Hello and thanks for joining me today, I know
the world is a bit crazy at the moment and
I hope you're doing OK.
Remember that one of the best ways to stop
the spread of the coronavirus is to stay home
as much as you can.
Over the last few days i’ve been at home
trying to understand this - which is richard
feynman’s PhD thesis.
Now I wanted to make a video on this topic
a long time ago but on my first attempt trying
to read it I found it really difficult and
put it in the too hard basket.
But now with a little bit more reading i think
i finally understand enough of it to share
the basic idea with you of what’s going
on.
So Richard Feynman was a physicist who won
the nobel prize for his work in electrodynamics.
The work that he won the prize for was done
a bit later on after submitting his thesis
but a lot of the foundation ideas were laid
down in here while he was a student.
There are no feynman diagrams in here though.
It was submitted in 1942 at Princeton University
and Feynman would have been about 24 years
old at the time he submitted this.
He was supervised by John Wheeler and the
title of the thesis is “Principles of Least
Action in Quantum Mechanics”.
So let’s have a little look at it.
There will be a link down to this document
in the description so you can read along at
home if you’d like to.
There will also be some links to various further
reading sources so that you can get more of
an explanation, more than I've got time to
go into today.
So let’s have a little start with the contents
page, we’ll read through what each section
is and give a brief overview.
At the start Feynman gives us an introduction
into some of the current problems with theories
of quantum electrodynamics.
These problems have got to do with infinities
popping up in calculations and Feynman's whole
approach here is to solve some of these problems
that he points out by trying to do away with
the need for a field.
And instead to describe everything as direct
interactions between particles.
If Feynman wants to be able to describe everything
without a field then he’ll need to come
up with his own formulation of quantum mechanics
and he does this by first coming up with a
new formulation in classical mechanics by
using the principle of least action.
And once he’s done that he needs to also
make it work for quantum mechanics and that
turns out to be a bit of a challenge.
Feynman liked to emphasise the value of approaching
old problems in a new way and that’s what
we really see him doing here in this work.
He’s coming up with a new way to approach
things.
Through his discussion of least action he
talks about the mathematical concept of a
functional which is really just a number which
depends on a function.
He talks about conservation of energy and
things like noether's theorem and he does
some examples for things like particles interacting
through an intermediate oscillator.
And down here in the part where he needs to
make it work for quantum mechanics he talks
about the lagrangian in quantum mechanics,
he talks about equations of motion.
Do note that some of the ideas that i'm going
to talk about and that are discussed in this
thesis, even some of the main motivating ideas,
were later discarded as not being really that
useful even by feynman himself when he goes
on to a lot more work on this and to win his
nobel prize.
So it's not necessarily true that we need
to do away with the concept of a field but
it is what Feynman was trying to do in this
thesis.
So let's take a closer look, here’s our
introduction and he says in here that one
of the problems with quantum electrodynamics
is that taken literally they predict infinite
values for many experimental quantities.
And from what i can tell it boiled down to
two problems.
One is that we have a self-interaction of
the charge with its own field and when we
do calculations regarding this they predict
that a pointlike electron would have an infinite
mass.
And this is a problem that exists even in
the classical theory.
And the second problem that i can tell is
that there seems to be something to do with
an infinite number of degrees of freedom of
this electromagnetic field.
So you can see feynman wants to solve both
of these problems by just doing away with
the idea of the field altogether.
In Fact he thought that the idea that a particle
could act on itself is infact a silly one
and instead suggests that electrons cannot
act on themselves, they can only act on other
electrons.
There’s no field, just a direct interaction
between charges.
Although with a delay.
For example he says that if an electron in
the sun were to shake, then an electron in
your eye would shake 8 minutes later.
There was one big obstacle to this idea however.
See when a charged particle such as an electron
is accelerated it emits electromagnetic radiation
but this radiation carries off energy and
momentum which by conservation must result
in work being done back on the original charge.
Back on the original electron.
It's referred to as radiation resistance,
radiation reaction, self-force, or the abraham-lorentz
force and you have to do extra work to account
for that energy.
This can be accounted for in calculations
if you have an electron acting on itself in
a field but since feynman is not allowing
that and he's only letting electrons act on
other electrons then the only possible source
to explain this radiation reaction is another
electron in the world.
So he considers two charges that interact
in such a way that the second charge, accelerated
by absorbing the radiation emitted by the
first charge then itself emits radiation which
reacts back upon the first but feynmans supervisor
pointed out that this could not explain radiation
reaction because it would come with a time
delay - the time required for light to pass
between the two particles whereas radiation
reaction was instantaneous.
But then in order to do away with the time
delay problem feynman and his supervisor come
up with this totally crazy theory that's called
the wheeler-feynman absorber theory and I
find it a bit hard to article because it just
sounds really out there but they suppose that
the return action by the absorber charge reaches
the source by advanced waves as well as by
the ordinary reflected waves.
So that the law of interaction acts backwards
in time as well as forwards in time.
