# Wave Function Collapse, Lorentz Invariance, and the Third Postulate of
Relativity

Edward J. Gillis<EMAIL_ADDRESS>
###### Abstract

The changes that quantum states undergo during measurement are both
probabilistic and nonlocal. These two characteristics complement one another
to insure compatibility with relativity and maintain conservation laws.
Nonlocal entanglement relations provide a means to enforce conservation laws
in a probabilistic theory, while the probabilistic nature of nonlocal effects
prevents the superluminal transmission of information. In order to explain
these measurement-induced changes in terms of fundamental physical processes
it is necessary to take these two key characteristics into account. One way to
do this is to modify the Schrödinger equation by adding stochastic, nonlinear
terms. A number of such proposals have been made over the past few decades. A
recently proposed equation based on the assumption that wave function collapse
is induced by a sequence of correlating interactions of the kind that
constitute measurements has been shown to maintain strict adherence to
conservation laws in individual instances, and has also eliminated the need to
introduce any new, ad hoc physical constants.

In this work it is shown that this modified Schrödinger equation is naturally
Lorentz invariant. It is further argued that the additional spacetime
structures that it requires provide a way to implement the assumption that
spacelike-separated operators (and measurements) commute, and that this
assumption of local commutativity should be regarded as a third postulate of
relativity.

##  I Introduction

If quantum theory is regarded as an objective description of the physical
world then it should be possible, at least in principle, to explain how
individual measurement outcomes are generated from fundamental processes.
Because quantum states change in a probabilistic and nonlocal manner during
measurements it is reasonable to suppose that these features will play key
roles in constructing such a fundamental explanation. One major approach to
this issue takes these features into account by adding stochastic, nonlinear
terms to the Schrödinger equation. These additional terms are designed to
induce the wave function to collapse to one of its several branches. The
nonlinearity is necessary in order to generate collapse, and the stochasticity
is required in order to prevent superluminal signaling, as shown in a work by
Gisin[1].

Gisin’s work was one of a number of proposed stochastic modifications of the
Schrödinger equation aimed at resolving the measurement problem[2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15]. Most of these proposals are designed to
collapse the state vector to either an approximate position state or to an
energy eigenstate. These attempted solutions have often been met with
skepticism because they introduce new, ad hoc physical constants and imply
small violations of conservation laws. A recent work has shown how to
eliminate these problematic features[16]. It is based on the idea that wave
function collapse is induced by the elementary interactions that establish
correlations between physical systems, and was motivated by the fact that
these correlating interactions play a central role both in the measurement of
physical quantities and in the instantiation and transmission of physical
information. In this work it will also be shown that this proposed equation
maintains Lorentz invariance in a natural way.

Since the proposed equation is Lorentz invariant it is reasonable to ask
whether it can be considered “genuinely relativistic”. In order to answer this
question it is necessary to deal with the nonlocal nature of wave function
collapse.

Concerns about nonlocality were first raised almost immediately after the
Schrödinger equation was proposed (by Einstein at the 1927 Solvay conference.
See also [17].) However, they were really sharpened by Bell when he showed
that the correlations between entangled systems that are separated by
spacelike intervals could not be explained by any account in which all
physical processes are restricted to propagate only within the light cone[18].
Bell’s argument relied on entanglement between two elementary systems.
Realistic measurement situations involve entanglement among many more systems,
some of which are macroscopic. This large scale entanglement magnifies the
effect of the nonlocality. This is due to the fact that measurement collapses
the system to an entire entangled branch of the wave function, and not simply
to an eigenstate of the measured system.

The nonlocal nature of these effects has been, by far, the biggest challenge
to developing a fully satisfactory account of quantum measurement because of
the apparent conflict with relativity. This challenge has often seemed
insurmountable because of the extreme reluctance to consider the possibility
that the nonlocal correlations implied by quantum theory might require that we
modify or supplement the metric structure of relativistic spacetime (which is
based on classical physics).

The reason that the nonlocal effects do not generate any manifest conflicts
with relativity is that they are fundamentally probabilistic. More
specifically they obey the Born probability rule[19]. The rule was discovered
empirically and was simply tacked on to quantum theory in an ad hoc manner.
There was no serious effort to integrate it into the mathematical structure of
the theory; nor was it immediately associated with relativity.

Relativistic quantum field theory deals with the nonlocal correlations in a
somewhat more formal manner. It assumes that spacelike-separated operators
commute (or anticommute), and this assumption implies the Born rule, thus
preserving the Lorentz invariance of the theory. In his text on quantum field
theory Weinberg is quite explicit about the fact that this is an additional
assumption that is necessary to maintain the relativistic character of the
theory when he states that the assumption of the commutativity of spacelike-
separated operators is made in order to preserve the Lorentz invariance of the
scattering matrix[20]. He specifically states that he is not linking local
commutativity with the notion of causality. It appears that his motivation for
emphasizing this point is that ‘causality’ is used both as a synonym for the
no-superluminal-signaling principle, and a shorthand for the idea that no
physical processes can propagate faster than light. He seems to rightly regard
the conflation of these two concepts as a mistake.

