I’ve been studying the properties of algebraic Spinors for a long time, but only recently have I actually started expressing such a Spinor in terms of components.
For a starting point, take the relationship between a Spinor and a Vector:
Solving this expression for S is possible, in terms of the original components of V, the magnitude of V, and an arbitrary complex exponential, or ‘gauge factor’.
There are two things to point out here.
The plus/minus means that there are 2 distinct classes of Spinors that can represent the vector V. The distinction between particles and anti-particles are an example of these 2 Spinor classes. Spinors can distinguish this case, where Vectors can not.
I’ve written the gauge factor (the complex exponential) in terms of phi, which perhaps we presume is a scalar, but there’s actually more degrees of freedom allowed than just a mere scalar. As long as we maintain that it multiplies on the right, the complex exponential can be any Unitary Spinor, (a Spinor which when multiplied by its Hermitian conjugate results in 1). This means that the phase angle is allowed to contain imaginary scalar and imaginary vector components.
Let’s list some examples of Spinors that represent physical quantities, (written without the gauge factor).
Spacetime Coordinate Spinor (s is the total spacetime interval of the given coordinate):
Momentum Spinor:
Lorentz Transform Spinor:
It is interesting to contemplate what the plus/minus classes represent for each of these physical quantities. In the case of the Momentum Spinor, it is clear that these are the positive and negative energy solutions. In the case of the Lorentz Transform, these are the chronous/anti-chronous transforms.
What do these classes signify in the case of the spacetime coordinate spinor?