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HOST: This episode is supported by Curiosity Stream.
Quantum field theory is stunningly
successful at describing the smallest scales of reality,
but its equations are also stunningly complex.
A lot of the genius in QFT's development
was in finding brilliant hacks to make
these equations workable.
The most famous of these are the incredible Feynman diagrams.
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The equations of quantum field theory
allow us to calculate the behavior of subatomic particles
by expressing them as vibrations in quantum fields.
But even the most elegant and complete formulations
of quantum field theory, like the Dirac equation or Feynman's
path integral, become impossibly complicated when
we try to use them on anything but the most simple systems.
But physicists tend to interpret "that's impossible" as "I dare
you to try," and try they did.
First, they expressed these impossible equations
in approximate, but solvable, forms.
Then they tackled the pesky infinities
that kept appearing in these new approximate equations.
Finally, the entire mess was ordered into a system
that mere humans could deal with using the famous Feynman
diagrams.
To give you an idea of how messy quantum field theory can be,
let's look at what should be a simple phenomenon-- electron
scattering, when two electrons repel each other.
In old-fashioned classical electrodynamics,
we think of each electron as producing
an electromagnetic field.
That field then exerts a repulsive force
on the other electron.
At least in the simplest cases, the Coulomb equation
governing this subatomic billiards shot
is really easy to solve.
But in quantum field theory, specifically
quantum electrodynamics, or QED, the story is very different.
We think of the electromagnetic field as existing everywhere
in space, whether or not there's an electron present.
Vibrations in the EM field are called photons,
what we experience as light.
The electron itself is just an excitation, a vibration
in a different field--
the electron field.
And the electron and EM fields are connected.
Vibrations in one can cause vibrations in the other.
This is how QED describes electron scattering.
One electron excites a photon, and that photon
delivers a bit of the first electron's momentum
to the second electron.
It's arguable exactly how real that exchanged photon is.
In fact, we call it a virtual photon,
and it only exists long enough to communicate this force.
There are other types of virtual particle
whose existence is similarly ambiguous.
We'll get back to those in another episode.
This is a good time to introduce our first Feynman diagram.
The brilliant Richard Feynman developed these pictorial tools
to organize the painful mathematics of quantum field
theory, but they also serve to give a general idea of what
these interactions look like.
In a Feynman diagram, one direction is the time--
in this case, up.
The other axis represents space, although the actual distances
aren't relevant.
Here we see two electrons entering in the beginning
and moving towards each other.
They exchange a virtual photon--
this squiggly line here--
and the two electrons move apart at the end.
But Feynman diagrams aren't really
just drawings of the interaction.
They're actually equations in disguise.
Each part of the Feynman diagrams
represents a chunk of the math.
Incoming lines are associated with the initial electron
states, and outgoing lines represent the final electron
states.
The squiggle represents the quantized fueled excitation
of the photon, and the connecting points,
the vertices, represent the absorption and emission
of the photon.
The equation you string together from this one diagram
represents all of the ways that two electrons
can deflect involving only a single virtual photon.
And from that equation, it's possible to perfectly calculate
the effect of that simple exchange.
Unfortunately, real electron scattering at a quantum level
is a good deal more complicated than this.
For that reason, this simple calculation
gives the wrong repulsive effect between two electrons.
If we observe two electrons bouncing off each other,
all we really see is two electrons going in
and two electrons going out.
The quantum event around the scattering is a mystery.
There are literally infinite ways
that scattering could have occurred.
In fact, according to some interpretations,
all infinite intermediate events that
lead to the same final result actually do happen, sort of.
We talked about this weirdness when we discussed the Feynman
path integral recently.
Just as with the path integral, to perfectly calculate
the scattering of two electrons, we
need to add up all of the ways the electrons can be scattered.
And this is where Feynman diagrams
start to come in handy, because they
keep track of the different families of possibilities.
For example, the electrons might exchange
just a single virtual photon, but they might also exchange
two, or three, or more.
The electrons might also emit and reabsorb a virtual photon.
Or any of those photons might do something crazy,
like momentarily split into a virtual anti-particle-particle
pair.
Those last two events are actually hugely complicating,
as we'll see.
With infinite possible interactions behind this one
simple process, a perfectly complete quantum field
theoretic solution is impossible.
But if you can't do something perfectly,
maybe near enough is good enough.
This is the philosophy behind perturbation theory,
an absolutely essential tool to solving quantum field theory
problems.
The idea is that if the correct equation is unsolvable,
just find a similar equation that you can solve, then make
small modifications to it--
perturb it-- so it's a bit closer to the equation
that you want.
It'll never be exact, but it might get you pretty close.
In the case of electron scattering,
the most likely interaction is the exchange
of a single photon.
Every other way to scatter the electrons
contributes less to the probability of the event.
In fact, the more complicated the interaction, the less it
contributes.
Here, Feynman diagrams are indispensable.
It turns out that the probability amplitude
of a particular interaction depends
on the number of connections, or vertices, in the diagram.
Every additional vertex in an interaction
reduces its contribution to the probability
by a factor of around 100.
