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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.
