# Relativistic Constraints on Interpretations of Quantum Mechanics

Wayne C. Myrvold
Department of Philosophy
The University of Western Ontario

(In Eleanor Knox and Alastair Wilson, eds., _The Routledge Companion to
Philosophy of Physics_ , Routledge, 2021.)

###### Contents

  1. 1 Introduction
  2. 2 The Quantum Measurement Problem
  3. 3 Relativistic Spacetimes
  4. 4 Quantum theories in relativistic spacetimes
  5. 5 Additional-Beables Theories
    1. 5.1 De Broglie-Bohm theory
      1. 5.1.1 Causal structure
      2. 5.1.2 Bohmian quantum field theories
    2. 5.2 Modal interpretations
      1. 5.2.1 Difficulties arising from relativistic causal structure
      2. 5.2.2 Difficulties arising from infinitely many degrees of freedom
  6. 6 Collapse Theories
    1. 6.1 Stochastic quantum state evolution
    2. 6.2 Collapse theories in Minkowski spacetime
      1. 6.2.1 Causality conditions
      2. 6.2.2 Collapse theory ontology
      3. 6.2.3 A problem: stability of the vacuum
      4. 6.2.4 Impossibility of a relativistic collapse theory using only standard degrees of freedom
    3. 6.3 Causal Quantum Theory
  7. 7 Everettian Interpretations
  8. 8 Non-Realist and Pragmatist Interpretations
  9. 9 Conclusion
  10. 10 Acknowledgment

##  1 Introduction

Though it is fairly noncontroversial that an empirically adequate quantum
theory must be a quantum field theory, and must be able to treat of the
relativistic regime of energies, much of the literature on interpretations of
quantum theory has been focussed on nonrelativistic quantum mechanics. There
is some justification for this. Nonrelativistic quantum mechanics is
conceptually and mathematically simpler, and does, after all, work very well
in a low-energy regime. Any solution of the measurement problem, or any
account of the ontology of quantum theory, must yield sensible results in this
regime, and so nonrelativistic quantum mechanics can serve as a testing-ground
for the viability of interpretations.

It should not be assumed, however, that an approach to the interpretation of
quantum mechanics will carry over straightforwardly to the context of a
relativistic quantum field theory. There are two sorts of challenges to be
met. One is adaptation of the approach to a theory with relativistic causal
structure. Another challenge arises from the fact that relativistic quantum
theories are _quantum field theories_ , that is, quantum theories of systems
with infinitely many degrees of freedom.

As we shall see below, each of these challenges can pose a serious difficulty
for an approach to the interpretation of quantum mechanics. We will focus on
four main avenues of approach: (i) additional beables theories, which include
“hidden-variables” theories and modal interpretations, (ii) dynamical collapse
theories, (iii) Everettian, or “many worlds” interpretations, and (iv) non-
realist interpretations, which deny that quantum states represent anything in
physical reality independent of considerations of agents and their beliefs.

Summarizing briefly, the conclusions that will be drawn:

(i) Additional beables theories face serious difficulties with relativistic
causal structure. This can be dealt with by introducing a dynamically
distinguished (albeit unobservable) foliation of spacetime, or else by making
the additional beables relational, so that some quantities, such as pointer
positions, that we think of as local beables, are not local beables after all.
Furthermore, modal interpretations that employ the spectral resolution of a
system’s quantum state to pick out the additional beables run into serious
trouble when extended to quantum field theories, as they run the risk of
picking out no non-trivial additional beables.

(ii) Dynamical collapse theories can be adapted, in a fairly straightforward
way, to relativistic causal structure. However, a collapse theory that
respects the full spacetime symmetry of Minkowski spacetime and its causal
structure can do so only at the price of introducing additional, nonstandard
quantum degrees of freedom that are unlike those that appear in familiar
quantum field theories.

(iii) Everettian interpretations fare better; they extend to the relativistic
context with little or no adjustment.

(iv) Some non-realist interpretations bypass difficulties with relativity by
limiting the scope of application of quantum mechanics.

##  2 The Quantum Measurement Problem

If a quantum theory is capable, in principle, of affording a complete
description of the physical goings-on in the world (or even just the
laboratory), then it must be possible to model an experimental apparatus as a
complicated array of quantum systems, and the interactions of the apparatus
with the system of interest as a process conforming to the laws of quantum
dynamical evolution. This, however, poses a problem, as typical interactions
will result in a quantum state that is a superposition of terms corresponding
to distinct experimental outcomes. The literature on what is (misleadingly)
known as the interpretation of quantum mechanics deals with how to make sense
of this.

Among approaches that attempt a realist construal of quantum theories, we can
distinguish three main avenues of approach. One avenue accepts the usual,
linear evolution of the quantum state, but does not saddle the quantum state
with the burden of fully representing the physical world, but, rather,
supplements it with additional structure. These are traditionally called
“hidden-variables theories,” but, as Bell (1981b, 1987a) has pointed out, this
is misleading, since, according to a theory like this, it is via these
additional variables that the world manifests itself. This is the reason we
are calling theories of this sort _additional beables theories_. These include
the de Broglie-Bohm pilot wave theory, and also modal interpretations.

Another avenue of attack modifies the dynamics of the quantum state so that,
in appropriate circumstances, a process closely approximating the textbook
collapse of the state vector occurs. This typically involves a stochastic
modification of the dynamics, resulting in an indeterministic theory. The
best-known theory of this type is the Ghirardi-Rimini-Weber (GRW) theory, on
which collapses are instantaneous, discrete events; there is a continuous
analog, known as _continuous spontaneous localization_ (CSL).

These two approaches modify the quantum formalism, either by adding additional
structure, or by modifying the dynamics. A third class of approaches retains
the linear quantum dynamics, with its attendant superpositions of
macroscopically distinct states, and attempts to construe a quantum state
description as a complete description. These are _Everettian_ , or _many-
worlds_ approaches to quantum theory.

An approach that does not fit neatly into these categories is the _relational
interpretation_ advocated by Carlo Rovelli. It is akin in some ways to
Everett’s original conception, which he called the _relative-state_
interpretation. It differs from it in not taking quantum states to be
representational. For more on this, see Laudisa and Rovelli (2019), and
references therein.

On any approach, empirical adequacy requires recovery of at least an
approximation to the quantum probabilities for outcomes of experiments,
insofar as these have been verified by experiment. This includes violations of
Bell inequalities. It might seem that this, by itself, leads to an unavoidable
conflict with relativistic causality. As we shall see, things are not so
simple!

##  3 Relativistic Spacetimes

Compatibility with special relativity will be a concern in this chapter, and
this amounts to the question of whether a theory can live happily in Minkowski
spacetime, without adjoining any additional spacetime structure, such as a
privileged relation of distant simultaneity. But, of course, special
relativity is false, and the spacetime in which we live and move and have our
being is not Minkowski spacetime, but, as general relativity teaches us, a
spacetime of variable and dynamic curvature. Minkowski spacetime is of
interest only as a local approximation, on scales at which curvature can be
ignored. And general relativity itself is, almost certainly, an approximation
to a deeper theory that incorporates quantum gravity.

For this reason, we will be interested, not only in Minkowski spacetime, but
in a wider class of relativistic spacetimes. The radical departure of special
relativity from previous conceptions of spacetime lies in its rejection of a
privileged global temporal order. In Galilean spacetime, for any event $p$,
any other event is either in the past of $p$, the future of $p$, or is
simultaneous with $p$, and the relation of simultaneity is, as one would
expect, an equivalence relation. In Minkowski spacetime, events that are
spacelike separated are neither to the past nor to the future of each other,
and, although there will always be some inertial reference frame whose
accompanying Lorentz coordinates assign the two events the same
$t$-coordinate, this is not simultaneity in any physically significant sense.

In Minkowski spacetime, define a relation $\prec$ that holds between spacetime
points $p$, $q$, when there is a future-directed curve from $p$ to $q$ that is
everywhere timelike or lightlike. This relation is irreflexive (no point
stands in that relation to itself), antisymmetric (if $p\prec q$ then it is
not the case that $q\prec p$), and transitive.

Now define the relation $\sim$ of temporal unconnectedness, that holds between
two points $p$ and $q$ if neither $p\prec q$ nor $q\prec p$. Distinct points
that are temporally unconnected are said to be _spacelike separated_. This
relation is reflexive and symmetric, but not transitive. On the contrary, for
any two points $p$, $q$ such that $p\sim q$, one can always find a third, $r$,
such that $p\sim r$ but $r\prec q$.