The critical point is that the assumption of local commutativity functions
effectively as a third postulate for relativity. Nevertheless, very little
consideration has been given to the possible implications this postulate has
for spacetime structure. The reasons for the extreme reluctance are fairly
obvious. The conventional picture of relativistic spacetime is very elegant,
and it beautifully captures our intuitive notion that causal processes
propagate through space in a continuous manner. Historically, both special and
general relativity preceded the full development of quantum theory, and ideas
about spacetime had become pretty firmly fixed by the time the Heisenberg and
Schrödinger equations were published[21, 22].

The problem is that this reluctance has left the logical structure of
contemporary physics in a very muddled state. The rules governing individual
measurement outcomes imply a type of change at odds with the unitary evolution
described by the Schrödinger equation, and there is no clear definition of the
range of applicability of the two distinct types of change.

The stochastic nonlinear modifications of the Schrödinger equation mentioned
above do explain how individual measurement outcomes are selected in accord
with the Born rule, and thereby provide a more complete and unified
mathematical framework for contemporary physics. Since these modifications
require adding structure to relativistic spacetime, let us consider the
reasons for such a move.

In the conventional view relativistic spacetime is a four-dimensional manifold
with a Lorentzian metric that defines a light cone structure. This framework
implements Einstein’s two postulates for relativity by prohibiting the
assignment of an absolute temporal order to events that are separated by a
spacelike interval. Why, then, did the advent of quantum theory necessitate
the introduction of a third postulate to maintain this prohibition? Prior to
the development of quantum theory the relativistic prohibition on temporal
order was strongly associated with the presumption that no physical processes
could propagate outside the light cone. But the nonlocal correlations implied
by quantum theory strongly suggest that there are physical effects that
propagate across spacelike intervals. To “explain” why these effects do not
result in any manifest inconsistencies with relativity the new theory simply
ruled by fiat that they had to respect the prohibition on temporal ordering.
As noted, this was done by requiring that spacelike-separated operators
commute. For the reasons mentioned a few paragraphs back there was no
inclination to consider the possibility that spacetime possessed additional
structure. In fact, the assumption of local commutativity was often conflated
with the limitation on the speed of light. The following quote from Gell-Mann,
Goldberger, and Thirring illustrates this point[23]:

> “The quantum mechanical formulation of the demand that waves do not
> propagate faster than the speed of light is, as is well known, the condition
> that the measurement of two observable physical quantities should not
> interfere if the points of measurement are space-like to each other…the
> commutators of two Heisenberg operators… shall vanish if the operators are
> taken at space-like points.” (italics added)

(This quotation was cited by Bell in his discussion of local
commutativity[24]).

As suggested by Weinberg’s very careful characterization of local
commutativity described above, the kind of conflation demonstrated in the
quotation is simply misguided. The assumption of the commutativity of
spacelike-separated operators entails substantially more than a limitation on
the speed of light. It implies that any physical effects across spacelike
intervals must not transmit physical information.111The use of the notion of
“physical information” in formulating the no-superluminal signaling principle
is based on the characterization of information as a reproducible and
referential property of physical systems given in [25]. This is the reason
that the assumption is made. It involves an implicit recognition that there
are nonlocal effects, and that there is the need to regulate them. But, this
kind of regulation ought to be explained, and not simply imposed by fiat.

As mentioned above Einstein’s postulate about the invariance of the speed of
light is implemented by attributing a light cone structure to spacetime. In
other words, this postulate is explained as a consequence of the fundamental
nature of spacetime. Should we not then also consider adding structure to
spacetime to explain how nonlocal effects propagate and how they are
regulated? This is exactly what stochastic collapse equations do. By assuming
a foliation of spacetime into spacelike surfaces, and invoking a stochastic
process (or processes) they provide the desired explanation. By making wave
function collapse and the Born rule follow from the fundamental equation of
the theory, they provide a coherent logical structure for contemporary
physics.

To assess the extent to which these proposals can be judged to be genuinely
relativistic we can start by stipulating that the primary criterion for a
relativistic theory is that the laws of physics have the same form in all
inertial frames. In particular, this means that they must be Lorentz
invariant. Assertions that a proposed account can be judged as “genuinely
relativistic” only if it does not add structure beyond the standard
relativistic metric cannot be accepted as a priori truths. They face a burden
of proof that must take into account the nonlocal aspects of quantum theory.
In order to reconcile relativity and quantum theory one must consider the
possibility that both these pillars of contemporary physics will have to be
modified. The evaluation of any particular proposal will then turn on whether
it is Lorentz invariant and whether the Lorentz invariance follows from
fundamental physical principles in a natural manner. It will be argued here
that the proposal in [16] meets these criteria.

The next section describes the basic assumptions employed by nonlinear, norm-
preserving stochastic collapse equations and illustrates how they work.
Section III reviews the proposal of [16], which eliminates the need to
introduce new ad hoc physical constants and insures that conservation laws are
respected in individual experiments. It then goes on to demonstrate that it
meets a critical test of Lorentz invariance in a very natural way. Section IV
addresses the question of what constitutes “serious” or natural Lorentz
invariance.

