# Lorentz invariance and quantum mechanics

Ward Struyve111Department of Physics and Astronomy, KU Leuven,
Belgium∗222Centre for Logic and Philosophy of Science, KU Leuven, Belgium

###### Abstract

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.

##  1 Introduction

In the non-relativistic domain, there exist precise versions of quantum
mechanics that do not suffer from the measurement problem. Examples are
Bohmian mechanics [1, 2, 3] and spontaneous collapse theories [4]. The
extension of these theories to the relativistic domain has proven challenging.
One aspect is to deal with the phenomenon of particle creation and
annihilation. Bohmian [5, 6, 7, 8] as well as spontaneous collapse models [9]
have been developed to account for this. However, the simplest formulation of
those models employs a preferred reference frame. Even though this frame may
not be detectable and the corresponding statistical predictions may be Lorentz
invariant, like in the Bohmian theories, the dynamical dependence on the frame
implies a violation of Lorentz invariance at the fundamental level.

Lorentz invariance at the fundamental level is hard to achieve. Only in recent
years, some (toy) models have been proposed [10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26]. It is hard because these theories must
necessarily be non-local, i.e., they must entail influences over space-like
distances, as a result of Bell’s theorem [27, 28, 29]. The non-locality does
not exclude Lorentz invariance. Lorentz invariance concerns a dynamical
symmetry and can hold for a non-local theory, examples being tachyonic
theories or the Wheeler–Feynman theory of electrodynamics. Nevertheless, in
the context of quantum mechanics, it seems difficult to marry the non-locality
with Lorentz invariance.

Prima facie, it might sound strange that a Lorentz invariant formulation is
difficult. After all, every textbook on relativistic quantum mechanics or
quantum field theory is concerned with such formulations. However, these
textbooks tend to ignore the collapse postulate. In standard quantum
mechanics, the wave function is supposed to collapse upon measurement, but if
the collapse is instantaneous, it requires the specification of a particular
time coordinate, thereby violating the Lorentz invariance.

When we say that a Lorentz invariant quantum theory is hard to devise, we mean
Lorentz invariance in what Bell called a “serious” way [30]. After all, there
are trivial ways to make space-time theories Lorentz invariant, even theories
that are not regarded as such, like Newtonian mechanics [31, 10, 26].

To see this, let us first recall what is meant by Lorentz invariance. A
physical theory concerns the specification of a set ${\mathcal{K}}$ of
kinematically allowed histories (e.g., particle world lines or space-time
fields) and a subset ${\mathcal{D}}\subseteq{\mathcal{K}}$ of dynamically
allowed histories that satisfy the laws of motion. For a theory in Minkowski
space-time, a Lorentz transformation will map possible histories to histories,
yielding a permutation of ${\mathcal{K}}$. A theory is Lorentz invariant iff
each Lorentz transformation $\Lambda$ maps dynamically allowed histories to
dynamically allowed histories, i.e., $\Lambda{\mathcal{D}}={\mathcal{D}}$.

Consider now Newtonian mechanics. Newtonian mechanics is naturally formulated
in Galilean space-time and the natural space-time symmetry group is that of
the Galilean transformations [32, 33]. However, the theory can trivially be
recast in Minkowski space-time in a Lorentz invariant way. To do this,
consider a Lorentz frame $F$ and postulate the Newtonian dynamics to hold in
this frame. As such, the theory is of course not Lorentz invariant. Namely,
denoting the set of dynamically allowed histories by ${\mathcal{D}}^{F}$
(i.e., the set of trajectories that form solutions of Newton’s equations in
the frame $F$), then for a Lorentz transformation $\Lambda$, we have that
$\Lambda{\mathcal{D}}^{F}\neq{\mathcal{D}}^{F}$. That is, the Lorentz
transformation of a solution to Newton’s equations is generically not a
solution. However, consider a reformulation where the frame is part of the
degrees of freedom, so that a kinematically allowed history is now $(F,X)$,
with a law such that $(F^{\prime},X^{\prime})\in{\mathcal{D}}$ iff there is a
Lorentz transformation $\Lambda$, such that
$(F^{\prime},X^{\prime})=\Lambda(F,X)$, with $X\in{\mathcal{D}}^{F}$. The
theory is Lorentz invariant since $\Lambda{\mathcal{D}}={\mathcal{D}}$ for all
Lorentz transformations $\Lambda$.

The example can perhaps be made less abstract as follows. A frame determines a
future causal unit-normal vector field $n^{\mu}$ which is normal to the
constant-time surfaces and is constant over space-time, i.e., it satisfies

$\partial_{\nu}n^{\mu}=0.$ (1)

Using this vector field $n^{\mu}$, Newton’s equations can be written
explicitly in Lorentz invariant form, with the familiar form of the equations
holding in the frame where $n^{\mu}=\delta^{\mu}_{0}$. The vector field
$n^{\mu}$ is introduced as an extra dynamical quantity, but clearly it
satisfies a Lorentz invariant law.

So, the trick is to introduce the frame (or the vector field $n^{\mu}$) as
extra space-time structure. Another possibility would be to let the frame be
determined by the matter distribution itself. In that case, there would be no
extra space-time structure. For example, Bell [30] (crediting Baumann)
suggested that Newtonian mechanics can be made Lorentz invariant by letting
the frame be the center-of-mass frame determined by the matter distribution
(assuming the latter is finite). Indeed, if this frame could be determined in
a Lorentz invariant way from the trajectories, then the theory would be
Lorentz invariant. However, it is not clear how this proposal is supposed to
work. In the first place, it is problematic to have a relativistic
generalization of the notion of center-of-mass frame that shares the simple
properties of the Newtonian one [34]. Furthermore, the Newtonian dynamics
determines a Newtonian center-of-mass frame (where the total Newtonian
momentum is zero), but does not necessarily admit a relativistic center-of-
mass frame, which is independent of the Lorentz frame in which it is defined,
and which is what is required for a Lorentz invariant formulation. Conversely,
a relativistic center-of-mass frame does necessarily determine a Newtonian
one. (For example, the relativistic center-of-mass frame could be defined as
the frame where relativistic total 3-momentum, i.e., the spatial part of the
total 4-momentum, of the particles vanishes. But the relativistic 3-momentum
differs from the Newtonian 3-momentum and therefore the latter generically
does not vanish in such a frame.) So, it is unclear how exactly to realize
Bell’s proposal. But perhaps there are other ways of letting the frame be
determined by the matter distribution.