All of this remains true in the curved spacetimes of general relativity,
provided that they are sufficiently well-behaved.111There is no need here for
a precise characterization of “well-behaved,” but global hyperbolicity
suffices for our purposes. This is a striking difference between relativistic
spacetimes and Galilean spacetime. In Galilean spacetime we have a relation
$\prec$ of temporal precedence that is irreflexive and antisymmetric, but the
corresponding relation of temporal unconnectedness is transitive, and hence
spacetime can be uniquely be partitioned into equivalence classes under this
simultaneity relation.

In this chapter, we will be concerned with spacetimes equipped with a relation
$\prec$ of temporal precedence, from which we define a relation $\sim$ of
temporal unconnectedness. We will assume that $\prec$ is irreflexive and
antisymmetric (no temporal loops). We will say that a spacetime’s temporal
structure is _causally relativistic_ iff for any two points $p$, $q$ such that
$p\sim q$, there exists a point $r$, such that $p\sim r$ and $r\prec q$. Note
that this is a condition on causal structure and not a condition of symmetry
under translations or boosts (which, of course, are additional conditions that
one might impose).

On the usual notion of causation, a cause must temporally precede its effects,
and so, if an event $p$ is a cause of an event $q$, we must have $p\prec q$.
It is this that motivates the claim that special relativity forbids action at
a distance; without it, there is no reason to think that causal relations
between spacelike separated events are incompatible with relativistic
spacetime structure. Given a causally relativistic spacetime, we will say that
a theory is _causally local_ if all causal relations that the theory affords
respect the condition that the cause be in the past of the effect and, hence,
that there are no cause-effect relations between temporally unconnected
events. There seems to be no harm in assuming that any spacetime point to the
past of a point $q$ is potentially a locus of an event that is a cause of $q$.
If we make this assumption, then the relations of temporal precedence and
potential causal influence coincide, and we may, as is usual in discussions of
spacetime structure, treat temporal structure and causal structure
interchangeably.

The condition of causal locality—that is, the condition that there be no
cause-effect relations between spacelike separated points—is independent of
symmetries such as boost symmetries. One can invent theories that are causally
local but not invariant under boosts, and one can invent theories that violate
the condition of causal locality without picking out a preferred reference
frame.

There is a tradition of invoking retrocausality—that is, backwards-in-time
causation—in explaining quantum phenomena. This approach has its roots in a
suggestion by O. Costa de Beauregard (1976) and has been championed by a
number of authors; see Friederich and Evans (2019) for an overview. The idea
is to make causal connectability time-symmetric by permitting cause-effect
relations in both temporal directions. On this approach, causal influences can
propagate along time-like or light-like lines in either temporal direction.
Assuming that causal connectability is transitive, this turns the relation of
causal connectability into the trivial relation that holds between any pair of
spacetime points. Any approach of this sort is, therefore, straightforwardly
causally nonlocal, in the sense we are using the phrase, as any two spacetime
points, including those that outside of each other’s lightcones, are causally
connected. As Kastner (2013, Ch. 6) has argued, an approach of this sort is
capable of respecting the symmetries of Minkowski spacetime.

##  4 Quantum theories in relativistic spacetimes

A quantum-mechanical theory is what results when one subjects a classical
theory of a system with finitely many degrees of freedom (such as, for
example, a finite number of particles) to quantization. A quantum field
theory, on the other hand, results from quantization of a classical theory
with infinitely many degrees of freedom. Introductory quantum field theory
textbooks typically contain, in their early chapters, arguments that a
relativistic quantum theory must be a quantum field theory. For a precise
version of one such argument, see Malament (1996).

There are, therefore, two sorts of difficulties one might encounter when one
seeks to extend to the relativistic domain an approach to the quantum
measurement problem that has been formulated in terms of nonrelativistic
quantum mechanics. One source of difficulties might lie in a adapting the
theory to relativistic causal structure. Another source of difficulties might
stem from differences between quantum mechanics and quantum field theories.

Typically, relativistic quantum field theories are expressed in the Heisenberg
picture, rather than the Schrödinger picture. The Schrödinger picture, or an
analogue of it, can, however, be formulated in a relativistic spacetime. On
this picture, we track the rate of change of an observable via a changing
state vector. Take a foliation of spacetime into spacelike hypersurfaces (most
conveniently, hyperplanes, but we can also consider nonflat surfaces), taken
to be smooth, so that there is, at any point of each of these hypersurfaces, a
vector normal to the surface. Integral curves of this vector field form a
timelike congruence. Pick one hypersurface $\sigma_{0}$, and consider a field
operator $\hat{\phi}(x)$. For each $x_{0}$ on $\sigma_{0}$, there will be a
family of operators $\hat{\phi}(x)$ corresponding to points on the integral
curve containing $x_{0}$, and, for any quantum state, one can consider how the
expectation value of $\hat{\phi}(x)$ changes as one follows these timelike
curves. This can be expressed, either in the Heisenberg picture, using the
same state vector and different operators $\hat{\phi}(x)$ for different points
on an integral curve, or in the Schrödinger picture, using the same operator
$\hat{\phi}(x_{0})$ for every spacetime point on the integral curve containing
$x_{0}$, and different state vectors for evaluating expectation values of
observables on different elements of our foliation.

There is also an analog of the interaction picture, called the _Tomonaga-
Schwinger picture_. On this picture, one writes the Lagrangian density as a
sum of a free-field Lagrangian and a term incorporating interactions:

$\mathcal{L}(x)=\mathcal{L}_{0}(x)+\mathcal{L}_{I}(x).$ (1)

The operators employed are solutions of the free-field equations, and one
associates, with each spacelike Cauchy surface $\sigma$ (whether flat or not),
a state vector $|\psi(\sigma)\rangle$. Evolution from a surface $\sigma$ to
another, $\sigma^{\prime}$, differing by an infinitesimal deformation about a
point $x$, satisfies the _Tomonaga-Schwinger equation_ (Tomonaga, 1946;
Schwinger, 1948):

$i\hbar
c\,\frac{\delta|\psi(\sigma)\rangle}{\delta\sigma(x)}=\mathcal{H}_{I}(x)|\psi(\sigma)\rangle.$
(2)

Integration of this equation yields, for any Cauchy surfaces $\sigma$,
$\sigma^{\prime}$, a unitary mapping from $|\psi(\sigma)\rangle$ to
$|\psi(\sigma^{\prime})\rangle$. This approach is not commonly discussed in
recent textbooks but can be found in some older texts; see, _e.g._ , Schweber
(1961).

##  5 Additional-Beables Theories

###  5.1 De Broglie-Bohm theory

The best-known additional-beables theory is the de Broglie-Bohm pilot wave
theory. On this theory, ordinary physical objects consist of point particles,
or corpuscles, whose motion is guided by the quantum wave function. The
dynamical laws of theory are the Schrödinger equation for the evolution of the
quantum wave function, and the guidance equation, which fixes the velocities
of Bohmian corpuscles.

$m_{i}\mathbf{v}_{i}(\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{n},t)=\boldsymbol{\nabla}_{i}S(\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{n},t),$
(3)

where $S/\hbar$ is the phase of the wave function:

$S(\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{n},t)=\hbar\>\mbox{Im}\>\log\Psi(\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{n},t).$
(4)

It is assumed that, at some time $t_{0}$, the probability density for
configurations of the corpuscles is given by the square of the wave function
amplitude. If this holds for some time $t_{0}$, the dynamics, consisting of
the Schrödinger equation and the guidance equation, ensures that it holds for
all time. This property is known as _equivariance_ of the Born-rule
distribution.

####  5.1.1 Causal structure

It can be seen from the guidance equation (3) that the velocity of any one of
the corpuscles depends on an instantaneous configuration, and may, in
principle, depend on the positions of arbitrarily many corpuscles, arbitrarily
far away. Thus, the guidance equation requires a distinguished relation of
distant simultaneity for its formulation.

As an illustration, consider a pair of spin-$1/2$ particles in the singlet
state, spatially separated and sequentially subjected to Stern-Gerlach
experiments for the same direction, say the $z$-direction, which we will
regard as the vertical direction. The quantum state can be written,

$|\Psi(t)\rangle=\psi_{+-}(\mathbf{x}_{1},\mathbf{x}_{2},t)\>|z^{+}\rangle_{1}|z^{-}\rangle_{2}+\psi_{-+}(\mathbf{x}_{1},\mathbf{x}_{2},t)\>|z^{-}\rangle_{1}|z^{+}\rangle_{2}$
(5)

Before the experiment, there is no correlation between position and spin-
state, and the two wavepackets overlap completely.

