The Feynman diagrams revolutionized particle physics by providing a simple system to sort out the infinite possibilities
when elementary particles interact.
This incredible simplicity provides some stunning insights into the nature of reality.
Feynman's Path integral shows us that to properly calculate the probability of a particle traveling between two points
we need to add up the contributions
from all conceivable paths between those points;
including the impossible paths. In fact we can go even further...
According to Feynman's approach to quantum mechanics, every conceivable
"Happening" that leads from a measured initial state to a measured final state; it does in a sense happen,
at least in the math.
To calculate the probability of any quantum system evolving between two states we need to sum over every conceivable
intermediate state.
This is impossible
because there are infinite possible intermediate states.
But as we discussed in our episode on solving impossible equations, the Feynman diagrams allow
physicists to quickly figure out which of the infinite possibilities
need to be considered to get an answer that's good enough.
Each diagram represents a family of interactions and tells us the equation
needed to calculate the contribution of that family to the total probability.
The miracle of Feynman diagrams is that an absurdly simple set of rules allows you to easily find all of the important
interactions.
Today, we're going to learn these rules.
Then you're going to apply them to do some quantum field theory (QFT) yourself.
There are Space Time t-shirts at stake.
We're going to stick to quantum electrodynamics, the first and most predictively powerful quantum field theory.
QED talks about the interaction of the Electron field with the electromagnetic field.
That means interactions between Electrons their antimatter counterparts, the Positron, and Photons.
In Feynman diagrams, we depict the electron as an arrow pointing forwards in time,
or the positron is an arrow pointing backwards in time.
We'll soon see the power of representing antimatter as time-reversed matter.
The photon is shown as a wavy line.
Time Direction is irrelevant for the photon.
Throw these on a plot of space versus time and we have a Feynman diagram.
A useless one. None of these particles are doing anything worth calculating. For this to be interesting
the electric and electromagnetic fields need to interact. This is where we start to see the power and simplicity of this approach.
Particle / Field interactions are represented as a vertex: a point where the lines representing the different particles come together.
It turns out that there's only one vertex that's possible in QED:
One with an arrow pointing in, an arrow pointing out, and a single photon connection. It looks like this.
This vertex alone represents six very different seeming interactions, and it can be used to construct
infinite Feynman diagrams! Let's look at the possibilities...
Oriented like this, with time increasing upwards, this vertex represents an initial electron that emits a photon;
after which both particles move off in opposite directions.
But if we rotate this vertex,
so that photon is coming in from Below,
we have a picture in which an electron absorbs
that incoming photon. The photon vanishes and its momentum is completely transferred to the electron.
Rotate again...
and the picture is of a photon coming in and giving up its energy to produce an electron-Positron Pair.
A process we call "Pair Production".
Rotate again, and now we have a
Positron absorbing a photon.
-And a positron emitting a photon .
...And finally, an electron and a positron annihilating each other to produce a photon.
And that's it.
That's all the ways that the electromagnetic and electron fields can interact.
Every single QED interaction is built from these.
But why only this interaction? Well, because of conservation laws.
Energy and Momentum conservation requires that particles not to vanish or appear from nothing which guarantees that if something goes in
then something else must come out.
Charge must also be conserved. If one electron or positron goes in then one electron or positron,
respectively, must leave. If an electron and positron both go in then their charges
Cancel; so a zero charge photon must leave.
Similarly if a photon creates a negatively charged electron it must also create a positively charged
Positron. There are other more complex ways in which in going outgoing particles can balance charge,
but as we'll see, all of these can be built up from this one vertex.
Before we look at those more complex interactions. Here's another important rule:
the overall interaction described by set of Feynman diagrams is
defined by the particles going you and the particles going out.
These are the particles that we actually measure; we know their properties for example their energy and momentum, and they obey
Einstein's Mass-Energy Equation. We say that these particles are "on the mass shell" or just: "on shell."
They sit on the shell structure you get when you plot Einstein's equation of Energy, Momentum and Mass.
On the other hand, everything that happens between in-going and outgoing tracks has
questionable reality. Each possible diagram, the results in the same in-going and outgoing particles, is a valid part of the
possibility space for that interaction.
The particles that have their entire existence
between vertices within the diagram,
but don't enter or leave,
are called "virtual particles". Their correspondence to anything resembling real particles is debatable...
They are also, by definition,
unmeasurable.
Otherwise they'd be one of our in-going or outgoing particles. These particles do not obey Mass-Energy equivalence.
So they are "off shell."
These particles aren't even limited by the speed of light or the direction of time.
Which leads to all sorts of fun...
Let's go back to the simple interaction we looked at in our recent episode:
Electron Scattering can be depicted as two electrons going into an interaction and then two electrons going out.
We know the momentum of the in-going and outgoing electrons.
Any combination of the fundamental three path vertex that can lead to this final result has to be considered.
Simple examples are the exchange of a single photon to transfer momentum between electrons, or the exchange of two or more photons.
But we can add as many of these vertices as we like:
Including, the electrons exchanging photons with themselves at different
stages in the process, or photons momentarily splitting into virtual electron-positron pairs.
