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.
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