$\psi_{+-}(\mathbf{x}_{1},\mathbf{x}_{2},t_{0})=\psi_{-+}(\mathbf{x}_{1},\mathbf{x}_{2},t_{0}).$
(6)

The outcome of both experiments is determined by the position of the particle
on which the experiment is done first. The Stern-Gerlach experiment on
particle 1 separates the wavepackets $\psi_{+-}$ and $\psi_{-+}$ along the
$z_{1}$-direction. Since Bohmian trajectories cannot cross, there will be a
plane or other two-dimensional surface in $\mathbf{x}_{1}$-space such that
particle 1 will go in the $+$ direction if it is initially located above this
plane, in the $-$ direction, if it is located below. After this experiment,
there is no longer any overlap, in configuration space, between the two
wavepackets, and the two particles are determinately located in only one of
them. The second particle responds to its Stern-Gerlach apparatus accordingly,
responding $+$ or $-$ depending on which wavepacket it is in.

This means that, if the experiment on particle 1 is done first, the outcome of
both experiments is determined by the initial position of particle 1. By
parity of reasoning, if the experiment on particle 2 is done first, the
outcome of both experiments is determined by the initial position of particle
2. These outcomes could differ, meaning that, for the same initial
wavefunction and initial configuration of the particles, the outcomes of the
experiments could differ depending on which experiment is done first. Thus,
the theory requires there to be a matter of fact about which experiment is
done first, even if the experiments are done at spacelike separation.

We might ask: is this an artefact of the way this theory is formulated, or is
it inherent in the very project of formulating a theory of this type, on which
particles have definite trajectories and probabilities over configurations are
to be given by the Born rule, with dynamics for the particles that ensures
that this probability distribution is equivariant?

As Berndl et al. (1996) have shown, the answer to this question is that a
theory of this type must, indeed, invoke a privileged foliation, as it is
impossible for the distribution postulate to be satisfied on all spacelike
hyperplanes.

Suppose that we have a pair of spin-$1/2$ particles, initially both located in
the same small region. The particles are then separated a large distance, and
each is passed through a Stern-Gerlach apparatus oriented in the
$x$-direction, which separates the wavepacket (in configuration space) along
the $x$-axis of the particle it’s interacting with. The beams are then
carefully recombined, with care taken not to let the particles interact with
anything that might induce decoherence, and then are passed through Stern-
Gerlach devices oriented in the $z$-direction. We suppose that all of this
splitting, recombining, and resplitting of the two beams occurs at spacelike
separation.

We consider four hypersurfaces (see Figure 1):

  1. 1.

A hypersurface $\alpha$, on which both particle 1 and particle 2 have their
wavepackets split into $x+$ and $x-$ wavepackets.

  2. 2.

A hypersurface $\beta$, on which both particle 1 and particle 2 have their
wavepackets split into $z+$ and $z-$ wavepackets.

  3. 3.

A hypersurface $\gamma$, on which the wavepackets are split along $z_{1}$ and
$x_{2}$ axes, respectively.

  4. 4.

A hypersurface $\delta$, on which the wavepackets are split along $x_{1}$ and
$z_{2}$ axes, respectively.

Particle $1$Particle $2$$\alpha$$\beta$$\gamma$$\delta$ Figure 1: The
hypersurfaces considered in the proof.

It is not, in fact, always possible to satisfy the distribution postulate
along all four of these hypersurfaces. Suppose that our pair of spin-$1/2$
particles is prepared in the Hardy state:

$\displaystyle|\psi\rangle_{H}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{12}}\left(|x^{+}\rangle_{1}|x^{+}\rangle_{2}-|x^{+}\rangle_{1}|x^{-}\rangle_{2}-|x^{-}\rangle_{1}|x^{+}\rangle_{2}-3|x^{-}\rangle_{1}|x^{-}\rangle_{2}\right)$
(7) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\left(|z^{+}\rangle_{1}|z^{-}\rangle_{2}-|z^{+}\rangle_{1}|z^{+}\rangle_{2}+|z^{-}\rangle_{1}|z^{+}\rangle_{2}\right)$
(8) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\left(|z^{-}\rangle_{1}|z^{+}\rangle_{2}-\sqrt{2}\>|z^{+}\rangle_{1}|x^{-}\rangle_{2}\right)$
(9) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\left(|z^{+}\rangle_{1}|z^{-}\rangle_{2}-\sqrt{2}\>|x^{-}\rangle_{1}|z^{+}\rangle_{2}\right)$
(10)

Let us write the states on these four hypersurfaces, with the position degrees
of freedom included, writing $|+\rangle$ for the particle being in the $+$
beam, $|-\rangle$ for its being in the $-$ beam.

$|\psi(\alpha)\rangle=\frac{1}{\sqrt{12}}\left(|x^{+}\rangle_{1}|+\rangle_{1}|x^{+}\rangle_{2}|+\rangle_{2}-|x^{+}\rangle_{1}|+\rangle_{1}|x^{-}\rangle_{2}|-\rangle_{2}\right.\\\
\left.-|x^{-}\rangle_{1}|-\rangle_{1}|x^{+}\rangle_{2}|+\rangle_{1}-3|x^{-}\rangle_{1}|-\rangle_{1}|x^{-}\rangle_{2}|-\rangle_{2}\right)$
(11)
$|\psi(\beta)\rangle=\frac{1}{\sqrt{3}}\left(|z^{+}\rangle_{1}|+\rangle_{1}|z^{-}\rangle_{2}|-\rangle_{1}-|z^{+}\rangle_{1}|+\rangle_{1}|z^{+}\rangle_{2}|+\rangle_{2}+|z^{-}\rangle_{1}|-\rangle_{1}|z^{+}\rangle_{2}|+\rangle_{1}\right)$
(12)
$|\psi(\gamma)\rangle=\frac{1}{\sqrt{3}}\left(|z^{-}\rangle_{1}|-\rangle_{1}|z^{+}\rangle_{2}|+\rangle_{2}-\sqrt{2}\>|z^{+}\rangle_{1}|+\rangle_{1}|x^{-}\rangle_{2}|-\rangle_{2}\right)$
(13)
$|\psi(\delta)\rangle=\frac{1}{\sqrt{3}}\left(|z^{+}\rangle_{1}|+\rangle_{1}|z^{-}\rangle_{2}|-\rangle_{2}-\sqrt{2}\>|x^{-}\rangle_{1}|-\rangle_{1}|z^{+}\rangle_{2}|+\rangle_{2}\right)$
(14)

If the distribution postulate is satisfied on $\alpha$, there is a probability
$1/12$ that both particles are in their respective $x^{+}$ beams. If the
postulate is satisfied on $\gamma$, then in such cases particle 1 must be in
its $z^{-}$ beam on $\gamma$. If the postulate is satisfied on $\delta$, then
particle 2 must be in its $z^{-}$ beam on $\delta$. But this means that, on
$\beta$, both particles will be in the $z^{-}$ beams, which should have
probability zero. Conclusion: the distribution postulate cannot be satisfied
along all foliations, and the dynamics for the particles must be choosy about
which, if any, foliation, it is with respect to which equivariance is to be
maintained.

The argument does not depend on preparing the Hardy state exactly, or that our
theory yield quantum probabilities exactly. On a theory of this sort, there
must be a joint probability distribution over the positions of the particles
at each point of intersection of the hyperplans considered that yields the
correct correlations on each of the four hyperplanes. As demonstrated by Fine
(1982a, b), violation of a Bell-type inequality for these variables is
equivalent to impossibility of a joint distribution of that sort.

For example: if there is a joint distribution over these variables, we will
have to have,

$Pr(x_{1}^{+}(\alpha),x_{2}^{+}(\alpha))\leq
Pr(x_{1}^{+}(\delta),z_{2}^{+}(\delta))+Pr(z_{1}^{+}(\gamma),x_{2}^{+}(\gamma))+Pr(z_{1}^{-}(\beta),z_{2}^{-}(\beta)),$
(15)

which is violated by the Hardy state.

A trajectory theory must choose a foliation along which the distribution
postulate is to be satisfied. Moreover, it is fairly easy to see that, in
order to maintain the distribution along this foliation, the dynamics must
permit causal dependence of the behaviour of particles on choice of experiment
at spacelike separation.

Consider three scenarios.

  1. A.

For particle 1, operations proceed as above, recombining the beams and
resplitting them into $z^{+}$ and $z^{-}$ wavepackets. Nothing is done to
particle 2 between $\alpha$ and $\beta$, and there is a split into $x^{+}$ and
$x^{-}$ wavepackets for particle 2 on both $\alpha$ and $\beta$.

  2. B.

Nothing is done to particle 1 between $\alpha$ and $\beta$, and it remains
split into $x^{+}$ and $x^{-}$ wavepackets on both $\alpha$ and $\beta$.
Recombining and resplitting proceed as above, for Particle 2.

  3. C.

Recombining and resplitting proceed as above, for both particles.

