Office chair philosophy: Spinors

I started to learn a bit about spinors, so this is my early take on them. Firstly

Pauli spinors:

The equation for rotating a vector v in 3D is

for unit quaternion q. Noting that v is placed in the i,j,k positions with zero w.

The equivalent in spinor notation is on 2x2 complex-valued matrices:

where the dagger symbol is the conjugate transpose.

U is an SU(2) matrix:

and V is a 'Pauli vector' which is derived from the vector v=(x,y,z):

is a direct equivalent of

because SU(2) matrices are isomorphic to unit quaternions. There is a direct translation and multiplication works the same.

Spinors fit in when we factor V into the outer product of two complex length-2 vectors, to give:

Each of those vectors is a two-component spinor, as it has two complex components, labelled with the xi letter for standard Pauli spinors.

Now unlike with the quaternion formula, we can now throw away the right half of the formula and work on just the left hand side, where a single spinor is oriented using an SU(2) matrix. We only need to add the conjugate transpose right hand side when we want to turn the result into a pure 3D rotation.

So what exactly are U and the Pauli spinor in this single product?

The spinor can be thought of as a quaternion -- it maps spinor (a+bi, c+di) to quaternion (a,b,c,d) -- but only a quaternion as a special isoclinic rotation, not as a rotator, it is the 2x2 SU(2) matrix which acts like a quaternion rotator, i.e. to rotate other objects.

The rotational axis of the spinor is just the ratio of the two complex components mapped onto the Riemann sphere (i.e. mapped stereographically).

I said 'special' isoclinic rotation as a quaternion (or an SU(2) matrix) does not rotate in an arbitrary pair of orthogonal planes, but in a pair such that one of the planes intersects the quaternion's w axis. For quaternions you are able to achieve an arbitrary 2-axis rotation of a 4D vector using:

The same is possible with the spinor notation. Allowing spinors to set the phases for any 2-axis 4D (or complex 2D) rotation.

So as with quaternions, there are three ways you can use these 2-component Pauli spinors:

1. for 3D rotations, using the formulae above.

2. for general 4D rotations (equivalently 2-axis complex 2D rotations)

3. natively to represent special isoclinic rotations in 4D (or complex 2D), as in:

In this native 4D space we can picture it in 3D as a vortex, with the spinor components giving the phase offset in the two toroidal rotation planes.

So 2-component spinors are useful whenever there are two orthogonal planes which are rotated equally (isoclinic). That includes light polarisation (phase space for up and right transverse directions is 4D isoclinic). This phenomenon is completely non-quantum but uses spinors to represent the light 2D phase.

Weyl spinors.

These are the same size as Pauli spinors, but we transform them with a 2x2 Lorentzian matrix SL(2,C) instead of the SU(2) matrix. The generators form two sub algebras: J+iK for left-chiral Weyl spinors and J-iK for right-chiral Weyl spinors. Each transforms in the SU(2) fashion, so the equivalent to the isoclinic rotation is presumably a 3D spatial rotation and a 3D boost. We can look at how a Weyl spinor transforms under normal 3D rotations, by showing the effect of rotations in the x, y and z axes:

This is exactly how they operate on the Pauli spinors above. If you write the spinor as (a+bi, c+di) then rotating in the z axis rotates the vectors (a,b) and (c,d) by equal amounts. Rotating in the y axis rotates the vector (a,c) and (b,d) by equal amounts, and rotating in x rotates the vectors (a,d) and (b,c) by equal amounts.

But they also transform under boosts (which Pauli spinors do not):

In this case it is the x axis that transforms as a real-valued matrix. For boosts in z we have a scale up on (a,b) and a scale down on (c,d). For boosts in x this acts diagonally as a scale up in (a+c, b+d) and a scale down in (a-c, b-d) or equivalently as an equal boost in (a,c) and (b,d). For the y axis the boost is in (a,d) and (b,c).

So as a 4D vector the Weyl twistor transforms as an isoclinic rotation for real-world rotations, and as an isoclinic boost for real-world boosts. Since there are 3 isoclinic axes in 4D (a,b), (a,c) and (a,d), it works fine in representing the Lorentz transformations as a double cover.

Treating the Weyl spinor (a+bi,c+di) as a 4D vector (a,b,c,d)=(t,x,y,z), rotations are points on a 3-sphere. The 2-sphere of equal sized boosts give the set:

which is just a 2-sphere in a,c,d, centred at (cosh(r/2),0,0) and with radius sinh(r/2) where r is the boost rapidity. So the set of boosts in a Weyl spinor is just the half-volume a>0 in a,c,d, and the rotations are the unit 3-sphere.

Where these two transformations meet represents a rotation followed by a boost that the return the spinor back to the identity (see above). They are therefore the Lorentz transformations that a Weyl spinor is completely invariant to.

This set of invariant transformation is two dimensional and paraboloidal, it is called the parabolic subgroup of Lorentz transformations, or more simply, the null rotations. Since Lorentz transformations are 6D and Weyl spinors are only 4D, it make sense that there is a 2-space of Lorentz transformations that equal the same Weyl spinor.

This happens when a pure rotation (theta, rotation axis k) and a pure boost (rapidity \xi, boost dir n) intersect. n and k are both unit length.

It occurs when:

n_x is the radius r of the circle of intersection here. k and n are both normalised, so the remaining degrees of freedom are the orientations (k_x,k_y) and (n_x,n_y), intersection occurs when they are are 90 degrees to each other in the x,y plane. Note that the rapidity and angle tend to equality as they get smaller, hence they are equal in the generators of parabolic transformations.