So the most probable interaction for electron scattering
is the simple case of one photon exchange with its two vertices.
A three-vertex interaction would contribute about 1%
of the probability of the main two-vertex interaction.
However, it turns out that for electron scattering,
there are no three-vertex interactions.
However, there are several interactions
that include four vertices, and each contributes about 1%
of 1% of the two-vertex interaction,
and this is true even though those complex interactions are
very different to each other.
They include exchanging two virtual photons,
or one electron emitting and reabsorbing a virtual photon,
or the exchanged photon momentarily exciting
a virtual electron-positron pair.
And more complicated interactions
add even less to the probability.
So with Feynman diagrams, you very quickly
get an idea of which are the important additions
to your equation and which you can ignore.
Perturbation theory, with the help of Feynman diagrams,
make the calculation possible, but that
doesn't mean we're done.
Including all of these weird intermediate states
really opens up a can of worms.
This is especially true for so-called loop interactions,
like when a photon momentarily becomes
a virtual particle-anti-particle pair
and then reverts to a photon again,
or when a single electron emits and reabsorbs the same photon.
This latter case can be thought of as the electron causing
a constant disturbance in EM field.
Electrons are constantly interacting
with virtual photons.
This impedes the electron's motion
and actually increases its effective mass.
The effect is called self-energy.
But if you try to calculate the self-energy correction
to an electron's mass using quantum electrodynamics,
you get that the electron has infinite extra mass.
This sounds like a problem.
To calculate the mass correction due to one of these self-energy
loops, you need to add up all possible photon energies,
but those energies can be arbitrarily large,
sending the self-energy-- and hence, the mass--
to infinity.
In reality, something must limit the maximum energy
of these photons.
We don't know what that limit is.
The answer probably lies within a theory of quantum gravity
which we don't yet have.
But just as with perturbation theory,
physicists found a cunning trick to get around
this mathematical inconvenience.
It's called renormalization.
Obviously, electrons do not have infinite mass,
and we know that because we've measured that mass,
although any measurement we make actually
includes some of this self-energy,
so our measurements are never of the fundamental or bare mass
of the electron, and that is where the trick lies.
Instead of trying to start with the unmeasurable fundamental
mass of the electron and solve the equations from there,
you fold in a term for the self-energy corrected
mass based on your measurement.
In a sense, you capture the theoretical infinite terms
within an experimental finite number.
This renormalization trick can be
used to eliminate many of the infinities that
arise in quantum field theory--
for example, the infinite shielding of electric
charge due to virtual particle-anti-particle pairs
popping into and out of existence.
However, you pay a price for renormalization.
For every infinity you want to get rid of,
you have to measure some property in the lab.
That means the theory can't predict
that particular property from scratch.
It can only make predictions of other properties
relative to your lab measurements.
Nonetheless, renormalization saved quantum field theory
from this plague of infinities.
Feynman diagrams successfully describe everything
from particle scattering, self-energy interactions,
matter-anti-media creation and annihilation,
to all sorts of decay processes.
We'll go further into the nuts and bolts
of Feynman diagrams in an upcoming challenge episode.
A set of relatively straightforward rules
governs what diagrams are possible,
and these rules make Feynman's doodles an incredibly powerful
tool for using quantum field theory to predict the behavior
of the subatomic world.
The results led to the standard model of particle physics.
In future episodes, we'll talk more about
what is now the most complete description we
have for the smaller scales of space time.
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Last week, we talked about Richard Feynman's
brilliant contribution to the development
of quantum field theory with his path integral formulation.
You guys had a lot to say.
Christian Haas asked how Feynman's path integral method,
which is compatible with special relativity,
can derive Schrodinger's equation when Schrodinger's
equation is not compatible.
So the deal is that Schrodinger's equation
is a special case of a more general formulation of quantum
mechanics.
In Schrodinger's equation, all of the particles
are tracked according to one universal master clock.
In Feynman's approach, each particle
is tracked according to its own proper time clock, which
can vary in its tick speed depending on how
fast the particle is traveling.
Derivation of Schrodinger from Feynman
requires approximating all of the separate proper time
coordinates to give a single time coordinate.
That approximation is OK at low speeds
but breaks when things get close to the speed of light.
[INAUDIBLE] points out that the final probability
for a particle journey is the square
of the length of the complex probability amplitude vector.
And yeah, that's right.
Probability is the square of the probability amplitude.
That's the Born rule right there.
However, the sense I wanted to relay
is that the total probability depends
on the length of the summed probability amplitudes--
so the square of the real plus the square
of the complex components.
[INAUDIBLE] also correctly points out
that the individual paths don't have different probability
amplitude lengths taken separately,
but rather, they're pointing in the complex vector
space, rotates so that each path adds differently
to the total probability.
[INAUDIBLE] points out that the wildly divergent paths
would require superluminal speeds
to reach their destination at the same time
as the straight line paths.
Yeah, that's right.
Feynman didn't limit particle velocity to the speed of light.
By applying the principle of least action
in the determination of probability amplitudes,
it turns out that the "crazy" paths, including
the superluminal ones, cancel out and add little
to the probability.