These are some paintings by Zdzisław Beksiński.
Solar eclipses on Mars look really different than they do here on Earth. And this one, created by the Martian moon Phobos, lasted just 40 seconds! Details here:
Above you see a 'standing wave', but the wave equation also has 'traveling wave' solutions:
The wave equation is 'linear', so we can add solutions and get new ones. In fact the standing wave I showed you is the sum of two traveling waves going in opposite directions:
In fact, every solution of the wave equation in 1d looks like this:
The wave equation in 2d space
But you can get more interesting solutions by adding up waves going in lots of different directions! Here's a nice example:
It starts as a little 'wave packet'. It spreads out, since it's really a sum of waves going in different directions.
And just so you remember: the wave equation in 3 dimensions is
But why am I telling you this? Because eventually I want to explain photons, which are wave solutions of the vacuum Maxwell's equations!
One of my favorite calculations: why electric fields make waves!
We use the vacuum Maxwell equations and an identity involving the Laplacian
But the vacuum Maxwell equations say more than just
You can see the proof of that fact here:
We say electromagnetic waves are 'transverse' because the fields point at right angles to the direction the wave is moving. People knew this before Maxwell, and spent a lot of time trying to explain it.Sound waves in air are 'longitudinal': the air vibrates along the direction the wave is moving, not at right angles to it. Sound in a solid can be either longitudinal or transverse.
So, back when scientists thought light was a vibration in a medium called 'aether', they struggled to understand why these vibrations are only transverse, never longitudinal. The aether would need to be an extremely rigid solid, because the speed of light is so high — and it would need to be completely incompressible, since there are no longitudinal waves of light!
What is a photon? This is a complicated question. But a single photon in empty space has a simple description: it's a solution of the vacuum Maxwell equations.
Yes, solutions of the classical Maxwell equations also describe quantum states of a single photon!
But wait: quantum states are described by vectors in a complex Hilbert
space. How do we multiply the quantum state of a photon by
If it has positive frequency, replace
Why don't we just replace replace
To get a Hilbert space of photon states we also need to choose an
inner product for solutions of the vacuum Maxwell equations.
Up to scale, there's just one good way to do this that's invariant
under all the relevant symmetries. The formula is a bit scary:
When the electric and magnetic fields change with time, they affect each other. But when they're unchanging, they don't! Then electromagnetism splits into two separate subjects, called electrostatics and magnetostatics.
The equations of electrostatics and magnetostatics look opposite from each other. But tomorrow I'll show we can study them in similar ways, using the electric 'scalar potential' and the magnetic 'vector potential'. There will be hints of some deeper math.
They're opposites. But there's a way to look at them that makes them very similar!
A great fact: a vector field on 3d Euclidean space has zero curl if and only if it's the gradient of some function.
So put in a minus sign just for fun and say the electric field is
Then electrostatics boils down to just one equation!
Another great fact: a vector field on 3d Euclidean space has zero divergence if and only if it's the curl of some vector field.
So say the magnetic field is
Now magnetostatics also boils down to just one equation!
Yet another great fact: there are different choices of
So write
If we do this, magnetostatics looks a lot like electrostatics!
In short: electrostatics and magnetostatics look very similar if we use a scalar potential to describe the electric field and a vector potential for the magnetic field. This is the start of a deeper understanding of electromagnetism, called 'gauge theory'.
Also, the three "great facts" I used are part of an important branch of math: De Rham cohomology. It gets more interesting on spaces with holes. Then these facts need to be adjusted to take the holes into account.
In electrostatics, the scalar potential
The nice picture here is by Geek3 on Wikicommons.
It helps to use quantum mechanics. Then
But we can also understand the vector potential
...plus whatever action it gets for other, non-magnetic, reasons.
As a quantum particle moves along a path, its phase rotates. By what
angle? By the action of the corresponding classical particle moving
along that path, divided by Planck's constant
But you can only compare phase changes for two paths with the same
starting point and ending point. So you can change
This is called gauge freedom.
Similarly, in classical mechanics you can only compare actions for two
paths if they start at the same point and end at the same two point.
But this 'gauge freedom' wasn't understood very well until quantum
mechanics came along. So
To understand the concept of 'curl', imagine water flowing with
velocity vector field
Alas, we need an arbitrary 'right-hand rule' or 'left-hand rule' to
convert the wheel's rotation into a vector! We usually say the wheel
rotates counterclockwise around the curl of
To avoid this arbitrary convention, we should use better math. Better math reveals that various things we'd been calling 'vectors' are not all the same. Using vectors to describe the curl is a hack! It uses an arbitrary rule... which changes into a different rule if we look at things in a mirror.