In any case, it is clear that Newtonian mechanics, as well as other space-time
theories, can be reformulated in a Lorentz invariant way. However, such
formulations cannot be considered as seriously Lorentz invariant. But what
exactly does “serious Lorentz invariance” mean? Bell found this hard to make
precise [30]. Since Bell, little has been said about this in the context of
quantum mechanics. On the other hand, there has been an extensive debate on
this for general relativity [35]. While Einstein motivated his field equations
by requiring invariance under general coordinate transformations, it was
argued by Kretschmann [36], very early on, that this motivation was physically
vacuous, since non-invariant theories could trivially be made invariant. There
are some attempts to make precise what serious general invariance could mean.
Here, we want to consider these in the context of quantum theory.

One of the best developed attempts is that of Anderson [37]. Anderson
distinguishes between the covariance group and the symmetry group of a theory.
A covariance group is a group of transformations that maps solutions of the
dynamical equations to solutions (i.e., leaves ${\mathcal{D}}$ invariant).
(This is how we characterized Lorentz invariance before.) The group action
defines an equivalence relation, with an equivalence class containing
solutions that are related by elements of the covariance group. The symmetry
group is defined as the largest subgroup of the covariance group that leaves
invariant the absolute objects of the theory. The absolute objects are
distinguished from the dynamical objects and are the geometric objects (on
some manifold) that appear, together with all their transformations under the
covariance group, in every equivalence class. Anderson’s analysis, presented
here in a nutshell, could serve as a possible candidate to characterize
serious Lorentz invariance. If the Lorentz group is a covariance group, then
it will also be a symmetry group if it leaves invariant all the absolute
objects of the theory. An object will be absolute if it is the same for each
solution, up to a Lorentz transformation. (One of these absolute objects is of
course the Minkowski metric.)

What does Anderson’s analysis entail for the Lorentz invariant formulation of
Newtonian mechanics? The covariance group is clearly the group of Lorentz
transformations. But $n^{\mu}$ (or $F$) is an absolute object, since for any
two possible solutions $(n^{\mu},X)$ and $(n^{\prime\mu},X^{\prime})$, there
is a Lorentz transformation that maps $n^{\mu}$ to $n^{\prime\mu}$. Since the
Lorentz group does not leave $n^{\mu}$ invariant, the symmetry group is not
the Lorentz group; the Lorentz invariance is not serious.

As has been observed before, Anderson’s approach is not without problems [35,
38], and similar problems will resurface in the present context.

Another possible characterization of serious Lorentz invariance is through a
relativity principle for (approximately) isolated subsystems [39, 40, 41, 42].
This is inspired by the following observation. Empirical evidence for
dynamical symmetries is usually obtained by considering transformations of
dynamically isolated subsystems within the universe (i.e., subsystems whose
motion does not depend on the environment), rather than of the universe
itself. Symmetry transformations of the whole universe will not lead to
observable differences, but a transformation of just an isolated subsystem —
leaving the environment invariant — may lead to a physically distinct
situation, albeit unobservable from within the subsystem itself. The prime
example is Galileo’s ship experiment which can be taken (when suitably
qualified) as empirical evidence for Galilean boost symmetries. Namely, the
situations where the ship is at rest or moving with respect to the shore are
physically distinct, but unobservable from within the ship.

A theory can now be said to satisfy a relativity principle for isolated
subsystems when solutions of the equations of motion can be transformed into
other (approximate) solutions by applying the transformations (Lorentz
transformations in the present case) to subsystems that are (approximately)
dynamically isolated. For a stochastic theory, also the probabilities for the
subsystem should be transformed and match the probabilities set by the theory
for the whole universe.

Note that we do not require that the subsystem be completely isolated. In
reality, this will usually not be the case, with some interaction with the
environment remaining. But it suffices that the interaction can be arbitrarily
small in principle. In order not to trivialize this principle, it should also
be understood that the theory should allow for approximately isolated
subsystems in the first place and that these can be of arbitrary complexity,
in the sense that if the dynamical laws allow for a particular system, then
they should also allow for multiple such systems in the universe, perhaps
separated far away from each other, that evolve approximately independently.

This relativity principle is admittedly somewhat imprecise, but this will
suffice for our purposes. Note that in our characterization of it, we did not
refer to empirical aspects, as is sometimes done (see, e.g., [40]). Our
formulation seems more up to the task of investigating serious Lorentz
invariance in the context of quantum mechanics since the latter’s statistical
character might entail that objects violating Lorentz invariance like a
preferred frame are unobservable.

Now let us apply this to the Lorentz invariant formulation of Newtonian
mechanics. An approximately isolated system will consist of particles that do
not (or only very weakly) interact with their environment, e.g., because of
substantial spatial separation. The history $(n^{\mu},X)$ of the universe can
then be decomposed as $(n^{\mu}_{s},X_{s};n^{\mu}_{e},X_{e})$, where $X_{s}$
and $X_{e}$ respectively denote the trajectories of the subsystem and
environment, and $n^{\mu}_{s}$ and $n^{\mu}_{e}$ are respectively the unit-
normal vector at the space-time region of the subsystem and environment. A
Lorentz boost of the subsystem $(n^{\mu}_{s},X_{s})$, to
$(n^{\prime\mu}_{s},X^{\prime}_{s})$, will lead to the history
$(n^{\prime\mu}_{s},X^{\prime}_{s};n^{\mu}_{e},X_{e})$ of the universe. This
is however not a solution of the dynamical equations, since the law (1) for
$n^{\mu}$ implies that it is constant over space-time, which is not the case
since $n^{\prime\mu}_{s}\neq n^{\mu}_{e}$. Hence, the Lorentz invariant
formulation of Newtonian mechanics does not satisfy the relativity principle
for isolated subsystems.

This should be contrasted with for example the Wheeler–Feynman theory, which
counts as Lorentz invariant in Anderson’s sense and satisfies the relativity
principle for subsystems (with a subsystem being dynamically isolated when it
is sufficiently distant from its environment).

Here, we will consider some Lorentz invariant Bohmian and spontaneous collapse
models. This will be done in a setting of a fixed number of non-interacting
Dirac particles, which makes the analysis relatively simple, avoiding the
complications of for example quantum field theory. While the particles are
non-interacting, the theories are still non-local because the wave function
may be entangled.

The many worlds theory is one of the other main approaches to solve the
measurement problem. However, in this theory, at least in the way usually
understood, and unlike in Bohmian mechanics or spontaneous collapse theories,
space-time and stuff in space-time are not considered fundamental, but
emergent. This requires a different discussion of space-time symmetries, which
we will not embark on here. We will only briefly consider some versions of
this theory that do have an ontology in space-time on the fundamental level.

We focus here on the possible meaning of serious Lorentz invariance. But this
discussion should be viewed in the broader context of what it could mean for a
theory to be fundamentally relativistic. We will return to this question in
the conclusion.