##  II Stochastic Collapse Equations

This section first provides an explanation of why stochastic collapse
equations need to add structures to spacetime, specifically a preferred
foliation associated with a stochastic process (or processes). This is
followed by an illustration of how these dynamic equations generate collapse
in accord with the Born rule. The literature cited earlier contains more
general and formal demonstrations of how these equations work; the purpose
here is just to provide an intuitive understanding based on a simplified case
that captures the essential features. The idea is to show that it is not
necessary to paste the measurement postulates onto quantum theory in an ad hoc
manner, but rather that they follow in a very natural way from a relatively
simple modification of the Schrödinger equation that takes into account the
fundamentally probabilistic nature of quantum theory as we currently
understand it.

###  II.1 The Need for Additional Spacetime Structure

Interactions play a critical role in stochastic collapse equations. In the
proposal of [16] that will be described in detail in Section III they play the
central role because it is assumed that it is interactions that actually
induce the collapse. In other proposals they are crucial to establishing the
large scale entanglement that allows nonlinear collapse to occur on
macroscopic scales while leaving the (almost) linear quantum behavior of
microscopic systems essentially undisturbed. Entanglement is a generic result
of interaction, as shown by Gemmer and Mahler and by Durt[26, 27]. The fact
that the changes that occur in measurements involve entire entangled branches
of the wave function implies that any proposal that treats wave function
collapse as a real physical process and seeks to explain it by relating it to
other fundamental processes must add structure to relativistic spacetime.

To see this we need to contrast the idea of wave function collapse as a
process with the conventional view that it is an instantaneous event that just
happens at some point during (or immediately after) a measurement. If there
are two or more spacelike-separated measurements involving some of the same
systems the final outcome is independent of the order in which the
instantaneous collapse is assumed to occur. This has to be the case in order
to insure compatibility with relativity. On the other hand, stochastic
collapse equations assume that the collapse process is mediated by
entanglement relations, and, therefore, it is essential that the entangled
branches be well defined at each stage of the process. Thus, it is necessary
that there be some means to determine whether other spacelike-separated
systems that have interacted with other branches of the measured wave function
are involved in the collapse. A foliation of spacetime into spacelike surfaces
provides the sequencing that determines the structure of a wave function when
collapse occurs.

The need for a method to sequence spacelike-separated interactions also arises
in derivations of the Born probability rule in collapse processes. Again, we
can contrast the conventional view of collapse as an instantaneous occurrence
with attempts to describe it as a process. In the conventional view one simply
assumes the Born rule (or some principle such as local commutativity that
implies it). As will be described below, when collapse is assumed to result
from small nonlocal changes that occur in the various branches of the wave
function it is essential to the derivation of the rule that the changes are
proportional to the relative amplitudes of the branches. Each such change
results in a redistribution of amplitudes among the branches. Therefore, the
sequence of changes, which can occur at spacelike separation, makes a
difference in the result.

For these reasons proposals to describe wave function collapse as a process
typically assume that additional spacetime structure such as a foliation is
required.

In addition to a foliation (or some similar structure) the other critical
feature that must be introduced is a stochastic process (or processes). As
mentioned earlier any nonlinear modification of the Schrödinger equation must
be stochastic in order to prevent superluminal signaling. The stochastic
process to be described here is based on the Wiener integral of a white noise
Gaussian process. This can be thought of as the continuous time limit of an
unbiased random walk with zero mean. As such, it scales with time as
$\sqrt{t}$. It is designated as $\xi(t)$, and its differential, which plays a
key role in the equations, is designated as $d\xi(t)$. In general, $\xi(t)$
can be complex. The process is governed by the rules of the It$\hat{o}$
stochastic calculus[28]: $d\xi^{*}d\xi=dt,\;\;\;dtd\xi=0.$

More general stochastic processes can be considered, and a number of proposals
employ a stochastic field which is a function of both space and time,
$\xi(x,t)$, rather than just time. There is also another form of the
stochastic calculus due to Stratonovich. See the references for details. A
good general reference is the text by Gardiner.[29]

###  II.2 Nonlinear, Norm-Preserving Stochastic Collapse Equations

The overall aim of this work is to show how relativity and the nonlocal
aspects of quantum theory can be encompassed in a unified mathematical
structure describing the dynamics of physical systems without the need to
introduce ad hoc rules that limit the applicability of the mathematics.
However, the general form of the stochastic collapse equations outlined here
(which includes the proposal described in Section III) is formulated in a
nonrelativistic framework. The main reason for this is to avoid mathematical
complexity. The assumption of a preferred reference frame that will be used
here should be seen as just the simplest special case of a randomly evolving
spacelike surface. The restriction to nonrelativistic quantum mechanics makes
it possible to illustrate the essential ideas of the proposal in [16] while
avoiding many of the complications involved in quantum field theory. It also
makes it possible to build on the substantial body of work developed in [1, 2,
3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15].

Measurements of quantum systems can have a very large number of possible
outcomes, but at the most basic level they come down to either a detection or
a failure to detect. The essentially binary character of measurement processes
means that at each stage the Hilbert space (of any number of dimensions) can
be decomposed into two orthogonal subspaces, and the state vector of the total
system can be represented as the sum of two components, one in each subspace.
Although the particular decomposition can vary during the process (thus,
allowing any number of possible outcomes) the fundamentally binary nature
makes it possible to model measurement processes as a random walk between two
alternatives.222The claim that measurement processes are essentially binary
might seem implausible when one considers many common-place measurements that
we make such as length, height, weight, etc. However, that is because we
typically describe these kinds of measurements by reference to extremely
complex human actions. Imagine eliminating any reference to humans and try to
design a robot to carry out the measurement. Eventually one must get back to
describing the fundamental physical processes involved. This is essentially
what stochastic collapse equations do.333Wave function collapse can occur
either inside or outside a laboratory, but for ease of explanation the
discussion here will focus primarily on typical measurement situations.

