It is obvious that when we introduce Lorentz violation we have to rethink causality - there is no universal light cone given by the metric that all fields must propagate within. Even with Lorentz violation we must certainly maintain some notion of causality, at least in concordant frames, since we know that our low energy physics is causal. Causality from a strict field theory perspective is usually discussed in terms of microcausality which in turn comes from the cluster decomposition principle: Physical observables at different points and equal times should be independently measurable. This is essentially a statement that physics is local. We now briefly review how microcausality arises from cluster decomposition. Let represent two observables for a field theory in flat space. In a particular frame, let us choose the equal time slice , such that and further assume that . The cluster decomposition principle then states that and must be independently measurable. This in turn implies that their commutator must vanish, . When Lorentz invariance holds there is no preferred frame, so the commutator must vanish for the surface of any reference frame. This immediately gives that whenever are spacelike separated, which is the statement of microcausality. Microcausality is related to the existence of closed timelike curves since closed timelike curves violate cluster decomposition for surfaces that are pierced twice by the curves. The existence of such a curve would lead to a breakdown of microcausality.
Lorentz violation can induce a breakdown of microcausality, as shown in . In this work, the authors find that microcausality is violated if the group velocity of any field mode is superluminal. Such a breakdown is to be expected, as the light cone no longer determines the causal structure and notions of causality based on “spacelike” separation would not be expected to hold. However, the breakdown of microcausality does not lead to a breakdown of cluster decomposition in a Lorentz violating theory, in contrast to a Lorentz invariant theory. Even if fields propagate outside the light cone, we can have perfectly local and causal physics in some reference frames. For example, in a concordant frame Lorentz violation is small, which implies that particles can be only slightly superluminal. In such a frame all signals are always propagated into the future, so there is no mechanism by which signals could be exchanged between points on the same time slice. If we happened to be in such a concordant frame then physics would be perfectly local and causal even though microcausality does not hold.
The situation is somewhat different when we consider gravity and promote the Lorentz violating tensors to dynamical objects. For example in an aether theory, where Lorentz violation is described by a timelike four-vector, the four-vector can twist in such a way that local superluminal propagation can lead to energy-momentum flowing around closed paths . However, even classical general relativity admits solutions with closed timelike curves, so it is not clear that the situation is any worse with Lorentz violation. Furthermore, note that in models where Lorentz violation is given by coupling matter fields to a non-zero, timelike gradient of a scalar field, the scalar field also acts as a time function on the spacetime. In such a case, the spacetime must be stably causal (cf. ) and there are no closed timelike curves. This property also holds in Lorentz violating models with vectors if the vector in a particular solution can be written as a non-vanishing gradient of a scalar.
Finally, we mention that in fact many approaches to quantum gravity actually predict a failure of causality based on a background metric  as in quantum gravity the notion of a spacetime event is not necessarily well-defined . A concrete realization of this possibility is provided in Bose-Einstein condensate analogs of black holes . Here the low energy phonon excitations obey Lorentz invariance and microcausality . However, as one approaches a certain length scale (the healing length of the condensate) the background metric description breaks down and the low energy notion of microcausality no longer holds.
In any realistic field theory one would like a stable ground state. With the introduction of Lorentz violation, one must still have some ground state. This requires that the Hamiltonian still be bounded from below and that perturbations around the ground state have real frequencies. It will again be useful to discuss stability from a field theory perspective, as this is the only framework in which we can speak concretely about a Hamiltonian. Consider a simple model for a massive scalar field in flat space similar to Equation (1),. If the energy for every mode is positive, then the vacuum state is stable.
As an aside, note that while the energy is positive in , it is not necessarily positive in a boosted frame . If , then for large momentum , yielding a spacelike energy momentum vector. This implies that the energy can be less than zero in a boosted frame. Specifically, for a given mode in , the energy of this mode in a boosted frame is less than zero whenever the relative velocity between and is greater than . The main implication is that if is large enough the expansion of a positive frequency mode in in terms of the modes of (one can do this since both sets are a complete basis) may have support in the negative energy modes. The two vacua and are therefore inequivalent. This is in direct analogy to the Unruh effect, where the Minkowski vacuum is not equivalent to the Rindler vacuum of an accelerating observer. With Lorentz violation even inertial observers do not necessarily agree on the vacuum. Due to the inequivalence of vacua an inertial detector at high velocities should see a bath of radiation just as an accelerated detector sees thermal Unruh radiation. A clue to what this radiation represents is contained in the requirement that only if , which is exactly the criteria for Čerenkov radiation of a mode . In other words, the vacuum Čerenkov effect (discussed in more detail in Section 6.5) can be understood as an effect of inequivalent vacua.
We now return to the question of stability. For the models in Section 3.1 with higher order dispersion relations ( with ) there is a stability problem for particles with momentum near the Planck energy if as modes do not have positive energy at these high momenta. However, it is usually assumed that these modified dispersion relations are only effective - at the Planck scale there is a UV completion that renders the fundamental theory stable. Hence the instability to production of Planck energy particles is usually ignored.
So far we have only been concerned with instability of a quantum field with a background Lorentz violating tensor. Dynamical Lorentz violating tensors introduce further possible instabilities. In such a dynamical theory, one needs a version of the positive energy theorem [252, 279] that includes the Lorentz violating tensors. For aether theories, the total energy is proportional to the usual ADM energy of general relativity . Unfortunately, the aether stress tensor does not necessarily satisfy the dominant energy condition (although it may for certain choices of coefficients), so there is no proof yet that spacetimes with a dynamical aether have positive energy. For other models of Lorentz violation the positive energy question is completely unexplored. It is also possible to set limits on the coefficients of the aether theory by demanding that the theory be perturbatively stable, which requires that excitations of the aether field around a Lorentz violating vacuum expectation value have real frequencies .
© Max Planck Society and the author(s)