For small angles the n_z is tiny, which also explains why the parabolic transformation's generators are a lateral rotation followed by a lateral-only orthogonal boost.

For the identity spinor (1;0) this null rotation is the matrix (1,g;0,1) where the complex g is the 2 degrees of freedom.

While the above equations were performed on an identity spinor, we can pre-rotate and post-rotate the spinor to any angle, so a spinor's 'paraboloid' of Lorentz transformations can be in any coordinate frame, and transformed to the inertial frame of the particle.

Summary of Weyl spinors

Weyl spinors represent a point travelling at the speed of light in a particular direction, with an internal phase around that direction and boost in that direction. The boost represents the momentum/energy of the particle.

The spinor can be parameterised as

The complex number d on the Riemann sphere is the direction vector. The phase around this direction is theta and the boost (and therefore the momentum) is r. We can see that a zero momentum particle is a zero spinor, which has no direction or phase. In the flagpole analogy d on the Riemann sphere is the pole direction, theta is the flag phase and r is the length of the pole.

What makes the spinor actually spin over time is the Weyl equation:

which acts to make the phase theta change with time, as well as propogating the particle wave at the speed of light.

The parameterisation above hides the internal description. Like with the Pauli spinor, the Weyl spinor is a rotation around two orthogonal axes. But it also has a non-unit length, which represents the boost part (or its momentum). This is really a boost in two orthogonal planes.

As with for Pauli spinors and wuaternions, the sandwich product doubles the rotation in one plane and cancels it in another. But in addition it doubles the boost in one plane and cancels it in another. So it is an isoclinic rotation and boost, with different values for rotation and boost.

The point of the isoclinic part is that it can cancel out different planes allowing the direction vector to be any.

Visually we can depict the theta parameter with the Hopf Fibration (1. left handed or 2. right handed), and we can depict the boost parameter as two options:

1: uniform scale

2: momentum direction upscale (and time upscale) with lateral plane downscale.

The first transformation in both cases represents the Weyl spinor (d,theta,r) and the second transformation is the conjugate inverse that cancels out the second planar rotation and boost, resulting in the double-covered rotation and double-strength boost.

For 6-DOF Lorentz transformations, the Weyl spinor supports just a direction (2D) a phase (1D) around the direction and boost (1D) in that direction. The fact that only 4 spinor components can represent an isoclinic rotation and also isoclinic boost represents the fact that these two operations are on the same pair of planes. The null space left over from this ambiguity is the spare extra 2-DOFs in the Lorentz transformations.

Shape

The set of Lorentz transformations for complex b is:

Looking at the effect of this for the massless particle moving with z=t, and in the moving frame, we have two ways of interpreting it.

1. If we consider spatially separated points from our particle, then the invariant set looks like a temporal behaviour:

The effect has cylindrical symmetry with no helicity
radial inwards velocity is z / r for lateral radius r, so it sucks inwards above the particle (that's going upwards), and expands below the particle.

2. If we consider points at a future time we get a sparial invariance:

points are invariant as though translations on a Riemann sphere with apex (infinity) at the particle point:
the particle is at the top (0,0,1) in this plot and the points on the sphere represent particles that were originally coincident but travel slower than light (the bottom point) or travel slightly laterally. The translations on the Riemann sphere represent different null rotations that affect particles going slower than light. Giving the neighbourhood below our upwards shooting particle an invariance depicted in this diagram.

Spin

The spin of a Weyl or Pauli spinor is 1/2. This means it turns twice to return to the same spot, just like an anti-twistor. At small scales linear momentum is wave number (cycles per metre) so angular momentum is cycles per turn, hence an anti-twistor or spinor has half a cycle per turn.

Here spin refers to its magnitude, but we can measure the spin in a direction, which is done by taking the dot product of the direction with the three spin generators to get a 2x2 matrix, which you then apply to the spinor to get its spin. If the direction is up then the matrix is sigma_3 which is (1/2,0;0,-1/2). The eigenvectors are (1,0) and (0,1) with eigenvectors (spin values) of 1/2 and -1/2.

The result is written as a|1/2> + b|-1/2> which means a superposition of the spin half (spin up) and spin minus a half (spin down). But really the a and b in this superposition just give the size of the first and second components respectively in the spinor. (for a different measured spin vector it uses a different combination of the spinor's 4 degrees of freedom as a and b).

When we say spin 1/2 or spin -1/2 we are now talking about the value, not the magnitude. A measured spin in a direction. There is one spin state for each row of the generator (Pauli) matrix. More cycles per turn (S) means more spin states: 2S+1. So there is a single spin state plus one for every half cycle.

For higher angular momentum you have more cycles per turn (such as the graviton at 2 cycles per turn), but my conjecture is that you can then start to have different numbers of cycles per turn in different directions. That is not possible with low numbers. So the classical limit can have different angular momenta in different directions.

Because we are treating the two complex numbers in a spinor as spin up and spin down, this designation just refers to which of the dual rotation planes is has an amplitude. The spin up and spin down therefore refer to the two orthogonal rotation planes in the spinor (which is a quaternion). We say that spin up and spin down are orthogonal.

So the clifford torus is a nice image for the Pauli spinor at least. There is a spin around each of the two axes, spin up and spin down. The opposite handed clifford torus is the conjugate spinor.