The outline of the paper is as follows. In the next section, we will first
consider Anderson’s analysis in the context of the Klein–Gordon theory,
rehearsing difficulties with this approach that will also resurface in the
context of quantum mechanics. In section 3, we turn to the Galilean invariance
of non-relativistic Bohmian mechanics, for which both Anderson’s criterion and
the relativity principle are satisfied. In sections 4 and 5, respectively,
Lorentz invariant Bohmian and spontaneous collapse models are discussed,
followed by a brief consideration of the many worlds theory in section 6. We
conclude in section 7.

##  2 The Klein–Gordon theory

A prime example of Anderson’s analysis, as well as its shortcomings, is that
of the Klein–Gordon equation in Minkowski space-time:

$\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\phi=0.$ (2)

This equation, with $\eta_{\mu\nu}$ the Minkowski metric, is Poincaré
invariant (i.e., invariant under Lorentz transformations as well as
translations in space and time). It can be cast in a generally invariant way
by replacing the Minkowski metric by a Lorentzian metric $g_{\mu\nu}$ and
demanding that the corresponding Riemann curvature tensor
$R_{\mu\nu\kappa\sigma}$ vanishes. The resulting equations are

$g^{\mu\nu}\partial_{\mu}\partial_{\nu}\phi=0,$ (3)
$R_{\mu\nu\kappa\sigma}=0.$ (4)

The latter entails that the metric is the Minkowski one up to a space-time
diffeomorphism. The solutions of (3) and (4) correspond to diffeomorphisms of
$(\eta_{\mu\nu},\phi)$ where $\phi$ is a solution of (3).333Basically the same
trick is played here as in making Newtonian mechanics Lorentz invariant.
Namely, the metric is considered to be part of the degrees of freedom, in
addition to the field, and then a new law is formulated that has as solutions
all the space-time diffeomorphisms of the Minkowkski metric and field
solutions of (3). As such, this theory has the same empirical content as (3).
But since the metric is always the Minkowski one up to a diffeomorphism, it is
an absolute object according to Anderson and hence the symmetry group is not
that of general coordinate transformations, but of the Poincaré group.

As suggested by Pitts [38], consider now replacing (4) by

$R_{\mu\nu\kappa\sigma}=\frac{k}{12}(g_{\mu\sigma}g_{\nu\kappa}-g_{\mu\kappa}g_{\nu\sigma}),$
(5)

with some constant $k$. This entails that the space-time is maximally
symmetric with constant curvature $k$ [43, p. 141]. This theory is not
equivalent to (2), since the space-time is not necessarily flat. Nevertheless
the extension could be made as minimal as possible, e.g., by restricting the
possible values of $k$, and leaving it as a contingency whether the world is
flat or not. In any case, the upshot is that the metric no longer qualifies as
an absolute object, since space-time diffeomorphisms do not change the
curvature.444Other examples of this sort are given by Torretti [44] and Norton
[35, p. 848]. Toretti considers Newtonian gravity where space has a constant
curvature. For a law that singles out a particular constant curvature, the
spatial geometry is an absolute object. But by allowing different spatial
geometries of constant curvature, it is no longer absolute. Norton extended
this example to Friedmann–Lemaître–Robertson–Walker space-times.

So, an absolute object in the theory can trivially be turned into a non-
absolute object by enlarging the space of solutions in such a way that it
contains solutions for which the object under consideration is not related by
an element of the covariance group. The new theory is no longer equivalent to
the original theory, but nevertheless if the actual world is described well
with the original theory, it will also be by the new theory.

Interestingly, Anderson’s analysis does not stop here. While the metric itself
is not an absolute object, part of it is (just like in general relativity), as
argued in [38], which means that the theory described by (5) is still not
generally invariant in Anderson’s sense.

##  3 Non-relativistic Bohmian mechanics

Before turning to the possibility of Lorentz invariant formulations of Bohmian
mechanics, it is instructive to recall the Galilean invariance of the non-
relativistic theory [45], and see how it satisfies Anderson’s criterion and
the relativity principle.

Bohmian mechanics describes point-particles whose motion depends on the wave
function. The wave function $\psi$ satisfies the usual Schrödinger
equation555Throughout, we assume units such that $\hbar=c=1$.

$\textrm{i}\partial_{t}\psi_{t}(x)=\left(-\sum^{N}_{k=1}\frac{\hbar^{2}}{2m_{k}}\nabla^{2}_{k}+V(x)\right)\psi_{t}(x)\,,\qquad
x=({\bf x}_{1},\dots,{\bf x}_{N}),$ (6)

while the motion of the particles, with positions ${\bf X}_{k}$,
$k=1,\dots,N$, is determined by

$\frac{d{\bf X}_{k}(t)}{dt}={\bf v}^{\psi_{t}}_{k}({\bf X}_{1}(t),\dots,{\bf
X}_{N}(t)),$ (7)

where

${\bf
v}^{\psi}_{k}=\frac{\hbar}{m_{k}}{\textrm{Im}}\frac{{\boldsymbol{\nabla}}_{k}\psi}{\psi}=\frac{1}{m_{k}}{\boldsymbol{\nabla}}_{k}S,\qquad\psi=|\psi|\textrm{e}^{\textrm{i}S/\hbar}.$
(8)

This theory is Galilean invariant in that a Galilean transformation will map
solutions $({\bf X}_{1}(t),\dots,{\bf X}_{N}(t),\psi_{t}({\bf
x}_{1},\dots,{\bf x}_{N}))$ of (6) and (7) to solutions. For example, for a
boost with velocity ${\bf v}$, the transformation reads

$\displaystyle{\bf X}_{k}(t)$ $\displaystyle\to{\bf X}^{\prime}_{k}(t)={\bf
X}_{k}(t)+{\bf v}t,$ (9) $\displaystyle\psi_{t}(x)$
$\displaystyle\to\psi^{\prime}_{t}(x)=\psi_{t}({\bf x}_{1}-{\bf v}t,\dots,{\bf
x}_{N}-{\bf v}t)\exp\left[\textrm{i}\sum^{N}_{k=1}m_{k}\left({\bf v}\cdot{\bf
x}_{k}-\frac{1}{2}v^{2}t\right)\right].$ (10)

Since there are no absolute objects (apart from those of Galilean space-time),
the theory is Galilean invariant according to Anderon’s analysis. The theory
also satisfies the relativity principle. To see this, consider a system
described by the coordinates ${\bf x}_{1},\dots,{\bf x}_{M}$ and an
environment described by the coordinates ${\bf x}_{M+1},\dots,{\bf x}_{N}$.
Assume further that the potential $V(x)$ contains no interaction terms between
the system and environment. Then a wave function which is initially a product
of the form

