The Classical Spinor

In the previous post we discussed the different relativistic significant quantities. We will now use these quantities to quantify physical parameters.

Consider a particle that is travelling on a path with arbitrary acceleration. At any point on the path, it is possible to find a co-moving reference frame, or a frame where the particle appears to be instantaneously at rest. We will call this the "rest frame", even though we must continually change this frame as the particle progresses.

The term "rest" in relativity does not mean "motionless", for even if the particle is not moving in space, it is moving in time. Strictly speaking it is moving in time at the speed of light. In the rest frame of the particle, the velocity is always along the time direction. Thus, we can represent the velocity of the particle with a scalar.

u = c

If we want to transform this velocity from the rest frame back to the frame where we are observing the particle, we need to apply a Lorentz transformation.

u' = LuL^† = cLL^†

As we have previously seen, a para-vector such as u' can be a composition of spinors. Thus, we can consider the spinor L to characterize the velocity of the particle. As part of his development of the Algebra of Physical Space, W. E. Baylis chooses to assign a special name to the spinor L. He calls it the Classical Spinor, or Classical Eigen-Spinor, and denotes it Λ. Baylis uses units where c = 1, but we aren't going to do that here. Therefore we need to include this factor in the definition of the classical spinor.

Λ = (c^1/2) L

This is called the classical spinor, because it is the spinor which is representative of the classical trajectory of the particle, and so in some respect representative of the particle itself. Using the classical spinor has manay advantages. For instance, we not only can determine the velocity of the particle, but we can also determine the spatial orientation of the coordinate system that the particle resides in.

We previously saw that the Lorentz transformation can be given in an exponential form, in terms of the generators of the transformation. We can likewise construct the classical spinor from generators, which are functions the proper time of the particle τ. For instance, we can express a particle that is spinning at a constant rate as

Λ = (c^1/2) e^{(1/2) ωτ}

In this case, the generator of the transformation is the vector ωτ. We call the vector ω the spatial rotation rate, and require that it is purely imaginary. The classical spinor can describe a particle with quantum-like spin, without the need of any quantum postulates. For instance, the classical spinor changes sign upon a full rotation, and requires two full rotations before the sign is restored. In this sense, a particle represented by a classical eigenspinor can be associated with a spin 1/2 particle, like an electron.

If we want a particle that can accelerate as well as spin, then we allow the rotation rate to have real as well as imaginary parts. If the rotation rate has arbitrary real and imaginary parts, we call it the space-time rotation rate, and designate it as Ω.

A classical eigenspinor with a constant space-time rotation rate represents a particle that is spinning and accelerating at a constant rate.

Λ = (c^1/2) e^{(1/2) Ωτ}

We can take a derivative with respect to τ in order to acheive what Baylis calls the equation of motion for the classical spinor.

dΛ/dτ = (1/2) ΩΛ

This equation ends up being valid, even when the space-time rotation rate is not constant.

So what are the possible values the Ω can take? Knowing that the magnitude of the 4-velocity of a particle is constant, we also know that the magnitude of the classical spinor is also constant. This constraint helps us determine the restriction on Ω.

ΛΛ = c e^{(1/2) (Ω + Ω) τ}

In order for Λ to be constant, we need the factor involving the exponent to go away. This will happen if the quantity in the exponent involving Ω is always zero. This quantity in the exponent just happens to represent the scalar portion of Ω.

There are two ways of interpreting this result. We may say that the physical quantity that we associate with Ω can not have a scalar portion, or we can say that the physical quantity can have a scalar portion, but the scalar portion does not contribute to Λ. Though Baylis uses the first interpretation, I personally prefer the second. If you want to take the second interpretation, then you need to modify the expression for the equation of motion of Λ.

dΛ/dτ = (1/2) <Ω>_VΛ

The reason I prefer this interpretation, is because it introduces a symmetry. The symmetry here is that we can freely change the scalar portion of Ω without changing the resulting trajectory of the particle. This symmetry should have physical consequences, as all symmetries do. If these physical consequences can be justified, then this form of the equation will be justified.