Zero-mass limit of a Dirac spinor with general spin orientation

TL;DR

This paper investigates the zero-mass limit of Dirac spinors with arbitrary spin orientation, clarifying their behavior and relation to chirality, which is often overlooked in textbooks and crucial for understanding particle physics.

Contribution

It provides a detailed derivation of Dirac spinors for general spin orientations and analyzes their zero-mass limit, connecting it to chirality eigenstates.

Findings

Zero-mass limit of general spin Dirac spinors differs from helicity states.

The behavior of spinors near zero mass has implications for particle physics.

Clarifies the relation between spin orientation and chirality in the massless limit.

Abstract

The helicity eigenstates which describe the fermions with a special spin orientation (parallel or antiparallel to the direction of momentum) provide considerable simplification in calculations. Hence, it is generally preferred to use helicity basis during the calculations in Relativistic Quantum Mechanics or Quantum Field Theory. Possibly because of the above reason, the Dirac spinors describing a general spin orientation have been ignored in many textbooks. Although the helicity eigenstates give almost complete understanding of the behavior of the free Dirac solutions, the zero-mass limit is one of its exception. The zero-mass behavior of the free spinor with general spin orientation and its relation with chirality eigenstates has been skipped in textbooks and hence it deserves a clear, detailed investigation. In this paper we obtain the free Dirac spinors describing a general spin orientation and examine their zero-mass limit. We also briefly discuss some of the implications of this small-mass behavior of the spinors on particle physics.