$\psi({\bf x}_{1},\dots,{\bf x}_{N})=\psi_{s}({\bf x}_{1},\dots,{\bf
x}_{M})\psi_{e}({\bf x}_{M+1},\dots,{\bf x}_{N})$ (11)

will remain a product at all times. As a consequence, the velocity field is

${\bf v}^{\psi_{t}}_{k}({\bf x}_{1},\dots,{\bf x}_{N})=\begin{cases}{\bf
v}^{\psi_{s,t}}_{k}({\bf x}_{1},\dots,{\bf x}_{M})&\text{if }\ k\leqslant
M,\\\ {\bf v}^{\psi_{e,t}}_{k}({\bf x}_{M+1},\dots,{\bf x}_{N})&\text{if }\
k>M.\end{cases}$ (12)

Hence, it is clear that the relativity principle holds. One can apply a
Galilean transformation of just the subsystem and obtain another solution of
the dynamical equations. The assumption that the potential contains no
interaction between the system and environment can be relaxed. It is
sufficient that the interaction be negligible, which will be the case when the
interaction potential falls off with distance and $\psi_{s}$ and $\psi_{e}$
have the bulk of their support on configurations corresponding to spatial
regions that are well separated. (Furthermore, since the velocity of a
configuration is only determined by the value of the wave function in the
neighbourhood of that configuration, it is actually sufficient that the wave
function obtains the form (11) in a sufficiently large region near the
configuration. One could use the notion of conditional wave function, which is
the natural definition of the wave function of a subsystem, to make this more
precise [45].)

##  4 Lorentz invariant Bohmian models

###  4.1 Dynamics

We will consider a few representative examples of Lorentz invariant Bohmian
models [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] in the simplified setting
of a fixed number of free Dirac particles [11, 18]. While there are no
interactions, there can still be entanglement, which is the source of the non-
locality.

In the multi-time picture, the wave function $\psi=\psi(x_{1},...,x_{N})$ is a
function on ${\mathcal{M}}^{N}$, with ${\mathcal{M}}$ Minkowski space-time,
and takes values in the $N$-particle spin space $({\mathbb{C}}^{4})^{\otimes
N}$. To simplify later discussions, we will not require complete anti-symmetry
of the wave function. The wave function satisfies the $N$ Dirac equations

$\textrm{i}\gamma^{\mu}_{k}\partial_{k,\mu}\psi-m_{k}\psi=0,$ (13)

with $k=1,...,N$. Here, $\gamma^{\mu}_{k}=I\otimes\cdots\otimes
I\otimes\gamma^{\mu}\otimes I\otimes\cdots\otimes I$ with the Dirac matrix
$\gamma^{\mu}$ at the $k$-th of the $N$ places. There are also $N$ particles
with worldlines $X_{k}(s)$, $k=1,\dots,N$, parameterized by $s$. In the models
we will consider, the equations of motion for the particles are of the form

${\dot{X}}_{k}(x)\propto
J^{\Sigma_{k,x},\psi}_{k}\left(X^{\Sigma_{k,x}}_{1},\dots,X^{\Sigma_{k,x}}_{N}\right).$
(14)

As illustrated in Fig. 1, ${\dot{X}}_{k}(x)$ is the unit tangent vector to the
$k$-th worldline at the space-time point $x$, $\Sigma_{k,x}$ is a space-like
hypersurface (to be specified below) for the $k$th particle that contains $x$
and $X^{\Sigma_{k,x}}_{i}$ is the crossing of the $i$-th worldline with the
surface $\Sigma_{k,x}$. The vector field $J^{\Sigma,\psi}_{k}$ is

$\left[J^{\Sigma,\psi}_{k}\right]^{\mu_{k}}(x_{1},\dots,x_{N})=J^{\mu_{1}\dots\mu_{N}}(x_{1},\dots,x_{N})n_{\mu_{1}}(x_{1})\dots\widehat{n_{\mu_{k}}(x_{k})}\dots
n_{\mu_{N}}(x_{N}),$ (15)

where $n^{\mu}$ is the unit vector field normal to $\Sigma$,

$J^{\mu_{1}\ldots\mu_{N}}(x_{1},\dots,x_{N})=\overline{\psi}(x_{1},\ldots,x_{N})\gamma_{1}^{\mu_{1}}\ldots\gamma_{N}^{\mu_{N}}\psi(x_{1},\ldots,x_{N}),$
(16)

and the hat on $n_{\mu_{k}}(x_{k})$ denotes that this factor should be
omitted.

Figure 1: The tangent vector $\dot{X}_{k}(x)$ of the $k$-th worldline at the
point $x$ depends on the surface $\Sigma_{k,x}$ and the crossings of this
surface with the other worldlines, cf. Eq. (14). (Only one spatial direction
is shown.) Figure 2: Trajectories and tangent vectors in the case of a
foliation, with a leaf $\Sigma$. In this case,
$\Sigma_{k,x}=\Sigma_{l,y}=\Sigma$.

So, the theory involves the particles, the wave and the surfaces
$\Sigma_{k,x}$. It remains to specify these surfaces. There are different
possibilities. The surfaces may be determined by the wave function or the
worldlines, or may satisfy some law, possibly also depending on the wave
function or particle positions. The surfaces may or may not determine a
foliation of space-time (see Fig. 2).

A foliation ${\mathscr{F}}$ can be completely characterized by its unit normal
vector field $n^{\mu}$. A unit vector field $n^{\mu}$ will determine a
foliation iff $n\wedge\textrm{d}n=0$, where $n=n_{\mu}\textrm{d}x^{\mu}$ is
the one-form associated to the vector field.

The following six choices for the surfaces $\Sigma_{k,x}$ are considered:

  1. (i)

The $\Sigma_{k,x}$ are the leaves of a foliation ${\mathscr{F}}$, with unit
normal vector field $n^{\mu}$, that satisfies [11, p. 2736]:

$\partial_{\mu}n^{\nu}=0.$ (17)

This law determines a foliation in terms of hyperplanes. (Given a particular
space-like hyperplane as an “initial” leaf of the foliation, the law
determines the whole foliation.)