The full story is rather long, but when we write
You may be wondering why I haven't said your favorite word yet —
'differential form', or 'bivector', or whatever. These kinds of math
are very important because they're better at distinguishing different
kinds of vector-like things. But the full story is a bit bigger. In
3-dimensional space, each different 3d irreducible representation of
Each of these different vector-like things forms a 3-dimensional
representation of the group of
Even better, let's work in a coordinate-free way. Let
Elements of
But
Elements of
In classical mechanics, velocity is a vector while momentum is a covector. We can identify covectors with vectors using an inner product, but they transform differently under general linear coordinate transformations. So, we often distinguish between them.
Next there's
Details here:
We can identify bivectors with vectors using an inner product and an
orientation on
If vectors have dimensions of length, bivectors have dimensions of length².
Next there's
In terms of units:
(In units where Planck's constant is 1, momentum has units of 1/length, because it's a covector.)
So far I've described four inequivalent 3-dimensional representations
of
But in fact, there are infinitely many 3d irreducible representations
of
For example there are 'pseudovectors', also known as 'axial vectors'.
These are just vectors with a modified action of
To transform a pseudovector by
which is
When people call pseudovectors 'axial vector', they often call ordinary vectors 'polar vectors'.
To get all 3d irreducible representations of
In physics 'densitizing' is a way to change the units of a quantity,
multiplying it by some power of length. I mentioned that vectors have
units of length. But if we densitize them as described, we get things
with units of length
As an exercise, figure out how to get the representation of
I'll give you a hint: you have to start with
We can convert bivectors in 3d space into vectors, and vice versa, if we have an inner product and also an 'orientation' on that space: a choice of what counts as right-handed. But not all 3d vector spaces come born with this extra structure! So in general, vectors and bivectors are useful for different things.
Similarly, you can multiply 3 vectors
If you've only learned about the dot product and cross product, here's some good news: this is the start of a bigger, ultimately clearer story! Part of this story uses multivectors:
This isn't quite historically accurate. Einstein did make the remarkable leap toward realizing that light comes in quanta in his 1905 paper. Indeed, he wrote:
According to this picture, the energy of a light wave emitted from a point source is not spread continuously over ever larger volumes, but consists of a finite number of energy quanta that are spatially localized at points of space, move without dividing and are absorbed or generated only as a whole.
But in 1921 the Nobel committee only recognized him for his work on the photoelectric effect, not mentioning quanta. Only in 1922 did Compton come up with evidence that convinced everyone photons exist — and the name photon was introduced still later, in 1926, by the chemist Gilbert Lewis.
For more, see:
By the way, Planck really didn't understand the meaning of his own mathematics — it took Einstein to take quanta seriously. As Planck said in 1931, his introduction of energy quanta in 1900 was "a purely formal assumption and I really did not give it much thought except that no matter what the cost, I must bring about a positive result." For more on this, see:
In quantum mechanics, the magnetic field says how much the phase of a charged particle rotates when you move that particle around a loop! (Not counting other effects.) This means that the magnetic field is fundamentally something you want to integrate over a surface.
So, while we often act like the magnetic field is a vector field, it's fundamentally a '2-form'. This is something you can integrate over an oriented surface. Converting a 2-form into a vector field forces you to use a 'right-hand rule'. And that's awkward.
You can integrate a 2-form over an oriented surface... but what is a 2-form, actually? It's linear map from bivectors to real numbers! A bivector is like a tiny piece of oriented area.
So how does the magnetic field eat a tiny piece of oriented area and give a number?
Here's how: if you move a charged particle around a tiny piece of oriented area, its phase changes by some tiny angle. And that angle is what the magnetic field tells you!
The electric and magnetic fields are very different when viewed as fields on space. But we can unify the electric and magnetic fields into a single field on spacetime: the electromagnetic field.
To do this, it helps to use 1-forms and 2-forms.
In quantum mechanics the magnetic field says how the phase of a
charged particle changes when you move it around a little loop in the
The electric field does the same for the
There's a lot more to say about this! To explain electromagnetism clearly, I would need to introduce quantum mechanics, and differential forms, and now the spacetime perspective — so, special relativity. So the job keeps getting bigger.
Nature is a unified whole.