We're going to start at the bottom of the staircase, which is basic examples of spinors in physics.

In the next four or five videos, I'm going to talk about two examples of how spinors come up in physics, the polarization of light waves and quantum spin states.

We're going to find that both of these phenomena can be described by spinors, in the form of two-by-one columns with complex number entries.

Afterward, we'll see how we can visualize these spinors as flagpoles on a sphere, called the Poincaré sphere or Bloch sphere.

In this video, we're going to discuss all the different polarizations of light waves and their corresponding spinors, which are called Jones vectors in this specific case.

And in the next video, we will discuss how to rotate between these polarizations using SU2 matrices.

So I'm hoping you're familiar with electric and magnetic fields.

The electric field is a set of vectors everywhere in space that point in the direction that a positive charge will be pulled in due to electric forces.

The magnetic field is another set of vectors everywhere in space that tell us which direction a compass will point due to magnetic forces.

According to the laws of electricity and magnetism, it's possible for electric and magnetic field vectors to vibrate back and forth together in a wave, called an electromagnetic wave.

The light we see is an example of electromagnetic waves.

Usually, electromagnetic waves are traveling waves, meaning they travel through space over time.

The image you're seeing here is a snapshot of an electromagnetic wave at a single instant in time.

But as time passes by, the wave will travel forward.

Electromagnetic waves are transverse waves.

This means the direction of the wave oscillations is always perpendicular to the direction of travel.

So if an electromagnetic wave is traveling in the z direction, the wave can oscillate in the x-y plane perpendicular to the direction of travel, but the wave cannot oscillate in the z direction, parallel to the direction of travel.

The polarization of a given electromagnetic wave is the geometric orientation of the wave.

We only use the electric field E to define the polarization of an electromagnetic wave, so we ignore the magnetic field when talking about polarization.

Let's look at an electromagnetic wave propagating in the z direction.

Again, here we're looking at a snapshot of a traveling wave at one instant in time.

When the electric field oscillates up and down along the y-axis, we call this vertical polarization.

If we rotate this wave by a quarter turn, it is now oscillating left and right along the x-axis, and we call this horizontal polarization.

If we rotate the wave by another quarter turn, we end up with vertical polarization again, although with the wave shifted by a half cycle compared to the original wave.

Let's look more closely at a vertically polarized wave.

Remember, the electric field is a vector field, meaning it is represented as an arrow at every point in space and time.

The arrow has components in the x, y, and z directions, which we can write like this, using a linear combination of the x, y, z basis vectors.

We can also write this as a three-component column like this.

For a vertically polarized wave, the electric field only exists along the y-axis, so the x and z components of the electric field go to zero.

If we were to write out the y-component of the electric field for this traveling wave, it would be an amplitude A times a cosine wave.

That depends on both time and space.

Here, t is time, and z is the location along the z-axis.

Omega is the angular frequency, and k is the angular wave number.

I find traveling waves are easiest to visualize on a spacetime diagram, with time moving into the future as we move upward.

Let's say that we have a wave that travels in the positive z direction over time.

We could draw out this traveling wave as a surface in our spacetime diagram.

The density of the waves in the spatial direction is given by the wave number, and the density of the waves in time is given by the frequency.

One thing we can do to modify our traveling wave is to add an extra phi value inside the sinusoid function.

You can think of this as indicating the wave's starting value at the origin z equals zero when time t equals zero.

This starting value is called a phase, and changing the phase value will shift the wave back and forth along the axis of travel.

Applying a phase shift of negative pi over two to a cosine will shift the wave ahead a quarter cycle, which is equivalent to a sine wave.

Now something that can help us represent waves is remembering Euler's formula, which tells us that we can write e to the power of the imaginary i times theta as cosine theta plus i times sine theta.

This is a useful property, because if we have e to the i theta and we want to add some angle phi to it, all we do is just multiply by e to the i times phi, and use the standard exponent rules to rewrite this as a single exponential with the exponents added.

If we then use Euler's formula to convert back to sine and cosine, we see that a phase phi has been added inside the sinusoid functions.

Now electromagnetic waves are described by real numbers.

But for convenience, we can pretend that our Ey component is actually a complex number, where the real part of this complex number is the actual electric field we observe in real life, with the imaginary part being ignored.

This means the full Ey component would have a real part with a cosine wave, and we can invent the fake imaginary part to be a sine wave, since we ignore it anyway.

Using Euler's formula, we can write this more compactly as a complex exponential.

This also means we can pull the phase phi out using exponent rules, and write it as a multiplicative factor, e to the i phi.

This allows us to write travelling waves as three separate multiplicative factors, the amplitude, times the phase, times the actual travelling wave.

So while the true electric field here is just the real part of this complex wave, we can write the electric field using complex exponentials for mathematical convenience.

As I said before, if we take this vertically polarized wave and rotate it a quarter turn, we get a horizontally polarized wave.

In this case, the wave only oscillates in the x direction, and so the y and z components of the field are zero.

Once again, the true electric field is given by a cosine, but for convenience, we can write it as a complex exponential, with the amplitude and phase written separately in front.

So to review, with the vertically polarized waves, the y component of the electric field is a travelling wave, and the x and z components are zero.

And with a horizontally polarized wave, the x component of the electric field is a travelling wave, and the y and z components are zero.

Now, we know from physical experiments that electromagnetic waves can be superimposed on top of each other.

