Negative Energies and Field Theory

TL;DR

This paper examines a fundamental inconsistency in quantum field theory related to vacuum energy, charge conservation, and gauge invariance, and discusses its resolution in quantum electrodynamics.

Contribution

It analyzes the conflict between vacuum energy assumptions and charge commutation relations, providing insights into gauge invariance in quantum electrodynamics.

Findings

Identifies a fundamental inconsistency in quantum field theory assumptions.

Shows how gauge invariance is maintained despite the conflict.

Provides a resolution within the framework of quantum electrodynamics.

Abstract

The assumption that the vacuum is the minimum energy state, invariant under unitary transformations, is fundamental to quantum field theory. However, the assertion that the conservation of charge implies that the equal time commutator of the charge density and its time derivative vanish for two spatially separated points is inconsistent with the requirement that the vacuum be the lowest energy state. Yet, for quantum field theory to be gauge invariant, this commutator must vanish. This essay explores how this conundrum is resolved in quantum electrodynamics.