On the Impossibility of a Poincare-invariant Vacuum State with Unit Norm

TL;DR

This paper proves that in quantum field theory, a translationally invariant vacuum state with finite norm cannot exist, highlighting a fundamental inconsistency in the axioms of QFT.

Contribution

It demonstrates the impossibility of constructing a Poincare-invariant vacuum state with finite norm within standard quantum field theory axioms.

Findings

Any translationally-invariant vector must have divergent or zero norm

A vacuum state cannot be both invariant and normalized

The axioms of QFT are internally inconsistent

Abstract

In the standard construction of Quantum Field Theory, a vacuum state is required. The vacuum is a vector in a separable, infinite-dimensional Hilbert space often referred to as Fock space. By definition the vacuum wavestate depends on nothing and must be translationally invariant. We show that any such translationally-invariant vector must have a norm that is either divergent or equal to zero. It is impossible for any state to be both everywhere translationally invariant and also have a norm of one. The axioms of QFT cannot be made internally consistent.