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Hey Crazies.
In a previous video, I explained that electric charge was a coupling property.
It measures how linked or coupled a particle is to the electromagnetic field.
This connection explains why electric charge is conserved
and, frankly, why the electromagnetic field even works.
But to understand it, we’re going to need a little quantum mechanics.
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Ok, so the ultimate question we’re trying to answer with this video is:
Why is charge conserved?
But we’re going to need to answer a few other questions first.
This video comes to you in four parts!
Part One! What do we mean by charge?
Charge is a fundamental property of particles.
These are the elementary particles.
Some of them have negative charge,
some of them have positive charge,
and some of them are electrically neutral.
They can combine to make composite particles, which can also carry electric charge.
You just total up the charge of the elementary particles they’re made of.
Charge is just a number.
But aren’t particles like electrons also called charges?
Yes, but that’s just people being lazy.
I don't know if you knew this, but humans are lazy.
When we call something like an electron a charge, that’s just short for charged particle.
I’ve honestly been guilty of it too,
but, for this video, I’m going to be more careful than that.
In this video, the word charge will always refer to the number, not the particle.
Part Two! What does it mean for something to be conserved?
A conserved quantity is just a number that stays the same over time.
Kind of like how environmental conservation is all about keeping the environment the same.
Noether’s theorem tell us that
all conserved quantities are the result of some invariance or symmetry in the universe.
If you can vary one condition, like where or when an event takes place,
and that event plays out in exactly the same way, that’s called invariance.
Noether’s theorem says there must be a conserved quantity.
No exceptions!
Linear momentum goes with location invariance.
Energy goes with time invariance.
Angular momentum goes with rotational invariance.
My stress goes with quarantine invariance.
[Screaming]
Every invariance has a conserved quantity.
Part Three! What invariance is causing charge to be conserved?
An invariance in the electromagnetic interaction.
Charged particles interact electrically and magnetically.
They do this over seemingly empty space, but not instantaneously.
This leads us to infer that maybe the space between them is filled with something,
something that carries that interaction between those particles.
Whatever that “something” might be, we have several ways of modeling it.
We could imagine it as some kind of field of forces.
The electric field exerts a force on charged particles.
The magnetic field does the same thing, but only if the particles are moving.
However, forces are not the only way to look at a situation.
Sometimes, energy and momentum can give you a deeper insight.
This is the electric potential. It’s a way for charged particles to provide energy to each other.
Is there a magnetic potential?
[Censor] yeah there is!
It fulfills a similar purpose, but for momentum instead of energy.
It’s a way for particles to exchange momentum with each other.
Fields are fine if you’re working with forces,
but potentials are better if you’re working with energy and momentum.
You know, things that tend to obey conservation principles.
And, as it turns out, the fields are invariant under certain changes in the potentials.
This is how they’re related.
“E” is the electric field,
“B” is the magnetic field,
“V” is the electric potential,
and “A” is the magnetic potential.
The potentials determine the fields, but not uniquely.
Say we take this electric potential and shift all the values by some constant.
The values might be different, but the changes across space are still the same.
Those changes are just the electric field. It was unaffected by the shift in potential.
That shift isn’t even limited to constants. It can vary across space and time.
It works as long as the shift is a rate of change across space in the magnetic potential
and a rate of change across time in the electric potential.
These fields are defined by these potentials,
but also these potentials,
or these potentials.
This “f” could be any function of space and/or time.
That “f” is something we call a gauge,
so this invariance is called a gauge invariance.
That gauge could be anything. It’s completely random.
Or is it?!
Part Four! What does an invariance in a field have to do with charged particles?
This is where quantum mechanics becomes important.
Little particles like electrons are inherently quantum mechanical.
They’re not really tiny spheres.
We might not know exactly what they are, but they’re definitely not tiny versions of this.
Their behavior is described by waves of probability, like this one that describes location.
These waves are completely unobservable, but we do get to see the square of the wave.
If the wave looks like this, its square looks like this.
We call it the probability density
because the area underneath tells us the probability that something will happen.
Another nice feature of these waves is that they have something called global phase invariance.
What’s a phase?
It’s just what we call a shift in a wave. It’s not that big of a deal.
Here’s a basic sine wave. It’s exactly where it’s supposed to be,
which means it has a phase of zero.
The size of any shift in that wave is called the phase.
If we’re talking about a quantum wave for something like an electron,
then a phase shift will not affect the wave’s probability density.
The likelihood of finding the electron in any particular place is unaffected by the shift.
If that phase shift is the same everywhere in space and time, we call it a global phase shift.
The real properties of all quantum particles are invariant under this kind of shift.
In the language of quantum mechanics, we’d write that like this,
where “phi” is the phase angle and “psi” is the quantum wave.
The complex squares of these two waves are the same, which means the probabilities are the same.
Is there something like that that only applies to charged particles?
[Censor] yeah there is!
Local Phase Invariance.
That’s when a different phase shift occurs at different places and different times.
Without some kind of control, it would take a wave that looks like this
and turn it into one that looks like this.
It’s a complete mess.
But not if we can tweak Schrodinger’s equation in just the right way.
That’s the equation that governs what the particle’s wave looks like in space
and how it evolves over time.
However, this form of it, the one that everybody knows,
only applies to electrically neutral, spin-zero particles.
Luckily, we can fix it with the electromagnetic potentials.
Remember those?
The adjusted Schrodinger equation looks like this.
Yes, it’s nastier, but it finally applies to charged particles with non-zero spin.
It still doesn’t apply to relativistic particles, but that’s a topic for another day.
Though, oddly enough, this adjustment does fix our phase shift problem.
The local phase shift turned our particle’s wave into this,
but the electromagnetic interaction turns it back into this.
The electromagnetic potentials give charged particles local phase invariance.
If a particle has electric charge, then it interacts with the electromagnetic field
and has local phase invariance.
All three of these statements are 100% equivalent.
Charged particles have local phase invariance.
Neutral ones don’t.
No exceptions!
But that middle statement, the one about interacting with the electromagnetic field,
it’s the key to this whole thing.
Do you remember that seemingly random gauge function from earlier?
Well, it’s the same as the quantum phase of the particle.
The gauge shift in the field is the phase shift in the particle.
Boom [Censor]!
So, wait, why is charge conserved?
Because of an invariance in the electromagnetic interaction.
Charged particles are inextricably linked to the electromagnetic field.
Quantum mechanics requires this.
That link is what it means to be electrically charged.
Any invariance in the field will be coupled with an invariance in the particle.
And, according to Noether’s theorem, any invariance results in a conserved quantity.
That conserved quantity is electric charge.
Conservation of charge shall not be violated!
So, hopefully that ties up all the loose ends.
I’m sure you’ll let me know in the comments if I forgot to mention something.
Thanks for liking and sharing this video.
Don’t forget to subscribe if you’d like to keep up with us.
And until next time, remember, it’s ok to be a little crazy.
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Feynstein 100 mentioned an old quote:
Time is what keeps everything from happening at once.
Yes, Wheeler said that,
but, technically, he was quoting a short story from 1921 called “The Time Professor.”
The more you know.
Anyway, thanks for watching!