Batalin-Vilkovisky quantization with an angular twist

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.