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.