  2. (ii)

The $\Sigma_{k,x}$ are the leaves of a foliation ${\mathscr{F}}$, with unit
normal vector field $n^{\mu}$, that satisfies [17]:

$\partial_{\mu}n_{\nu}-\partial_{\nu}n_{\mu}=0.$ (18)

This law implies that $n^{\mu}\partial_{\mu}n_{\nu}=0$. Hence, the vector
field $n^{\mu}$ does not change in the direction normal to the surfaces.
(Again, given a particular space-like Cauchy666A Cauchy surface crosses every
inextensible curve just once. A space-like hypersurface in not necessarily a
Cauchy surface. hypersurface as an initial leaf of the foliation, the law
determines the whole foliation.777Specified as such, the foliation may have
kinks, i.e., points where the leaves are not smooth. The Bohmian dynamics can
still be defined for such foliations [46], but such details do not concern us
here. The distance between two surfaces, in the normal direction, will be
constant along the surface.)

  3. (iii)

The $\Sigma_{k,x}$ are the leaves of a foliation ${\mathscr{F}}$, with unit
normal vector field $n^{\mu}$, that satisfies [20]:

$n\wedge\textrm{d}n=0.$ (19)

This implies that there is a foliation, which is not further restricted.
(Given an initial space-like Cauchy surface, there are many foliations that
include it as a leaf.)

  4. (iv)

The $\Sigma_{k,x}$ are the leaves of a foliation that is covariantly
determined by the wave function [11, 18]. That is, the law specifies a map
$\psi\to{\mathscr{F}}$, such that the diagram

$\begin{CD}\psi{}@>{}>{}>{\mathscr{F}}^{\psi}{}\\\
@V{U_{\Lambda}}V{}V@V{}V{\Lambda}V\\\
\psi{}^{\prime}@>{}>{}>{\mathscr{F}}^{\psi{}^{\prime}}\end{CD}$ (20)

is commutative. $\Lambda$ is a Lorentz transformation, with
${\mathscr{F}}^{\prime}=\Lambda{\mathscr{F}}^{\psi}{}$ the corresponding
natural transformation of the foliation and $\psi{}^{\prime}=U_{\Lambda}\psi$
is the corresponding transformation of the wave function, i.e.,

$\psi{}^{\prime}(x_{1},\dots,x_{N})=(U_{\Lambda}\psi)(x_{1},\dots,x_{N})=D_{\Lambda}\otimes\cdots\otimes
D_{\Lambda}\psi(\Lambda^{-1}x_{1},\dots,\Lambda^{-1}x_{N}),$ (21)

where $D_{\Lambda}$ is the representative of $\Lambda$ in the representation
of the Lorentz group for the Dirac theory [47].

An example is the map $\psi\to P^{\mu}$, with $P^{\mu}$ the total energy-
momentum 4-vector determined by $\psi$ [18]. This vector satisfies
$\partial_{\nu}P^{\mu}=0$, so that it determines a foliation in terms of
hyperplanes normal to it.888So, this foliation determines a center-of-mass
frame for the wave function. As such, this proposal is akin to Bell’s attempt
to formulate a Lorentz invariant version of Newtonian mechanics using the
center-of-mass frame determined by the particles.

Explicitly, this vector is given by

$P^{\mu}=\sum^{N}_{k=1}\int_{\Sigma}d\sigma_{\nu}(x)T^{\mu\nu}_{k}(x),$ (22)

where

$T^{\mu_{k}\nu_{k}}_{k}(x_{k})=\int_{\Sigma}d\sigma_{\nu_{1}}(x_{1})\dots\widehat{\int_{\Sigma}d\sigma_{\nu_{k}}(x_{k})}\dots\int_{\Sigma}d\sigma_{\nu_{N}}(x_{N})\\\
\overline{\psi}(x_{1},\ldots,x_{N})\gamma_{1}^{\nu_{1}}\ldots\gamma_{k-1}^{\nu_{k-1}}\frac{\textrm{i}}{2}\left(\overleftrightarrow{\partial}^{\mu_{k}}\gamma_{k}^{\nu_{k}}+\gamma_{k}^{\mu_{k}}\overleftrightarrow{\partial}^{\nu_{k}}\right)\gamma_{k+1}^{\nu_{k+1}}\dots\gamma_{N}^{\nu_{N}}\psi(x_{1},\ldots,x_{N}),$
(23)

with $\Sigma$ some arbitrary space-like hypersurface and
$\overleftrightarrow{\partial}^{\mu}=\frac{1}{2}(\overrightarrow{\partial}^{\mu}-\overleftarrow{\partial}^{\mu})$.999For
a single particle, $T^{\mu\nu}(x)$ is the symmetrized energy-momentum tensor.
This expression does not depend on the choice of $\Sigma$, since the
divergence of the integrand with some $\partial_{\nu_{i}}$ gives zero [47, p.
58]. It also implies that $T^{\mu_{k}\nu_{k}}_{k}(x_{k})$ does not depend on
the $x_{i}$, $i\neq k$.

  5. (v)

The $\Sigma_{k,x}$ are the surfaces orthogonal to

$P^{\mu}_{k}=\int_{\Sigma}d\sigma_{\nu}(x)T^{\mu\nu}_{k}(x).$ (24)

For each $k$, the constancy of the $P^{\mu}_{k}$ determines a foliation in
terms of hyperplanes. But this foliation might be different for different
$k$.101010For a completely anti-symmetric wave function, all the $P^{\mu}_{k}$
are equal and the law reduces to the one considered in (iv). So, it is not
necessarily the case that the velocity fields are determined by a single
foliation. It might be the case, as illustrated in Fig. 3, that
$\Sigma_{k,x}\neq\Sigma_{l,y}$, where $y$ is the crossing of $\Sigma_{k,x}$
with the $l$-th worldline.

  6. (vi)

The $\Sigma_{k,x}$ are the hyperplanes orthogonal to ${\dot{X}}_{k}(x)$.
Generically, these surfaces do not determine a foliation (not even for a fixed
$k$).

Figure 3: Trajectories and tangent vectors in the case of theories (v) and
(vi). Situation where the surfaces do not determine a foliation, since
$\Sigma_{k,x}\neq\Sigma_{l,y}$, even though $y\in\Sigma_{k,x}$.

Let us first mention some properties of these theories. In each case, the
single-particle law is

${\dot{X}}^{\mu}(x)\propto J^{\mu}(x)=\overline{\psi}(x)\gamma^{\mu}\psi(x).$
(25)

So, the possible trajectories are the integral curves of the 4-current
$J^{\mu}$, which does not depend on any surface $\Sigma_{k,x}$.

The dynamics is non-local, since for an entangled wave function the velocity
of a particle will depend on the positions of the other particles along the
relevant space-like hypersurface $\Sigma$. This means that the velocity does
not only depend on what happened in the past light cone, but also by what
happens outside it. Such non-locality is unavoidable by Bell’s theorem.

