian you
just integrate over all space,

00:45:28.460 --> 00:45:35.385 align:middle line:84%
essentially sum over all
the degrees of freedom, OK?

00:45:35.385 --> 00:45:39.600 align:middle line:90%


00:45:39.600 --> 00:45:41.530 align:middle line:90%
Just leave them at that.

00:45:41.530 --> 00:45:42.745 align:middle line:90%
Yes?

00:45:42.745 --> 00:45:44.620 align:middle line:84%
AUDIENCE: Is it, so are
we implicitly summing

00:45:44.620 --> 00:45:46.630 align:middle line:84%
over the little a's
for the Hamiltonian?

00:45:46.630 --> 00:45:47.830 align:middle line:90%
PROFESSOR: Yeah, exactly.

00:45:47.830 --> 00:45:49.970 align:middle line:84%
Yeah, yeah, it's
just like you sum--

00:45:49.970 --> 00:45:52.240 align:middle line:90%
so if you treat the--

00:45:52.240 --> 00:45:57.160 align:middle line:84%
yeah, yeah, yeah, yeah,
so the a is summed.

00:45:57.160 --> 00:45:58.360 align:middle line:90%
So I will always--

00:45:58.360 --> 00:46:01.280 align:middle line:84%
yeah, I forgot to mention--
good, this is a great question.

00:46:01.280 --> 00:46:05.890 align:middle line:84%
So I always assume this
Einstein convention in the sense

00:46:05.890 --> 00:46:10.270 align:middle line:84%
that all the repeated indices
are assumed to be summed.

00:46:10.270 --> 00:46:14.110 align:middle line:84%
So no matter how many components
are there, you just sum over a.

00:46:14.110 --> 00:46:17.630 align:middle line:90%
So a is summed here.

00:46:17.630 --> 00:46:25.740 align:middle line:90%


00:46:25.740 --> 00:46:28.430 align:middle line:90%
Good?

00:46:28.430 --> 00:46:29.360 align:middle line:90%
Other questions?

00:46:29.360 --> 00:46:33.745 align:middle line:90%


00:46:33.745 --> 00:46:34.245 align:middle line:90%
Good?

00:46:34.245 --> 00:46:40.890 align:middle line:90%


00:46:40.890 --> 00:46:43.035 align:middle line:84%
So now we can talk about
equation of motion.

00:46:43.035 --> 00:46:53.970 align:middle line:90%


00:46:53.970 --> 00:46:58.510 align:middle line:84%
So it's the same thing
as classical mechanics.

00:46:58.510 --> 00:47:02.220 align:middle line:90%


00:47:02.220 --> 00:47:05.070 align:middle line:84%
So classical field theory is
just like classical mechanics.

00:47:05.070 --> 00:47:08.140 align:middle line:84%
You generalize to infinite
number of degrees of freedom.

00:47:08.140 --> 00:47:10.840 align:middle line:84%
So now with each
point in space you

00:47:10.840 --> 00:47:12.465 align:middle line:84%
associate with some
degrees of freedom.

00:47:12.465 --> 00:47:20.230 align:middle line:90%


00:47:20.230 --> 00:47:22.390 align:middle line:84%
OK, so now let's look at
the equation of motion.

00:47:22.390 --> 00:47:26.840 align:middle line:90%


00:47:26.840 --> 00:47:31.440 align:middle line:84%
OK, so again, you just do
the variation of the action,

00:47:31.440 --> 00:47:34.430 align:middle line:84%
extremize the
action to be 0, OK?

00:47:34.430 --> 00:47:37.160 align:middle line:84%
So the action is,
again, just defined

00:47:37.160 --> 00:47:41.570 align:middle line:84%
by the dt over the
Lagrangian, OK?

00:47:41.570 --> 00:47:45.280 align:middle line:90%


00:47:45.280 --> 00:47:50.920 align:middle line:84%
So the S is the
dt L and then can

00:47:50.920 --> 00:47:58.240 align:middle line:84%
be written as the four-integral
dt and dx of the density, OK?

00:47:58.240 --> 00:47:59.560 align:middle line:90%
So now let me just write--

00:47:59.560 --> 00:48:02.940 align:middle line:90%


00:48:02.940 --> 00:48:06.120 align:middle line:84%
let me now just write it in
a more relativistic notation,

00:48:06.120 --> 00:48:10.180 align:middle line:84%
just assuming there's one time
and one spatial derivatives,

00:48:10.180 --> 00:48:10.680 align:middle line:90%
OK?

