Some remarks on point split commutators

TL;DR

This paper investigates the impact of point splitting the Hamiltonian in quantum field theory, revealing that it can lead to states with lower energy than the vacuum, thus requiring a redefinition of the vacuum state.

Contribution

It demonstrates that point splitting the Hamiltonian affects the vacuum energy and necessitates a new vacuum definition in free fermion fields in 1+1 dimensions.

Findings

Quantum states with less energy than the normal vacuum can exist.

Point splitting the Hamiltonian influences the vacuum structure.

Vacuum state must be redefined when Hamiltonian is point split.

Abstract

Point splitting has been suggested as a way to deal with anomalous commutators in quantum field theory. It has been pointed out by D.G. Boulware[4] that in order to obtain a mathematically consistent theory the Hamiltonian operator must be point split also. We will examine the effect of point splitting the Hamiltonian for a free fermion field in 1-1D space-time. It will be shown that when the Hamiltonian operator is point split then quantum states will exist with less energy than the normal vacuum state. This requires the vacuum state to be redefined.