Avoiding Haag's theorem with parameterized quantum field theory

TL;DR

This paper proposes a parameterized quantum field theory formulation using an additional fifth path parameter, which avoids Haag's theorem and provides a consistent foundation for the practical perturbative approach in quantum field theory.

Contribution

It introduces a fifth path parameter into quantum field theory to circumvent Haag's theorem, enabling a consistent perturbative expansion for scattering processes.

Findings

Haag's theorem can be avoided with the parameterized formalism.

The Dyson perturbation expansion can be reproduced in the new framework.

Provides a mathematically consistent foundation for practical quantum field theory.

Abstract

Under the normal assumptions of quantum field theory, Haag's theorem states that any field unitarily equivalent to a free field must itself be a free field. Unfortunately, the derivation of the Dyson series perturbation expansion relies on the use of the interaction picture, in which the interacting field is unitarily equivalent to the free field but must still account for interactions. Thus, the traditional perturbative derivation of the scattering matrix in quantum field theory is mathematically ill defined. Nevertheless, perturbative quantum field theory is currently the only practical approach for addressing scattering for realistic interactions, and it has been spectacularly successful in making empirical predictions. This paper explains this success by showing Haag's Theorem can be avoided when quantum field theory is formulated using an invariant, fifth path parameter in addition to the usual four position parameters, such that the Dyson perturbation expansion for the scattering matrix can still be reproduced. As a result, the parameterized formalism provides a consistent foundation for the interpretation of quantum field theory as used in practice and, perhaps, for better dealing with other mathematical issues.