00:48:10.680 --> 00:48:15.660 align:middle line:84%
So the mu-- now, yeah, I
combine the space and the time

00:48:15.660 --> 00:48:18.860 align:middle line:90%
derivative together, OK?

00:48:18.860 --> 00:48:23.000 align:middle line:84%
And you can straightforwardly
generalize to, say, involving

00:48:23.000 --> 00:48:26.190 align:middle line:90%
more spatial derivatives.

00:48:26.190 --> 00:48:27.030 align:middle line:90%
Good?

00:48:27.030 --> 00:48:33.520 align:middle line:84%
So here, so let's
just do the variation.

00:48:33.520 --> 00:48:36.720 align:middle line:84%
So we want 0 equal
to delta S. So you

00:48:36.720 --> 00:48:39.480 align:middle line:84%
want the variation
of this S to be 0.

00:48:39.480 --> 00:48:46.210 align:middle line:84%
So now let's just vary
S. So you have d4 x.

00:48:46.210 --> 00:48:49.980 align:middle line:84%
Now, just remember this
Langrangian density is just

00:48:49.980 --> 00:48:52.710 align:middle line:84%
an ordinary function
because of the locality

00:48:52.710 --> 00:48:55.830 align:middle line:84%
just an ordinary function of
phi_a and its derivatives, OK?

00:48:55.830 --> 00:49:00.180 align:middle line:84%
And we can just
do the- variation

00:49:00.180 --> 00:49:01.950 align:middle line:90%
in a straightforward way.

00:49:01.950 --> 00:49:07.080 align:middle line:84%
We can just write a partial
L, partial phi a and delta phi

00:49:07.080 --> 00:49:22.370 align:middle line:84%
a plus partial L partial mu phi
a and delta partial mu phi a,

00:49:22.370 --> 00:49:22.870 align:middle line:90%
OK?

00:49:22.870 --> 00:49:32.220 align:middle line:90%


00:49:32.220 --> 00:49:35.850 align:middle line:84%
And in this case,
so the delta is

00:49:35.850 --> 00:49:38.070 align:middle line:90%
a variation and the partial mu.

00:49:38.070 --> 00:49:40.600 align:middle line:84%
So these two operations are
independent of each other.

00:49:40.600 --> 00:49:42.080 align:middle line:90%
So you can exchange them, OK?

00:49:42.080 --> 00:49:43.870 align:middle line:84%
So you can put
extra delta inside.

00:49:43.870 --> 00:49:47.775 align:middle line:84%
So this is the same as
partial mu delta phi_a, OK?

00:49:47.775 --> 00:49:52.300 align:middle line:90%


00:49:52.300 --> 00:49:56.280 align:middle line:84%
So now you can integrate by
parts of the second term,

00:49:56.280 --> 00:50:00.000 align:middle line:84%
OK, because here is a little
bit awkward because involving

00:50:00.000 --> 00:50:01.630 align:middle line:90%
the partial mu of delta phi a.

00:50:01.630 --> 00:50:04.190 align:middle line:84%
So again, here the
repeated indices,

00:50:04.190 --> 00:50:05.370 align:middle line:90%
they're all summed, OK?

00:50:05.370 --> 00:50:08.340 align:middle line:90%
So you should sum them.

00:50:08.340 --> 00:50:10.775 align:middle line:84%
So now we can do
integration by parts here.

00:50:10.775 --> 00:50:14.320 align:middle line:90%


00:50:14.320 --> 00:50:20.780 align:middle line:84%
So then you get the d4 x, and
you get partial L partial phi

00:50:20.780 --> 00:50:31.450 align:middle line:84%
a minus partial mu partial L
partial partial mu phi a, OK?

00:50:31.450 --> 00:50:37.970 align:middle line:84%
And then the whole thing,
delta phi a and then

00:50:37.970 --> 00:50:38.945 align:middle line:90%
path boundary terms.

00:50:38.945 --> 00:50:44.250 align:middle line:90%


00:50:44.250 --> 00:50:45.830 align:middle line:84%
So the boundary
terms come from you

00:50:45.830 --> 00:50:49.840 align:middle line:84%
do integration by
parts here, OK?

00:50:49.840 --> 00:50:53.370 align:middle line:84%
You're doing integration
by parts here.