So it's possible to add horizontally and vertically polarized waves together like this.

Now notice that the two polarizations each have their own amplitude and their own phase.

But the actual travelling wave portion is identical in both, so we can factor it out and write it outside the column like this.

Now here I want to state the main idea of this video.

When it comes to studying the polarizations of waves, all the information we need to look at is contained inside this column, the amplitudes and the phases.

The actual travelling wave portion of the formula can be ignored, since it's the same for all components.

We can even go one step further and ignore the z component altogether, since we can always choose coordinates where the z direction is the direction of wave propagation.

And therefore the z component of the electric field will always be zero.

The resulting two-by-one column of complex numbers is called a Jones vector, and it tells us everything we need to know about the polarization of a given wave.

Any light wave polarization can be written as a linear combination of a horizontally polarized wave and a vertically polarized wave, each with complex numbers in front denoting their respective amplitudes and phases.

Moving forward, I'm going to write this horizontally polarized wave of amplitude 1 using the vector capital H, and I'll write this vertically polarized wave with amplitude 1 using the vector capital V.

So using the idea of Jones vectors, we can create new polarizations by selecting different values for the amplitude and phase parameters.

For example, we can set both phases to zero, so that the horizontal and vertical polarizations are in phase with each other, and then set both amplitudes to one.

If we think of H as a vector along the x-axis and V as a vector along the y-axis, this new Jones vector points diagonally to the upper right.

This is called diagonal polarization, denoted with a capital D.

This is what we would get if we took a horizontally polarized wave and rotated it counterclockwise by 45 degrees.

Although, according to Pythagoras, the Jones vector H plus V has an amplitude of the square root of 2, so we usually divide the components by the square root of 2 to force the amplitude of D to be 1.

It's also possible to set the amplitude to plus 1 for H and negative 1 for V, and this We call this anti-diagonal polarization A.

This is what we would get if we took a horizontally polarized wave and rotated it clockwise by 45 degrees.

Once again, we normalize this by dividing by the square root of 2.

So we can get the D and A polarizations by adding H and V together in the right amounts.

In fact, we can make the wave's polarization have any angle on this circle, if we set the coefficients in front of H and V properly.

But there are actually more polarizations that we still haven't seen yet.

Let's again set both amplitudes to 1, and set the horizontal phase to be 0, and set the vertical phase to be pi over 2, which is a phase of a quarter wave cycle.

This polarization is more difficult to visualize, so let's bring back our column notation with the traveling wave to understand it better.

Let's pretend we're stuck at the position z equals 0, and the angular frequency of the wave is 1, and we're watching the wave travel around the xy plane over time.

What would this look like?

We can combine the exponentials, then use Euler's formula, and then take the real part to see that the electric field is given by a cosine, and a cosine with a phase factor of a quarter cycle, which is really the same thing as a negative sign.

At time equals 0, this gives the coordinates 1, 0.

Then at t equals pi over 2, we get the coordinates 0, negative 1.

Then at t equals pi, we get negative 1, 0.

At t equals 3 pi over 2, we get 0, 1.

And at t equals 2 pi, we get 1, 0 again.

So in this wave, the electric field vector travels around in a circle in a clockwise direction over time.

Bringing back the z-axis, if we take our left hand and point our thumb in the direction of wave travel along the z-axis, the electric field vector at the origin z equals 0 will follow the direction of our curled fingers.

Since the wave follows a circle according to our left hand, we call it left circular polarized.

This wave would have a helix shape in three dimensions.

Wikipedia user Dave3457 has made a great animation of this that he's put into the public domain.

We denote left circular polarized waves as capital L.

It's also more common to write e to the i pi over 2 as just i, which is a quarter turn in the complex plane.

And once again, we normalize by dividing by the square root of 2.

If we repeat this process from the start, but instead giving the vertical polarization a phase of negative pi over 2, we get a helix wave that corkscrews in the opposite direction, matching our right hand when the right thumb points in the direction of wave travel.

So we call this right circular polarized, denoted by capital R.

Now you'll notice in this video, I've written my traveling waves with a positive time term and a negative z term.

Some textbooks use the opposite sign convention, with a positive z term and a negative time term.

And as a result, the Jones vectors for circularly polarized waves have opposite signs on the So make sure you know which sign convention you're using when talking about circularly polarized waves.

So to summarize this video, we initially discovered the Jones vectors h and v, which represent horizontally and vertically polarized traveling waves.

We then found that writing them in different linear combinations could give us new polarizations of light.

Like diagonal and anti-diagonal polarizations, and the left and right circular polarizations.

We're going to see in the next video that these Jones vectors that represent wave polarizations are actually spinors.

At this point, there's no particular reason why we would decide these pairs of complex numbers are special objects that deserve the special name of spinner.

But in the next video, we're going to see how we can rotate these various polarizations into each other using special matrices called SU2 matrices.

And this will reveal a special angle doubling relationship between physical space and the space of wave polarizations.

We'll see that one rotation in physical space corresponds to two full rotations in the polarization space.

The Jones vectors that live in polarization space require two full rotations to get back to the starting point.

And this is exactly what we would expect from spinors.

In the meantime, before the next video, I'd suggest trying to draw out the shapes of various polarizations for the Jones vectors shown on the screen here.

If you feel confused, just use the strategy I showed earlier of taking z equals zero and omega equals one, and plugging in different time values to see how the wave moves around in the xy plane.

The answers are linked in the description.