To see how these equations work we can begin with a simple theorem about the
relative probabilities of a random walk ending at either of the two end
points. If the walk continues for enough steps it will eventually finish at
one or the other of the end points. Label the end points as $0$ and $1$, and
suppose that the walk begins at a point, $p$, between them. It will be shown
that the probability of ending at $1$ is $p$ and that the probability of
ending at $0$ is $1-p$. The step size is labeled $\delta$. It is allowed to
vary anywhere between $0$ and the distance to the nearest end point:
$0\,\leq\,\delta\,\leq\,p$, $0\,\leq\,\delta\,\leq\,1-p$. Label the
probability of reaching $1$ as $Pr(p)$. Because the walk is assumed to be
unbiased we get: $Pr(p)\,=\,1/2Pr(p-\delta)+1/2Pr(p+\delta)$. Because this
relationship holds for all values of $p$ and $\delta$ it is linear. Therefore,
the probability of reaching $1$ is $p$, and the probability of reaching $0$ is
$1-p$. What follows is an attempt to present an intuitive explanation of how
nonlinear, norm-preserving stochastic collapse equations map onto this simple,
binary picture.444A good general discussion of these equations can be found in
the paper by Adler and Brun[12].

The binary character of measurement processes noted above allows us to
illustrate the essential operation of stochastic collapse equations using a
simple two state system. Consider a system with a wave function $\psi$ and
Hamiltonian, $\mathbf{H}$. The system evolves under the action of the
Hamiltonian and of a stochastic collapse operator that is constructed from a
self-adjoint operator, $\mathbf{O}$, with two eigenstates, $|x\rangle$ and
$|y\rangle$, associated with eigenvalues $\mathbf{a}$ and $\mathbf{b}$. The
eigenstates of the operator, $\mathbf{O}$, define the collapse basis. The
nonlinear stochastic operator is defined as:
$\,\mathcal{O}\;\equiv\;k[\mathbf{O}-\langle\,\psi|\mathbf{O}|\psi\,\rangle]\,$,
where $\langle\,\psi|\mathbf{O}|\psi\,\rangle$ is the expectation value of the
operator, $\mathbf{{O}}$, in the state, $\psi$, and $k$ is a constant that
helps to determine the scale of the collapse effects and also insures that
$\mathcal{O}$ is dimensionless. For example, if $\mathcal{O}$ is based on the
position operator $k$ could determine the range of the collapse effect.

The modified stochastic Schrödinger equation is defined as:

$d\psi\,=\,\frac{-i}{\hbar}\mathbf{{H}}\,\psi\,dt\,+\,\mathcal{O}\,\,\psi\,\sqrt{\gamma}\,d\xi(t)\,-\,\frac{1}{2}\mathcal{O}^{2}\,\psi\,\gamma\,dt.$
(1)

The first term on the right represents the standard Schrödinger evolution. The
primary stochastic action is described by the middle term. The parameter,
$\gamma$, determines the rate at which the stochastic operator acts. The
square root operator is applied to the rate parameter, $\gamma$, because it
works in conjunction with the stochastic differential, $d\xi$, which scales as
$\sqrt{dt}$ (as described above). In the definition of the stochastic
operator, $\mathcal{O}$, the subtraction of the expectation value,
$\langle\,\psi|\mathbf{O}|\psi\,\rangle$, acts, as in the Gram-Schmidt
procedure, to insure that the stochastic modification of $\psi$ is orthogonal
to the existing wave function. The small orthogonal addition to the wave
function slightly alters the norm. This alteration is compensated for by the
third term on the right which involves $\mathcal{O}^{2}$.

The action of the stochastic term is quite small in comparison to that of the
Hamiltonian. So, in order for it to be effective in generating collapse it
needs to act in a manner that is essentially independent of the Hamiltonian.
This can be achieved in several ways, through appropriate choices for the rate
parameter, the operator, $\mathbf{O}$, and possibly other parameters.

The stochastic term is designed to drive the system to one of the eigenstates
of the operator, $\mathbf{O}$. In this simple example the wave function is
represented as:
$\psi\,=\,\alpha|x\rangle+\beta|y\rangle$, with
$\alpha^{2}\,+\,\beta^{2}\,=\,1$. To simplify the example these amplitudes can
be taken as real and positive with no loss of generality. The action of the
self-adjoint operator, $\mathbf{O}$, on the wave function is:
$\mathbf{O}\psi\,=\,\mathbf{a}\alpha|x\rangle+\mathbf{b}\beta|y\rangle$, and
its expectation value is
$\langle\,\psi|\mathbf{O}|\psi\,\rangle\,=\,\mathbf{a}\alpha^{2}\,+\,\mathbf{b}\beta^{2}$.
The action of the stochastic operator on the wave function can be expanded as:

$\begin{array}[]{ll}\mathcal{O}\,\,\psi\,=\,k\\{\mathbf{a}\alpha|x\rangle+\mathbf{b}\beta|y\rangle-(\mathbf{a}\alpha^{2}\,+\,\mathbf{b}\beta^{2})\big{[}\,\alpha|x\rangle+\beta|y\rangle\,\big{]}\\}&\\\
\,=\,k\\{\
\alpha\,[\mathbf{a}(1-\alpha^{2})\,-\,\mathbf{b}\beta^{2}]|x\rangle\,+\,\beta\,[\mathbf{b}(1-\beta^{2})\,-\,\mathbf{a}\alpha^{2}]|y\rangle\\}&\\\
\,=\,k\\{\alpha\,\beta\,(\mathbf{a}-\mathbf{b})\big{[}\beta|x\rangle\,-\,\alpha|y\rangle\big{]}\\}.\end{array}$
(2)

So the middle term of 1 can be written as:

$k\\{\alpha\,\beta\;\big{[}\beta|x\rangle\,-\,\alpha|y\rangle\big{]}\;(\mathbf{a}-\mathbf{b})\sqrt{\gamma}\,d\xi(t)\\}.$
(3)

The expression in square brackets can be recognized as a normalized vector
that is orthogonal to $\psi$. As long as the rate parameter, $\gamma$, is
independent of $\alpha$ and $\beta$ the only dependence on the amplitudes
(aside from the orthonormal state vector) is the term, $\alpha\beta$.

In this form and with the simplifying assumptions described above it is
possible to trace the evolution of the wave function through Hilbert space
under the influence of the stochastic operator. Since $\alpha$ and $\beta$ are
assumed to be real and positive the evolution can be modeled as a random walk
along the arc joining $x$ and $y$ axes (corresponding to the eigenstates
$|x\rangle$ and $|y\rangle$).555In this simplified picture with $\alpha$ and
$\beta$ real and positive the process can be pictured on a standard Cartesian
graph with x and y axes. It is very straightforward to transfer the analysis
to the Bloch sphere afterward. The state,
$\psi\,=\,\alpha|x\rangle\,+\,\beta|y\rangle$, lies on the arc; the orthogonal
state, $\beta|x\rangle\,-\,\alpha|y\rangle$, is tangent to the arc and it
drives the state to one or the other of the eigenstates in infinitesimal
steps. The magnitude and direction of the steps are determined by the
coefficient of the tangent state vector,
$\alpha\,\beta\;k(\mathbf{a}-\mathbf{b})\sqrt{\gamma}\,d\xi(t).$

The dependence on $\alpha$ and $\beta$ means that the step size varies as the
wave function, $\psi$, traverses the arc under the influence of the stochastic
operator. Therefore, the calculation of the distance to the end points,
$|x\rangle$ and $|y\rangle$, must take this variation into account. This can
be done by introducing a parameter, $\theta$, with $\alpha\,=\,\cos\,\theta$
and $\beta\,=\,\sin\,\theta$, and integrating the coefficient of the tangent
state vector along the arc:

$k(\mathbf{a}-\mathbf{b})\sqrt{\gamma}\,d\xi(t)\;\int\,\cos\,\theta\,\sin\,\theta\,d\theta\;=\;A\sin^{2}\,=A\beta^{2}.$
(4)

where $A$ is some term independent of $\alpha,\beta$ and $\theta$. If we
associate the two eigenstates with the end points of a random walk as
described earlier, with $|x\rangle$ corresponding to $0$ and $|y\rangle$
corresponding to $1$, then the position of $\psi$ along the arc can be
parameterized as $\beta^{2}$. As shown earlier, this is the probability that
the random walk ends at $|y\rangle$, and $\alpha^{2}\,=\,1-\beta^{2}$ is the
probability that it ends at $|x\rangle$. So the Born rule follows from the
basic structure of the collapse equation in a straightforward manner. It is
also worth noting that the random walk terminates because the term,
$\alpha\beta\,=\cos\,\theta\,\sin\,\theta$ goes to $0$ as $\psi$ approaches
one of the end points.666The fact that $\alpha\,\beta$ approaches $0$ at the
end points also creates a problem (the “tails problem”) in that the walk does
not end in a finite number of steps. This problem will not be dealt with in
detail here, but I will offer a speculative solution later.

As stated at the beginning of this section what has been shown here is that,
if we are willing to countenance some additional fundamental structure for
relativistic spacetime and incorporate the probabilistic character of quantum
theory at the fundamental level it is possible to modify the Schrödinger
equation in a fairly simple way so that it yields the measurement postulates
as dynamic consequences. In this way contemporary physical theory is rendered
much more coherent.

With this background we can now review the proposed equation described in
[16], and see how it maintains Lorentz invariance.

##  III Interaction-Induced Wave Function Collapse

Measurements consist of interactions that establish correlations between
physical systems. Correlations are established through the exchange of
conserved quantities. Given the probabilistic nature of quantum theory, the
generation of stable information and its transmission depend on these
correlating interactions. These considerations are what motivated and guided
the construction of the equation described below. Roughly speaking, the idea
is that the magnitude of the collapse effect associated with an interaction is
proportional to the amount of correlation that is generated.