00:50:53.370 --> 00:50:56.970 align:middle line:84%
So the boundary term will be
proportional to delta phi a,

00:50:56.970 --> 00:50:57.630 align:middle line:90%
OK?

00:50:57.630 --> 00:50:58.380 align:middle line:90%
Yes.

00:50:58.380 --> 00:51:00.920 align:middle line:84%
AUDIENCE: So could
you define delta?

00:51:00.920 --> 00:51:03.410 align:middle line:84%
PROFESSOR: Delta's just the
arbitrary variation, yeah,

00:51:03.410 --> 00:51:04.670 align:middle line:90%
just arbitrary variation.

00:51:04.670 --> 00:51:08.620 align:middle line:90%


00:51:08.620 --> 00:51:09.550 align:middle line:90%
Other questions?

00:51:09.550 --> 00:51:12.120 align:middle line:90%


00:51:12.120 --> 00:51:16.140 align:middle line:84%
OK, so the boundary--
so we always

00:51:16.140 --> 00:51:19.800 align:middle line:84%
assume we have the boundary
conditions, OK, and

00:51:19.800 --> 00:51:22.020 align:middle line:84%
so that the boundary
term vanishes, OK?

00:51:22.020 --> 00:51:25.810 align:middle line:84%
So here I will not
go into to detail.

00:51:25.810 --> 00:51:32.030 align:middle line:84%
So we always choose
delta phi a so

00:51:32.030 --> 00:51:35.450 align:middle line:84%
that boundary
terms vanishes, OK?

00:51:35.450 --> 00:51:45.190 align:middle line:84%
The term comes from
integration by parts, OK?

00:51:45.190 --> 00:51:52.655 align:middle line:84%
And the second, just remember
repeated indices are summed.

00:51:52.655 --> 00:51:56.875 align:middle line:90%


00:51:56.875 --> 00:51:57.375 align:middle line:90%
OK?

00:51:57.375 --> 00:52:00.030 align:middle line:90%


00:52:00.030 --> 00:52:06.520 align:middle line:84%
And now, then we don't have
to worry about boundary terms.

00:52:06.520 --> 00:52:07.770 align:middle line:90%
So the boundary term vanishes.

00:52:07.770 --> 00:52:12.080 align:middle line:84%
Then now we just have
this has to be 0.

00:52:12.080 --> 00:52:14.940 align:middle line:84%
But this has to be
0 for any variation.

00:52:14.940 --> 00:52:21.520 align:middle line:84%
So this delta x mu can
be any function of x, OK?

00:52:21.520 --> 00:52:22.900 align:middle line:90%
Can be any function of x.

00:52:22.900 --> 00:52:26.050 align:middle line:84%
The only way this can happen
is that this prefactor

00:52:26.050 --> 00:52:27.370 align:middle line:90%
must vanish, OK?

00:52:27.370 --> 00:52:36.830 align:middle line:84%
So this implies that the delta
L partial phi a minus partial

00:52:36.830 --> 00:52:46.670 align:middle line:84%
mu partial L partial partial
mu phi a, [INAUDIBLE],, OK?

00:52:46.670 --> 00:52:51.260 align:middle line:84%
So this is the general equation
of motion for classical field

00:52:51.260 --> 00:52:51.915 align:middle line:90%
theory, OK?

00:52:51.915 --> 00:53:04.600 align:middle line:90%


00:53:04.600 --> 00:53:05.100 align:middle line:90%
Good?

00:53:05.100 --> 00:53:06.035 align:middle line:90%
Any questions on this?

00:53:06.035 --> 00:53:12.320 align:middle line:90%


00:53:12.320 --> 00:53:18.020 align:middle line:84%
OK, so from now on, we will
make two restrictions, OK?

00:53:18.020 --> 00:53:20.700 align:middle line:90%


00:53:20.700 --> 00:53:26.910 align:middle line:84%
Just for simplicity,
also they're the most--

00:53:26.910 --> 00:53:29.190 align:middle line:84%
we make two restrictions
just for simplicity.

00:53:29.190 --> 00:53:31.800 align:middle line:90%


00:53:31.800 --> 00:53:36.330 align:middle line:84%
Yeah, most of our discussion
can be generalized

00:53:36.330 --> 00:53:38.910 align:middle line:90%
beyond those situations.

00:53:38.910 --> 00:53:43.650 align:middle line:84%
Yeah, so the first
is that we restrict

00:53:43.650 --> 00:53:56.620 align:middle line:84%
to field theories which are
translationally invariant, OK?