Suppose that we have dynamics according to which, if the probabilities for
positions of particles match the Born-rule probabilities on $\alpha$, they do
so on $\beta$, for all three scenarios. Suppose, also, in scenarios $A$ and
$B$, the particle to which nothing is done does not hop from one of its
wavepackets to another. That is, in scenario $A$, if particle 2 is in the
$x^{+}$ beam on $\alpha$, it is in the $x^{+}$ beam on $\beta$.

These conditions can be satisfied only if, for some initial configurations on
$\alpha$, transition probabilities—that is probabilities of a particle ending
up in a certain position on $\beta$, given the configuration on $\alpha$—for
one particle depend on what is being done to the other particle.

To see this, suppose the contrary. Suppose that, for any configuration on
$\alpha$, the probability that particle 1 ends up in the $z^{+}$ beam on
$\beta$ is the same, whether scenario $A$ or $C$ is in effect, and that the
probability that particle 2 ends up in the $z^{+}$ beam on $\beta$ is the
same, whether scenario $B$ or $C$ is in effect. Then, since, on Scenario $A$,
particle 2 is in the same beam on $\alpha$ and $\beta$, if the Born rule is
satisfied on $\beta$, it is also satisfied on $\gamma$. If transition
probabilities for particle 1 are the same in scenarios $A$ and $C$, it follows
that, in scenario $C$, the Born rule is satisfied on $\gamma$.

Running the same argument with the roles of particles 1 and 2 reversed yields
the conclusion that, if in scenario $B$ the Born rule is satisfied on $\beta$,
and if transition probabilities for particle 2 are the same in scenarios $B$
and $C$, it follows that, in scenario $C$, the Born rule is satisfied on
$\delta$. But, as we have seen, in scenario $C$, the Born rule cannot be
satisfied on all four hypersurfaces.

####  5.1.2 Bohmian quantum field theories

Even though there are good reasons to think that an additional-variables
theory must employ a preferred foliation, the question still arises whether
there could be a theory of this sort that recovers the empirical content of
relativistic quantum field theories. This would, of course, entail that the
preferred foliation is empirically inaccessible. Two avenues of approach
suggest themselves for extending Bohmian theories to the context of quantum
field theories. One is to retain a basic particle ontology, and permit
particle creation and annihilation. The other is to take classical field
configurations as the basic ontology, and to provide a non-classical dynamics
for the evolution of these field configurations. For an overview of these
options, see Struyve (2011). Struyve concludes that the field approach works
best for bosonic fields, but is problematic when one attempts to extend it to
fermionic fields. For fermionic fields, the particle approach seems more
appropriate.

The task of extending additional-beables theories to the context of quantum
field theories remains a work in progress. However, there do not seem to be
in-principle obstacles to the creation of a theory of this sort that recovers
the phenomena that are currently handled by quantum field theories.

###  5.2 Modal interpretations

Modal interpretations supplement the quantum state with additional beables. A
variety of interpretations along these lines have been proposed; see Lombardi
and Dieks (2017) for an overview. Many of these share the feature that the
nature of the additional beables depends on the quantum state. Given a quantum
system $\alpha$, which may be a subsystem of a larger system, one considers
the reduced state of $\alpha$, obtained by tracing out the degrees of freedom
of the rest of the system. If the state of the larger system is a pure quantum
state, the reduced state of $\alpha$ will be pure if $\alpha$ is not entangled
with the rest of the system, mixed, if it is. One resolves the density
operator representing the reduced state into its spectral components:

$\rho_{\alpha}=\sum_{i}w_{i}\>P_{i},$ (16)

where $\\{P_{i}\\}$ are mutually orthogonally projections. In the non-
degenerate case, in which all nonzero coefficients $\\{w_{i}\\}$ are distinct,
this resolution is unique. The _basic modal rule_ assigns definite values 1 or
0 to these projections; one of these projections represents a property
definitely possessed by the system, the others, definitely absent. The
probability that $P_{i}$ is the possessed property is $w_{i}$.

####  5.2.1 Difficulties arising from relativistic causal structure

Can modal interpretations respect relativistic causal structure, or do they
require a distinguished relation of distant simultaneity? Dickson and Clifton
(1998) proved, for a broad class of modal interpretations, that is impossible
to furnish the theory with transition probabilities in such a way as to yield
Born-rule probabilities for possessed values on arbitrary hyperplanes. Dickson
and Clifton’s proof relies on an assumption concerning the transition
probabilities for possessed values, the assumption they call “stability,” but,
as Arnztenius (1998) pointed out, the stability requirement is dispensable and
the core of the proof concerns the nonexistence of certain joint distributions
yielding the appropriate Born probabilities as marginals. Taking these proofs
as a starting point, Myrvold (2002, 2009) adapted the argument of Berndl et
al. (1996), outlined above, to show that any modal interpretation, so long as
the possessed values of a system are capable of being regarded as local
beables, cannot have Born-rule probabilities for these possessed values
satisfied on arbitrary spacelike hyperplanes. The argument at the end section
5.1.1, above, can be straightforwardly adapted to show that these theories,
also, must permit causal dependence of the behaviour of the additional beables
on choice of experiment at spacelike separation.

Undeterred, Berkovitz and Hemmo(2005a,2005b,2006) have proposed a relational
modal interpretation. The interpretation assigns properties to systems that
are not thought to be intrinsic to that system, but are possessed by it only
relative to some other system. In a relativistic context, these relational
values may depend on what hypersurface is used to define a global state of the
universe, and it can happen, in the presence of entanglement at spacelike
separation, that some property of a system, at a single position on its
worldline, may have different values relative to different hypersurfaces.
Thus, the possessed values are not to be thought of as local beables, and, as
the authors argue in some detail, the no-go theorems don’t apply.

####  5.2.2 Difficulties arising from infinitely many degrees of freedom

In addition to the difficulties for modal interpretations arising from
relativistic causal structure, there is another difficulty for attempts to
extend modal interpretation to the context of relativistic quantum field
theories, arising from infinitude of degrees of freedom.

There is, first, the issue of extending the basic modal rule to the context of
a relativistic quantum field theory. We want to be able to apply the rule to
local subsystems of the universe. If we consider a bounded region $R$ of
spacetime, and the algebra of operators $\mathcal{A}(R)$ formed from fields in
$R$, there is a difficulty in applying the basic modal rule, since, as pointed
out by Dieks (2000), these local algebras will not contain either a density
operator representing the state, or projections onto finite-dimensional
subspaces. Clifton (2000) provided a version of the modal rule that makes
sense in the context of algebraic quantum field theory, and also showed that
there exists a whole host of states for which the rule picks out no definite
observables other than multiples of the identity (that is, observables that
have the same value no matter what the state is). Earman and Ruetsche (2005)
extend Clifton’s result, and discuss the significance of these results for
modal interpretations of quantum theory.

These trivialization results mean that a modal interpretation of this sort
cannot respect relativistic causal structure and symmetries while
accomplishing the goal of providing a solution to the measurement problem. The
purpose of a modal interpretation is to have observables such as apparatus
pointer observables take on definite values in physically realistic states.
Obviously, this is not accomplished if no observables are definite other than
multiples of the identity.

##  6 Collapse Theories

A dynamical collapse theory modifies the evolution of the quantum state. One
approach that might be tried would be to replace the deterministic, unitary
evolution of the quantum state by a deterministic and non-unitary evolution.
There are arguments to the effect that, given plausible assumptions, this
would lead to the possibility of superluminal signalling (Gisin, 1989; Simon
et al., 2001).222These assumptions may be questioned; see Kent 2005b. For this
reason, in this section we focus on stochastic collapse theories.

On a theory such as this, the dynamical laws do not specify a unique future
state, given the state at a time $t_{0}$; rather, which state will be realized
is, on such a theory, a matter of chance. The idea is to specify a
probabilistic law that approximates the usual unitary, deterministic evolution
within those domains where we have good evidence for it, but departs
sufficiently from it in some situations—which should include experiments that
lead to recorded outcomes—so as to closely approximate the textbook collapse
postulate, according to which some definite outcome obtains at the end of an
experiment, with probabilities as to which outcome is obtained given by the
Born rule. One theory of this sort, with unitary evolution punctuated by
discrete jumps, is the Ghirardi-Rimini-Weber (GRW) theory, referred to by the
authors as _Quantum Mechanics with Spontaneous Localization_ (QMSL) (Ghirardi
et al., 1986). Another is the _Continuous Spontaneous Localization_ (CSL)
theory (Pearle 1989, Ghirardi, Pearle, and Rimini 1990).

Among the experiments to be accounted for are tests of Bell inequalities.
Since our theory is required to produce correct probabilities for the outcomes
of such experiments, we do not expect the probabilistic law to be one on which
spacelike separated events are probabilistically independent. It might seem
obvious, at first, that this dooms compatibility with relativity from the
start.