While there are influences over space-like distances, the trajectories
themselves are never space-like. That is because the tangent vector
${\dot{X}}_{k}(x)$ is always time-like or light-like, i.e.,
$\left[J^{\Sigma,\psi}_{k}\right]^{\mu_{k}}\left[J^{\Sigma,\psi}_{k}\right]_{\mu_{k}}\geqslant
0$ for all $k$ and choices of space-like hypersurface $\Sigma$ [11]. (At least
in theory (i), the particles typically do not reach the light-speed [48].)

All these theories are also Lorentz invariant (or, in Anderson’s terminology,
the Lorentz group forms the covariance group); a Lorentz transformation
$\Lambda$ maps solutions $(X_{1},\dots,X_{N},\psi)$ of the dynamical equations
to solutions $(\Lambda X_{1},\dots,\Lambda X_{N},U_{\Lambda}\psi)$, with
$U_{\Lambda}\psi$ defined in (21).

The theories (i)-(iii) contain extra space-time structure, namely the
foliation. The trajectories depend on this foliation, but the foliation itself
is independent of the particles. In the theories (iv)-(vi), there is no extra
space-time structure. The surfaces $\Sigma_{k,x}$ are completely determined by
the wave function in theories (iv) and (v) and by the worldlines in theory
(vi).

In the theories (i)-(iv), the surfaces $\Sigma_{k,x}$ are part of a single
foliation of space-time. For these theories, there is a notion of quantum
equilibrium and it can be argued that these theories agree with the usual
quantum predictions [11, 49, 26]. Namely, in quantum equilibrium, the density
of crossings of the trajectories with a leaf $\Sigma$ of the foliation with
normal field $n^{\mu}$ is given by

$\rho^{\Sigma,\psi}(x_{1},\dots,x_{N})=J^{\mu_{1}\dots\mu_{N}}(x_{1},\dots,x_{N})n_{\mu_{1}}(x_{1})\dots
n_{\mu_{N}}(x_{N}),$ (26)

relative to the volume defined by the induced metric on $\Sigma$, and
evaluated on $\Sigma^{N}$. (In the case $\Sigma$ is a hyperplane,
$\rho^{\Sigma,\psi}$ equals the familiar expression $\psi^{\dagger}\psi$, with
$\psi$ the wave function on that hyperplane.) The crossings with a space-like
Cauchy surface $\Sigma^{\prime}$ that is not a leaf in the foliation will
generically not be distributed according to $\rho^{\Sigma^{\prime},\psi}$
[10]. Nevertheless, it can be shown that actual measurements along an
arbitrary space-like Cauchy surface will give results in accordance with the
Born rule. This also means that the foliation itself is unobservable. So, we
are in this prima facie awkward situation where the foliation plays a
dynamical role for the trajectories, but is absent in the statistical
predictions of the theory. (In quantum non-equilibrium, the foliation would be
observable.)

In the theories (v) and (vi), on the other hand, the surfaces are
(generically) not part of a single foliation.111111In [12, 13, 14, 15, 19]
other laws are considered that are not based on a foliation and which hence
have the problem of statistical transparency. For example, in [14, 19], the
surfaces are given by the future light-cones and the contraction in (15) is
with the tangent vectors of the particles rather than the normals to the
surface (which do not exist in this case). This not only implies that it is
hard to establish existence of solutions (especially in the case of theory
(vi), where the stipulation of the surfaces is not even independent of (14),
which might make the theory overconstrained), it also implies that it is hard
to perform a statistical analysis and to compare the theory with the usual
quantum predictions. Hence, it is not at all clear to what extent the
predictions of these theories agree with those of quantum theory. (In the non-
relativistic limit, assuming that there is a frame where the velocities of all
the particles are small compared to the light speed, the surfaces
$\Sigma_{k,x}$ nearly agree and presumably the predictions of non-relativistic
quantum mechanics are recovered.)

###  4.2 Anderson’s criterion in terms of absolute objects

According to Anderson’s definition, the vector field $n^{\mu}$ and hence the
corresponding foliation are absolute objects in the theory (i), but not in the
theories (ii) and (iii). Namely, in the theory (i), just as in the Lorentz
invariant formulation of Newtonian mechanics, the constancy over space-time of
the vector field $n^{\mu}$ entails that any two such fields are related by a
Lorentz transformation. This is not the case in theories (ii) and (iii),
because the law for $n^{\mu}$ allows for vector fields (and hence foliations)
that are not connected by a Lorentz transformation. As such, the group of
Lorentz transformations counts as the symmetry group in the theories (ii) and
(iii), but not in the theory (i).

However, this conclusion seems counterintuitive and the concern is the same as
the one voiced in section 2 in relation to the Klein–Gordon theory. The
theories (ii) and (iii) extend the set of solutions of theory (i). So, if the
actual world is described by theory (i), it will also be by theories (ii) and
(iii). That the actual world is described by a foliation in terms of
hyperplanes, rather than curved surfaces, is then just a contingent fact. It
then seems odd that the vector field is absolute in theory (i), but not in
theories (ii) and (iii).

The original motivation of Anderson was actually to capture the idea that
absolute objects act on the other objects but are not acted back on [37, p. 73
and 83]. In this case, Anderson’s definition fails to capture that idea. In
each of the theories (i)-(iii), the foliation acts on the particles, without
back-reaction. However, as already observed by Norton [35], the notion of
action without back-reaction is hard to make precise in a sensible way. Here
too, in the context of Bohmian mechanics, this idea seems immediately
problematic. The wave function acts on the particles but without back-
reaction. But it does not seem to make sense to regard it as an absolute
object.

The theories (iv)-(vi) contain no extra space-time structure like a foliation;
there is just the wave and the particles. As such, there are no absolute
objects (other than the Minkowski metric). However, there still is a
dynamically distinguished foliation in the case of (iv). It is along the
leaves of this foliation that the non-locality will be instantaneous.

So, in summary, adopting Anderson’s principle as a criterion for serious
Lorentz invariance would mean that only theory (i) would not count as
seriously Lorentz invariant. But this seems at odds with the intuition that
theories (i)-(iii) seem on par concerning possible serious Lorentz invariance.

###  4.3 Relativity principle for subsystems

To describe an isolated subsystem, consider a product wave function121212If
the form (27) holds on a surface $\Sigma_{1}\times\cdots\times\Sigma_{N}$,
with the $\Sigma_{i}$ Cauchy surfaces (e.g., $\Sigma_{i}$ constant-time
surfaces in some Lorentz frame), then it will hold on ${\mathcal{M}}^{N}$.
Note that this form cannot obtain if the wave function is completely anti-
symmetric. However, such a form could hold locally on
$\Sigma_{1}\times\cdots\times\Sigma_{N}$, which would be sufficient to carry
out a similar analysis.