The degree of correlation between two systems that is generated during an
interaction depends on the extent to which the interaction changes the
individual state of each system. This, in turn, depends on the strength of the
interaction and the resistance of each of the systems to a change of state.
The strength is measured by the interaction potential energy. This is assumed
to depend on the separation between the two systems and to decrease as the
separation increases. For systems $j$ and $k$ it will be indicated as
$\mathbf{{V}_{jk}}$. The resistance to change depends on the mass of the
systems, $m_{j}$ and $m_{k}$. The effective mass of elementary systems such as
electrons in bound states is altered by the binding interactions. So atoms,
molecules, and larger complex structures are treated as single systems with a
total mass and a net charge (or electric multipole moment).

So the collapse operator is based on the interaction potential energies,
$\mathbf{{V}_{jk}}$, and is proportional to the ratio,
$\mathbf{{V}_{jk}}/(m_{j}+m_{k})$. As mentioned in Section II the stochastic
operator must be dimensionless. To convert the denominator to an energy it is
multiplied by the square of the speed of light, $c^{2}$. This is the only
nonarbitrary speed, it eases the way to a relativistic generalization, and it
eliminates the need to introduce an arbitrary constant, $k$. It is also
necessary, as in Section II, to subtract the expectation value,
$\langle\,\psi|\mathbf{{V}_{jk}}|\psi\,\rangle$. So the collapse operator
associated with the interaction between systems $j$ and $k$ is:

${\mathcal{V}}_{jk}\;\equiv\;\frac{\mathbf{{V}_{jk}}-\langle\,\psi|\mathbf{{V}_{jk}}|\psi\,\rangle}{(m_{j}+m_{k})c^{2}}.$
(5)

In most proposed collapse equations the rate or frequency parameter, $\gamma$,
is chosen in an ad hoc fashion to insure that collapse occurs on an
appropriate scale. However, since it is assumed here that collapse effects are
induced by the physical processes that establish correlations between systems
it is possible to define the frequency parameter in terms of the rate at which
the correlations are generated. Since correlations are established through the
exchange of conserved quantities, and since these exchanges are associated
with variations in the interaction potentials, $\mathbf{{V}_{jk}}$ we can
define a rate parameter associated with each interaction in terms of the rate
at which these potentials vary:

$\gamma_{jk}\;\equiv\;\frac{\parallel\frac{d}{dt}\langle\,\psi|\mathbf{{V}_{jk}}|\psi\,\rangle\parallel}{\parallel\langle\,\psi|\mathbf{{V}_{jk}}|\psi\,\rangle\parallel_{max}}.$
(6)

The numerator in this expression is obviously well defined at every stage of
the interaction. The fact that the denominator is also well defined is based
on a couple of observations. First, since the Born probability rule follows
from the basic structure of collapse equations the expectation value for the
full collapse equation is the same as for the Schrödinger equation associated
with it. Second, the maximum of the expectation value over the course of the
interaction can be projected by integrating the Schrödinger equation. This can
be done because the interaction potentials are functions of position, and
hence, time-independent. It is true that both numerator and denominator can
vary over the course of the interaction, but the crucial point is that they
are well defined at every stage. The fact that both numerator and denominator
pick out only the interacting components of the the wave function insures that
the rate parameter is independent of the amplitude of these components, and
that is crucial for guaranteeing compliance with the Born probability rule.
The norm of both numerator and denominator is taken in order to insure the
positivity of the expression for $\gamma_{jk}$.777This formulation of the rate
parameter is simpler and more general than the expression that was used in
[16]. Note also that the expression for the denominator can be integrated in
both directions in time.

A critical implication of this formulation for $\gamma_{jk}$ is that when
multiplied by $dt$ it integrates to a value of order $1$ over the course of
the interaction. This is also true for the expression
$\sqrt{\gamma_{jk}}d\xi$. This fact allows one to treat each interaction as a
discrete event, and makes it possible to estimate the scale on which collapse
occurs and the duration of the collapse process. This kind of analysis shows
how the scale and duration depend on the average strength of the interactions
involved. For this purpose it is useful to have an estimate of the average
interaction rate and duration. This can be done by examining the interaction
term in the Schrödinger equation,

$\frac{d\psi}{dt}\,=\,-\,\frac{i\mathbf{{V}_{jk}}}{\hbar}\,\psi.$ (7)

This shows that the average rate is proportional to the interaction strength

$\gamma_{jk}\,\approx\,\frac{\Delta\mathbf{{V}_{jk-max}}}{\hbar}.$ (8)

where $\Delta\mathbf{{V}_{jk-max}}$ is the maximum difference in potential
energy over the course of the interaction. The characteristic interaction time
is the inverse of this quantity:

$dt_{int}\,\equiv\,\frac{\hbar}{\Delta\mathbf{{V}_{jk-max}}},$ (9)

For two electrons separated by the Bohr radius the duration would be about
$2.5*10^{-17}$ seconds. Rates and durations for other interactions can be
scaled from this estimate, taking into account the mass and charge of the
systems involved.