Things are not so simple, however. Bell himself said that, when he first
encountered the GRW theory, “I thought I could blow it out of the water, by
showing it was _grossly_ in violation of Lorentz invariance” (Bell, 1989, p.
17). A careful examination led him to the opposite conclusion, namely, that,
though couched in terms of nonrelativistic quantum mechanics, the theory
exhibits “a residue, or at least an analogue, of Lorentz invariance” (Bell
1987a, p. 10 in Bell 1987b, pp. 206–207). The idea is this. On the usual,
Schrödinger, dynamics, for two (or more) systems, widely separated from each
other and not interacting with each other, evolution of each system is
independent, and we can employ separate time variables for each system. The
Schrödinger dynamics do not require any notion of synchronization of the time
variables. This is a feature that Bell calls _relative time translation
invariance_. The question he addressed is: for systems like that, does the GRW
theory retain relative time translation invariance, or do its probabilistic
laws require there to be a matter of fact about the temporal order of events
at a distance? Bell’s verdict: the probability of any sequence of GRW jumps is
relative time translation invariant.

> I am particularly struck by the fact that the model is as Lorentz invariant
> as it could be in the nonrelativistic version. It takes away the ground of
> my fear that any exact formulation of quantum mechanics must conflict with
> fundamental Lorentz invariance (Bell 1987a, p. 14, in Bell 1987b, p. 209).

In his Trieste lecture, delivered in November 1989, Bell discussed the
prospects for a genuinely relativistic version of a dynamical collapse theory,
and concluded that the difficulties encountered by Ghirardi, Grassi, and
Pearle in producing a genuinely relativistic version of CSL a theory that
would be “Lorentz invariant, not just for all practical purposes but deeply,
in the sense of Einstein, eliminating entirely any privileged reference system
from the theory” (Bell, 2007, p. 2931), were “Second-Class Difficulties,”
technical difficulties, and not deep conceptual ones (see also Bell 1989, p.
25).

A relativistic collapse theory, if it is to yield even an approximation to the
usual quantum probabilities, will include correlations between events at
spacelike separation that violate Bell inequalities. The Bell inequalities can
be derived from the condition that correlations be _locally explicable_. This
has two parts: that the correlations be explicable in a certain sense, and the
explanation be local. The explicability condition was taken by Bell to be the
condition that correlations between events that are not in a direct cause-
effect relation with each other be attributable to a common cause (see Bell
1981a, p. C2–55, in Bell 1987b, p. 152). This condition is violated in
relativistic collapse theories, which involve probabilistic correlations
between spacelike separated events that are _not_ attributable in the usual
way to events in their common past. Unlike cause-and-effect relations as
usually conceived, the relation of probabilistic correlation between these
distant events is symmetric, and does not require a temporal order between the
events. For this reason, it is misleading to assimilate this relation to the
causal relation and refer to it as nonlocal “influence” or “action at a
distance.” See Myrvold (2016) for further discussion.

As we shall see, there are difficulties associated with constructing a fully
relativistic collapse theory. These difficulties arise, not because a collapse
theory cannot be formulated in the absence of a distinguished relation of
distant simultaneity, but rather from the problem of constructing a
relativistic collapse theory with a stable vacuum state. In order to see how
all this works, we start with a few words about collapse theories in general.

###  6.1 Stochastic quantum state evolution

It is generally accepted that any physically realizable change of the state of
a quantum system must be a _completely positive_ mapping of the system’s state
space into itself. Any such mapping can be represented by a set $\\{K_{i}\\}$
of operators, such that the density operator $\rho$ representing the state
undergoes the change

$\rho\rightarrow\sum_{i}K_{i}\>\rho\>K_{i}^{\dagger},$ (17)

where

$\sum_{i}K_{i}^{\dagger}K_{i}\leq\mathds{1}.$ (18)

If $\sum_{i}K_{i}^{\dagger}K_{i}=\mathds{1}$, the operation is called a
_nonselective operation_ ; if $\sum_{i}K_{i}^{\dagger}K_{i}<\mathds{1}$, it is
_selective_. Such a representation of a completely positive mapping of the
state space into itself is called a _Kraus representation_ of the mapping; and
the operators $\\{K_{i}\\}$, _Kraus operators_.

Selective operations reduce the trace-norm of the density operator, but this
is not an issue, as normalization is only a convention. With an unnormalized
density operator, we compute the expectation value of an observable
represented by an operator $A$ via

$\langle A\rangle_{\rho}=\mbox{Tr}(\rho A)/\mbox{Tr}(\rho).$ (19)

In the simplest case, the set of Kraus operators is a singleton, and the
evolution is deterministic; this includes the case of unitary evolution.
However, one can also consider stochastic processes. Suppose that we has a set
of operators $\\{K_{i}\\}$, such that the state vector undergoes a stochastic
transition of the following form. For some $i$,

$|\psi\rangle\Rightarrow|\psi^{\prime}\rangle=K_{i}|\psi\rangle/\|K_{i}|\psi\rangle\|,$
(20)

with the probability for which transition it undergoes given by

$p_{i}=\|K_{i}|\psi\rangle\|^{2}.$ (21)

Since these probabilities must sum to unity for every vector $|\psi\rangle$,
we must have $\sum_{i}K_{i}^{\dagger}K_{i}=\mathds{1}$. For each $i$, we have
a selective operation. One of these yields the actual state. The transition to
the mixture of these candidate states, which corresponds to the proposition
that some one of these transitions has occurred, without specification of
which, is given by weighted mean of these.

$\rho\rightarrow\bar{\rho^{\prime}}=\sum_{i}p_{i}\>\frac{K_{i}\rho
K_{i}^{\dagger}}{\mbox{Tr}(K_{i}\rho K_{i}^{\dagger})}.$ (22)

In (20), we indexed the possible state transitions via a discrete index $i$.
We can also consider stochastic processes of a more general sort. Let
$\langle\Gamma,\mathcal{F},\mu\rangle$ be a probability space.333That is,
$\Gamma$ is some set, $\mathcal{F}$ is a $\sigma$-algebra of subsets of
$\Gamma$ containing both the empty set $\emptyset$ and $\Gamma$ itself, to be
regarded as the measurable subsets of $\Gamma$, and $\mu$ is a non-negative
countably additive set function on $\mathcal{F}$ with $\mu(\Gamma)=1$. Suppose
that we have a family of operators $\\{K_{\gamma}\\}$, for $\gamma\in\Gamma$,
such that

$\int_{\Gamma}K^{\dagger}_{\gamma}K_{\gamma}\,d\mu(\gamma)=\mathds{1}.$ (23)

These can serve as the operators that induce our state transitions.

This gives us a rather general schema for a stochastic process in a Hilbert
space. For the moment, we will presume that we have a unique global time
function; extension to a relativistic spacetime will be considered in the next
subsection.

Any theory satisfying the following conditions will give us stochastic
evolution of the state vector.

  1. 1.

We assume a measure space $\langle\Gamma,\mathcal{F},\mu\rangle$. The elements
of $\Gamma$ are to be thought of as possible complete histories, specifying
events for all times. For each time $t$ there is a $\sigma$-algebra
$\mathcal{F}_{t}$, whose elements are to be thought of as sets of possible
histories up to time $t$. For $t<s$, we will have
$\mathcal{F}_{t}\subseteq\mathcal{F}_{s}\subseteq\mathcal{F}$.

  2. 2.

Given a history up until time $t$, and a state vector $|\psi(t)\rangle$, the
state at a later time $t+s$ is a random variable, defined as follows.

    1. (a)

We assume a family of operators $K_{\gamma}(s;t)$, which depend only on events
up until time $s$ (that, is, if two histories $\gamma,\gamma^{\prime}$ agree
up to time $s$, then $K_{\gamma}(s;t)=K_{\gamma^{\prime}}(s;t)$), such that

$\int_{\Gamma}K^{\dagger}_{\gamma}(s;t)K_{\gamma}(s;t)\,d\mu=\mathds{1}.$

    2. (b)

For some $\gamma$,

$|\psi(t+s)\rangle=\frac{K_{\gamma}(s;t)|\psi(t)\rangle}{\|K_{\gamma}(s;t)|\psi(t)\rangle\|}.$

    3. (c)

The probability distribution for the realized outcome is given by

$\mbox{Prob}(\gamma\in
F)=\int_{F}\|K_{\gamma}(s;t)|(\rangle\psi(t)\|^{2}\,d\mu.$

for any $F$ in $\mathcal{F}_{s}$.