$\psi(x_{1},\dots,x_{N})=\psi_{s}(x_{1},\dots,x_{M})\psi_{e}(x_{M+1},\dots,x_{N}).$
(27)

The wave functions $\psi_{s}$ and $\psi_{e}$ will then separately satisfy the
many particle Dirac equations (13). Since

$J^{\Sigma,\psi}_{k}(x_{1},\dots,x_{N})=\begin{cases}J^{\Sigma,\psi_{s}}_{k}(x_{1},\dots,x_{M})\rho^{\Sigma,\psi_{e}}(x_{M+1},\ldots,x_{N})&\text{if
}\ k\leqslant M,\\\
\rho^{\Sigma,\psi_{s}}(x_{1},\ldots,x_{M})J^{\Sigma,\psi_{e}}_{k}(x_{M+1},\dots,x_{N})&\text{if
}\ k>M,\end{cases}$ (28)

the corresponding particle equations (14) can be written as

${\dot{X}}_{k}(x)\propto\begin{cases}J^{\Sigma_{k,x},\psi_{s}}_{k}(X^{\Sigma_{k,x}}_{1},\dots,X^{\Sigma_{k,x}}_{M})&\text{if
}\ k\leqslant M,\\\
J^{\Sigma_{k,x},\psi_{e}}_{k}(X^{\Sigma_{k,x}}_{M+1},\dots,X^{\Sigma_{k,x}}_{N})&\text{if
}\ k>M.\end{cases}$ (29)

For the subsystem consisting of the particles $1,\dots,M$ to be isolated, also
the particle dynamics should decouple. That is, the evolution of the particles
$1,\dots,M$ should not depend on the wave function $\psi_{e}$ or the particle
positions $X_{M+1},\dots,X_{N}$. Because of the form (29), there can only be
such a dependence if the surface $\Sigma_{k,x}$ has it. For the theories we
consider, this only happens in the case (iv), where the surface $\Sigma_{k,x}$
depends on the universal wave function $\psi$. (In theory (v), the vectors
$P^{\mu}_{k}$, $k=1,\dots,M$, and hence the corresponding surfaces
$\Sigma_{k,x}$, only depend on $\psi_{s}$.) So, in the case of theory (iv),
there are no isolated subsystems, except in the special case where the wave
function $\psi_{s}$ is completely separable, i.e.,

$\psi_{s}(x_{1},\dots,x_{M})=\psi_{1}(x_{1})\dots\psi_{M}(x_{M}),$ (30)

because then the velocity of each particle is of the form (25) and does not
depend on a surface $\Sigma_{k,x}$. So, if the wave function of the subsystem
is of the form (30), then the dynamics of the particles $1,\dots,M$ does not
depend on the other particles. If it is not of that form, then the dynamics of
(some of) the particles $1,\dots,M$ will depend on $\psi_{e}$. Namely, if the
wave function $\psi_{s}$ is entangled, then the non-local influences between
the particles $1,\dots,M$ will be instantaneous along the leaves of the
foliation determined by the total wave function $\psi$. As a result, the
relativity principle will hold for subsystems for which the wave function is
of the form (30), but not for those not of that form. Because if $\psi_{s}$ is
of the form (30), then for a given solution of the dynamics one can always
apply any Lorentz transformation on just this subsystem to obtain a new
possible solution, because the particle trajectories are the integral curves
of a 4-vector of the form (25). (These solutions are even exact in this case
because of the simplified setup of non-interacting particles.) As such, the
relativity principle only holds for a very limited set of subsystems. It does
not hold for systems of arbitrary complexity (as characterized above) needed
for a proper application of the principle.

So, apart from theory (iv), there exist isolated subsystems and Lorentz
transformations can be considered of these subsystems that leave the
environment invariant. As is easy to see, applying such a transformation leads
to a new solution only in the theories (ii), (iii), (v) and (vi), but not in
theory (i). In the latter case, the problem is exactly the same as in the
Lorentz invariant formulation of Newtonian mechanics. As with the theory (iv),
the relativity principle holds merely for subsystems whose wave function is
completely separable.

In summary, the theories (ii), (iii), (v) and (vi) satisfy the relativity
principle for isolated subsystems. The difference with Anderson’s principle is
that now also theory (iv) fails to qualify as seriously Lorentz invariant. As
with Anderson’s principle, it seems counterintuitive that theory (i) does not
qualify as seriously Lorentz invariant, while theories (ii) and (iii), which
enlarge the set of solutions of theory (i), do. (One could actually enlarge
the solution space of (iv) too, to have a theory that satisfies the relativity
principle, by stipulating that in the case of a separable wave function (27),
the particle dynamics for the subsystems depends on surfaces that are
determined by the energy-momentum 4-vector either corresponding to $\psi_{s}$
or $\psi$. But such a move does not seem amenable to the case where
interactions are included and the separability of the universal wave function
into subsystem and environment wave function is never exact.)

##  5 Lorentz invariant spontaneous collapse models

From the Lorentz invariant spontaneous collapse models [21, 22, 25], let us
consider the one of Tumulka [21] here, since it is also formulated for non-
interacting Dirac particles, like the Bohmian theories above. It will appear
that both criteria for serious Lorentz invariance are satisfied.

In this model, the ontology in space-time is given by particular space-time
points called flashes, which can be understood as the locations and times
where collapses happen.131313Other choices are possible for the space-time
ontology, like a matter density [24]. The flashes are denoted by $X^{i}_{k}$,
where $k=1,\dots,N$ is the particle label and $i=1,\dots,n_{k}$ labels the
flashes for the $k$-th particle. The probability distribution of these flashes
is determined by a wave function $\psi$ satisfying the multi-time Dirac
equations (13). To each generation of flashes a new “collapsed” wave function
can be associated, but these need not to be considered to specify the law for
the flashes. For a wave function $\psi$ and initial flashes
$X^{0}_{1},\dots,X^{0}_{N}$, the joint probability distribution for the
flashes $X^{i}_{k}$ to occur in the volume elements $d^{4}x^{i}_{k}$ is

$P^{\psi}(X^{i}_{k}\in
d^{4}x^{i}_{k};k=1,\dots,N;i=1,\dots,n_{k})=\langle\psi_{0}|E^{(n_{1})}\otimes\cdots\otimes
E^{(n_{N})}|\psi_{0}\rangle,$ (31)

where $\psi_{0}$ is the restriction of $\psi$ to some surface
$\Sigma_{1}\times\cdots\times\Sigma_{N}$, with $\Sigma_{i}$, $i=1,\dots,N$
arbitrary space-like Cauchy surfaces. The $E^{(n_{k})}$ are operators acting
only on the $k$-th particle subspace [21, p. 836]. Their explicit expression
does not concern us here. For our purposes it suffices to note the following
two properties.

