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 [176]. 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 [206]. 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. [272
]) 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 [121] as in quantum gravity the notion of a spacetime event is not necessarily well-defined [239]. A concrete realization of this possibility is provided in Bose-Einstein condensate analogs of black holes [40]. Here the low energy phonon excitations obey Lorentz invariance and microcausality [270]. 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),
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 [104]. 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 [158
].
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