00:53:56.620 --> 00:53:58.465 align:middle line:84%
Means that there's
no special point.

00:53:58.465 --> 00:54:01.440 align:middle line:90%


00:54:01.440 --> 00:54:06.370 align:middle line:84%
Means there's no special
spacetime point in this theory,

00:54:06.370 --> 00:54:06.870 align:middle line:90%
OK?

00:54:06.870 --> 00:54:10.013 align:middle line:84%
If you do experiment
here in Boston,

00:54:10.013 --> 00:54:12.180 align:middle line:84%
it's the same as you do
experiment in Washington DC,

00:54:12.180 --> 00:54:13.380 align:middle line:90%
et cetera, OK?

00:54:13.380 --> 00:54:16.110 align:middle line:84%
And if your theory is not
translation invariant,

00:54:16.110 --> 00:54:18.058 align:middle line:84%
and then when you do
experiment in Boston,

00:54:18.058 --> 00:54:20.100 align:middle line:84%
then it will be different
from when you do it in,

00:54:20.100 --> 00:54:23.170 align:middle line:90%
say, in New York.

00:54:23.170 --> 00:54:28.420 align:middle line:84%
And the second is that we
assume it's a Lorentz invariant.

00:54:28.420 --> 00:54:30.150 align:middle line:84%
So in some condensed
matter applications,

00:54:30.150 --> 00:54:33.780 align:middle line:84%
which you don't have
Lorentz symmetry, and then

00:54:33.780 --> 00:54:38.310 align:middle line:90%
you don't have Lorentz symmetry.

00:54:38.310 --> 00:54:43.770 align:middle line:84%
But yeah, but similar techniques
we are going to talk about

00:54:43.770 --> 00:54:47.270 align:middle line:90%
can be applied.

00:54:47.270 --> 00:54:49.670 align:middle line:84%
And we will also elaborate
a little bit more

00:54:49.670 --> 00:54:51.770 align:middle line:90%
on both of these aspects.

00:54:51.770 --> 00:54:58.920 align:middle line:90%


00:54:58.920 --> 00:55:02.610 align:middle line:84%
OK, so now let me give
you some simple examples

00:55:02.610 --> 00:55:06.030 align:middle line:84%
of the classical
field series, OK,

00:55:06.030 --> 00:55:07.770 align:middle line:84%
which satisfy these
two condition.

00:55:07.770 --> 00:55:47.490 align:middle line:90%


00:55:47.490 --> 00:55:51.690 align:middle line:84%
Examples, so the first
one is the Maxwell theory.

00:55:51.690 --> 00:55:55.590 align:middle line:90%


00:55:55.590 --> 00:56:06.160 align:middle line:84%
So for Maxwell's theory,
the dynamical variable,

00:56:06.160 --> 00:56:11.450 align:middle line:84%
this 4-vector
potential A mu, OK?

00:56:11.450 --> 00:56:14.180 align:middle line:84%
So now I'm using the
four-vector notation.

00:56:14.180 --> 00:56:18.720 align:middle line:84%
So I will often suppress
the indices on x.

00:56:18.720 --> 00:56:22.580 align:middle line:84%
That means this
is a 4-vector, OK?

00:56:22.580 --> 00:56:26.030 align:middle line:84%
And then from the A mu we can
define the field strengths.

00:56:26.030 --> 00:56:35.650 align:middle line:90%


00:56:35.650 --> 00:56:37.810 align:middle line:84%
And the E and B
then can be obtained

00:56:37.810 --> 00:56:42.940 align:middle line:84%
from this F mu nu OK, can
be obtained from F mu nu

00:56:42.940 --> 00:56:50.080 align:middle line:84%
And then the action for the E
& M then can be obtained by--

00:56:50.080 --> 00:56:59.820 align:middle line:90%


00:56:59.820 --> 00:57:01.680 align:middle line:90%
OK, that's this, OK?

00:57:01.680 --> 00:57:11.250 align:middle line:84%
So this is the action for the
E & M. without charged matter,

00:57:11.250 --> 00:57:12.030 align:middle line:90%
OK?

00:57:12.030 --> 00:57:15.570 align:middle line:84%
So if you do the
variation here, OK,

00:57:15.570 --> 00:57:18.060 align:middle line:84%
to get the equation of motion,
then you get the Maxwell

00:57:18.060 --> 00:57:21.210 align:middle line:90%
equations without sources, OK?