On a theory of this sort, given a state $|\psi(t_{0})\rangle$, for any later
time $t_{1}=t_{0}+s$ there will be an actual state vector
$|\psi(t_{1})\rangle$, and its corresponding pure-state density operator

$\rho(t_{1})=|\psi(t_{1})\rangle\langle\psi(t_{1})|.$ (24)

There will also be what may be called an _ensemble_ , or _mean_ density
operator, which is a weighted average of the various possibilities for
$\rho(t_{1})$, weighted by their respective probabilities.

$\bar{\rho}(t_{1};t_{0})=\int_{\Gamma}K_{\gamma}(t_{1};t_{0})\,\rho(t_{0})\,K^{\dagger}_{\gamma}(t_{1};t_{0})\,d\mu.$
(25)

###  6.2 Collapse theories in Minkowski spacetime

####  6.2.1 Causality conditions

It is useful to work within what may be called the _stochastic Tomonaga-
Schwinger picture_. We start with field operators that are solutions of the
standard Heisenberg-picture equations (these may be free or interacting
fields). We associate with each spacelike Cauchy surface $\sigma$ a state
vector $|\psi(\sigma)\rangle$. Our theory should deliver a stochastic
evolution of $|\psi(\sigma)\rangle$ to $|\psi(\sigma^{\prime})\rangle$
whenever $\sigma^{\prime}$ is nowhere to the past of $\sigma$. This will go
much as it did in §6.1.

  1. 1.

We assume a measure space $\langle\Gamma,\mathcal{F},\mu\rangle$. The elements
of $\Gamma$ are to be thought of as possible complete histories, specifying
events for all times. For each spacelike Cauchy surface $\sigma$ there is a
$\sigma$-algebra $\mathcal{F}_{\sigma}$, whose elements are to be thought of
as sets of histories to the past of $\sigma$. For two Cauchy surfaces
$\sigma,\tau$, with $\tau\preceq\sigma$, we will have
$\mathcal{F}_{\tau}\subseteq\mathcal{F}_{\sigma}\subseteq\mathcal{F}$.

  2. 2.

Given two Cauchy surfaces $\sigma,\tau$, with $\tau\preceq\sigma$, let
$\delta$ be the spacetime region between them. Given a state vector
$|\psi(\tau)\rangle$, the state vector $|\psi(\sigma)\rangle$ is a random
variable, defined as follows.

    1. (a)

We assume a family of operators $\\{K_{\gamma}(\delta)\\}$, which depend only
on events in the past of $\sigma$ (that, is, if two histories
$\gamma,\gamma^{\prime}$ agree on events in the past of $\sigma$ then
$K_{\gamma}(\delta)=K_{\gamma^{\prime}}(\delta)$), such that

$\int_{\Gamma}K^{\dagger}_{\gamma}(\delta)K_{\gamma}(\delta)\,d\mu=\mathds{1}.$

We call these _evolution operators_ for the region $\delta$.

    2. (b)

For some $\gamma$,

$|\psi(\sigma)\rangle=\frac{K_{\gamma}(\delta)|\psi(\tau)\rangle}{\|K_{\gamma}(\delta)|\psi(\tau)\rangle\|}.$

    3. (c)

The probability distribution for the realized outcome is given by

$\mbox{Prob}(\gamma\in
F)=\int_{F}\|K_{\gamma}(\delta)|\psi(\tau)\rangle\|^{2}\,d\mu.$

for any $F$ in $\mathcal{F}_{\sigma}$.

Here, again, we can define a mean density operator for the state on $\sigma$,
given a state vector on $\tau$, with $\tau\subseteq\sigma$.

$\bar{\rho}(\sigma;\tau)=\int_{\Gamma}K_{\gamma}(\delta)\rho(\tau)K^{\dagger}_{\gamma}(\delta)\,d\mu.$
(26)

Consider two Cauchy surfaces, $\tau$, $\sigma$, which coincide everywhere
except on the boundaries of two bounded regions $\delta_{1}$, $\delta_{2}$,
where $\sigma$ is to the future of $\tau$ (see Figure 2). We must have a
unique law of stochastic evolution from $\tau$ to $\sigma$, through
$\delta_{1}\cup\delta_{2}$, and this should coincide with the composition of
the evolution through $\delta_{1}$ and the evolution through $\delta_{2}$, in
either order. The necessary and sufficient condition for this is,

  1. 3.

Evolution operators corresponding to spacelike separated regions commute.

$\sigma$$\delta_{1}$$\delta_{2}$$\tau$ Figure 2: Spacelike separated regions
between two spacelike hypersurfaces.

One difference between a theories of this sort and a theory involving
deterministic, unitary evolution, is that, for calculating probabilities of
outcomes of experiments performed in some spacelike slice $\alpha$ that is
common to two Cauchy surfaces $\tau$, $\sigma$, it can make a difference
whether the calculation is made using $|\psi(\tau)\rangle$ or
$|\psi(\sigma)\rangle$. It is this feature that has given rise to concerns
that a relativistic collapse theory must either be incoherent or exhibit
radical contextualism about possessed properties. These concerns are
misplaced; more on this, below, in subsection 6.2.2. However, it _would_ be
problematic if probabilities for events in $\alpha$ differed depending on
whether one used $|\psi(\tau)\rangle$ or $\bar{\rho}(\sigma)$, which is the
probabilistically weighted mixture of various possibilities for
$|\psi(\sigma)\rangle$, given the state on $\tau$. Moreover, if the evolution
from $\tau$ to $\sigma$, through a region $\delta$, spacelike separated from
$\alpha$, depended on a choice of experiments to be made, then, if different
choices yielded different mixtures $\bar{\rho}(\sigma)$, the choice of
experiment could be used for superluminal signalling. We, therefore, require
$\bar{\rho}(\sigma)$ to yield the same probabilities for outcomes of
experiments as $|\psi(\tau)\rangle$, for any allowable evolution through
$\delta$. The necessary and sufficient condition for this is (Arias et al.,
2002),

  1. 4.

Evolution operators pertaining to a region $\delta$ commute with all operators
representing observables at spacelike separation from $\delta$.

The conditions 3, 4 ensure compatibility with relativistic causal structure.

####  6.2.2 Collapse theory ontology

If the region between two Cauchy surfaces $\tau$, $\sigma$ is a bounded
spacetime region $\delta$, then the transition from the state on $\tau$ to the
state on $\sigma$ can be attributed to events within $\delta$. Quantum states,
on the other hand, are in some sense “nonlocal”—a better term is
_nonseparable_. They cannot be regarded as supervening on local matters of
fact. A quantum state encodes, not merely local matters of fact, but also
correlations between distant regions.

This is reflected in the way that a collapse affects the global state. Let
$\tau$ and $\sigma$ be two spacelike hypersurfaces, that coincide everywhere
except on the boundaries of a bounded region $\delta$, where $\sigma$ lies to
the future of $\tau$. Let $\alpha$ be a bounded region that lies in both
$\tau$ and $\sigma$, and hence is at spacelike separation from $\delta$ (see
Figure 3). The two states can differ on probabilities assigned to observables
pertaining to $\alpha$. For example, consider the familiar EPR-Bohm scenario.
Suppose the state on $\tau$ involves an entangled pair of spin-$1/2$
particles, one of which is located (to the extent that it can be located)
within $\alpha$, and suppose that in $\delta$ an experiment is performed on
the other one, yielding, on $\sigma$, a near eigenstate of spin in some
direction ${a}$ that is correlated by the state on $\tau$ with spin of the
other particle in direction ${b}$. Then the state on $\tau$ is a near-
eigenstate of ${b}$-spin for the second particle. Thus, the two states
$|\psi(\tau)\rangle$ and $|\psi(\sigma)\rangle$ differ on probabilities
assigned to experiments performed in the forward domain of dependence of
$\alpha$.

$\alpha$$\tau$$\delta$$\sigma$ Figure 3: A region common to two spacelike
hypersurfaces.

Relativistic causality precludes events occurring in a region $\delta$ from
affecting _local_ beables pertaining to regions at spacelike separation from
$\delta$. If one thought that the differing probabilities assigned to
observables pertaining to $\alpha$ by the states $|\psi(\tau)\rangle$ and
$|\psi(\sigma)\rangle$ were local beables, this would give rise to a serious
worry about the conceptual coherence of a relativistic collapse theory (see,
_e.g._ Maudlin 1994, p. 209 for worries along these lines).

But the reduced states of $\alpha$ obtained from the global states
$|\psi(\tau)\rangle$ and $|\psi(\sigma)\rangle$ are not obviously local
matters of fact about $\alpha$, and, indeed, what these reduced states are
depends on events at spacelike separation from $\alpha$. What _can_ , and
_should_ be regarded as the intrinsic state of the spacetime region $\alpha$,
that is, its state stripped of implicit references to goings-on at a distance
from $\alpha$, is the past light cone state. This is the limiting state
obtained from a sequence of spacelike hypersurfaces that converge on the past
light cone of $\alpha$. This state will reflect collapse events to the past of
$\alpha$, but not collapse events at spacelike separation.

