# Wave Function Collapse, Correlating Interactions, and Conservation Laws

Edward J. Gillis111email<EMAIL_ADDRESS>
###### Abstract

The assumption that wave function collapse is induced by correlating
interactions of the kind that constitute measurements leads to a stochastic
collapse equation that does not require the introduction of any new physical
constants and that is consistent with conservation laws. The collapse operator
is based on the interaction (potential) energy, with a variable timing
parameter related to the rate at which individual interactions generate the
correlations. The approximate localization of physical systems follows from
the distance-dependent nature of the interaction potentials. The equation is
consistent with strict conservation of momentum and orbital angular momentum,
and it is also consistent with energy conservation within the accuracy allowed
by the limited forms of energy that can be described within nonrelativistic
theory. The possibility of extending the proposal to a fully relativistic
version is discussed.

##  1 Introduction

Stochastic, nonlinear modifications of the Schrödinger equation have been
proposed in order to explain measurement outcomes in terms of fundamental
physical processes, and to describe how the apparently linear evolution of
systems consisting of a few elementary particles is altered when large numbers
of interacting particles become involved. In these accounts wave function
collapse and the Born probability rule are consequences of the basic
mathematical structure of the theory, rather than ad hoc postulates, seemingly
at odds with that structure. This approach has been developed over a number of
years in works by various authors[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13].

The need to use an equation that is nonlinear in order to describe wave
function collapse follows from the fact that projection of a state vector to
one of its components is a nonlinear operation.222A recent work by Mertens, et
al.[14] has also shown that nonlinearity is required for the derivation of the
Born rule. The need for a nonlinear equation to be stochastic was demonstrated
by Gisin who showed that any deterministic nonlinear modification would lead
to superluminal signaling[8]. Most of the proposals seek to collapse the
measured system either to an approximate position state or to an energy
eigenstate. Typically, they require the introduction of new physical
constants. For example, position-based approaches usually include distance and
timing parameters to govern the range and rate of the collapse effects.

The approach described here originated as an attempt to reconcile relativity
with the nonlocal aspects of quantum theory by taking the prohibition of
superluminal information transmission as the fundamental principle that these
theories have in common. Consideration of the way in which information is
physically instantiated and transmitted led to the hypothesis that the
interactions that establish correlations between systems, such as those
involved in measurements, are responsible for inducing wave function
collapse[15]. This hypothesis was formalized in a stochastic collapse equation
in [16] that used two-particle interaction potentials as the basis for the
stochastic operator. In that equation the strength of the collapse effects was
determined by the ratio of the interaction energy to the total relativistic
energy of the two particles. The timing parameter was assumed to be a new
constant that could be chosen to insure that collapse occurred on a time scale
consistent with our macroscopic experience.

The need to introduce such a new constant can be eliminated by tying the
collapse rate to the speed with which individual interactions separate the
wave function into orthogonal branches in the process of generating
correlations. It can be precisely quantified in terms of the rate at which the
potential energy changes during the course of the interaction relative to the
total energy of the interacting systems. It is defined in such a way that it
integrates to a value of order, $1$, over the course of the interaction, and
goes to zero when the branching is complete or when the two particles settle
into a stationary state.333This condition holds for those interactions that
contribute significantly to the collapse process. For other interactions the
integrated value can be much less. This insures that the collapse equation
reduces to the ordinary nonrelativistic Schrödinger equation when applied to
stationary states or to noninteracting systems.

Besides eliminating the need to introduce new physical constants there are
some other key differences between the equation presented here and most
previous proposals. Most approaches have assumed that the correlations between
the measured system and the measuring apparatus are established before the
collapse, and that the collapse occurs on a time scale that is much shorter
than that on which ordinary Hamiltonian evolution is effective. In contrast,
the process proposed here is assumed to work in an incremental fashion with
small transfers of amplitude between distinct branches of the wave function
associated with each elementary interaction. Because these incremental
stochastic effects are several orders of magnitude smaller than the changes
induced by standard Schrödinger evolution it works in parallel with the
Hamiltonian as the correlations are being established. In addition, because
the collapse operator is based on the inter-particle potential energy its
outcome cannot be readily characterized in terms of an eigenfunction of any
single type of operator (such as position or energy). This is due to the fact
that the potential energy varies continuously over the course of an entangling
interaction, and the integrated value typically varies from one interaction to
the next. The collapse basis in each instance is defined by the correlating
interactions.444These branches tend to coincide with the collapse basis
described in the decoherence literature[17, 18, 19, 20], but the collapse is
brought about, primarily, by the interactions between the elementary and other
microscopic systems that constitute the measurement apparatus, and, in
general, is not dependent on the coupling of the apparatus to the environment.

These distinguishing factors require a different type of proof that the
proposed equation is effective in bringing about collapse. As described above,
the process works by shifting amplitude between entangled branches depending
on the level of interaction potential energy in each branch of the wave
function at each stage of the process. So the proof presented here consists,
essentially, of an explicit description of the collapse process as a random
walk between these developing branches.

The next section provides a heuristic description of how the collapse process
works as illustrated by a simple example. It then lays out the key assumptions
that are made and discusses some relevant issues related to entanglement. In
Section 3 the collapse operator and equation are defined, and the way in which
the relevant parameters are determined is described. These parameters
determine both the rate at which collapse occurs and the size of the systems
necessary to bring about collapse. Section 4 describes how the wave function
evolves in configuration space in measurement situations. It focuses on the
way in which the tensor product structure of the Hilbert space and the basis
in which to view entanglement relations are defined by the interactions that
generate the correlations. Section 5 gives a detailed description of the
collapse process in configuration space and a proof that it conforms with the
Born probability rule. Section 6 discusses the fact that virtually all
elementary systems are entangled to some extent with other systems with which
they have interacted, and the implications that this entanglement has for the
definition of conserved quantities and our understanding of conservation laws.
Based on this discussion Section 7 goes on to demonstrate that the proposed
collapse equation is consistent with strict conservation of momentum and
angular momentum in individual experiments. Section 8 shows that, within the
limits permitted by the somewhat restricted characterization of energy in
nonrelativistic theory, consistency with energy conservation is also
maintained. Section 9 discusses the possibility of extending the proposal to
cover relativistic situations. Section 10 is a summary.

##  2 Motivation, Approach, and Assumptions

The original motivation for assuming that wave function collapse is induced
directly by correlating interactions stems from the role that these
interactions play in the physical instantiation and transmission of
information. Although wave function collapse is a nonlocal effect that appears
to be at odds with relativity, the fact that it is a probabilistic phenomenon
prevents such effects from carrying any information. Thus, it makes sense to
focus on the fundamental physical processes involved in enforcing this limit
in order to reconcile the “two fundamental pillars of contemporary
theory”[21].

The emphasis on entangling interactions has important additional benefits.
First, the interactions pick out the basis into which the wave function
collapses, and they also help define the tensor product structure of the
Hilbert space[22, 23, 24, 25] which is crucial in characterizing the structure
of entanglement relations. Second, it allows one to use the strength of the
interactions and the duration of the entangling process to eliminate the need
to introduce any new physical constants. Finally, it allows the construction
of a collapse equation that respects conservation laws in individual
instances.

Several objections have been raised to the proposed equation based on
misunderstandings of how an interaction-induced collapse process works. For
this reason it is helpful to begin with a heuristic description such a process
in a very simple measurement situation. Suppose that the wave function of an
elementary particle is split into two spatially separated branches, and that a
detector is placed in the path of one of these branches. When one of the
branches encounters the detector it initiates a sequence or cascade of
interactions that correlate the states of the particle to those of the
detector. Most collapse proposals assume that these correlations are generated
prior to the actual collapse.555In many accounts the sequence of individual
interactions between the measured system and the the apparatus is idealized by
introducing a single effective interaction Hamiltonian. The viewpoint offered
here is that this sort of idealization misses the most crucial steps in the
collapse process, and creates the impression that the collapse operator must
compete with the Hamiltonian, rather than work in parallel with it. In
contrast, the assumption made here is that collapse proceeds incrementally as
the entangling interactions are taking place. Specifically, it is assumed that
with each such interaction a small amount of amplitude is transferred either
into or out of the interacting branch based on some random process. With a
sufficient number of interactions eventually all of the amplitude is
transferred into either the interacting or the noninteracting branch of the
wave function.

The question that naturally arises is what happens if detectors are placed in
the paths of both branches. In these situations the process can work just as
it does in the case of a single detector, as long as one assumes that the
interactions are not perfectly synchronized in both time and strength. Such an
assumption is extremely plausible for interaction sequences of sufficient
length.

This intuitive idea must now be formalized in a stochastic collapse equation
that covers the wide range of possible measurement situations, and also
accounts for collapse in naturally occurring settings. The remainder of this
section will set the stage for this by describing the assumptions, framework,
and notation that will be used in carrying out this task.

Although the motivation for this proposal is a desire to reconcile relativity
and quantum theory the formulation presented here is nonrelativistic. This
simplifies the handling of the nonlocal aspects of wave function collapse and
makes it possible to analyze the collapse process in configuration space. The
configuration space picture is also helpful in explaining how the process
respects conservation laws. The nonrelativistic formulation also entails that
the kinetic and interaction energies are much less than the total relativistic
energy.

The total system is assumed to consist of a very large number of subsystems.
Much of the discussion will focus on elementary particles or other very small
subsystems, but we must also allow for the possibility of elementary systems
organizing themselves into much larger, even macroscopic, subsystems that can
interact with other subsystems as a unit. Individual subsystems will be
labeled with subscripts, $j,k,l,...$ Space coordinates will be labeled as
$x_{j},y_{j},z_{j}$, and their vector representation will be represented as
$\mathbf{w}_{\textbf{j}}$.