00:57:21.210 --> 00:57:23.700 align:middle line:84%
You get the vacuum
Maxwell equations.

00:57:23.700 --> 00:57:25.320 align:middle line:84%
You get the vacuum
Maxwell equations.

00:57:25.320 --> 00:57:28.170 align:middle line:90%


00:57:28.170 --> 00:57:31.140 align:middle line:84%
And so you see that
this action have

00:57:31.140 --> 00:57:34.110 align:middle line:84%
the form we mentioned
here is corresponding

00:57:34.110 --> 00:57:38.700 align:middle line:84%
to a local theory, OK, a single
integral of space time, OK?

00:57:38.700 --> 00:57:42.370 align:middle line:90%


00:57:42.370 --> 00:57:49.000 align:middle line:84%
And the second example
is Einstein gravity

00:57:49.000 --> 00:57:53.540 align:middle line:84%
is another classical
field theory, OK?

00:57:53.540 --> 00:57:55.650 align:middle line:84%
So let me just write
down the action.

00:57:55.650 --> 00:57:58.880 align:middle line:90%
So this is written as G Newton.

00:57:58.880 --> 00:58:04.720 align:middle line:90%


00:58:04.720 --> 00:58:06.250 align:middle line:90%
So G is the metric.

00:58:06.250 --> 00:58:10.560 align:middle line:84%
Yeah, anyway, if you don't
know the Einstein theories, OK?

00:58:10.560 --> 00:58:12.760 align:middle line:90%
OK, so R is a Ricci scalar.

00:58:12.760 --> 00:58:15.730 align:middle line:84%
And if you don't know
the Einstein series, OK.

00:58:15.730 --> 00:58:17.230 align:middle line:84%
I'm just write it
down just to show

00:58:17.230 --> 00:58:19.600 align:middle line:84%
that this is a local
theory involving

00:58:19.600 --> 00:58:25.880 align:middle line:84%
only a single spacetime
integral of some quantities.

00:58:25.880 --> 00:58:30.230 align:middle line:84%
And the simplest
quantum field theories

00:58:30.230 --> 00:58:40.500 align:middle line:84%
are called the scalar
field theories because--

00:58:40.500 --> 00:58:42.570 align:middle line:90%
I think I erased it--

00:58:42.570 --> 00:58:45.780 align:middle line:84%
because this involving
the vector field, OK?

00:58:45.780 --> 00:58:48.960 align:middle line:84%
So this is a vector field
because at each spacetime point

00:58:48.960 --> 00:58:49.980 align:middle line:90%
there's a vector.

00:58:49.980 --> 00:58:54.900 align:middle line:84%
Here, the Einstein gravity, the
dynamical variable is a tensor.

00:58:54.900 --> 00:58:56.920 align:middle line:84%
So it's even something
more complicated.

00:58:56.920 --> 00:58:58.650 align:middle line:84%
So the simplest
case is a scalar,

00:58:58.650 --> 00:59:01.920 align:middle line:84%
which at each spacetime point,
you just have a single value

00:59:01.920 --> 00:59:04.320 align:middle line:90%
quantity, OK?

00:59:04.320 --> 00:59:07.110 align:middle line:84%
And so let's just consider
the simplest case.

00:59:07.110 --> 00:59:23.030 align:middle line:84%
Just consider a real value
scalar field phi, OK?

00:59:23.030 --> 00:59:26.340 align:middle line:84%
So this is just
take a real value.

00:59:26.340 --> 00:59:29.060 align:middle line:84%
You can also consider
the complex value, OK?

00:59:29.060 --> 00:59:30.870 align:middle line:84%
But the simplest is
just a real value.

00:59:30.870 --> 00:59:33.650 align:middle line:84%
So at each spacetime point
you have a single value, OK?

00:59:33.650 --> 00:59:37.500 align:middle line:84%
And so that's the
dynamical variable.

00:59:37.500 --> 00:59:42.800 align:middle line:84%
And so in real life there are
many examples of scalar fields.

00:59:42.800 --> 00:59:46.740 align:middle line:84%
So for example, the Higgs,
OK, the Higgs field,

00:59:46.740 --> 00:59:49.970 align:middle line:84%
which is a celebrated
Higgs field, which

00:59:49.970 --> 00:59:58.040 align:middle line:84%
was discovered a number of years
ago, is a real scalar field.