This has implications for what the local beables of the theory could be.
Ghirardi and Grassi (1994, p. 419) proposed a criterion for property
attribution in the context of relativistic collapse theories. If $A$ is a
local observable with support in $\alpha$, a system possesses a property $A=a$
when the expectation value of $P_{a}$, the projection onto the subspace $A=a$,
is extremely close to 1, evaluated with respect to the past light cone state
of $\alpha$.

Ghirardi, Grassi, and Benatti (1995) proposed, as an appropriate local beable
for collapse theories, a smeared mass density, with the smearing taken over
lengths scales large compared to the atomic scale but small on a macroscopic
scale. For a relativistic theory, the corresponding quantity would be a
smeared energy-momentum density, whose $00$ component is mass-energy density,
which, in a nonrelativistic context, is dominated by the rest mass. Combining
the choice of mass density as local beable with the past light cone criterion
for property attribution, the mass density that in a relativistic context is
taken to be a local beable in a region $\alpha$ is the mass density defined by
the state on the past light cone of $\alpha$ (Tumulka, 2007; Bedingham et al.,
2014). See Myrvold (2016, 2018, 2019) for further discussion of ontological
issues in collapse theories.

####  6.2.3 A problem: stability of the vacuum

There is a difficulty with constructing a physically sensible collapse theory
that respects the full symmetries of Minkowski spacetime. This has to do with
the stability of the vacuum. A relativistic collapse theory must have a stable
vacuum, which is not achievable via a straightforward extension of CSL to a
relativistic quantum field theory.

If the theory were to produce excitations of the vacuum, it would have to
furnish a probability distribution over possible excitations. The difficulty
is the nonexistence of a bounded Lorentz-invariant measure over possible
excitations.

This is readily seen in the simplest case, a free theory.444This paragraph is
based on the discussion in Pearle (2015, p. 2). Let $\Delta$ be some subset of
momentum space, and suppose that our collapse theory yields a certain
probability $p(\Delta)$ per unit time of producing from the vacuum a particle
with momentum in $\Delta$. Lorentz invariance requires that there be the same
probability for producing a particle with its momentum in any Lorentz boost of
this volume.

This requirement cannot be satisfied. The problem is that there is no bounded
invariant measure on the mass-shell consisting of all four-momenta of a given
magnitude $m$. Any probability measure on the mass-shell, therefore, must
break Lorentz symmetry. In the vacuum state, which is a Lorentz invariant
state, there is nothing that could be used to define such a measure. To
generalize beyond free theories, the issue is the absence of a bounded
Lorentz-invariant measure over the space of possible excitations of the
vacuum.

The requirement of a strictly stable vacuum applies only to theories that
respect the full set of spacetime symmetries of Minkowski spacetime. One
could, for example, have a theory on a lattice, with relativistic causal
structure. The rate of energy increase could, for such a theory, be kept
finite. Whether it could be kept below the threshold of current observations
would depend on the lattice spacing and other parameters of the theory; see
Ghirardi, Grassi and Pearle (1990, p. 1297) for discussion.

####  6.2.4 Impossibility of a relativistic collapse theory using only
standard degrees of freedom

Though relativistic causality does not preclude the formulation of a
relativistic collapse theory, there is a tension between the requirements of
relativistic causality on a collapse theory and the need for a stable vacuum.
It is possible to avoid an outright contradiction, but only at the cost of
introducing nonstandard degrees of freedom, that do not obey anything like the
usual laws of quantum evolution.

Here’s the problem. Suppose we have a collapse theory with a stable vacuum
state. Consider evolution from one surface $\tau$ to another, $\sigma$,
through a bounded spacetime region $\delta$. Stability of the vacuum means if
the state on $\tau$ is the vacuum state, subsequent evolution is
deterministic, in that there is only a single outcome that has nonzero
probability, namely, maintenance of the vacuum. Deterministic evolution
starting from vacuum means that, except perhaps for a set with zero
probability of being realized, all collapse operators
$\\{K_{\gamma}(\delta)\\}$ map the vacuum into multiples of the same vector.
It can be shown (see Myrvold 2017 for details) that this will also hold for
all standard states, where standard states are those that can be obtained by
operating on the vacuum with operators of the sort that appear in standard
quantum field theories. That is, for standard quantum field theories, the
constraints imposed by relativistic causal structure, plus the constraint,
imposed by Lorentz invariance, of a stable vacuum, are satisfied at the
expense of rendering the theory deterministic, and thus incapable of
suppressing the sorts of superpositions of macroscopically distinct states
that collapse theories are meant to suppress.

This is not a no-go result for relativistic collapse theories _tout court_.
Bedingham (2011a, b) and Pearle (2015) utilize a nonstandard field first
introduced by Pearle (1993). This “index” or “pointer” field does not obey
anything like an ordinary law of evolution; instead, its value at any point is
independent of its value at any other point. The vacuum state $|\Omega\rangle$
of this theory is the ground state of the standard fields _and_ this pointer
field. Standard operators operating on the vacuum state leave the pointer
field in its ground state. On Bedingham’s theory the pointer field couples to
the standard fields, so that, if there are excitations of the standard fields,
these produce excitations of the pointer fields. The theory tends to collapse
towards eigenstates of smeared pointer-field operators, with accompanying
collapse towards eigenstates of smeared matter fields.

One might also reject the requirement of having a theory that produces
collapses in Minkowski spacetime. Okon and Sudarsky (2014, 2016, 2018) have
proposed, in the context of semi-classical gravity, a modification of the CSL
theory on which the collapse rate depends on local curvature. One might
consider a modification of this proposal, according to which collapse rate
goes to zero as curvature goes to zero (Bengochea et al., 2020). If there is a
viable theory of this sort, it might satisfy the constraint of having a stable
vacuum in Minkowski space, while producing collapse in the presence of matter.

###  6.3 Causal Quantum Theory

Standard quantum theory predicts correlations between outcomes of spacelike
separated experiments that violate Bell inequalities. The crucial assumption
of derivation of Bell inequalities is that the correlation be _locally
explicable_ , that is, that they can be explained by reference to conditions
in the past lightcones of the experiments, such that a full specification of
these conditions screens off the correlations. Experimental demonstrations of
violations of Bell inequalities are usually taken to indicate correlations
that are not locally explicable are a genuine physical phenomenon (see
Myrvold, Genovese, and Shimony 2019 for an overview, and a discussion of the
conditions required to reach this conclusion).

There is a loophole, however, identified by Kent (2005a), and called by him
the _collapse locality loophole_. Tests of Bell inequalities that aim to
establish nonlocality take care to ensure that the experiments involved take
place at spacelike separation from each other, from choice of experimental
setting to registration of result. A judgment of whether this condition is
satisfied requires, therefore, a judgment of when an experiment is over.
Typically it is assumed that this occurs once the result has been permanently
recorded in some macroscopically readable device. However, as Kent observes,
on some proposals, the quantum state will remain in a superposition
corresponding to distinct outcomes beyond this point. One such proposal is the
suggestion that state reduction takes place only when the uncollapsed state
involves a superposition of sufficiently distinct gravitational fields (Diósi,
1987; Penrose, 1989, 1996). Another is Wigner’s suggestion that conscious
awareness of the result is required to induce collapse (Wigner, 1961).

Kent introduces a class of theories, which he calls _causal quantum theories_
, on which collapses are localized events, collapse probabilities are
conditioned only on events in the collapse’s past light cone, and, as a
consequence, collapses at spacelike separation from each other are
probabilistically independent, conditional on events in their past lightcones.
A theory of that sort differs in its predictions from standard quantum
mechanics, but a decisive test requires experiments that are completed at
spacelike separation from each other.

On a theory of the sort proposed by Diósi and Penrose, on which collapse
occurs only when differing results correspond to sufficiently distinct
gravitational fields, the condition that the experiments be concluded at
spacelike separation had not been enforced in any of the experimental tests of
Bell inequalities as of the time of writing (2005). The experiment of Salart
et al. (2008) closed the loophole for the particular proposals of Penrose and
Diósi, though, as Kent (2018) pointed out, altering the Penrose-Diósi
threshold by a few orders of magnitude would render these proposals compatible
with the results of this experiment.

No experiment to date has addressed the collapse locality loophole if the
collapse condition is taken to be awareness of the result by a conscious
observer. See Kent (2018) for proposals of ways in which causal quantum theory
could be subjected to more stringent tests.