Interactions will be modeled by conservative (hence, distance-dependent), two-
particle potentials,
$\mathbf{\hat{V}_{jk}}(\mathbf{w}_{j},\mathbf{w}_{k})\,=\,\mathbf{\hat{V}_{jk}}(|\mathbf{w}_{j}\,-\,\mathbf{w}_{k}|)$,
that fall off with increasing distance. The term, ’particle’, is to be
interpreted as applying to any subsystem interacting as a unit. Since I will
be assessing the effect of the collapse operator on conservation laws it is
also necessary to rule out external potentials. All systems, even macroscopic
ones, are considered to be quantum systems. There is no classical boundary.

By basing the collapse operators on the potentials, which are smooth functions
of distance that decrease in magnitude as distance increases, we can insure
that that the collapse results in the approximate localization of subsystems.
The special status of the position basis follows from the assumption that
collapse is tied to distance-dependent interactions. The close connection
between the interaction basis and the position basis will also facilitate the
description of the collapse process and the maintenance of conservation laws
in configuration space.

It is also assumed that it is possible to measure any (reasonable) observable.
Since measurements consist of sets of interactions, and since interactions are
distance dependent, this implies that measurements typically begin with the
spatial separation of the wave function of the target system into branches
characterized by distinct values of the quantity associated with the
observable being measured. This leads to the splitting of the wave function of
the total system into well separated regions of configuration space. The
random walk described in Section 5 takes place between these regions.

##  3 Stochastic Collapse Equation

Like a number of previous proposals, the collapse equation described here is
based on the Wiener process, also known as Brownian motion. The derivative of
this process is white noise, which is usually indicated in the literature as
$dW$, $dB$, or $d\xi$. A fairly standard form for these equations is:666Some
recent proposals employ colored noise rather than white noise, and incorporate
the stochastic processes into a classical noise field. See [26, 27, 28].

$d\psi\,\,=\,(-i/\hbar)\mathbf{\hat{H}}\,\,\psi\,dt\,+\,\Big{[}\sum_{k}\sqrt{\gamma}(\mathbf{\hat{L}_{k}}-\langle\,\mathbf{\hat{L}_{k}}\,\rangle)d\xi_{k}-\,\frac{1}{2}\sum_{k}\gamma(\mathbf{\hat{L}_{k}}-\langle\,\mathbf{\hat{L}_{k}}\,\rangle)^{2}dt\Big{]}\psi.$
(3.1)

The first term on the right represents the ordinary Schrödinger evolution,
governed by the Hamiltonian, $\mathbf{\hat{H}}$. The $\mathbf{\hat{L}_{k}}$
are Lindblad operators which are almost always taken to be self-adjoint;
hence, they correspond to observables. This schema allows for multiple
independent Wiener processes, which can be real or complex.
$\langle\,\mathbf{\hat{L}_{k}}\,\rangle$ is the expectation value of
$\mathbf{\hat{L}_{k}}$ in the state, $\psi$,
$\langle\psi|\mathbf{\hat{L}_{k}}|\psi\rangle$, and its presence makes clear
the nonlinearity of the equation. The parameter, $\gamma$, determines the
effectiveness of the process in bringing about collapse, and, in most
proposals, incorporates new physical constants. The stochastic processes,
$\xi_{k}$, are assumed to have zero mean, and the differentials, $d\xi_{k}$,
obey the It$\hat{o}$ stochastic calculus rules:

$d\xi^{*}_{j}d\xi_{k}=dt\delta_{jk},\;\;\;dtd\xi_{k}=0.$ (3.2)

These conditions on the differentials determine the units,
$d\xi_{k}\,\sim\,(dt)^{\frac{1}{2}}$, and they are responsible for the factor,
$dt$, in the final term on the right. The middle term,
$\sum_{k}\sqrt{\gamma}(\mathbf{\hat{L}_{k}}-\langle\,\mathbf{\hat{L}_{k}}\,\rangle)d\xi_{k}$,
is primarily responsible for bringing about the collapse of the state vector.
In most other proposed equations it tends to concentrate the state on
eigenstates of the observables, $\mathbf{\hat{L}_{k}}$[10]. (The approach
taken here differs in that the set of possible outcomes is determined by which
interactions are most relevant in a particular situation.) The nonunitary
nature of the operator described by middle term necessitates an adjustment to
insure that the resulting vector has unit norm. This is provided by the third
term,
$-\,\frac{1}{2}\sum_{k}\gamma(\mathbf{\hat{L}_{k}}-\langle\,\mathbf{\hat{L}_{k}}\,\rangle)^{2}dt$.

The collapse equation proposed here will use just a single collapse operator,
$\hat{\mathcal{V}},$ and a single Wiener process, $\xi$. In order to implement
the hypothesis that wave function collapse is induced by those interactions
that generate entanglement the stochastic operator will be based on
conservative, two-particle interaction potentials,
$\mathbf{\hat{V}_{jk}}(\textbf{w}_{j},\textbf{w}_{k})$, where $\textbf{w}_{j}$
and $\textbf{w}_{k}$ indicate the coordinates of systems $j$ and $k$ in
configuration space. The requirement that the potential functions are
conservative implies that they are functions only of the separation between
the systems:
$\mathbf{\hat{V}_{jk}}(\mathbf{w}_{j},\mathbf{w}_{k})\,=\,\mathbf{\hat{V}_{jk}}(|\mathbf{w}_{j}\,-\,\mathbf{w}_{k}|)$.

The emphasis here is on correlating interactions. So we want to multiply the
potential by some measure of the effectiveness of the interaction in
establishing a correlation between the systems involved. Since this depends on
the extent to which the interaction changes the states of the systems
involved, the expression,
$\mathbf{\hat{V}_{jk}}\,-\,\langle\,\mathbf{\hat{V}_{jk}}\,\rangle$, will be
divided by the sum of the effective masses, $(m_{j}+m_{k})$, of the
interacting systems. For the present purpose the most straightforward way to
determine an effective mass is to aggregate particles in bound states such as
atoms, molecules, and other complex structures into single systems with a net
charge and a total mass.

To maintain overall consistency in the dimension of the stochastic operator it
is necessary to transform the effective mass in the denominator into an energy
by multiplying by the square of some velocity. The only nonarbitrary choice is
$c$, the speed of light. As will be shown, multiplying
$\mathbf{\hat{V}_{jk}}\,-\,\langle\,\mathbf{\hat{V}_{jk}}\,\rangle$ by
$1\,/\,(m_{j}+m_{k})c^{2}$ sets the size of the amplitude shifts associated
with individual interactions to a very reasonable value, and insures that the
stochastic effect is a minor perturbation relative to the changes induced by
standard Schrödinger evolution. It also anticipates the eventual extension of
this proposal to a relativistic version.

The other critical piece of the stochastic operator is the rate parameter,
$\gamma$. As stated earlier the single parameter, $\gamma$, will be replaced
by multiple variable rate parameters, $\gamma_{jk}$, each of which is
associated with the interaction between system, $j$, and system, $k$. These
are tied to the speed with which individual interactions separate the wave
function into orthogonal branches in the process of generating correlations.
They are effective only during the brief period in which the correlation is
being established, and they tend to zero afterward. They are defined below.

The individual terms,
$[\sqrt{\gamma_{jk}}(\mathbf{\hat{V}_{jk}}-\langle\mathbf{\hat{V}_{jk}}\rangle)]\;/\;[(m_{j}+m_{k})c^{2}]$,
will be abbreviated as $\hat{\mathcal{V}}_{jk}.$ The single collapse operator,
$\hat{\mathcal{V}},$ is then defined as: $\sum_{j<k}\hat{\mathcal{V}}_{jk}.$
The proposed collapse equation can be represented as either:

$d\psi\,=\,(-i/\hbar)\mathbf{\hat{H}}\,\psi\,dt\,+\,\sum_{j<k}\hat{\mathcal{V}}_{jk}\,\,\psi\,d\xi(t)-\,\frac{1}{2}(\sum_{j<k}\hat{\mathcal{V}}_{jk})\,(\sum_{m<n}\hat{\mathcal{V}}_{mn})\,\psi\,dt;$
(3.3)

or, more compactly, as:

$d\psi\,=\,(-i/\hbar)\mathbf{\hat{H}}\,\psi\,dt\,+\,\hat{\mathcal{V}}\,\psi\,d\xi(t)-\,\frac{1}{2}\hat{\mathcal{V}}^{2}\,\psi\,dt.$
(3.4)

It is important to emphasize that the amplitude transfer brought about by each
$\hat{\mathcal{V}}_{jk}$ applies across the entire configuration space and not
only to the systems, $j$ and $k$. Other systems are affected by the transfer
to the extent that they are entangled with either $j$ or $k$. Because the
potentials, $\mathbf{\hat{V}_{jk}}$, are distance-dependent the amplitude
transfers occur between localized regions of configuration space.

To complete the collapse proposal it is necessary to specify the rate
parameters, $\gamma_{jk}$. These will be defined in terms of the rate at which
the potential energy is changing during the interaction between system, $j$,
and system, $k$. They vanish for stationary states since the interaction
potential is constant in these situations. They also tend to zero when the two
systems become widely separated.

