The standard model
of particle physics is the most successful, most
accurate physical theory ever developed, describing
with stunning accuracy the fundamental quantum
building blocks of our universe. But even more stunning is how it
was discovered, by peering deep into the symmetries of reality. As far as we can
tell, mathematics is the language in which
the universe is written. Our laws of physics are
equations of motion tuned by the fundamental constants. Previously, we've talked
a bit about the symmetries of these equations and how they
lead us to conserved quantities like energy and momentum. But that's just the tip of
the theoretical iceberg. These symmetries can be portals
into entirely new aspects of reality. The most amazing example of
this is the standard model of particle physics. Today, I'm going to
open the first portal of the standard model
and show you the origin of the electromagnetic field. To appreciate the
theoretical whirlwind that is the standard model,
we need to introduce the idea of a gauge theory. In simple terms,
a gauge theory is one that has mathematical
parameters or degrees of freedom that can be
changed without affecting the predictions of the theory. An example would be
a ball rolling down a hill under a constant
gravitational acceleration. The speed of the ball at
the bottom of the hill depends on its
change in altitude. But it doesn't matter what we
define to be altitude zero-- the bottom of the
hill, sea level, even the center of the earth--
for the equations of motion of the ball, the altitude
zero point is irrelevant. It's what we call a gauge
freedom or a gauge symmetry. And we say that the
equations of motion are invariant to that parameter. That's a pretty basic example. But it turns out that
these gauge symmetries are an important feature of
most of our physical theories describing the universe. Newton's laws of
motion and gravity, Maxwell's equations
for electromagnetism, Einstein's general
relativity, and of course, the standard model. We're not quite sure
why this is the case, but it seems to be a trend. A theory that has
these gauge symmetries is called a gauge theory. Today, we're going to look at
the simplest of the symmetries of the standard model. The standard model is ultimately
based on quantum field theory, but we're going to use
the Schrodinger equation. That's the most basic equation
of motion of quantum mechanics. It describes the evolution
of the wave function, which is the mathematical
object that contains all the information about a
particular physical system. We can never see
the underlying wave function of, say, a particle. The best we can do
is make a measurement of physical observables,
like position or momentum. The wave function can
represent different observables and it determines
the distribution of possible results
of measurement of those observables. In this episode, we'll be
talking about the position wave function. OK. Pay attention to
this bit of math. It'll be important. The square of the magnitude
of this wave function tells us the
probability distribution of a particle's position. The position that we observe
when we look at the particle is picked randomly
from that distribution. This step of squaring
the wave function is called the Born rule. And this innocuous seeming step
introduces a simple symmetry that has profound implications. Let's see what happens when
we square the wave function. The function is an oscillation
in quantum possibility, moving through space and time. It's no simple wave. It's literally complex in
the mathematical sense. It has two components, one
real and one imaginary. These components oscillate
in sync with each other, but they're offset, shifted
in phase by a constant amount. Phase is just the
wave's current state in its up-down oscillation. When we apply the Born
rule, what we're doing is squaring these two waves
and adding them together. But it turns out that this
value doesn't depend on phase. The magnitude squared of the
real and imaginary components stays the same, even as those
components move up and down. It's that magnitude squared
that we can observe. It determines the
particle's position. The phase itself is
fundamentally unobservable. You can shift
phase by any amount and you wouldn't change
the resulting position of the particle,
as long as you do the same shift to both the
real and imaginary components. In fact, as long as
you make the same shift across the entire wave
function, all the observables are unchanged. We call this sort of
transformation a global phase shift. And it's analogous to
transforming our altitude zero point up or down by the
same amount everywhere. The equations of
quantum mechanics have what we call
global phase invariance. Global phase is a gauge
symmetry of the system. Let's push a little further to
see how far this symmetry goes. This time, we'll shift the
phase by different amounts at different
locations, while still keeping the real
and imaginary shifts the same at each location. This position
dependent phase shift is called a local phase shift,
instead of a global phase shift. We'll try this because,
well, we already know that the magnitude squared
of the wave function should still stay the same
under local phase shifts. Let's see what this
would look like. A global phase shift looks like
this, where all points move by the same amount. However, if we do a
local phase shift, say, only this point here,
only that location changes, as if it were part
of the shifted wave, making a discontinuous spike. If you allow this sort
of local phase shift, you can change each
point in a different way and really mess up
the wave function. That shouldn't change
our probabilities for the positions
of the particles, but what about observables
besides positions? According to the basic
Schrodinger equation, we just ruined everything. Among other things,
messing with local phase really screws up our prediction
for the particle's momentum. See, momentum is related to the
average steepness of the wave function. Change the shape of
that wave function with local phase
shifts and you actually break conservation of momentum. Local phase is not
a gauge symmetry of the basic
Schrodinger equation. OK. That was a bust. I guess we're done here. All right. Wait just a second. Just for funsies, maybe we can
