Batalin-Vilkovisky quantization with an angular twist

TL;DR

This paper develops two noncommutative quantum field theories on $$-Minkowski space using Batalin-Vilkovisky formalism and harmonic analysis, revealing distinct UV behaviors and UV/IR mixing phenomena.

Contribution

It introduces two inequivalent noncommutative scalar field theories with novel spectral decompositions and analyzes their ultraviolet divergences and mixing effects.

Findings

The braided theory exhibits usual logarithmic UV divergences and no UV/IR mixing.

The standard theory shows UV/IR mixing with non-analytic behavior at exceptional momenta.

Both theories relate spectral decompositions to eigenmodes of the Klein-Gordon operator.

Abstract

We construct cubic scalar field theory on $\lambda$-Minkowski space by combining the Batalin-Vilkovisky formalism with harmonic analysis, and produce two inequivalent noncommutative quantum field theories. The braided theory is based on a braided $L_\infty$-algebra whereby covariance dictates a spectral decomposition into cylindrical Bessel functions that diagonalise the angular Drinfel'd twist; in this theory we find the usual logarithmic ultraviolet divergences and confirm the absence of UV/IR mixing. The standard noncommutative theory is based on a classical $L_\infty$-algebra; in this theory we relate the spectral decompositions into plane wave and cylindrical harmonic eigenmodes of the Klein-Gordan operator, we verify the planar equivalence theorem, and we demonstrate a periodic form of UV/IR mixing in which non-planar correlators are generically ultraviolet finite but become non-analytic on an infinite lattice of exceptional momenta.