The rate of change of potential energy is:

$\frac{d}{dt}\,\langle\,\psi|\,\mathbf{\hat{V}_{jk}}\,|\psi,\rangle\;=\;\frac{-i}{\hbar}\Big{(}\,\langle\,\psi|\,\Big{[}\mathbf{\hat{H}_{jk}},\mathbf{\hat{V}_{jk}}\Big{]}\,|\psi\rangle\,\Big{)},$
(3.5)

where $\Big{[}\mathbf{\hat{H}_{jk}},\mathbf{\hat{V}_{jk}}\Big{]}$ is the
commutator of $\mathbf{\hat{H}_{jk}}$ and $\mathbf{\hat{V}_{jk}},$ and the
nonrelativistic Hamiltonian is:

$\mathbf{\hat{H}_{jk}}\;\;=\;\;-\frac{\hbar^{2}}{2m}(\mathbf{\nabla_{j}}^{2}\,+\,\mathbf{\nabla_{k}}^{2})\;+\;\mathbf{\hat{V}_{jk}}.$
(3.6)

This yields:

$(-\frac{i\hbar}{2m})\,\langle\,\psi|\,(\mathbf{\nabla_{j}}^{2}\mathbf{\hat{V}_{jk}}\,+\,\mathbf{\nabla_{k}}^{2}\mathbf{\hat{V}_{jk}}\;+\;2(\mathbf{\nabla_{j}}\mathbf{\hat{V}_{jk}}\mathbf{\nabla_{j}}\,+\,\mathbf{\nabla_{k}}\mathbf{\hat{V}_{jk}}\mathbf{\nabla_{k}})\,|\psi\rangle\,.$
(3.7)

The nonrelativistic limit on energy entails a minimum for the term,
$|\mathbf{w}_{j}\,-\,\mathbf{w}_{k}|$. This, in turn, implies that the
integration over $\psi$ eliminates the terms,
$\mathbf{\nabla_{j}}^{2}\mathbf{\hat{V}_{jk}}$ and
$\mathbf{\nabla_{k}}^{2}\mathbf{\hat{V}_{jk}}.$ The result is:

$||\,\int\,\psi^{*}\,\Big{[}(\mathbf{\hat{\nabla}_{j}}\mathbf{\hat{V}_{jk}}\,\cdot\,(\frac{-i\hbar}{m})\,\mathbf{\nabla_{j}}\psi)\,+\,(\mathbf{\hat{\nabla}_{k}}\mathbf{\hat{V}_{jk}}\,\cdot\,(\frac{-i\hbar}{m})\,\mathbf{\nabla_{k}}\psi)\Big{]}\,||.$
(3.8)

This expression includes the norm ($||...||$) in order to insure a positive
value. (All integrals are over the entire configuration space unless otherwise
noted.)

Some further refinement of this formula is required. As described earlier
typical measurement situations begin with the branching of the wave function
of an elementary or other very small system, $j$, into spatially separated
regions. Only one of these branches encounters system, $k$, and interacts with
it. It is essential for the derivation of the Born probability rule that the
rate parameter, $\gamma_{jk}$, be independent of the amplitude of the
interacting branch. For this reason we want to pick out just the interacting
portion of the wave function. This also allows one to define $\gamma_{jk}$ in
such a way that it integrates to a value of order, $1$, over the duration of
the interaction.

The interacting portion of the wave function can be picked out by using a
weighting factor,
$\frac{\mathbf{\hat{V}_{jk}}\,}{\langle\,\mathbf{\hat{V}_{jk}}\,\rangle}$,
reflecting the extent to which each segment of the wave function is involved
in the interaction. To insure that $\gamma_{jk}$ has the dimensions of inverse
time and that it integrates to a value of order, $1$, the rate of change of
potential energy will be divided by the total nonrelativistic energy of the
interacting systems. Since the kinetic energy of the center of mass of the
interacting systems adds an arbitrary offset to the total energy it is
necessary to eliminate it by specifying the energy in the center-of-mass
frame. The center-of-mass velocity can be identified by integrating the
relevant momentum operators over the interacting portion of the wave function
and dividing by the the combined mass:

$\vec{\mathbf{u}}\;=\;\int\,\Big{(}\frac{\mathbf{\hat{V}_{jk}}\,}{\langle\,\mathbf{\hat{V}_{jk}}\,\rangle}\,\Big{)}\,\psi^{*}\,\,(\frac{-i\hbar}{mj+m_{k}})\,[\,\mathbf{\nabla_{j}}\psi\,+\,\mathbf{\nabla_{k}}\psi\,].$
(3.9)

The wave function in the center-of-mass rest frame can then be represented as:

$\psi_{cm}\;=\;\Big{(}\frac{\mathbf{\hat{V}_{jk}}\,}{\langle\,\mathbf{\hat{V}_{jk}}\,\rangle}\,\Big{)}\,\psi\,\,e^{i[-\,\vec{\mathbf{u}}\cdot\vec{\mathbf{w}}_{j}\frac{m_{j}}{\hbar}\,-\,m_{j}\mathbf{u}^{2}\frac{t}{2\hbar}\,-\,\vec{\mathbf{u}}\cdot\vec{\mathbf{w}}_{k}\frac{m_{k}}{\hbar}\,-\,m_{k}\mathbf{u}^{2}\frac{t}{2\hbar}]}.$
(3.10)

With this expression the relevant measure for the total energy is:

$\langle\,\psi_{cm}\,|\,\Big{(}\,\hat{\mathbf{H}}_{jk}\,-\,E_{0}\,\Big{)}\,|\,\psi_{cm}\,\rangle,$
(3.11)

where $E_{0}$ is the ground state energy. (It should be noted that the terms,
$\frac{\mathbf{\hat{V}_{jk}}\,}{\langle\,\mathbf{\hat{V}_{jk}}\,\rangle}$,
$\vec{\mathbf{u}}$, and, hence, $\psi_{cm}$ can vary over the duration of the
interaction.) Putting together 3.8 through 3.11 we arrive at the definition
for $\gamma_{jk}$ :

$\gamma_{jk}\;\equiv\;\Big{[}\,||\,\int\,\psi^{*}\,\Big{[}(\mathbf{\hat{\nabla}_{j}}\mathbf{\hat{V}_{jk}}\,\cdot\,(\frac{-i\hbar}{m})\,\mathbf{\nabla_{j}}\psi)\,+\,(\mathbf{\hat{\nabla}_{k}}\mathbf{\hat{V}_{jk}}\,\cdot\,(\frac{-i\hbar}{m})\,\mathbf{\nabla_{k}}\psi)\Big{]}\,||\,\Big{]}\;\Big{/}\;\Big{[}\,\langle\,\psi_{cm}\,|\,\Big{(}\,\hat{\mathbf{H}}_{jk}\,-\,E_{0}\,\Big{)}\,|\,\psi_{cm}\,\rangle\,\Big{]}.$
(3.12)

Our primary interest is in the integrated value of $\gamma_{jk}$ for those
pairs of systems that interact most strongly since these interactions are
chiefly responsible for inducing the collapse. It should be clear that in
these cases the integrated value has a maximum of order, $1$, since, in the
center-of-mass frame one expects an approximately complete changeover from
potential to kinetic energy (or vice versa). In a transition from a free state
to a free state the value would be about $2$. Transitions to bound states
would be about $1$. (Radiative effects are neglected since this is a
nonrelativistic formulation.) Since $\int\,\gamma_{jk}\,dt\;\sim\;1$ it is
also clear that
$\parallel\,\int\,\sqrt{\gamma_{jk}}\,d\xi(t)\parallel\;\sim\;1.$ In more
weakly interacting cases the change in potential energy is much smaller than
the average kinetic energy over the course of the interaction, and so the
integrated value of $\gamma_{jk}$ and the effect on the amplitude are much
less.

For the cases of primary interest, the fact that $\gamma_{jk}$ integrates to
about $1$ allows us to estimate the size of the amplitude shift associated
with each interaction. The shift will be roughly equal to the ratio of
interaction energy to total relativistic energy. The upper limit for this
value can be estimated by considering electrostatic potentials and electron-
electron interactions since these have the lowest mass of any particles in a
nonrelativistic theory. The upper limit on the interaction energy must be
large enough to accommodate typical measurement situations, while respecting
the nonrelativistic constraint, $\mathbf{\hat{V}}_{jk}\,\ll\,mc^{2}$. The
ratio of potential energy to relativistic energy for an electron in the ground
state of a hydrogen atom is equal to the square of the fine structure
constant, $\alpha^{2}\,\approx\,5.33*10^{-5}$. This number provides a
convenient reference, but it is not quite large enough to cover all
measurements of interest.777In photomultipliers the electrons involved in the
measurement process are accelerated up to several hundred electron volts.
Since one would not expect all of this energy to be concentrated in a single
electron-electron interaction this limit should be sufficient. Therefore, it
is reasonable to consider interaction energy ratios of roughly one order of
magnitude above this reference,
$\mathbf{\hat{V}}_{jk}\;/\;[(m_{j}+m_{k})c^{2}]\sim\,10\alpha^{2}\,\sim\,5*10^{-4}$.
Energy ratios much above this value can no longer be described with simple
distance-dependent potentials, and relativistic corrections would need to be
taken into account.

It is also of interest to estimate the the ratio of the change in the wave
function due to the stochastic operator relative to that induced by the
Hamiltonian over the course of the interaction. This ratio can be expressed
as:

$\Big{[}\int\,\hat{\mathcal{V}}_{jk}\,\,\psi\,d\xi(t)\Big{]}\;/\;\Big{[}\int\,\frac{\mathbf{\hat{V}}_{jk}}{\hbar}\,\psi\,\delta
t\Big{]}$ (3.13)

Since $\hat{\mathcal{V}}_{jk}$ is defined as
$[\sqrt{\gamma_{jk}}(\mathbf{\hat{V}_{jk}}-\langle\mathbf{\hat{V}_{jk}}\rangle)]\;/\;[(m_{j}+m_{k})c^{2}]$,
and $\parallel\,\int\,\sqrt{\gamma_{jk}}\,d\xi(t)\parallel\;\sim\;1,$ this
ratio is approximately $[1\,/\,(m_{e}+m_{e})c^{2}]*[\hbar\,/\,\delta t].$ The
time interval, $\delta t$, is inversely proportional to the $\frac{3}{2}$
power of the interaction energy, and the mass of the electron, $m_{e}$, has
again been used to determine an upper limit.