Finally, the full collapse operator is obtained by summing over the operators
for every pair, $j,k$:

${\mathcal{V}}\;\equiv\;\sum_{j<k}{\mathcal{V}}_{jk}.$ (10)

The proposed collapse equation takes the form:

$\begin{array}[]{ll}d\psi\,=\,(-i/\hbar)\mathbf{{H}}\,\psi\,dt\,+\,\sum_{j<k}{\mathcal{V}}_{jk}\,\,\psi\,\sqrt{\gamma_{jk}}d\xi(t)&\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\,\frac{1}{2}(\sum_{j<k}{\mathcal{V}}_{jk})^{2}\,\psi\,\gamma_{jk}dt.\end{array}$
(11)

A detailed proof that 11 results in collapse with the correct probabilities is
given in [16], along with estimates of the scale and duration of collapse
processes. As indicated these depend on the average strength of the
interactions involved. Such processes can involve anywhere from about $10^{8}$
to $10^{16}$ elementary interactions. Since many of the interactions can be
occurring in parallel the durations are typically very small fractions of a
second as can be seen from the estimates of the durations associated with
individual interactions given above.

The claim that conservation laws hold exactly in individual instances of
collapse obviously runs counter to the prevailing presumption that
conservation laws hold only on average in quantum theory. However, this
presumption is based on an artificial division of the world into classical and
quantum systems, and also on an overly idealized concept of elementary
physical systems being in strictly factorizable states. In order to properly
assess the status of conservation laws in quantum theory it is necessary to
treat all systems, both macroscopic and microscopic, as quantum systems, and
also to recognize that all these systems have a history of interaction with
other systems (which include preparation apparatuses). As emphasized in [26,
27] interaction generates entanglement. Therefore, the idealization of
elementary systems as being in purely factorizable states is never fully
realized in practice. This is pointed out by the authors of [26] where they
say:

> “Thus it is, strictly speaking, unjustified to describe a particle in a box,
> which is part of an interacting quantum system, by a wave-function”.

In other words, the common textbook example of a particle in a box ignores the
small amount of entanglement that results from the interaction between the
particle, the box and whatever apparatus was used to prepare the system. The
interactions involved in both the preparation and measurement of quantum
systems are conservative interactions. When these interactions induce
branching of the wave function they lead to a different distribution of
conserved quantities among the interacting systems in the various branches,
but they do not alter the total amount (up to normalization of the branches.)
Based on these kinds of observations a number of articles in recent years have
demonstrated that conservation laws hold exactly in individual instances, and
not just when applied to ensembles of identical measurement situations. In
addition to [16] these include [25, 30, 31, 32, 33, 34, 35].

In discussing the status of conservation laws with reference to stochastic
collapse equations there are two issues that must be addressed. First, some of
the collapse proposals imply small violations of conservation laws even with
regard to statistical averages. For example, consider the most widely known
collapse proposal, the continuous spontaneous localization (CSL) model[36,
37]. It implies small violations of energy conservation as described in [38].
One of the main reasons for these violations is that this proposal uses a
stochastic field, $\xi(x,t)$, which is a function of both position and time,
as opposed to a single, global stochastic process, $\xi(t)$, which is a
function only of time. The stochastic field induces spatial variations in the
wave function - in effect, pumping energy into it. Other proposals, such as
one of the variants discussed in [1] use a single, global stochastic process,
and base the stochastic operator on the Hamiltonian. In this way it maintains
energy conservation on average, but still violates it in individual instances.

Because the proposed equation in [16] uses a single global stochastic process
it avoids the kinds of statistical violations of conservation laws just
described. The reason that it is able to maintain strict conservation in
individual measurement situations is that it is based on the assumption that
the amplitude shifts between branches that bring about collapse are generated
by conservative interactions. So these interactions are responsible both for
the splitting of the wave function into branches and for the redistribution of
amplitude among the branches. As stated above subsequent to the splitting
under ordinary Schrödinger evolution conserved quantities are identical in
each branch (up to normalization of the branch). Thus, the amplitude shifts
between branches do not alter any of these quantities, and the eventual
selection (or elimination) of a particular branch leaves the surviving wave
function with the same values that it had prior to the chain of interactions
(including preparation) that led to the collapse.

It is shown in [16] that the proposed equation conserves momentum and angular
momentum exactly. Because the proposal is formulated in a nonrelativistic
framework it is only able to conserve energy within the accuracy allowed by
the limited forms of energy describable in nonrelativistic theory.

Experimental consequences of the proposal are also discussed in the earlier
work. These deal with very small discrepancies in the correlations between
entangled systems predicted by conventional quantum theory and the equation
described above. They are a result of the nonlinearity of the equation.

This review of the proposal in [16] is intended as background for the
demonstration that 11 is Lorentz invariant. The general question of Lorentz
invariance hangs over any proposal that deals with the measurement problem.
But there is a particular concern that arises in connection with the type of
collapse equation discussed here because they are formulated in a preferred
reference frame. There is essentially zero probability that any reference
frame in which one chooses to analyze a collapse process will coincide with
the preferred frame. The critical question is whether the magnitude of the
collapse effects is independent of the frame in which they are considered.

The sort of discrepancies in predictions of correlations mentioned two
paragraphs back occur in any nonlinear collapse equation. (Other experimental
deviations from conventional theory may also occur.) If the predicted
magnitude of the collapse effects differs from one frame to the next there
will be, in principle, observable differences from the preferred frame and the
proposal will be in explicit conflict with relativity. So the task here is to
show that the magnitude of the collapse effects predicted by equation 11 is
independent of the reference frame in which they are viewed.

