Lorentz-Covariant Spin Operator for Spin 1/2 Massive Fields As a Physical Observable

TL;DR

This paper derives a relativistic-covariant spin operator for massive spin-1/2 fields from space-time symmetries, clarifying its properties and relation to known spin operators in quantum field theory.

Contribution

It introduces a unique, covariant spin operator for massive fields based on Poincare symmetry and space inversion, unifying different spin descriptions.

Findings

The derived spin operator is covariant and generates the SU(2) little group.

It reduces to the Foldy-Wouthuysen spin for positive energy states.

The operator uniquely describes the spin of massive fields in relativistic quantum theory.

Abstract

We derive a relativistic-covariant spin operator for massive case directly from space-time symmetry in Minkowski space-time and investigate the physical properties of a derived spin operator. In the derivation we require only two conditions: First, a spin operator should be the generator of the SU(2) little group of the Poincare group. Second, a spin operator should covariantly transform under the Lorentz transformation. A space inversion transformation is shown to play a role to derive a unique relativistic-covariant spin operator, we call the field spin operator, whose eigenvalue labels the spin of a massive (classical) field that provides the irreducible representation space of the Poincare group. The field spin becomes the covariant spin in the covariant Dirac representation, which is shown to be the only spin that describes the Wigner rotation properly in the covariant Dirac representation. Surprisingly, the field spin also gives the non-covariant spin, which is the FW spin for the positive energy state. We also show that the field spin operator is the unique spin operator that generate the (internal) SU(2) little group transformation of the Poincare group properly.