The interaction energy limit described above can be used to estimate the
relevant time interval, $\delta t$, by considering the time it takes for the
potential energy to change from $0.1$ of its maximum value to its maximum
value (or vice versa). The maximum speed of an electron at the energy limit is
about $7*10^{6}m/s$ and the distance over which the change in
$\mathbf{\hat{V}_{jk}}$ takes place is about equal to the Bohr radius,
$\approx\,5*10^{-11}m$. So $\delta t$ is about $10^{-17}$ seconds. The
resulting ratio is:

$[1\,/\,(m_{j}+m_{k})c^{2}]*[\hbar\,/\,\delta
t]\;\approx\;[1\,/\,10^{-13}]*[10^{-34}\,/\,10^{-17}]\;=\;10^{-4}.$ (3.14)

The smallness of this ratio shows that the largest stochastic effect from a
single interaction is a fairly small nonlinear perturbation on the
Hamiltonian. Both of the ratios derived here play key roles in the discussion
in Section 5 which shows how 3.3 brings about the collapse of the wave
function. In preparation for that demonstration the next section deals with
some issues involved in the evolution of the wave function and the generation
of entanglement relations in measurement situations.

##  4 Evolution and Entanglement in Configuration Space

As just shown the effects of the stochastic operator from a single interaction
are minor perturbations on the standard Schrödinger evolution. So, to explain
how 3.3 induces complete collapse it is helpful to first review the way in
which the wave function evolves in typical measurement situations under the
influence of the Hamiltonian. This background is assumed, either explicitly or
implicitly, by every serious attempt to understand what happens during quantum
measurements.

The interactions that constitute measurements are distance dependent. For this
reason measurements intended to determine a particular quantity typically
begin by splitting the subject wave function into branches characterized by
different values of that quantity. For example, to measure momentum one might
send the subject particles through a magnetic field that bends the
trajectories in various directions. To measure spin along a particular axis
one can also use a magnetic field to separate up and down states. In some
cases, of course, the separation occurs simply as a result of the evolution of
the wave function.

Some or all of the branches then proceed into devices in which the subject
particle can initiate a set of interactions that result in a final state of
the device that is macroscopically distinct from its initial state. The
arrangement of interactions that carry out this task must be designed in such
a way that an elementary or other microscopic system is capable of completely
altering the state of a macroscopic system.

In many accounts the detailed interactions involved in correlating the state
of an elementary system with that of the measurement apparatus are ignored.
Many stochastic collapse proposals reduce this process to a single step
described by a very simple Hamiltonian. In a similar fashion, most of the
literature on decoherence assumes that it is the interactions between the
apparatus and the “environment” that are most relevant in explaining the
appearance of particular outcomes.

In contrast, in this proposal the individual interactions that take place
between the measured system and the measurement apparatus, along with those
within the apparatus, play the crucial role in bringing about the collapse of
the wave function.888This is not meant to rule out the possibility that
subsequent interactions between the apparatus and environment might help to
finish the process.

The critical role that individual, localized interactions play in the
development of distinct branches of the wave function is evident in Mott’s
analysis of $\alpha$-ray tracks[29]. By examining the evolution of the wave
function in configuration space he showed how these interactions lead to
correlations between the positions of the atoms that are ionized by the
passage of the $\alpha$-particle. These correlations are responsible for the
well defined tracks that are seen in these experiments. Since Mott’s paper
numerous studies have shown how the establishment of correlations between
elementary systems leads to the suppression of interference between various
segments of the wave function. The reason for this suppression can be easily
understood by considering the description of the evolving system in
configuration space. The interactions lead to the separation of the wave
function into essentially disjoint regions. This point has been elaborated in
several discussions of the branching process in configuration space in
connection with the de Broglie-Bohm theory.[30, 31, 32, 33, 34]

Because the representation of the wave function in configuration space
exhibits the clear separation of the branches, it provides a convenient way to
understand how the measurement basis is selected. It also enables a detailed
description of the collapse process according to 3.3. It is, therefore, worth
examining this representation in more detail.

Each point in configuration space represents a classical configuration of the
total system. The regions in which the wave function differs significantly
from zero contain the configurations that are consistent with the prior
evolution of the system, in particular, with the interaction history. This
history determines how conserved quantities have been exchanged during
interactions, and the extent to which the states of various subsystems are
correlated.

The correlations play a key role in defining the entanglement structure of the
wave function, but entanglement also depends on how the total system is
decomposed into subsystems. In other words, it depends on the choice of a
coordinate system in configuration space. For the purpose of analyzing the
evolution of the wave function in measurement situations the natural choice
for a coordinate system is the one used to describe distance-dependent
interaction potentials, and is often referred to simply as the position basis.
This corresponds to a decomposition of the total system into interacting
subsystems. That is the decomposition that will be used here. (Alternate
decompositions that decouple evolution equations by eliminating any reference
to interactions between systems will be reviewed briefly in the next section.
These decompositions eliminate entanglement by changing the tensor product
structure of the Hilbert space.)

The separate localized segments of the wave function reflect the entanglement
structure defined by the position basis. The branches of systems that have
interacted and exchanged conserved quantities are represented in the same
region of configuration space and their complementary branches are in a
different region.

The entanglement relations can be examined in a little more detail by
considering the effect of an interaction between two elementary systems.
Suppose that system, $j$, is the subject of the measurement, and that a branch
of this system interacts with system, $k$, in a region around
$(\mathbf{w}_{j},\mathbf{w}_{k})$. The systems exchange momentum and energy
during the interaction, and, as a result, their trajectories are altered in a
coordinated manner reflecting the entanglement that now exists between them.
The components of the systems that interacted around
$(\mathbf{w}_{j},\mathbf{w}_{k})$ are now linked and will be represented in a
region of configuration space around
$(\mathbf{w}^{\prime}_{j},\mathbf{w}^{\prime}_{k})$. When either of these
components interacts with another system, $l$, it extends the chain of
entanglement relations. In this manner further measurement-like interactions
build a large structure of correlated components that define a macroscopically
distinct state of the total system, represented in a disjoint, localized
region in configuration space.

Consider the density, $\psi^{*}\psi$, corresponding to the system wave
function in just this entanglement region. Under ordinary Schrödinger
evolution, governed only by the Hamiltonian, it will have some integrated
value, say, $\mu^{*}\mu$. To demonstrate that 3.3 is effective in bringing
about the collapse of the wave function it must be shown that, as a result of
the stochastic action associated with each of the elementary correlating
interactions that constitute the measurement, the integrated value,
$\mu^{*}\mu$, is altered to either $1$ or to $0$. This is done in the next
section.

##  5 Collapse of the Wave Function

To begin the demonstration it is necessary to calculate the change in the
quantity, $\psi^{*}\psi$, induced by each correlating interaction according to
3.3. We first restrict 3.3 to a single interaction:

$d\psi\,=\,(-i/\hbar)\mathbf{\hat{H}}\,\psi\,dt\,+\,\hat{\mathcal{V}}_{jk}\,\,\psi\,d\xi(t)-\,\frac{1}{2}\hat{\mathcal{V}}_{jk}^{2}\,\psi\,dt.$
(5.1)

The adjoint equation is:

$d\psi^{*}\,=\,(+i/\hbar)\mathbf{\hat{H}}\,\psi^{*}\,dt\,+\,\hat{\mathcal{V}}_{jk}\,\,\psi^{*}\,d\xi^{*}(t)-\,\frac{1}{2}\hat{\mathcal{V}}_{jk}^{2}\,\psi^{*}\,dt.$
(5.2)

The change in the product, $\psi^{*}\psi$, is:

$d(\psi^{*}\psi)\;=\;(\psi^{*}+d\psi^{*})(\psi+d\psi)\,-\,(\psi^{*}\psi)\;=\;d\psi^{*}\psi\,+\psi^{*}d\psi\,+d\psi^{*}d\psi.$
(5.3)

To calculate the change we must take into account the It$\hat{o}$ calculus
rules regarding the way in which differentials are treated. In ordinary
calculus a term like $d\psi^{*}d\psi$ that is quadratic in differentials would
be discarded. But the Wiener process, $\xi(t)$, varies as $\sqrt{t}$, since it
is analogous to a random walk. It is this relationship that is responsible for
the It$\hat{o}$ rule stated in 3.2, ($d\xi^{*}d\xi=dt$), and it also implies
that the final term in 5.3, $d\psi^{*}d\psi$, must be retained.

Plugging 5.1 and 5.2 into 5.3 yields:

$\begin{array}[]{ll}d(\psi^{*}\psi)\;=\;\frac{i}{\hbar}\Big{[}\,(\mathbf{\hat{H}}\,\psi^{*})\psi\,-\,\psi^{*}(\mathbf{\hat{H}}\psi)\,\Big{]}\,dt&\\\
-\psi^{*}(\frac{1}{2}\hat{\mathcal{V}}_{jk}\hat{\mathcal{V}}_{jk})\,\psi\,dt\,-\psi^{*}(\frac{1}{2}\hat{\mathcal{V}}_{jk}\hat{\mathcal{V}}_{jk})\,\psi\,dt\,+\,\psi^{*}\,\hat{\mathcal{V}}_{jk}\hat{\mathcal{V}}_{jk}\psi\,d\xi^{*}d\xi&\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\;\;\psi^{*}\,\hat{\mathcal{V}}_{jk}\psi\,d\xi^{*}\,+\,\psi^{*}\,\hat{\mathcal{V}}_{jk}\psi\,d\xi.\end{array}$
(5.4)

Since $d\xi^{*}d\xi=dt$ we get:

$\;\;\;d(\psi^{*}\psi)\;=\;-\,\mathbf{\nabla}\cdot\Big{[}(\frac{i\hbar}{2m})\,(\,\psi\mathbf{\nabla}\psi^{*}\,-\,\psi^{*}\mathbf{\nabla}\psi)\Big{]}\,dt\;+\;(\psi^{*}\,\hat{\mathcal{V}}_{jk}\,\psi)\,(d\xi^{*}\,+\,d\xi).$
(5.5)

The first term on the right is just the negative of the divergence of the
probability current, which represents the change in $\psi^{*}\psi$ under
ordinary Schrödinger evolution.

