User:Singnear/sandbox

A spinor is a representation of a spinor field (the simplest example of a spinor field is the electromagnetic field) in Minkowski space.

In this article, we will explain how spinors describe the electromagnetic field in terms of quaternions.

Let’s consider the electromagnetic field in Minkowski space. It can be described by a vector field, which corresponds to the 3-dimensional vector of the electromagnetic fields (i.e. the electric field and the magnetic field). It can be expressed as

\[ F = q A \]

where \(q\) is an electric charge, and \(A\) is the magnetic vector potential. The vector field \(\mathbf{F}\) can be expressed in terms of a spinor field, \(\mathbf{F} = \mathbf{A} \mathbf{X} \), where \(\mathbf{A} = 0 = \mathbf{X}\) is the “true” vector field and the spinors \(\mathbf{X}\) and \(\mathbf{A}\) are the “quatitoned” vector field.

If we consider the component of the quaternion \(q A\) in the direction of u, it is simply a quaternion of the form

\[ \frac{1}{2} \begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4} \end{bmatrix} \begin{bmatrix} a_{1}\\ a_{2}\\ a_{3}\\ a_{4} \end{bmatrix} \]

where \(x_{i}\) are the components of the vector.