00:59:58.040 --> 01:00:01.880 align:middle line:84%
And also pions, OK,
the pion particle

01:00:01.880 --> 01:00:06.310 align:middle line:84%
can be considered as
excitations of a scalar field.

01:00:06.310 --> 01:00:15.530 align:middle line:84%
Anyway, so simplest family
of a scalar field action

01:00:15.530 --> 01:00:29.300 align:middle line:90%
is the volume form, OK?

01:00:29.300 --> 01:00:34.820 align:middle line:90%


01:00:34.820 --> 01:00:39.060 align:middle line:84%
So here I just write
this action down.

01:00:39.060 --> 01:00:42.050 align:middle line:84%
But essentially, you can argue
this is the most general action

01:00:42.050 --> 01:00:48.340 align:middle line:84%
you can write down, which
are compatible with locality,

01:00:48.340 --> 01:00:54.770 align:middle line:84%
OK, with locality and
with translation symmetry

01:00:54.770 --> 01:00:56.330 align:middle line:90%
and the Lorentz symmetry, OK?

01:00:56.330 --> 01:00:59.960 align:middle line:84%
So here you see the Lorentz
indices are contracted.

01:00:59.960 --> 01:01:02.600 align:middle line:84%
The Lorentz indices in the
derivative are contracted.

01:01:02.600 --> 01:01:06.660 align:middle line:84%
So that guarantees the
series Lorentz invariant.

01:01:06.660 --> 01:01:09.600 align:middle line:90%
And V just some function of phi.

01:01:09.600 --> 01:01:14.100 align:middle line:84%
V here is just some
function of phi.

01:01:14.100 --> 01:01:19.140 align:middle line:84%
So this is essentially
the simplest theory

01:01:19.140 --> 01:01:22.560 align:middle line:84%
you can write down
based on localities, OK?

01:01:22.560 --> 01:01:26.120 align:middle line:84%
So say we can take V
phi, in general, we

01:01:26.120 --> 01:01:28.560 align:middle line:84%
can take it to be polynomial,
yeah, just some function

01:01:28.560 --> 01:01:33.965 align:middle line:90%
of phi, OK?

01:01:33.965 --> 01:01:39.830 align:middle line:90%


01:01:39.830 --> 01:01:45.920 align:middle line:84%
So when you have a
translation symmetry,

01:01:45.920 --> 01:01:48.740 align:middle line:84%
means that in this theory
if you do experiment

01:01:48.740 --> 01:01:52.430 align:middle line:90%
it's the same everywhere.

01:01:52.430 --> 01:01:58.880 align:middle line:84%
That implies that V phi-- so
here there's no parameter here.

01:01:58.880 --> 01:02:05.260 align:middle line:90%
V(phi) then must-- yeah, sorry.

01:02:05.260 --> 01:02:17.820 align:middle line:84%
All parameters in V(phi)
must be constant, OK?

01:02:17.820 --> 01:02:22.425 align:middle line:84%
They cannot depend on space
time, OK, must be constant.

01:02:22.425 --> 01:02:25.197 align:middle line:90%


01:02:25.197 --> 01:02:27.280 align:middle line:84%
So if you have some
parameters depend on the space

01:02:27.280 --> 01:02:31.690 align:middle line:84%
time, then of course,
different space time point

01:02:31.690 --> 01:02:33.250 align:middle line:90%
will behave differently.

01:02:33.250 --> 01:02:37.660 align:middle line:84%
And they will not be
translation invariant, OK?

01:02:37.660 --> 01:02:40.888 align:middle line:84%
So let's try to
work out this theory

01:02:40.888 --> 01:02:42.055 align:middle line:90%
a little bit in more detail.

01:02:42.055 --> 01:02:50.670 align:middle line:90%


01:02:50.670 --> 01:02:52.760 align:middle line:84%
So let's find this
Lagrangian density--

01:02:52.760 --> 01:03:01.620 align:middle line:84%
let's try to find its
momentum and the Hamiltonian.

01:03:01.620 --> 01:03:04.670 align:middle line:90%


01:03:04.670 --> 01:03:07.400 align:middle line:90%
So it's momentum density.

01:03:07.400 --> 01:03:11.090 align:middle line:84%
Here there's only a single
field, so there's no index.

01:03:11.090 --> 01:03:15.750 align:middle line:84%
So pi x would be take
derivative of this.