First, for a Lorentz transformation $\psi^{\prime}=U_{\Lambda}\psi$,
$X^{\prime i}_{k}=\Lambda X^{i}_{k}$, which transforms $d^{4}x^{i}_{k}$ into
$d^{4}x^{\prime i}_{k}$, the joint probability distribution satisfies

$P^{\psi}(X^{i}_{k}\in d^{4}x^{i}_{k};k=1,\dots,N;i=1,\dots,n_{k})=\\\
P^{\psi^{\prime}}(X^{\prime i}_{k}\in d^{4}x^{\prime
i}_{k};k=1,\dots,N;i=1,\dots,n_{k}).$ (32)

This implies that the theory is invariant under Lorentz transformations of the
whole universe. Namely, given a dynamically possible history $(\psi$,
$X^{i}_{k})$ and the probability distribution $P^{\psi}$ for the flashes given
by (31), then a Lorentz transformation yields another dynamically possible
history $(\psi^{\prime}$, $X^{\prime i}_{k})$, with probability distribution
$P^{\prime}$ for the flashes given by

$P^{\prime}(X^{\prime i}_{k}\in d^{4}x^{\prime
i}_{k};k=1,\dots,N;i=1,\dots,n_{k})=P^{\psi}(X^{i}_{k}\in
d^{4}x^{i}_{k};k=1,\dots,N;i=1,\dots,n_{k}).$ (33)

From (32), it follows that $P^{\prime}=P^{\psi^{\prime}}$ and hence that
Lorentz invariance holds.

Second, if the wave function is separable, i.e.,

$\psi(x_{1},\dots,x_{N})=\psi_{s}(x_{1},\dots,x_{M})\psi_{e}(x_{M+1},\dots,x_{N}),$
(34)

then because of the form (31) of the probabilities,

$P^{\psi}(X^{i}_{k}\in d^{4}x^{i}_{k};k=1,\dots,N;i=1,\dots,n_{k})=\\\
P^{\psi_{s}}(X^{i}_{k}\in d^{4}x^{i}_{k};k=1,\dots,M;i=1,\dots,n_{k})\\\ \cdot
P^{\psi_{e}}(X^{i}_{k}\in d^{4}x^{i}_{k};k=M+1,\dots,N;i=1,\dots,n_{k}).$ (35)

Together with the first property and applying a similar argument as for
Lorentz transformations of the whole universe, it is clear that the relativity
principle for subsystems is satisfied.

Furthermore, the theory has no absolute objects (beyond the Minkowski metric)
and so the symmetry group is the Lorentz group according to Anderson’s
criterion.

##  6 Many worlds theory

Bohmian mechanics and spontaneous collapse theories concern an ontology in
space-time,141414This is sometimes referred to as the primitive ontology [50]
or the local beables [51]. like the particles or flashes or mass density,
whose dynamics depends on the wave function. In the many worlds theory [52],
the ontology is given completely by the wave function (or more abstractly by
some state vector in a Hilbert space). There is no ontology in space-time on
the fundamental level. There is not even space-time on the fundamental level.
Rather, space-time and stuff in space-time are considered to emerge from the
wave function. Hence, possible dynamical symmetries of the many worlds theory
cannot have anything to do with space-time symmetries, at least not on the
fundamental level, but only with how they emerge.

There are versions of the many worlds theory that do take space-time and stuff
in space-time as fundamental. For example, one option is to postulate a tensor
field
${\mathcal{M}}^{\mu\nu}(x)=\langle\psi|{\widehat{T}}^{\mu\nu}(x)|\psi\rangle$,
with ${\widehat{T}}^{\mu\nu}(x)$ the energy-momentum tensor operator [50]
(which equals $\sum_{k}T^{\mu\nu}_{k}(x)$ in the case of the non-interacting
Dirac theory). Another option is the notion of state in algebraic quantum
field theory that can be associated to each space-time region [53]. These
ontologies do not introduce absolute objects and they satisfy a relativity
principle for isolated subsystem. So, according to both Anderson’s principle
and the relativity principle, these versions of the many worlds theory qualify
as seriously Lorentz invariant.

##  7 Conclusion

We have considered two possible ideas for characterizing what it could mean to
have “serious Lorentz invariance” and found that there are both Bohmian and
spontaneous collapse theories that qualify as seriously Lorentz invariant
according to both characterizations. Nevertheless, some Bohmian models that do
not seem seriously Lorentz invariant are not identified as such, suggesting
the need for a more stringent criterion.

In any case, Lorentz invariance is just one property that one could demand
from a fundamentally relativistic theory. Others have been considered, see
e.g. [54] for a recent discussion. For example, one could demand that a
relativistic theory only contains local beables. This would make any theory
with a non-local beable like the wave function non-relativistic. Another
possible demand is that the theory be local. But as such it would be in
violation with the implications of Bell’s theorem and hence a truly
relativistic theory would seem impossible. Still others could be considered.
For example, in the context of experimental tests, Bell’s locality condition
is sometimes viewed as the conjunction of parameter independence and outcome
dependence, and it has often been argued that Bohmian theories are less
relativistic than spontaneous collapse theories because they violate parameter
dependence [55, 56, 57, 58, 59, 60, 26]. But, on the other hand, as argued in
[54], spontaneous collapse theories are less relativistic than Bohmian
theories in that they are non-local even for a single particle, unlike Bohmian
theories.

These criteria and the ones we have considered should perhaps not be seen as
strict demands for a relativistic quantum theory, but could indicate what
aspects — in addition to non-locality — may need to be given up in passing
from classical to quantum physics [18]. Because of the different predictions
of spontaneous collapse models and no-collapse theories, like Bohmian
mechanics, this is not a purely metaphysical question. Perhaps it is simply
the case that Lorentz invariance merely holds at the statistical level and not
on the fundamental level. Nevertheless, given the success of relativistic wave
equations in standard quantum mechanics (which figure unaltered in Bohmian
theories), it is worth exploring to what extent Lorentz invariance is
possible.

##  8 Acknowledgments

This work is supported by the Research Foundation Flanders (Fonds
Wetenschappelijk Onderzoek, FWO), Grant No. G0C3322N. It is a pleasure to
thank David Albert, Detlef Dürr, Sheldon Goldstein, Travis Norsen, Roderich
Tumulka and Nino Zanghì for useful discussions over the years, as well as
Guido Bacciagaluppi, Fred Muller and Sylvia Wenmackers for useful comments on
earlier drafts.

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