The Schrödinger term and the stochastic term,
$(\psi^{*}\,\hat{\mathcal{V}}_{jk}\,\psi)\,(d\xi^{*}\,+\,d\xi),$ operate in
essentially complementary and independent ways in measurement-like situations.
The change induced in $\psi^{*}\psi$ by the Hamiltonian is relatively large
and local, while the change resulting from the stochastic term is relatively
small and nonlocal. As described in the previous section, an effective
measurement process is one in which the individual interactions are arranged
so that they clearly separate the wave function into disjoint regions of
configuration space. The separation between the various branches is several
orders of magnitude greater than the effective range of the measurement
interactions.

The change induced by the Hamiltonian on the interacting portion of the wave
function is, by definition, confined to the interaction region. Thus, the
action of the Hamiltonian on each branch is effectively independent of its
action on other branches, and since it is unitary it is also independent of
the amplitude of each branch.

The effect of the stochastic operator is to transfer amplitude between the
branch(es) undergoing interaction and the disjoint, noninteracting branches.
The change induced by the individual transfers in the value of $\psi^{*}\psi$
integrated across each branch is given by 5.3:
$(\psi^{*}\,\hat{\mathcal{V}}_{jk}\,\psi)\,(d\xi^{*}\,+\,d\xi)$. The direction
of the transfer depends on the sign of the product of
$\hat{\mathcal{V}}_{jk}\,(d\xi^{*}\,+\,d\xi)$, and its magnitude was discussed
in Section 3. Because the term, $\sqrt{\gamma_{jk}}\,d\xi$, integrates to
approximately $1$, the size of each transfer depends on the ratio of the
interaction energy to the total relativistic energy of the systems involved,
along with the value of $\psi^{*}\psi$. Thus, with each individual correlating
interaction there is a transfer of amplitude either into or out of the
interacting portion of the wave function. Again, it is important to note that
the amplitude transfer applies to the total system \- not only to the
interacting subsystems. Thus, the entire branch of the wave function in which
the interaction takes place has its amplitude altered by the stochastic
action.999Although this point seems obvious it has been missed by several
critics. Since the wave function is assumed to be normalized and equations of
the form, 3.1, are norm-preserving the integrated value of $\psi^{*}\psi$ is
limited by $0$ and $1$. Therefore, with a sufficient number of correlating
interactions this integrated value must approach one of these limits.
Measurement processes typically involve enough individual interactions of
adequate strength to insure complete collapse. More detailed quantitative
estimates will be provided below.

Despite the straightforward manner in which 3.3 brings about collapse there
have been several claims that it is incapable of doing so. These are dealt
with here.

It has been objected that the collapse process could be frustrated if
measurement-like interactions are occurring in several branches
simultaneously. However, in situations in which there is a sufficient number
of interactions the probability of this is vanishingly small, as can be seen
by considering the numerical estimates from Section 3. The estimate for the
maximum ratio of interaction energy to total relativistic energy in
nonrelativistic situations was about $5*10^{-4}$. The amplitude transfer
process has the character of a random walk, with the step size determined by
this ratio. The endpoints of the random walk are $0$ and $1$ for the
integrated values of $\psi^{*}\psi$. The typical number of steps required to
reach an endpoint is the square of the inverse of the step size. Given the
maximum value for the step size just quoted, this means that the smallest
number of individual interactions to complete the process would be in excess
of $10^{6}$. The white noise that helps to determine the direction of the
steps varies continuously. This means that to frustrate the collapse the
interactions in disjoint branches of the wave function would have to be almost
perfectly synchronized in both time and interaction strength over a walk of
this length. An individual interaction with this strength has a duration of
about $10^{-17}$ to $10^{-16}$ seconds and a range of less than $10^{-10}$
meters. The branches are separated by at least a few orders of magnitude
greater than the effective range of the interactions. Variations in
propagation delays and the complexity of propagation patterns make it nearly
impossible to achieve the synchronization that would be needed. Interactions
of lower strength would have longer durations and expanded ranges, but the
required number of steps increases with the square of the inverse interaction
strength. So the necessary synchronization remains a virtual impossibility.

Another objection that has been raised is that in the cases in which the
collapse process is effective it would necessarily reduce the wave function to
a single point, implying an infinite increase in energy. This objection is
based on a misunderstanding of the way in which 3.3 differs from most collapse
equations. Most proposals are designed to collapse the wave function to an
eigenfunction of one particular observable, for example, either position or
total energy. Since the stochastic operator in 3.3 is based on distance-
dependent interaction potentials it has been alleged that it must reduce the
wave function to an eigenfunction of potential energy, that is, a single
point. As discussed previously, the effect of the operator relative to the
Hamiltonian is very small, and the duration of its action is extremely short.
Moreover, this effect is smeared over the interacting portion of the wave
function by the ordinary Hamiltonian evolution. There is simply no way that
collapse to a single point can occur.101010The extent to which the wave
function is reshaped by the collapse operator is examined in section 8.

One additional criticism of the current proposal is that because it conserves
momentum (as will be shown in Section 7) it cannot be effective in bringing
about collapse. The argument is that when momentum is conserved the center of
mass evolves unitarily. This feature is alleged to prevent collapse since
collapse equations are necessarily nonunitary.

This objection fails because, as long as the dynamics governing the system
insures conservation of momentum, the motion of the center of mass is
completely decoupled from the evolution of the individual components that make
up the system. Equations that redefine the the system into independently
evolving subsystems (such as in simple two-body problems) alter the tensor-
product structure of the Hilbert space by eliminating interactions between the
components, and changing the entanglement structure. This change in
entanglement structure has been discussed by Jekníc-Dugíc, Arsenijevíc, and
Dugíc, and by Thirring, Bertlmann, Köhler, and Narnhofer[22, 23, 24, 25]. The
authors of [24] note that most discussions of entanglement implicitly assume a
decomposition of the total system into interacting subsystems:

> “Thus, it’s only the interaction, which we consider to determine the density
> matrix, or the measurement set-up, which fixes the factorization.”

The center of mass, regarded as an individual subsystem, is completely
disentangled from the remainder of the system. It is not subject to any
interactions or external potentials, and so it evolves freely and unitarily.
But this implies nothing about whether the rest of the system also evolves in
a unitary fashion.

This concludes the proof that 3.3 is effective in inducing wave function
collapse in typical measurement processes. Note that the reference to the way
in which the wave function evolves in these processes has been used in order
to facilitate this demonstration. However, it should be clear that generally
similar branching processes also occur in natural settings, outside the
laboratory. In fact, they are ubiquitous. Thus, one would expect wave function
collapse, induced in this manner, to occur all the time.

What remains is to show that the probability of collapse to a particular
branch of the wave function complies with the Born rule. Consistency with the
Born Rule follows essentially from the general form of 3.1. This can be shown
in detail for 3.3 as follows.

Assuming an adequate number of sufficiently strong correlating interactions
the process described by 3.3 can be characterized as a one-dimensional random
walk with variable step size that ends with an integrated value of
$\psi^{*}\psi$ over the interacting component of the the wave function equal
to either $0$ or $1$. If one assumes that this integrated value at some stage
of the process is equal to $\mu^{*}\mu$ then to establish the Born rule we
need to show that the probability that the end value will be $1$ is
$\mu^{*}\mu$.

We begin with a lemma that applies to simple one-dimensional random walks.
Label one end point as $0$ and the other as $1$, and let the distance between
them be divided into $n$ equal intervals. An elementary theorem states that
from a point, $m$, the probability of the walk terminating at $1$ is
$\frac{m}{n}$. An analogous result holds if steps of variable size are
allowed. From a point, $p$, with $0\,\leq\,p\,\leq\,1$ the probability of
reaching $1$ is $p$. To see this, let $\delta$ be a variable increment that is
less than or equal to the distance to the nearest end point:
$0\,\leq\,p-\delta\,\leq\,p\,\leq\,p+\delta\,\leq\,1$. Designate the
probability of reaching $1$ as $Pr(p)$. We have $Pr(0)\,=\,0$, $Pr(1)\,=\,1$,
and $Pr(p)\,=\,\frac{1}{2}[Pr(p-\delta)]\,+\,\frac{1}{2}[Pr(p+\delta)]$. Since
the last condition holds for all values of $p$ and $\delta$, $Pr(p)$ must be
linear. Given the boundary conditions it follows that $Pr(p)\,=\,p.$ Since a
Wiener process is the scaling limit of a one-dimensional random walk, this
result holds whether the walk is regarded as a discrete or continuous process.

In the collapse process $p$ is replaced by the value of $\psi^{*}\psi$
integrated over the interacting component, $\mu^{*}\mu$. The corresponding
value for the complementary (noninteracting) component is
$\nu^{*}\nu\,=1\,-\,\mu^{*}\mu$. What must be shown is that the step size at
each stage of the collapse process is less than or equal to both $\nu^{*}\nu$
and $\mu^{*}\mu$.

