Boost modes for a massive fermion field

TL;DR

This paper constructs a complete set of boost modes for a 2D fermion field using the Wightman function, revealing singularities at the light cone that affect the completeness of the mode set.

Contribution

It introduces a method to generate solutions of relativistic wave equations from the Wightman function and analyzes the singularities of boost modes for a fermion field.

Findings

Boost modes contain delta-function singularities at the light cone.

Excluding a single mode with arbitrary boost quantum number renders the set incomplete.

The approach links the Wightman function to the completeness of mode solutions.

Abstract

We have shown that Wightman function of a free quantum field generates any complete set of solutions of relativistic wave equations. Using this approach we have constructed the complete set of solutions to 2d Dirac equation consisting of eigenfunctions of the generator of Lorentz rotations (boost operator). It is shown that at the surface of the light cone the boost modes for a fermion field contain $\delta$-function of a complex argument. Due to the presence of such singularity exclusion even of a single mode with an arbitrary value of the boost quantum number makes the set of boost modes incomplete.