This supposedly doesn't violate causality
because there’s a solution to maxwell's
equations that allows a solution which is
slightly delayed in time as well as one that
is slightly advanced in time.
Maybe check out the wikipedia page for this
to see a bit of the maths there.
Feynman went on to show that this can account
for radiation reaction if you use a mixture
of 50/50 advanced waves to the normal reflected
waves.
Feynman mentions that here in the introduction
to his thesis that he’s using this theory
developed by john wheeler and himself and
that under this theory the field becomes nothing
more than a mathematical construction that
you can use to aid in some of the problem
solving here.
We can see down here that his theory does
contain a symmetrical nature with respect
to past and future.
And hes using half the retarded waves plus
half the advanced solution.
What he does in the next chapter is to develop
an action function which involves only the
coordinates of the particles with no mention
of fields being made.
So let's go and have a look at that next chapter.
So I read that when feynman was in high school,
after class one day his teacher said to him
“you look bored, I want to tell you something
interesting” feynman was told about the
principle of least action and he apparently
found it so fascinating that whenever it came
up he would chose to work on it.
And we see it feature very prominently here
in his thesis.
For those who haven’t encountered it before
i’ll try to give a very brief introduction
to the principle of least action but also
to the concept of a lagrangian.
So we define a lagrangian to be a function
of the positions and the velocities of all
particles in a system.
We write the lagrangian as a function of the
total energy so in fact it is the kinetic
energy of the system minus the potential energy.
Now there are many possible paths but only
one true path which is taken by a system.
And which true path is it?
How do we find it out?
Well that is how the principle of least action
comes in.
So for each possible path, we can assign a
number called the action and it’s written
as S which is an integral from the initial
time to the final time of the lagrangian with
respect to time.
The principle of least action says that the
actual path taken by the system is an extremum
of S. So often it is the minimum of S. So
we’re saying that if I can find the minimum
of this function here, I can find the actual
path taken by the system.
I’ve put a link to my favorite notes on
classical mechanics in the description to
again allow you to read a bit more about this
here because I can't go too much into depth
but what we can do is flip through some of
the pages here where fyenman talks himself
about some of these principles, he does i
guess a few proofs and talks about the mathematics
behind it.
His mention of noether's theorem is here in
regards to the conservation of energy and
constants of motion.
But all you need to know here is that feynman
is trying to come up with a new way to describe
the path that a system will follow.
You can see some of his big expressions for
the action here.
So he will be wanting to find the extremum
of this function to describe what's happening
with his particles.
And that takes us to the third section in
the thesis which is applying his theory of
least action to quantum mechanics.
Now he says that it's usually kind of easy
to go from a quantum description to a classical
one by just letting hbar approach zero but
it is not so easy to go in the reverse.
To create a quantum description from a classical
one.
He tries many things along the way and I think
ultimately what helps him out is a paper that
he finds by Dirac which concerns the analogue
of the lagrangian and the action in quantum
mechanics.
So Dirac essentially describes a way that
the lagrangian might pop up in quantum mechanics
and Feynman really extends that work and shows
actually how it happens.
It seems that he does that by replacing some
classical components with an exponential and
this exponential is e to the i S /hbar where
S is the action that he’s been talking about
previously.
Doing some more maths he gets this expression
here.
In what feynman called the hamilton view,
a differential equation gives you the field
at the next moment based on what’s happening
in the field at the current moment but instead
feynman doesn't have that approach.
He has a thing that describes the whole character
of the path through all of space and time.
Something that the particle did in the past
is going to affect its future.
This is what feynman liked about his approach,
it gives a more complete view of the whole
view of the particle.
Feynman's approach should be equivalent to
other approaches like the schrodinger equation
and he does make some comments on the fact
that there are many ways to seemingly approach
the same problem but that that is not necessarily
a bad thing.
In fact he says of nature that “perhaps
a thing is simple if you can describe it fully
in several different ways without immediately
knowing that you are describing the same thing.''
His work leads on to the fact that for each
possible way that a particle can go from one
point to another in spacetime there's an amplitude
which is given by e to the i/hbar times the
action for the path.
And you can add up amplitudes from various
paths.
There were still a few problems and things
that needed to be figured out before feynman
would go on to win the nobel prize.
It seems that the half-advanced and half-reflected
waves thing wasn't really used much more into
the future.
His initial expression for the action wasn't
used.
And he abandoned the idea that charges do
not act on themselves.
But the idea of this path integral formulation
of quantum mechanics was useful and all of
it really became the foundation for which
he built on to go and do a lot more work.
So thanks for watching, I hope you gained
a little bit of insight from this into what
is a really dense document and something that's
quite difficult to understand.
All I hope is that it's inspired you to maybe
want to read a bit more about this yourself.
Check out Feynman's nobel address which is
something that really helped me to understand
some of these ideas and I've still got a lot
of reading and understanding to do of my own.
Stay safe in this crazy world of ours and
I'll see you next time.
This video was made possible by viewers who
support me on Patreon.
In particular, today's patron cat of the day,
Simon.
Thanks for your support.
