Representation Dependence of Superficial Degree of Divergences in Quantum Field Theory

TL;DR

This paper explores how the superficial degree of divergence in quantum field theories can depend on the representation chosen, especially between Hermitian and non-Hermitian forms, and introduces a class of non-Hermitian theories with physically acceptable metrics.

Contribution

It demonstrates the representation dependence of divergence degrees and introduces a new class of non-Hermitian quantum field theories with explicit metric operators and different coupling constant dimensions.

Findings

Divergences originate from various interaction terms regardless of spacetime dimensions.

Non-Hermitian theories can be physically acceptable with explicit metric operators.

Different representations can alter the perceived renormalizability of a theory.

Abstract

In this work, we investigate a very important but unstressed result in the work of Carl M. Bender, Jun-Hua Chen, and Kimball A. Milton ( J.Phys.A39:1657-1668, 2006). In this article, Bender \textit{et.al} have calculated the vacuum energy of the $i\phi^{3}$ scalar field theory and its Hermitian equivalent theory up to $g^{4}$ order of calculations. While all the Feynman diagrams of the $i\phi^{3}$ theory are finite in $0+1$ space-time dimensions, some of the corresponding Feynman diagrams in the equivalent Hermitian theory are divergent. In this work, we show that the divergences in the Hermitian theory originate from superrenormalizable, renormalizable and non-renormalizable terms in the interaction Hamiltonian even though the calculations are carried out in the $0+1$ space-time dimensions. Relying on this interesting result, we raise the question, is the superficial degree of divergence of a theory is representation dependent? To answer this question, we introduce and study a class of non-Hermitian quantum field theories characterized by a field derivative interaction Hamiltonian. We showed that the class is physically acceptable by finding the corresponding class of metric operators in a closed form. We realized that the obtained equivalent Hermitian and the introduced non-Hermitian representations have coupling constants of different mass dimensions which may be considered as a clue for the possibility of considering non-Renormalizability of a field theory as a non-genuine problem. Besides, the metric operator is supposed to disappear from path integral calculations which means that physical amplitudes can be fully obtained in the simpler non-Hermitian representation.