Photon spin operator and Pauli matrix

TL;DR

This paper develops a formalism for photon spin using polarization basis and Pauli matrices, showing that the spin components commute and relate directly to the Stokes vector along the wave vector.

Contribution

It introduces a representation of photon spin via a quasi-unitary transformation linking polarization vectors and spinors, clarifying the relation to Pauli matrices and Stokes parameters.

Findings

Photon spin components commute mutually.

Photon spin angular momentum equals the Stokes vector component along the wave vector.

Spin operator is represented by Pauli matrices in the polarization basis.

Abstract

Any polarization vector of a plane wave can be decomposed into a pair of mutually orthogonal base vectors, known as a polarization basis. Regarding this decomposition as a quasi-unitary transformation from a three-component vector to a corresponding two-component spinor, one is led to a representation formalism for the photon spin. The spin operator $\hat{\boldsymbol \gamma}$ defined on the space of unit spinors, referred to as the Jones space, has only component along the wave vector and is represented by one of the Pauli matrices in the commonly used polarization basis. It is deformed by the quasi-unitary transformation from the spin operator that is defined on the space of unit polarization vectors, referred to as the Pancharatnam space. On the basis of this theory, it is shown that the Cartesian components of spin operator $\hat{\boldsymbol \gamma}$ are mutually commutative and the spin angular momentum in units of $\hbar$ is exactly the component of the Stokes vector along the wave vector.