According to equation 11 during the brief time in which an interaction
generates a correlation between two systems it also transfers amplitude
between the interacting and noninteracting branches of the wave function. The
magnitude of the transfer is:

${\mathcal{V}}_{jk}\sqrt{\gamma_{jk}}d\xi(t)\,\psi\;\equiv\;\frac{\mathbf{{V}_{jk}}-\langle\,\psi|\mathbf{{V}_{jk}}|\psi\,\rangle}{(m_{j}+m_{k})c^{2}}\,\sqrt{\gamma_{jk}}d\xi(t)\,\psi.$
(12)

We want to show that this magnitude is the same whether it is calculated in
the rest frame of the laboratory in which the interaction takes place or in
the preferred frame in which the collapse equation is formulated.

Consider, first, the term, $\sqrt{\gamma_{jk}}d\xi(t)$, that governs the rate
at which the transfer takes place. As argued above, this term integrates to a
value of order, $1$, over the course of the interaction. As viewed from the
preferred frame the duration of the interaction is increased by the
relativistic time dilation factor, and the rate parameter, $\gamma_{jk}$, is
reduced by the same proportion. But, from the way in which the rate parameter
is constructed it is clear that it will still integrate to the same value over
the course of the interaction. So we need only show that the expression,
$\frac{\mathbf{{V}_{jk}}-\langle\,\psi|\mathbf{{V}_{jk}}|\psi\,\rangle}{(m_{j}+m_{k})c^{2}}$
is invariant.

To demonstrate this it is necessary to show that the relativistic
transformation of the numerator is the same as that of the denominator. The
numerator is based on the interaction potential energy. The elementary
interactions that occur in nonrelativistic quantum theory are electromagnetic.
Therefore, the potential energy is an electromagnetic potential. From a
relativistic point of view this potential is the timelike component of an
electromagnetic four-potential, and transforms accordingly. The denominator
consists of the sum of the relativistic energies of the interacting systems.
The relativistic energy is the timelike component of the energy-momentum four
vector. So the denominator transforms in the same way as the numerator, and,
hence, the expression for the collapse term is Lorentz invariant.

In Section II it was shown that stochastic collapse equations provide a very
natural extension of the mathematical formalism of quantum theory and that
they entail the measurement postulates as straightforward consequences of the
fundamental equation. This approach eliminates the need to insert them into
the logical structure of the theory at some very vaguely defined point. In
this section it was shown that it is possible to formulate such an equation
that respects conservation laws in individual measurement processes without
introducing any new, ad hoc physical constants, and is also Lorentz invariant.
The next section will consider whether this general approach can be considered
to respect “serious” Lorentz invariance.

##  IV “Serious” Lorentz Invariance

What is meant by “serious” Lorentz invariance? One common attitude is that no
proposal that requires adding structure to spacetime beyond a Lorentzian
metric can be considered to be genuinely relativistic. It implies that any new
physics must work within the limited framework that was erected in a classical
context prior to the development of quantum theory.

To evaluate this viewpoint it is necessary to take into account the
fundamentally probabilistic and nonlocal nature of quantum theory as we
currently understand it. As argued in Section I these characteristics
necessitated the introduction of an additional postulate for theoretical
physics - local commutativity. Because the new theory was such a large break
from the past, and because many of its consequences only became apparent over
time the central role that this additional postulate plays in contemporary
physics has not been fully appreciated.

The Lorentzian metric and light cone structure that we use to characterize
relativistic spacetime were adopted in order to provide a fundamental physical
explanation for Einstein’s two original postulates for relativity. The
assumption of local commutativity deals with the novel aspects of quantum
theory by forbidding the transmission of stable physical information across
spacelike intervals. It is intended to prevent any manifest violations of
relativity and should be regarded as a third postulate. But, just as the
earlier structures were designed to explain how the original postulates are
enforced we should be willing to entertain the possibility of adding structure
to explain how this third postulate is enforced. That is exactly what a
spacetime foliation associated with a stochastic process does.

The adoption of this sort of additional stochastic structure transfers the
probabilistic nature of quantum theory from the macroscopic to the elementary
level. This turns what were ad hoc postulates at the macro level into
straightforward consequences of the mathematical description of elementary
processes. The probabilistic nature of elementary correlating interactions
also explains why a preferred reference frame (or spacetime foliation) remains
hidden, and explains why the description of spacetime at the macroscopic level
is limited to the standard relativistic account involving just a Lorentzian
metric. This is why accounts involving these sorts of additional stochastic
structures ought to be regarded as respecting serious Lorentz invariance.

Because the proposal described here is formulated in a nonrelativistic
framework it must be considered as incomplete. But since it can, arguably, be
regarded as Lorentz invariant there do not appear to be any serious conceptual
obstacles to a full relativistic account. Such an account would encompass
quantum field theory, and might well offer solutions to some of the key
problems with this more limited proposal. As argued in [16] there is reason to
believe that such an extension could resolve the small discrepancies with
energy conservation that are implied by equation 11. One might also hope that
a relativistic account might provide a solution to the tails problem
(mentioned in a footnote in section II). With the transition from
distinguishable to indistinguishable particles the suppression of the very
small amplitudes in the wave function tail below the level of vacuum
fluctuations might point the way to a resolution. But these are just
speculations that must be addressed in future work.

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