The stochastic change in $\psi^{*}\psi$ from a single correlating interaction
is given by 5.5:
$(\psi^{*}\,\hat{\mathcal{V}}_{jk}\,\psi)\,(d\xi^{*}\,+\,d\xi)$. The
individual terms, $\hat{\mathcal{V}}_{jk}$, were defined in Section 3 as
$[\sqrt{\gamma_{jk}}(\mathbf{\hat{V}_{jk}}-\langle\mathbf{\hat{V}_{jk}}\rangle)]\;/\;[(m_{j}+m_{k})c^{2}]$.
We first calculate the value of
$(\mathbf{\hat{V}_{jk}}-\langle\mathbf{\hat{V}_{jk}}\rangle)$ across the
interacting and the noninteracting component of $\psi$. The value of
$\mathbf{\hat{V}_{jk}}$ across the noninteracting component is essentially
zero. Although the value varies somewhat across the interacting component, it
is substantially different from that of the noninteracting portion. Designate
the interacting component as $I$, and the orthogonal component as $O$. The
average value of $\mathbf{\hat{V}_{jk}}$ over $I$ is
$\overline{\mathcal{V}^{I}_{jk}}\;\equiv\;[\,\langle\,I\,|V_{jk}|\,I\,|\rangle\,]\;/\;[\,\langle\,I\,|\,I\,\rangle\,]$.
The average of $\mathbf{\hat{V}_{jk}}$ over the entire wave function is then:
$\langle\mathbf{\hat{V}_{jk}}\rangle\;=\,\mu^{*}\mu\,\overline{\mathcal{V}^{I}_{jk}}$.
The average value of
$(\mathbf{\hat{V}_{jk}}-\langle\mathbf{\hat{V}_{jk}}\rangle)$ across the
interacting component is
$\overline{\mathcal{V}^{I}_{jk}}\,(1\,-\mu^{*}\mu\,)\;=\;\nu^{*}\nu\,\overline{\mathcal{V}^{I}_{jk}}\,$.
The corresponding value averaged over the orthogonal component is
$-\,\mu^{*}\mu\,\overline{\mathcal{V}^{I}_{jk}}\,$. Using these averaged
values, the expression, $(\psi^{*}\,\hat{\mathcal{V}}_{jk}\,\psi)$, can be
written as:
$[\,\langle\,I\,|\,\nu^{*}\nu\,\overline{\mathcal{V}^{I}_{jk}}\,|I\,\rangle\;-\;\langle\,O\,|\mu^{*}\mu\,\overline{\mathcal{V}^{I}_{jk}}\,|O\,\rangle\,]\;/\;[(m_{j}+m_{k})c^{2}].$
Since $\overline{\mathcal{V}^{I}_{jk}}$ is constant over the wave function
this can be represented as:
$\overline{\mathcal{V}^{I}_{jk}}\,[\nu^{*}\nu\,\langle\,I\,|\,I\,\rangle\;-\;\mu^{*}\mu\,\langle\,O\,|O\,\rangle]\;/\;[(m_{j}+m_{k})c^{2}].$

We have $\langle\,I\,|\,I\,\rangle\;=\,\mu^{*}\mu$ and
$\langle\,O\,|\,O\,\rangle\;=\,\nu^{*}\nu$. Therefore, it is clear that the
stochastic operator associated with each individual interaction multiplies
each of the two orthogonal components by equal and opposite amounts:
$\pm\;\mu^{*}\mu\,\nu^{*}\nu\,\overline{\mathcal{V}^{I}_{jk}}\;/\;[(m_{j}+m_{k})c^{2}].$
Since this relationship holds for all pairs of systems, $j,k$, if multiple
interactions are taking place at the same time we simply sum over them. The
full stochastic effect on the various branches at each time is given by:

$\pm\;(\mu^{*}\mu\,\nu^{*}\nu)\,\mathbf{\sum_{jk}}\,(\psi^{*}\;\overline{\mathcal{V}^{I}_{jk}}\;/\;[(m_{j}+m_{k})c^{2}]\,\sqrt{\gamma_{jk}}\;\psi)\,(d\xi^{*}(t)\,+\,d\xi(t)).$
(5.6)

The estimate for the maximum ratio of
$\overline{\mathcal{V}^{I}_{jk}}\;/\;[(m_{j}+m_{k})c^{2}]$ from Section 3 was
$5*10^{-4}$. The terms, $\sqrt{\gamma_{jk}}$, are tied to the rate at which
the interaction proceeds and, when conjoined with the terms,
$d\xi^{*}(t)\,+\,d\xi(t))$, are designed to integrate to a value of about $1$
over the course of the interaction. The stochastic process, $\xi(t)$, has a
variance of $t_{2}-t_{1}$ over the period from $t_{1}$ to $t_{2}$. Therefore,
to insure that the norm of the sum,
$\mathbf{\sum_{jk}}\,(\psi^{*}\;\overline{\mathcal{V}^{I}_{jk}}\;/\;[(m_{j}+m_{k})c^{2}]\,\sqrt{\gamma_{jk}}\;\psi)\,(d\xi^{*}(t)\,+\,d\xi(t))$,
is less than or equal to $1$, we need only to take time increments, $dt$, that
are sufficiently small. Since $\mu^{*}\mu$ and $\nu^{*}\nu$ are both less than
or equal to $1$ this will guarantee that the total changes in the branches of
$\psi^{*}\psi$ given by 5.6 are less than or equal to $1$. Thus, 3.3
reproduces the Born rule.

It should be clear that the number of correlating interactions in a typical
measurement instrument during a measurement process is sufficient to bring
about nearly complete collapse. Instruments consist of well in excess of
$10^{23}$ particles and a substantial fraction of these are involved in the
chain or cascade of interactions that result in macroscopic differences in the
state of the device. With an average step size of $s$ the number of steps
required to complete a random walk between the point $0$ and $1$ is on the
order of $1/s^{2}$. The estimates of Section 3 showed that the maximum step
size is about $5*10^{-4}$. Measurement interactions could be a few orders of
magnitude less than this and still be effective. Measurements are often
initiated with individual interaction energies comparable to those of visible
photons, and these would give interaction energy ratios in the range of
$10^{-6}$ to $10^{-5}$. However, subsequent interaction energies in the
process are frequently enhanced by the instrument. It seems reasonable to take
the range of $10^{-7}$ to $10^{-6}$ as a lower limit for interaction energy
ratios in most measurement situations.

The other key factor in determining the step size is the product,
$\mu^{*}\mu\nu^{*}\nu$. As $\mu^{*}\mu$ approaches either $0.99$ or $0.01$
this term shrinks to a magnitude of about $0.01$, reducing the step size by a
couple of orders of magnitude. With the estimates from the previous paragraph
this would set an upper limit on the number of steps required at about
$10^{16}$ to $10^{17}$. It is important to note that, as the measurement
proceeds, the great majority of these interactions would be occurring in
parallel - not sequentially. Weaker interactions would take about $10^{-11}$
seconds to complete, but since so many would be taking place at the same time
the overall completion time for the process would be some small fraction of a
second.

The way in which 3.3 insures consistency with conservation laws in individual
situations will be reviewed in the next few sections.

##  6 Conserved Quantities Are Shared Across Entangled Branches

A proper understanding of conserved quantities in quantum theory must start
with a recognition of the fact that all systems are quantum systems. The
idealized simplifications that are used to analyze particular types of
situations and the classical boundaries that are placed around these
situations must be seen as just pragmatic approximations. The use of such
approximations is dictated by the impossibility of tracking all the
interactions that couple a “system” to its environment. But the fact that we
cannot do a detailed calculation of the effect of all of these interactions
does not prevent us from drawing general conclusions about their effect by
applying some fundamental results of quantum theory.

One such key result is that entanglement is a generic effect of the
interaction between systems. This has been demonstrated by Gemmer and
Mahler[35] and by Durt[36, 37]. An important implication of this result is
that there is almost always some small amount of entanglement between an
elementary system that has been “prepared” in an (approximate) eigenstate of
some observable and the preparation apparatus.

A photon interacting with a beam-splitter provides a simple example of such
entanglement between a system and apparatus. (This example has been discussed
in unpublished works by this author[38] and by Marletto and Vedral[39].)
During the interaction the reflected branch of the photon exchanges momentum
with the beam-splitter, and alters its state by a tiny amount. The extremely
narrow wave function representing the beam-splitter in position space implies
an extremely large spread in the momentum representation of the beam-
splitter’s state. Thus the exchange of momentum with the photon branch alters
that state by a very tiny amount. The magnitude of the inner product between
the slightly altered mirror state and the pre-interaction state of the
apparatus can be represented as $|1-\delta|$, where $\delta\ll 1$, but
$\delta>0$. This nonzero $\delta$ implies some (very small) amount of
entanglement between the photon and the beam-splitter. If, for the sake of
simplicity, one assumes that the photon branches have equal amplitude, the
entanglement, in terms of $\delta$, can be approximated as
$(\delta/2)|1-\log(\delta/2)|$[38]. It is important to note that momentum is
strictly conserved within each branch of the entangled wave function.

If the photon is subsequently detected in one of the branches then, because it
is entangled with the beam-splitter, the resulting collapse of the wave
function encompasses not only the photon but also the correlated state of the
beam-splitter. No matter which branch is detected the total momentum of the
photon, beam-splitter and measurement apparatus is the same after the
detection as it was before the initial photon-mirror interaction.