##  7 Everettian Interpretations

Everettian interpretations make use of minimal machinery: they leave the
usual, unitary dynamics untouched, and add nothing to the quantum state. The
complications arise in connection with painting a sensible picture of the
world with this minimal palette, and with making sense of probabilities or a
working substitute for them.

Because of this minimality, worries about compatibility with relativity are
also minimal. If the theory to be interpreted is a relativistic quantum field
theory, then there is no question about the dynamics being compatible with
relativistic causal structure; it is. As there are no additional beables,
there is no question about whether their dynamics is compatible with
relativity. Some proponents of Everettian interpretations have taken this to
be one of the chief attractions of interpretations of this sort (see, _e.g._
Vaidman 1995, 2016, 2018).

Given a bounded subset $\alpha$ that is part of many Cauchy surfaces $\sigma,$
$\sigma^{\prime}$, _etc._ , we can consider the reduced state of
$\alpha$,555Those who want to insist that states are to be assigned only to
regions with nonempty interior may take this to mean: the state of the causal
domain of dependence of $\alpha$. evaluated with respect to various Tomonaga-
Schwinger state vectors $|\psi(\sigma)\rangle$,
$|\psi(\sigma^{\prime})\rangle$. If the evolution from one Cauchy surface to
another is unitary, and if the generators of evolutions through regions
spacelike separated from $\alpha$ commute with operators representing
observables pertaining to $\alpha$, then the reduced state of $\alpha$ will be
the same whether calculated from $|\psi(\sigma)\rangle$ or
$|\psi(\sigma^{\prime})\rangle$ or any Cauchy surface containing $\alpha$. So,
in that sense, on any quantum-state monist theory with deterministic, unitary
dynamics, operations performed at a distance have no effect on states of
affairs in $\alpha$.

Is there nonlocality, in any sense, in an Everettian theory? It is not
immediately clear how to pose the question. Unlike, say, the case of
additional beables theories, we cannot ask (at least in any straightforward
way) whether a choice of a parameter setting at one location affects the
outcome of an experiment performed at another, because, on an Everettian
theory, talk of _the outcome_ of an experiment is unwarranted—experiments do
not have unique outcomes. If the usual notions of causal dependence, such as
act-outcome correlations, presuppose unique outcomes, then they cannot even be
formulated in an Everettian context—and _ipso facto_ they cannot be violated.
Instead of saying that Everettian theories are causally local, perhaps it
would be more apt to say: it is not the case that they are causally nonlocal.

Brown and Timpson (2016) have made a stronger claim about Everettian
interpretations. Not only is there no dynamical nonlocality, they claim, there
are no nonlocal correlations. The claim is that, in the Everettian context,
talk of correlated outcomes of experiments makes sense only when the outcomes
of the distant experiments are compared, at which time both experiments are in
the past. A similar argument has been made of which quantum theories are but a
special case, by Brassard and Raymond-Robichaud (2013,2019). Analysis of this
claim is beyond the scope of this chapter; we note only that compatibility of
Everettian interpretations with relativity does not ride on whether a claim of
this sort is accepted or rejected.

##  8 Non-Realist and Pragmatist Interpretations

There is a long tradition of denying that the ontology of the physical world
contains anything corresponding to a quantum state. In recent years, views of
this sort have been championed by advocates of a position called, by its
proponents, _Quantum Bayesianism_ , or _QBism_ (Caves, Fuchs and Schack 2002;
Fuchs, Mermin, and Schack 2014; Fuchs and Schack 2015). This is one variant of
a class of views commonly referred to as $\psi$_-epistemic_ views.666See
Harrigan and Spekkens (2010) for a discussion. A somewhat related, but
different, view is the pragmatist view of quantum mechanics defended by
Richard Healey (2012, 2017a, 2020). See, also, Healey (2017b) for an overview
of related views.

For Healey, quantum chances for events are assigned much as they have been in
the discussion, above, of collapse theories. The chance of an event, say, an
outcome of an experiment, is indexed by the spacetime event from which the
chance is to be ascribed. The chance $Ch_{x}(e_{A})$ of event $e_{A}$ is, in
effect, calculated using what in a dynamical collapse theory would be the past
light cone state, which incorporates experimental outcomes that have occurred
in the past light cone of $x$. On Healey’s view, “Any agent who accepts
quantum theory and is (momentarily) located at space-time point $x$ should
match credence in $e_{A}$ to $Ch_{x}(e_{A})$,” where $Ch_{x}(e_{A})$ is the
chance evaluated conditional on events in the past light cone of the point $x$
(Healey 2016, p. 179). Healey regards these chances as objective, in the sense
that it is an objective fact that these chances are optimal credences for an
agent that has epistemic access to all and only the facts in the past light
cone of $x$. But they are not to be thought of as localized physical
magnitudes.

> As we saw, the chance of outcome $e_{A}$ does not attach to it in virtue
> solely of its physical description: the _chances_ of $e_{A}$ attach also in
> virtue of its space-time relations to different space-time locations. Each
> such location offers the epistemic perspective of a situated agent, even in
> a world with no such agents. The existence of these chances is independent
> of the existence of cognizers. But it is only because we are not merely
> cognizers but physically situated agents that we have needed to develop a
> concept of chance tailored to our needs as informationally deprived agents.
> Quantum chance admirably meets those needs: an omniscient God could describe
> and understand the physical world without it (Healey 2016, p. 183; see also
> Healey 2017c, pp. 175–176).

In Healey (2020), Healey argues that this view should be regarded as a form of
scientific realism.

In their statement of QBism, Fuchs, Mermin, and Schack (2014) take a
considerably more radical stance. First, they not only deny that quantum
probabilities represent features of physical reality; for them, these
probabilities are not objective in any sense—they are “personal judgments,”
not backed up by objective facts. Second, the scope of application of quantum
mechanics is severely limited. In ordinary practice, physicists routinely use
quantum mechanics to calculate probabilities about what will be observed by
other people if they perform certain experiments, whether or not the agent
carrying out the calculation will ever be aware of the results (which might,
for example, not be realized within the agent’s lifetime). In contrast, Fuchs
_et al._ restrict an agent’s assignment of probabilities to propositions
concerning the agent’s future experience.

The class of propositions to which the agent assigns probabilities all concern
events along her own worldline, and therefore, contains no spacelike separated
events.

> when any agent uses quantum mechanics to calculate“[cor]relations between
> the manifold aspects of [her] experience,” those experiences cannot be
> space-like separated. Quantum correlations, by their very nature, refer only
> to time-like separated events: the acquisition of experiences by any single
> agent. Quantum mechanics, in the QBist interpretation, cannot assign
> correlations, spooky or otherwise, to space-like separated events, since
> they cannot be experienced by any single agent. Quantum mechanics is thus
> explicitly local in the QBist interpretation.
>
> And that’s all there is to it (Fuchs, Mermin, and Schack, 2014, pp.
> 750–751).

There is no action at a distance, on this view, because, there are no
distances between events that the agent assigns probabilities to. The
spacetime of events to which the theory is applied is, in effect, a one-
dimensional timelike worldline (or else a narrow world-tube). There is no
superluminal signalling, because there is no signalling, of any kind (unless
writing a note to your future self is said to be sending a signal to your
future self).

Approaches that deny the reality of quantum states bypass some of the problems
faced by realist approaches by limiting the scope of what is taken to be
physical. In my opinion, however, such approaches must be regarded as
promissory notes. If one denies that quantum states represent anything in
physical reality, this raises the question of what _does_ constitute physical
reality. It does not seem, for example, that QBists wish to deny that distant
places and other agents exists; they merely claim that such things are outside
the scope of legitimate application of quantum mechanics. This raises the
question of what physics _does_ apply to them. If one says nothing at all,
then ipso facto one says nothing that conflicts with relativity. The question
still remains whether whatever it is that _does_ exist can coexist peacefully
with relativity.

##  9 Conclusion

There is a temptation to say that, because of quantum nonlocality, as revealed
by Bell’s theorem, that quantum theory is flatly incompatible with relativity.
As we have seen, things are not so simple. It is only additional-beables
theories, such as the de Broglie-Bohm theory and related theories, that must
violate relativistic causal structure (at least, as long as the additional
beables are local beables). Collapse theories can respect relativistic causal
structure; unlike the additional-beables theories, they need no distinguished
relation of distant simultaneity. Full relativistic invariance, however, is
possible only at the cost of introducing nonstandard degrees of freedom. For
Everettian theories, relativity poses no special problem. For non-realist
interpretations, it is difficult to render a verdict, as proponents of such
views are usually less clear about what they do think _is_ in the world than
about what they think is _not_.

##  10 Acknowledgment

Support for this research was provided by Graham and Gale Wright, who sponsor
the Graham and Gale Wright Distinguished Scholar Award at the University of
Western Ontario.

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