[MUSIC PLAYING] 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. [MUSIC PLAYING] 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. This episode is brought to
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"The Ultimate Formula" gives a really nice
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and use the promo code spacetime during
the sign-up process. As always, a huge thanks to
our supporters on Patreon. This week, we'd like to
give an extra big thanks to Eugene Lawson, who's
contributing at the Big Bang level. Eugene, it's an
incredible help, and you make our jobs much easier. And another reminder to current
or would-be Patreon patrons-- we made some "Space
Time" eclipse glasses-- super handy for not
going blind watching the great American
eclipse in August. We're sending out a set to
every Patreon contributor at the $5 level
or above, and that includes anyone who
signs up at or increases to the $5 level in
the month of July, at least until we
run out of glasses. It'll be first
come, first served. 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.
This is Awesome. The last few episodes building up to this one, along with the promised ramp up into the Standard Model, really makes this channel feel like a curriculum. I feel like I'm actually getting educated when I watch. I love it!