Because of the generic nature of the relationship between interaction and
entanglement this conservation of the relevant quantities among “system”,
preparation apparatus, and measurement instrument holds in virtually all
situations. The fact that these quantities are exchanged and shared across
branches of entangled systems through interactions implies that definite
values of conserved quantities can be assigned to these entangled branches -
not just to elementary subsystems that are assumed to be in factorizable
eigenstates of the relevant observable.

We have no trouble in assigning definite values of conserved quantities to
branches of entangled systems when those systems consist of a small number of
elementary particles such as singlet states, correlated photons generated
during parametric down conversion, or the kinds of states typically considered
in discussions of quantum computation. But, we lose sight of the fact that
well-defined quantities can be distributed across branches when those branches
include somewhat larger subsystems.

The aim of this and the next two sections is to show that the proposed
collapse equation (3.3)is consistent with the conservation laws of standard,
unitary quantum theory in individual measurement situations in regard to
momentum, angular momentum, and energy. Given this, the foregoing
considerations about entanglement require a careful examination of exactly
what those laws imply. A quantity, $\mathbf{q}$, is conserved if the
observable, $\mathbf{\hat{Q}}$, with which it is associated commutes with the
evolution operator, $\mathbf{\hat{H}}$, for arbitrary states, $|\psi\rangle$:
$\mathbf{\hat{H}}\mathbf{\hat{Q}}|\psi\rangle\;=\;\mathbf{\hat{Q}}\mathbf{\hat{H}}|\psi\rangle\,$.
When this condition holds the time derivative of $\mathbf{q}$, integrated over
the state, $|\psi\rangle$, is zero:
$(d/dt)\langle\,\psi\,|\mathbf{\hat{Q}}|\psi\rangle\;=\;\frac{i}{h}\,\langle\,\psi\,|\mathbf{\hat{H}}\mathbf{\hat{Q}}\,-\,\mathbf{\hat{Q}}\mathbf{\hat{H}}|\psi\rangle\,$.

Note that these laws are not restricted to situations in which the the state,
$|\psi\rangle$, is (assumed to be) in an eigenstate of the operator,
$\mathbf{\hat{Q}}$. The argument presented earlier in this section implies
that the systems under consideration are almost never in exact eigenstates. To
impose such a restriction would render the conservation laws essentially
vacuous.

The question of whether the conservation laws hold in a particular situation
depends not only on the structure of equation 3.3, but also on the initial
conditions that are assumed. The claim being made here is only that 3.3 is
consistent with these laws. What will be shown in the next two sections is
that the proposed equation guarantees that, within each entangled
branch,111111The tensor product structure and the basis in which entangled
branches are defined is assumed to be that which is picked out by the
interactions. This basis coincides with the position basis in configuration as
described in Section 4. the relevant quantities, $\mathbf{q}$, are conserved,
when the branch is normalized. In order to establish overall consistency it
must be assumed that the normalized quantities in different (affected)
branches are equal. The remainder of this section will explain why this
assumption is reasonable for a wide range of initial conditions.

A key background assumption of the proposed collapse equation is that the
“system” under consideration consists of a very large number of subsystems
that have been interacting over an extended period. For such a system the
properties of any elementary subsystem are determined essentially by its
interactions with other subsystems, with the most recent interactions having
the greatest effect. Branching processes are initiated by these conservative
interactions. Consider again the example of the photon and beam-splitter. It
is clear that the interaction between them does not introduce any difference
in momentum between the reflected and transmitted branch. The momentum change
in the reflected branch of the photon is offset by a momentum change in one of
the (slightly different) beam-splitter states. The total momentum in each
(normalized) branch is the same after the interaction as it was prior to it.
The same kind of analysis can be applied to any conservative interaction,
involving any conserved quantity. Therefore, after an extended period of
interaction one would expect that the total value of these quantities would be
essentially the same in all of the branches.

This argument can be extended to cover varying configurations within each
branch. As the wave function evolves under the influence of the Hamiltonian it
continually recombines the configurations. Thus, what distinguishes the
configurations at different points is the distribution of various quantities
among the subsystems - not the total value of those quantities computed across
all subsystems. In other words, after a sufficient number of interactions
along with ordinary wave propagation, the total value of a conserved quantity
in a particular configuration is very close to the average value of that
quantity calculated across the branch. Therefore, in what follows it will be
assumed that the total value of conserved quantities is the same at each point
in configuration space where the wave function has nonzero amplitude.

With this assumption in the next two sections it will be shown that 3.3 is
consistent with strict conservation of momentum, angular momentum, and energy
in individual measurements. The demonstrations for momentum and angular
momentum are fairly straightforward, and they hold exactly. These cases are
dealt with in Section 7. Because of nonrelativistic limitations the situation
with energy is somewhat more complicated. It is dealt with separately in
Section 8.

##  7 Momentum and Angular Momentum

In this section I will show that the stochastic operator defined in section 3
maintains the normalized value of momentum and angular momentum at each point
in configuration space. In other words the change in these quantities induced
by the collapse equation is directly proportional to the change that is
induced in the squared amplitude at that point:

$d(\psi^{*}\mathbf{\hat{Q}}\psi)\,/\,(\psi^{*}\mathbf{\hat{Q}}\psi)\;=\;d(\psi^{*}\,\psi)\,/\,(\psi^{*}\,\psi).$
(7.1)

Together with the assumption described at the end of Section 6 this implies
that these quantities are strictly conserved under evolution governed by 3.3.

The change in $d(\psi^{*}\,\psi)$ was displayed in equation 5.5:
$d(\psi^{*}\psi)\;=\;-\,\mathbf{\nabla}\cdot\Big{[}(\frac{i\hbar}{2m})\,(\,\psi\mathbf{\nabla}\psi^{*}\,-\,\psi^{*}\mathbf{\nabla}\psi)\Big{]}\,dt\;+\;(\psi^{*}\,\hat{\mathcal{V}}_{jk}\,\psi)\,(d\xi^{*}\,+\,d\xi).$
The steps leading up to 5.5 can be repeated to compute
$d(\psi^{*}\mathbf{\hat{Q}}\psi)$:

$\begin{array}[]{ll}d(\psi^{*}\,\mathbf{\hat{Q}}\,\psi)\;=\;\frac{i}{\hbar}\Big{[}\,(\mathbf{\hat{H}}\,\mathbf{\hat{Q}}\,\psi^{*})\psi\,-\,\psi^{*}(\mathbf{\hat{Q}}\,\mathbf{\hat{H}}\psi)\,\Big{]}\,dt&\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\psi^{*}(\frac{1}{2}\hat{\mathcal{V}}\hat{\mathcal{V}})\,\,\mathbf{\hat{Q}}\,\psi\,dt\;-\;\psi^{*}\,\hat{\mathbf{Q}}\,(\frac{1}{2}\hat{\mathcal{V}}\hat{\mathcal{V}})\,\psi\,dt\,\par+\;\psi^{*}\,\hat{\mathcal{V}}\hat{\mathbf{Q}}\hat{\mathcal{V}}\,\psi\,d\xi^{*}d\xi&\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\;\;\psi^{*}\,\hat{\mathcal{V}}\hat{\mathbf{Q}}\,\psi\,d\xi^{*}\;+\;\psi^{*}\,\hat{\mathbf{Q}}\hat{\mathcal{V}}\,\psi\,d\xi&\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\;\;-\frac{1}{2}\psi^{*}[(\hat{\mathcal{V}})^{2}\mathbf{\hat{Q}}\,+\,\hat{\mathbf{Q}}(\hat{\mathcal{V}})^{2}]\psi\,dt\;+\;\psi^{*}(\hat{\mathcal{V}}\hat{\mathbf{Q}}\hat{\mathcal{V}}\,)\psi\,dt&\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\;\;\psi^{*}\,\hat{\mathcal{V}}\hat{\mathbf{Q}}\,\psi\,d\xi^{*}\;+\;\psi^{*}\,\hat{\mathbf{Q}}\hat{\mathcal{V}}\,\psi\,d\xi.\end{array}$
(7.2)

The first term in 7.2 represents the change due to ordinary Schrödinger
evolution. By assumption $\mathbf{\hat{H}}$ commutes with $\mathbf{\hat{Q}}$,
but commutativity of the operators only implies that
$\mathbf{\hat{H}}\mathbf{\hat{Q}}\,=\,\mathbf{\hat{Q}}\mathbf{\hat{H}}$, when
they are integrated over the entire wave function. We need a somewhat stronger
condition. We need to show that the change in $(\psi^{*}\mathbf{\hat{Q}}\psi)$
matches the change in $(\psi^{*}\,\psi)$ at each point in configuration space:
$d(\psi^{*}\mathbf{\hat{Q}}\psi)\;\sim\;d(\psi^{*}\,\psi).$ For the quantities
under consideration here this can be seen by observing that the probability
current,
$(\frac{i\hbar}{2m})\,(\,\psi\mathbf{\nabla}\psi^{*}\,-\,\psi^{*}\mathbf{\nabla}\psi)\,$,
is simply (except for the factor, $m$, in the denominator) the expression for
momentum in a slightly disguised form. In a manner of speaking the quantities,
momentum and angular momentum, “ride along” with the changes in
$(\psi^{*}\,\psi)$ on the probability current. Thus, the changes due to
Schrödinger evolution in both $(\psi^{*}\mathbf{\hat{Q}}\psi)$ and
$(\psi^{*}\,\psi)$, decouple from those induced by the collapse operator.

