Dimensional reduction of Carroll-Field-Jackiw theory

RESUMO

This article investigates the dimensional reduction of the massive Carroll-Field-Jackiw (CFJ) theory from $(3+1)D$ to $(2+1)D$ dimensions with the objective of constructing and analyzing the resulting effective field theory, termed pseudo-Carroll-Field-Jackiw (PCFJ). The theoretical framework relies on quantum field theory, pseudo-quantum electrodynamics (PQED) and Lorentz-violating models. The methodological procedure projects the three-dimensional dynamics onto the plane by rigorously integrating out the gauge field degrees of freedom along the transverse momentum. Subsequently, the study calculates the static interaction potential, applies both the generalized optical theorem and the Källén-Lehmann spectral representation to test probability conservation. The results reveal that the effective theory exhibits an intricate non-local structure, governed by nested pseudo-differential operators, which generates a static potential that transitions from a Coulomb-like behavior at short distances to a long-range decay modified by a topological mass. The research analytically proves that the integration of the extra-dimensional topology introduces ineliminable branch cuts in momentum space, which preclude the causal definition of asymptotic states and determine the breakdown of unitarity. The work concludes that this quantum instability emerges from the direct interplay between the spatial projection and the pre-existing Lorentz-violating background in the bulk.