Causality and CPT violation from an Abelian Chern–Simons-like term

Abstract

We study a class of generalized Abelian gauge field theories where CPT symmetry is violated by a Chern-Simons-like term which selects a preferred direction in spacetime. Such Chern-Simons-like terms may either emerge as part of the low-energy effective action of a more fundamental theory or be produced by chiral anomalies over a nonsimply connected spacetime manifold. Specifically, we investigate the issues of unitarity and causality. We find that the behaviour of these gauge field theories depends on whether the preferred direction is spacelike or timelike. For a purely spacelike preferred direction, a well-behaved Feynman propagator exists and microcausality holds, which indicates the possibility of a consistent quantization of the theory. For timelike preferred directions, unitarity or causality is violated and a consistent quantization does not seem to be possible. and ǫ µνρσ is the completely antisymmetric Levi-Civita symbol, normalized to ǫ 0123 = +1.(Our conventions, with = c = 1, will be given in more detail later on.)The Abelian Chern-Simons-like term (1.2) is characterized by a real mass parameter m and a real symmetry-breaking "vector" k µ of unit length, which may be spacelike (k 2 = +1) or timelike (k 2 = −1) but is fixed once and for all (hence, the quotation marks around the word vector). Strictly speaking, k µ can also be "lightlike" (k 2 = 0), but the present paper considers only the extreme cases, spacelike or timelike k µ . As long as k µ and m = 0 are fixed external parameters (coupling constants), both Lorentz and CPT invariance are broken, but translation invariance still holds. Note that the Lagrangian term (1.2) is called Chern-Simons-like, because a genuine topological Chern-Simons term exists only in an odd number of dimensions [14].

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Erratum: Dimensional reduction of a Lorentz- andCPT-violating Maxwell-Chern-Simons model [Phys. Rev. D67, 125011 (2003)]

“…Investigations concerning the unitarity, causality and consistency of a QFT endowed with violation of Lorentz and CPT symmetries (induced by a Chern-Simons term) were carried out by Adam & Klinkhamer [5]. As result, it was verified that the causality and unitarity of this kind of model can be preserved when the fixed (background) 4-vector is space-like, and spoiled whenever it is time-like or null.…”

Lorentz-violating SO(3) model: Discussing unitarity, causality, and ’t Hooft-Polyakov monopoles

“…In the case of the action corresponding to the 3-component, the authors of [3], [8], [11] study the implications on the unitarity and causality of the theory in the cases where, for small magnitudes, c µ is time-like and space-like. The analysis shows that the behavior of these gauge field theories depends drastically on the space-time properties of c µ .…”