01:03:15.750 --> 01:03:21.170 align:middle line:84%
So this is L. So this thing
inside the bracket is L,

01:03:21.170 --> 01:03:28.690 align:middle line:84%
this partial L partial phi dot,
OK, time derivative of phi.

01:03:28.690 --> 01:03:32.830 align:middle line:84%
And so if you look at here,
here the time derivative

01:03:32.830 --> 01:03:36.100 align:middle line:84%
is just phi dot squared,
OK, using this four vector

01:03:36.100 --> 01:03:37.250 align:middle line:90%
notation.

01:03:37.250 --> 01:03:42.130 align:middle line:84%
And if you expand it, and then
the momentum density just phi

01:03:42.130 --> 01:03:52.400 align:middle line:84%
dot x, OK, because just
because partial mu phi

01:03:52.400 --> 01:03:59.570 align:middle line:84%
partial mu phi is equal to
minus phi dot squared plus, OK?

01:03:59.570 --> 01:04:02.610 align:middle line:90%


01:04:02.610 --> 01:04:05.550 align:middle line:84%
And then you can find
the Hamiltonian density.

01:04:05.550 --> 01:04:10.870 align:middle line:90%


01:04:10.870 --> 01:04:17.080 align:middle line:84%
Pi phi dot minus L, and then
you find that this is given by--

01:04:17.080 --> 01:04:28.410 align:middle line:84%
we express in terms
of momentum, OK?

01:04:28.410 --> 01:04:39.800 align:middle line:84%
And the equation of
motion, if you apply here,

01:04:39.800 --> 01:04:44.120 align:middle line:84%
then you find given by partial
square equal to partial V

01:04:44.120 --> 01:04:48.140 align:middle line:90%
partial phi equal to 0, OK?

01:04:48.140 --> 01:04:50.670 align:middle line:84%
And again, here I'm using
a shorthand notation.

01:04:50.670 --> 01:04:56.090 align:middle line:84%
Partial square is defined to
be partial mu partial mu, OK?

01:04:56.090 --> 01:05:01.830 align:middle line:84%
So this is the same as minus
partial t squared plus v

01:05:01.830 --> 01:05:03.240 align:middle line:90%
squared, OK?

01:05:03.240 --> 01:05:08.510 align:middle line:90%


01:05:08.510 --> 01:05:25.160 align:middle line:84%
So here let me also give you
some simple examples of the v.

01:05:25.160 --> 01:05:26.390 align:middle line:90%
In the simplest case--

01:05:26.390 --> 01:05:34.120 align:middle line:90%


01:05:34.120 --> 01:05:42.673 align:middle line:84%
so in the simplest case, is
you take v just, of course,

01:05:42.673 --> 01:05:43.590 align:middle line:90%
it will be a constant.

01:05:43.590 --> 01:05:47.300 align:middle line:84%
But a constant does not
do anything because the--

01:05:47.300 --> 01:05:49.520 align:middle line:84%
yeah, so the
simplest case is just

01:05:49.520 --> 01:05:50.855 align:middle line:90%
v phi equal to linear term.

01:05:50.855 --> 01:05:54.710 align:middle line:90%


01:05:54.710 --> 01:05:56.450 align:middle line:84%
And from time
translation symmetry,

01:05:56.450 --> 01:05:58.400 align:middle line:90%
this f has to be a constant, OK?

01:05:58.400 --> 01:06:01.100 align:middle line:84%
It cannot be depend
on space time.

01:06:01.100 --> 01:06:03.680 align:middle line:84%
If this depend on space
time, and then you

01:06:03.680 --> 01:06:10.490 align:middle line:84%
will violate the space
translation symmetry, OK?

01:06:10.490 --> 01:06:14.990 align:middle line:84%
So when you have these, then
your equation of motion,

01:06:14.990 --> 01:06:21.600 align:middle line:84%
given by partial squared equal
to f, OK? f is a constant.

01:06:21.600 --> 01:06:23.690 align:middle line:90%
So in this case--

01:06:23.690 --> 01:06:26.210 align:middle line:84%
so essentially you have
some kind of thing, which--

01:06:26.210 --> 01:06:29.770 align:middle line:84%
some kind of a constant,
which is source for this phi.

01:06:29.770 --> 01:06:32.390 align:middle line:84%
So in this case, we say the
system have external force.

