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A spacetime-independent variable is introduced which characterizes a Lorentz-invariant self-sustained quantum vacuum. For a perfect (Lorentz-invariant) quantum vacuum, the self-tuning of this variable nullifies the effective energy density which enters the low-energy gravitational field equations. The observed small but nonzero value of the cosmological constant may then be explained as corresponding to the effective energy density of an imperfect quantum vacuum (perturbed by, e.g., the presence of thermal matter).

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].

A modified-gravity theory is considered with a four-form field strength F , a variable gravitational coupling parameter G(F ), and a standard matter action. This theory provides a concrete realization of the general vacuum variable q as the four-form amplitude F and allows for a study of its dynamics. The theory gives a flat Friedmann-Robertson-Walker universe with rapid oscillations of the effective vacuum energy density (cosmological "constant"), whose amplitude drops to zero asymptotically. Extrapolating to the present age of the Universe, the order of magnitude of the average vacuum energy density agrees with the observed near-critical vacuum energy density of the present universe. It may even be that this type of oscillating vacuum energy density constitutes a significant part of the so-called cold dark matter in the standard Friedmann-Robertson-Walker framework. PACS numbers: 04.20.Cv, 98.80.Jk, 95.35.+d, 95.36.+x

There is a unique Lorentz-violating modification of the Maxwell theory of photons, which maintains gauge invariance, CPT, and renormalizability. Restricting the modified-Maxwell theory to the isotropic sector and adding a standard spin-1 2 Dirac particle p ± with minimal coupling to the nonstandard photon γ, the resulting modified-quantum-electrodynamics model involves a single dimensionless "deformation parameter," κ tr . The exact tree-level decay rates for two processes have been calculated: vacuum Cherenkov radiation p ± → p ± γ for the case of positive κ tr and photon decay γ → p + p − for the case of negative κ tr . From the inferred absence of these decays for a particular high-quality ultrahigh-energy-cosmic-ray event detected at the Pierre Auger Observatory and a well-established excess of TeV gamma-ray events observed by the High Energy Stereoscopic System telescopes, a two-sided bound on κ tr is obtained, which improves by eight orders of magnitude upon the best direct laboratory bound. The implications of this result are briefly discussed.

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In a fermionic quantum vacuum, the parameters kµ of a CPT-violating Chern-Simonslike action term induced by CPT-violating parameters of the fermionic sector depend on the universality class of the system. As a concrete example, we consider the Dirac Hamiltonian of a massive fermionic quasiparticle and add a particular term with purelyspacelike CPT-violating parameters bµ = (0, b). A quantum phase transition separates two phases, one with a fully-gapped fermion spectrum and the other with topologicallyprotected Fermi points (gap nodes). The emergent Chern-Simons "vector" kµ = (0, k) now consists of two parts. The regular part, k reg , is an analytic function of |b| across the quantum phase transition and may be nonzero due to explicit CPT violation at the fundamental level. The anomalous (nonanalytic) part, k anom , comes solely from the Fermi points and is proportional to their splitting. In the context of condensed-matter physics, the quantum phase transition may occur in the region of the BEC-BCS crossover for Cooper pairing in the p-wave channel. For elementary particle physics, the splitting of Fermi points may lead to neutrino oscillations, even if the total electromagnetic ChernSimons-like term cancels out.