On the metric of the space of states in a modified QCD

TL;DR

This paper investigates the state space metric in a modified gauge-invariant QCD with massive fermions, revealing a mix of positive and negative norm states and discussing implications for asymptotic freedom and Lee-Wick theories.

Contribution

It introduces a quantization approach for a modified QCD model showing indefinite metric properties and analyzes the nature of quark states, contrasting with standard QCD.

Findings

Presence of massless and massive propagation modes.

Massive modes have positive metric, massless modes have negative norm.

Negative metric massless modes do not necessarily undermine the theory.

Abstract

The form of the resulting Feynman propagators in a proposed local and gauge invariant QCD for massive fermions suggests the existence of indefinite metric associated to quark states, a property that might relate it with the known Lee-Wick theories. Thus, the nature of the asymptotic free quark states in the theory is investigated here by quantizing the quadratic part of the quark action. As opposite to the case in the standard QCD, the free theory does not show Hamiltonian constraints. The propagation modes include a family of massless waves and a complementary set of massive oscillations. The theory can be quantized in a way that the massive modes show positive metric and the massless ones exhibit negative norms. It is remarked that, since QCD is expected to not exhibit gluon or quark asymptotic states, the presence of negative metric massless modes does not constitute a definite drawback of the theory. In addition, the fact that the positive metric quark states are massive, seems to be a good feature of the model, being consistent with the approximate existence of asymptotically free massive states in high energy processes.