Fermionic scalar field

TL;DR

This paper investigates the spin-statistics connection for scalar fields, revealing issues with negative norm states when imposing fermionic relations and proposing a fermionic symmetry approach to address them.

Contribution

It introduces a novel fermionic symmetry framework for scalar fields to resolve negative norm state issues in quantization.

Findings

Imposing anti-commutation relations leads to negative norm states.

Introducing fermionic symmetry creates a doublet structure.

The system becomes trivial with only the vacuum state remaining.

Abstract

We reexamine the connection between spin and statistics through the quantization of a complex scalar field, using the formulation with the property that the hermitian conjugate of canonical momentum for a variable is just the canonical momentum for the hermitian conjugate of the variable. Starting from an ordinary Lagrangian density and imposing the anti-commutation relations on the field, we find that the difficulty stems from not the ill-definiteness (or unboundedness) of the energy and the breakdown of the causality but the appearance of states with negative norms. It is overcome by introducing an ordinary scalar field to form a doublet of fermionic symmetries, although the system becomes empty leaving the vacuum state alone. These features also hold for the system with a spinor field imposing the commutation relations on.