Unitary Positive-Energy Representations of Scalar Bilocal Quantum Fields

TL;DR

This paper classifies superselection sectors of scalar bilocal quantum fields in four or more dimensions using unitarity constraints, confirming gauge group dualities and highlighting the role of Hamiltonian choices in fixing representations.

Contribution

It provides a novel classification of superselection sectors for scalar bilocal fields via Lie algebra methods, complementing existing algebraic approaches.

Findings

Superselection sectors correspond to duals of U(N) and O(N) gauge groups.

Models without scalar fields of dimension D-2 are sufficient under conformal invariance.

Appropriate Hamiltonian choices can fully determine the representation theory.

Abstract

The superselection sectors of two classes of scalar bilocal quantum fields in D>=4 dimensions are explicitly determined by working out the constraints imposed by unitarity. The resulting classification in terms of the dual of the respective gauge groups U(N) and O(N) confirms the expectations based on general results obtained in the framework of local nets in algebraic quantum field theory, but the approach using standard Lie algebra methods rather than abstract duality theory is complementary. The result indicates that one does not lose interesting models if one postulates the absence of scalar fields of dimension D-2 in models with global conformal invariance. Another remarkable outcome is the observation that, with an appropriate choice of the Hamiltonian, a Lie algebra embedded into the associative algebra of observables completely fixes the representation theory.