As long as the final result is the same any of these are possible.
Part of the beauty of Feynman diagrams is that each of these diagrams themselves represents
an infinite number of specific interactions.
To start with, each of the particle paths are actually infinite paths as well as infinite possibilities for particle Momenta.
We have to consider even impossible faster-than-light paths.
And this is really important: for any particle besides the in-going and outgoing "On Shell" particles...
any Energy, speed, and even direction in time is possible. This last point is bizarre,
but really powerful. For example: for two electrons exchanging a single photon
it doesn't matter if we draw the photon going from the first to the second, or the second to the first,
even though this seems like a very different interaction.
We can think of the differences just being the photon travelling forward in time in one case,
and backwards in the other. The math describing the transfer covers both cases.
Let's look at an even weirder example of this.
This is "Compton Scattering" - an incoming Electron and an incoming photon bounce off each other.
One way that can happen is for the electron to emit a new photon and later absorb the old
incoming photon. In that intermediate stage between vertices the electron is a virtual particle;
Which means we include all possible
paths it might take as long as they lead to producing the same final electron and photon.
That includes paths backwards in time.
Mathematically, a time-reversed electron looks exactly like a positron.
Like this...
The same particles go in and out, but now the interactions look very different;
instead of an electron emitting and then absorbing a photon...
we have on one side that incoming photon creating an Electron-Positron pair.
That new Electron becomes our outgoing electron, but the positron
annihilates with the incoming Electron to produce the outgoing photon.
These may seem like wildly different processes, but in the math represented by Feynman diagrams, they're exactly the same.
The interpretation of the interactions is irrelevant; all we care about is the topology of the diagram...
in other words: How are the vertices connected to each other?
This fact makes Feynman diagrams and incredibly powerful tool in simplifying quantum field theory calculations,
vastly reducing the number of contributing interactions that need to be separately solved.
The interpretation of antimatter as time-reversed matter is one that some, including Richard Feynman, took quite seriously.
We're gonna delve deep into that idea in an episode very soon...
But for now, I want to give you a chance to play with Feynman diagrams yourselves.
So, I have a challenge question for you.
When an electron and a positron interact electromagnetically,
we call it "Bhabha Scattering."
It's an interesting case: the most important Feynman diagrams for Bhabha Scattering are the two cases involving a single virtual photon,
and they include two vertices each.
Those diagrams seem to describe very very different events,
but they lead to exactly the same results.
Use the rules I described in this episode to draw both of the two vertex diagrams for Bhabha Scattering
and describe what's happening in each of these vertices.
Then I want you to try to draw all the possible four-vertex diagrams. For the latter,
don't bother with what we call the "self energy diagrams," in which electrons or positrons
emit and then reabsorb the photon.
Send your neatly drawn Feynman diagrams to PBS space time at gmail.com within one week of the release of this episode
include in the Subject line the words:
"Feynman diagram challenge" because we filter by subject line. We'll randomly choose five correct answers to win a spacetime t-shirt...
and that includes a choice from brand new t-shirt designs.
One will be an exclusive for challenge winners and Patreon supporters:
Introducing the Mighty
Astro Chicken von Neumann, Conqueror of the Galaxy.
And available to everyone including Challenge winners:
the "Heat Death of the Universe is Coming", a fun reminder in t-shirt form of the eventual cold, dark end of
Space-Time.
To all of our Patreon supporters, thanks so much for helping us keep the lights on,
and a very special thanks to Mayank Mehrotra,
for supporting us at the Quasar Level.
Mayank, thanks so much for choosing one of the infinite improbable paths that led you to join us.
And also one last call for anyone wanting one of our special
Space Time Eclipse-Glasses
If you sign up on patreon at the $5 level or above, or increase your pledge to $5 or above in July,
we will send you a set; at least until we run out of glasses. For those who are signed up,
we're going to be emailing you through your Patreon account to get shipping details. And don't worry,
we have glasses set aside for everyone so far.
Last week we talked about the history and danger of space militarization;
you guys had some great comments...
SuperPhillip asks: How you "drop" one of these 'Project Thor' Tungsten rods?
Surely would retain the same orbit as its satellite. Well, good point,
it would. In fact, you need to give it at least a little nudge to send it towards the ground,
but that nudge is far less powerful than whatever you need to launch to orbit.
Therefore it's far less detectable.
Lewinham asks what the real problem is with filling Low-earth orbit with Debris.
So the issue is that stuff in low Earth orbit is moving at around
8Km/s. That means an impact with even a tiny piece of junk is potentially catastrophic.
As space junk builds up, a runaway effect is possible in which more and more satellites are destroyed,
exponentially increasing the amount of debris. It's a real risk. It even has a name: The Kessler syndrome.
Currently there are a couple of thousand operational satellites,
but several hundred-thousand pieces of space junk.
Currently around one satellite is destroyed every year by space junk, but that could increase dramatically if this domino effect begins.
Myrmidon, points out what was perhaps the greatest tragedy in the entire
sorry history of space militarization: the really inexcusable
failure to nickname the strategic defense initiative not "Star Wars"... but, "Ronald Ray-gun."