  Bohmian mechanics and spontaneous collapse models are theories that overcome
the quantum measurement problem. While they are naturally formulated for
non-relativistic systems, it has proven difficult to formulate Lorentz
invariant extensions, primarily due to the inherent non-locality, which is
unavoidable due to Bell's theorem. There are trivial ways to make space-time
theories Lorentz invariant, but the challenge is to achieve what Bell dubbed
``serious Lorentz invariance''. However, this notion is hard to make precise.
This is reminiscent of the debate on the meaning of general invariance in
Einstein's theory of general relativity. The issue there is whether the
requirement of general invariance is physically vacuous (in the sense that any
space-time theory can be made generally invariant) or whether it is a
fundamental physical principle. Here, we want to consider two of the more
promising avenues that have emerged from that debate in order to explore what
serious Lorentz invariance could mean. First, we will consider Anderson's
approach based on the identification of absolute objects. Second, we will
consider a relativity principle for isolated subsystems. Using these criteria,
we will evaluate a number of Lorentz invariant Bohmian models and a spontaneous
collapse model, finding that the latter satisfies both criteria, while there
are some Bohmian models that violate the criteria. However, some Bohmian models
that satisfy both criteria still do not seem seriously Lorentz invariant. While
these notions may hence still not capture exactly what serious Lorentz
invariance ought to be, they clarify what aspects of relativity theory (in
addition to locality) may need to be given up in passing from classical to
quantum theory.