change the Schrodinger equation to find a version that really
is invariant to local phase shifts. To do that, we need to alter
the part of the Schrodinger equation that gives us the
momentum of a particle, the momentum operator. After all, momentum is
what got screwed up. It turns out that we can
add a mathematical term to the momentum operator
that's specially designed to undo any mess we make to
the phase of the wave function. If we choose this
term correctly, it absorbs any local changes
we make to the phase. And what is that extra term? Well, it's something we
call a vector potential. I won't go into that right
now, but the important and absolutely bizarre thing
about this mathematical entity is that it looks like
something very familiar. It looks exactly like the
type of vector potential that you would have
in the presence of an electromagnetic field. So we've discovered that
the only way for particles to have local
phase invariance is for us to introduce a new
fundamental field that pervades all of space. And it turns out that
field already exists, and it's the
electromagnetic field. This is totally crazy. We just rediscovered
electromagnetism by insisting on a gauge symmetry
that we had no right to expect to exist in the first place. But we didn't just
rediscover the EM field, we learned a ton about it. By discovering how it fits
into the Schrodinger equation, we've unlocked its
quantum behavior. And now we know how it interacts
with particles of matter to give them this symmetry. We also learned about the
origin of electric charge, which we now see as a coupling turn. Any particle that has
this kind of charge will interact with
and be affected by the electromagnetic field
and be granted local phase invariance. But the reverse is also true. In order to have this particular
type of local phase invariance, particles must possess
electric charge. By the way, applying
Noether's theorem tells us there is a
conserved quantity associated with any symmetry. In this case, the symmetry
is local phase invariance and the conserved quantity
is electric charge. At this point, we only need
a couple of extra steps to produce the full
description of electromagnetism in the quantum world. Quantum electrodynamics, or QED. First, we need to upgrade
the Schrodinger equation to the Dirac equation so
it works with Einstein's special relativity. And we talked about
that in this episode. Then, we need to apply quantum
principles to our field, like considering its
internal or self energy and allowing quantized
oscillations in the field itself. Those oscillations in our
new electromagnetic field turn out to be the photon. But what about all those
fundamental particles without electric charge? Neutral particles
like neutrinos. To understand those, we'll need
to go beyond the Schrodinger equation and to explore
new gauge symmetries. It turns out that
local phase invariance is just the simplest
of the larger suite of gauge symmetries
of the standard model. Those symmetries are obtusely
named, U1, SU2 and SU3, and they predict the
fields that give rise to electromagnetism, the weak
and the strong nuclear forces, respectively. The fields that arise from
these gauge symmetries are called gauge
fields, and they all have their associated
oscillations, their associated particles. These are, the gauge bosons,
the photon for electromagnetism, the W and Z bosons for
the weak interaction, and the gluon for the
strong interaction. Together, they govern
the interactions of the metaparticles
of the standard model. And we'll come back to
that in future episodes. Perhaps the greatest
mystery here is not the nature
of the quantum field nor the connection
between symmetry and the fundamental
forces, perhaps it's the fact that by pure
exploration of mathematics, delving many layers
of abstraction deeper than our
capacity for intuition, we are led to true discoveries
about physical reality. And following those
mathematical labyrinths reveals physical theory with
stunning predictive power, like the standard model
of particle physics. Mathematics truly seems to
be the language in which the universe is written. We should be amazed that
we can learn that language and through it,
comprehend the underlying nature of space time. Last time on Space
Time Journal Club, we looked at a new result
potentially detecting a particle beyond the standard
model, the sterile neutrino. Let's see what you had to say. Sebastian Elytron says
that the standard model won't change for unverified
discoveries, it has standards. Totally agreed. And no one, not even
the researchers, are claiming the actual
discovery of this new particle, yet. April put it well in her
response to your comment, a 6.1 sigma level
of significance doesn't mean for sure that
sterile neutrinos exist. It could mean there's some
other physical process that we don't understand yet. Regardless, it will definitely
mean something happened. Will it be something
interesting? That remains to be seen. Richard Brockman and
badly drawn turtle point out the danger of
combining multiple experiments to increase the significance
of your results. If you have enough
experiments to choose from, you can just select from those
with the high significance until it takes you over the
five sigma significance hurdle. In the case of the
mini bird experiment, they combined their 4.8
sigma fermion result with the 3.6 sigma result from
the only other substantially similar experiment that has
ever been performed, that one at Los Alamos. So there's some
justification because there weren't any very
similar experiments with lower significance. On the other hand, it seems
like they didn't integrate the constraints to sterile
neutrinos derived from the ice cube experiment or from the
cosmic microwave background radiation. It's much harder to
incorporate those because the experiments
were so different. Still, it's a worthwhile
note of caution. A few of you point out that if
you build a wall of lead, one likely, you'd think, to
try to stop neutrinos, you would just collapse
into a black hole. I think you're right. Call off the project,
guys, no more wall.
I can't get over the weird perspective that has him standing on his toes the entire time