01:06:32.390 --> 01:06:38.720 align:middle line:90%


01:06:38.720 --> 01:06:47.220 align:middle line:84%
OK, we call this f external
force because if f is nonzero,

01:06:47.220 --> 01:06:49.430 align:middle line:90%
then phi cannot be 0, OK?

01:06:49.430 --> 01:06:50.930 align:middle line:90%
Phi has to be nonzero.

01:06:50.930 --> 01:06:54.770 align:middle line:84%
And so in a sense, the phi
will be always be excited.

01:06:54.770 --> 01:06:57.320 align:middle line:90%
It will always be generate--

01:06:57.320 --> 01:07:00.800 align:middle line:84%
nonzero phi will always be
generated if s is nonzero, OK?

01:07:00.800 --> 01:07:04.310 align:middle line:84%
So here we call it
the external force.

01:07:04.310 --> 01:07:09.540 align:middle line:84%
And so normally we
like to consider phi--

01:07:09.540 --> 01:07:13.123 align:middle line:84%
normally we will
consider-- normally

01:07:13.123 --> 01:07:14.540 align:middle line:84%
we are interested
in the situation

01:07:14.540 --> 01:07:15.832 align:middle line:90%
there is no external force, OK?

01:07:15.832 --> 01:07:18.960 align:middle line:90%
The system develop on its own.

01:07:18.960 --> 01:07:22.700 align:middle line:84%
So in that case, so if we
forget about the external force,

01:07:22.700 --> 01:07:30.330 align:middle line:84%
then the simplest situation will
be phi equal to quadratic, OK?

01:07:30.330 --> 01:07:33.420 align:middle line:84%
It's some polynomial,
phi squared, OK?

01:07:33.420 --> 01:07:36.990 align:middle line:84%
So that's the next
simplest function of phi.

01:07:36.990 --> 01:07:40.650 align:middle line:84%
And m squared has to be
a constant, OK, again,

01:07:40.650 --> 01:07:43.940 align:middle line:90%
from the translation symmetry.

01:07:43.940 --> 01:07:48.490 align:middle line:84%
So in this case, then you
have the equation of motion,

01:07:48.490 --> 01:07:52.270 align:middle line:84%
partial square phi equal
to m square phi squared,

01:07:52.270 --> 01:07:53.800 align:middle line:90%
m square phi equal to 0.

01:07:53.800 --> 01:07:58.980 align:middle line:90%


01:07:58.980 --> 01:08:02.377 align:middle line:84%
So this is a very
famous equation.

01:08:02.377 --> 01:08:04.210 align:middle line:84%
So this is called the
Klein-Gordon equation.

01:08:04.210 --> 01:08:10.550 align:middle line:90%


01:08:10.550 --> 01:08:15.160 align:middle line:84%
And the next time you
will see very quickly why

01:08:15.160 --> 01:08:17.350 align:middle line:90%
this is a famous equation, OK?

01:08:17.350 --> 01:08:23.680 align:middle line:84%
And yeah, yeah, so this would
correspond to the simplest

01:08:23.680 --> 01:08:26.529 align:middle line:90%
field theory we can consider.

01:08:26.529 --> 01:08:29.080 align:middle line:84%
And then you can consider more
complicated field theories

01:08:29.080 --> 01:08:33.970 align:middle line:84%
with V(phi) to be some higher
polynomial or more complicated

01:08:33.970 --> 01:08:36.520 align:middle line:90%
functions.

01:08:36.520 --> 01:08:40.120 align:middle line:84%
But keep in mind that
this square is special.

01:08:40.120 --> 01:08:43.430 align:middle line:84%
Because it's square, this
is a linear equation.

01:08:43.430 --> 01:08:46.340 align:middle line:84%
But whenever you have
something which is not

01:08:46.340 --> 01:08:50.479 align:middle line:84%
linear or quadratic, and you
will get a nonlinear equation.

01:08:50.479 --> 01:08:52.670 align:middle line:84%
And then the story
becomes complicated, OK?

01:08:52.670 --> 01:08:54.800 align:middle line:84%
Nonlinear equations
are always complicated.

01:08:54.800 --> 01:08:58.399 align:middle line:84%
And anyway, so this
would be the simplest

01:08:58.399 --> 01:09:01.160 align:middle line:90%
theory we will consider, OK?

01:09:01.160 --> 01:09:03.820 align:middle line:90%
OK, so let's stop here.

01:09:03.820 --> 01:09:09.000 align:middle line:90%