Therefore, we are left with the task of showing that the change in
$(\psi^{*}\mathbf{\hat{Q}}\psi)$ resulting from the collapse operator,
$-\frac{1}{2}\psi^{*}[(\hat{\mathcal{V}})^{2}\mathbf{\hat{Q}}\,+\,\hat{\mathbf{Q}}(\hat{\mathcal{V}})^{2}]\psi\,dt\;+\;\psi^{*}(\hat{\mathcal{V}}\hat{\mathbf{Q}}\hat{\mathcal{V}}\,)\psi\,dt\;+\;\;\psi^{*}\,\hat{\mathcal{V}}\hat{\mathbf{Q}}\,\psi\,d\xi^{*}\;+\;\psi^{*}\,\hat{\mathbf{Q}}\hat{\mathcal{V}}\,\psi\,d\xi$,
is proportional to $(\psi^{*}\,\hat{\mathcal{V}}\,\psi)\,(d\xi^{*}\,+\,d\xi)$.
This can be done by showing that

$\hat{\mathbf{Q}}\hat{\mathcal{V}}\,\psi\;=\;\hat{\mathcal{V}}\,\hat{\mathbf{Q}}\psi$
(7.3)

at each point in configuration space.

The only spatial dependence in the collapse operator, $\hat{\mathcal{V}}$, is
in the terms representing the conservative interaction potentials,
$\mathbf{\hat{V}_{jk}}(\mathbf{w}_{j}\,-\,\mathbf{w}_{k})$. Since this implies
that
$\mathbf{\hat{\nabla}_{j}}\,\mathbf{\hat{V}_{jk}}\;=\;-\mathbf{\hat{\nabla}_{k}}\,\mathbf{\hat{V}_{jk}}$
it is quite simple to show that the relation, 7.3, holds for the two operators
in which we are interested here, momentum and orbital angular momentum.

The momentum operator can be expanded as:

$\mathbf{\hat{P}}\,\equiv\,-i\hbar\sum_{i}\,\mathbf{\hat{\nabla}_{i}}.$ (7.4)

Because the action of $\mathbf{\hat{\nabla}_{j}}$ cancels that of
$\mathbf{\hat{\nabla}_{k}}$ when acting on $\mathbf{\hat{V}_{jk}},$ and given
the fact that $\mathbf{\hat{\nabla}_{i}}\,\mathbf{\hat{V}_{jk}}\;=\;0$ for
$i\,\neq\,j,\;\;i\,\neq\,k$ it is clear that the operator, $\mathbf{\hat{P}}$,
simply passes through $\hat{\mathcal{V}}$, and acts only on the wave function,
$\psi$. So the normalized value of momentum is conserved at every point in
configuration space.

The orbital angular momentum operator can be represented as:

$\mathbf{\hat{L}}=-i\hbar\sum_{i}\,\mathbf{w_{i}}\mathbf{\times}\mathbf{\hat{\nabla}_{i}}.$
(7.5)

Explicit expansions for the x-component for systems, $j$, and $k$, are as
follows:

$\mathbf{\hat{L}_{x}}(\mathbf{w_{j}})\;=\;-i\hbar(y_{j}\partial{\mathbf{\hat{V}_{jk}}}/\partial{z_{j}}-z_{j}\partial{\mathbf{\hat{V}_{jk}}}/\partial{y_{j}}).$
(7.6)
$\mathbf{\hat{L}_{x}}(\mathbf{w_{k}})\;=\;-i\hbar(y_{k}\partial{\mathbf{\hat{V}_{jk}}}/\partial{z_{k}}-z_{k}\partial{\mathbf{\hat{V}_{jk}}}/\partial{y_{k}}).$
(7.7)

The action of $\mathbf{\hat{L}_{x}}$ on each $\mathbf{\hat{V}_{jk}}$ is given
by:

$\begin{array}[]{ll}[\mathbf{\hat{L}_{x}}(\mathbf{w_{j}})\,+\,\mathbf{\hat{L}_{x}}(\mathbf{w_{k}})]\mathbf{\hat{V}_{jk}}\;=\;-i\hbar[(y_{j}\,+\,y_{k})(\partial{\mathbf{\hat{V}_{jk}}}/\partial{z_{j}}\,+\,\partial{\mathbf{\hat{V}_{jk}}}/\partial{z_{k}})&\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\;(z_{j}\,+\,z_{k})(\partial{\mathbf{\hat{V}_{jk}}}/\partial{y_{j}}\,+\,\partial{\mathbf{\hat{V}_{jk}}}/\partial{y_{k}})].\end{array}$
(7.8)

Since we have both:

$(\partial{\mathbf{\hat{V}_{jk}}}/\partial{z_{j}}\,+\,\partial{\mathbf{\hat{V}_{jk}}}/\partial{z_{k}})\;=\;0;\;\;\;\;\;\;(\partial{\mathbf{\hat{V}_{jk}}}/\partial{y_{j}}\,+\,\partial{\mathbf{\hat{V}_{jk}}}/\partial{y_{k}})\;=\;0,$
(7.9)

it is clear that $\mathbf{\hat{L}}$, like $\mathbf{\hat{P}}$, effectively
operates only on the wave function, $\psi$, and not on $\hat{\mathcal{V}}$.
So, 3.3 also strictly conserves orbital angular momentum at a point.

This completes the demonstration that 3.3 is consistent with strict
conservation of momentum and angular momentum in individual measurement
situations. The issue of energy conservation is dealt with in the next
section.

##  8 Energy and Nonrelativistic Limitations

Conventional nonrelativistic quantum theory accurately characterizes the
status of energy conservation for stationary states and in situations with
freely evolving noninteracting particles. In these states there is no
radiation, and even though the theory does not represent relativistic
corrections to kinetic energy associated with mass increase, there is no
breach of conservation since the correction terms do not change.

The proposed collapse equation agrees with conventional theory in these cases
since it reduces to the Schrödinger equation when there is no exchange of
energy between systems. This is obvious when there is no interaction, and the
collapse operator also goes to zero in stationary states since it includes the
terms, $\gamma_{jk}$, which are based on the rate of change of potential
energy, which is zero in these situations.

In all other situations when there is an exchange of energy between systems
there is some inaccuracy in the conventional theory’s prediction of perfect
conservation of kinetic-plus-potential energy. This is due to the fact that it
lacks the means to represent some forms of energy. These include radiation,
relativistic corrections to the kinetic energy formula, antiparticles, and
interactions that cannot be fully characterized with scalar potentials.

The collapse operator in 3.3 does induce a change in energy at each point in
configuration space that is proportional to the change in $\psi^{*}\psi$ just
as in the cases of momentum and angular momentum analyzed in the previous
section. This accounts for the primary change in energy that occurs in
measurement situations,121212There are also energy changes induced by the
measurement apparatus, but these do not pose any problem for energy
conservation. and this change is consistent with strict energy conservation.

However, in addition to correctly accounting for the change in energy in the
measured subsystem the proposed collapse equation does predict some small
deviations from strict conservation of energy. This is due to the fact that it
is based on the conventional nonrelativistic theory and that it is designed to
act in exactly the situations in which the conventional theory fails to
account for all of the energy changes. What will be shown here is that these
deviations originate from exactly the same types of interactions that lead to
small errors in the predictions of the conventional theory. The fact that the
deviations predicted by the collapse proposal are qualitatively and
quantitatively similar to the inaccuracies of the conventional theory strongly
suggests that they should be regarded as a consequence of the nonrelativistic
formulation of both theories, rather than as an indication of a real problem
with energy conservation.

The change induced in a particular quantity by the collapse process was given
in equation 7.2. In parallel with the calculations of the previous section the
effects of the collapse operator on energy can be evaluated by considering the
relationship between $\mathbf{\hat{H}}\,\hat{\mathcal{V}}_{jk}$ and
$\hat{\mathcal{V}}_{jk}\,\mathbf{\hat{H}}.$ Since it is clear that, at each
point in configuration space,
$\mathbf{\hat{V}}\hat{\mathcal{V}}\;=\;\hat{\mathcal{V}}\mathbf{\hat{V}},$ the
only deviations from perfect conservation that can arise are those involving
the kinetic energy terms. Because the kinetic energy operator involves the
second derivative the expansion of
$-\frac{\hbar^{2}}{2m}\mathbf{\nabla}^{2}\,\hat{\mathcal{V}}_{jk}$ is more
complicated than the earlier calculations:

$\begin{array}[]{ll}(-\frac{\hbar^{2}}{2m}\mathbf{\nabla}^{2})\,\hat{\mathcal{V}}_{jk}\;\;=\;\;\hat{\mathcal{V}}_{jk}(-\frac{\hbar^{2}}{2m}(\mathbf{\nabla_{j}}^{2}\,+\,\mathbf{\nabla_{k}}^{2}))&\\\
\;\;\;\;\;\;\;\;\;\;\;\;+\;\;(-\frac{\hbar^{2}}{2m})\,\\{\,(\mathbf{\nabla_{j}}^{2}\hat{\mathcal{V}}_{jk}\,+\,\mathbf{\nabla_{k}}^{2}\hat{\mathcal{V}}_{jk}\;+\;2(\mathbf{\nabla_{j}}\hat{\mathcal{V}}_{jk}\,\cdot\,\mathbf{\nabla_{j}}\,+\,\mathbf{\nabla_{k}}\hat{\mathcal{V}}_{jk}\,\cdot\,\mathbf{\nabla_{k}})\\}.\end{array}$
(8.1)

As in the cases of momentum and angular momentum the changes in energy induced
by the Hamiltonian and the collapse operator decouple. The collapse-induced
changes can be computed by substituting $\mathbf{\hat{H}}$ for
$\mathbf{\hat{Q}}$ in the relevant part of 7.2, using 8.1, and expanding:

$\begin{array}[]{ll}d(\psi^{*}\,\mathbf{\hat{Q}}\,\psi)_{C}\;=\;\;\psi^{*}\,\hat{\mathcal{V}}\mathbf{\hat{H}}\,\psi\,\,(d\xi^{*}\,+\,d\xi)&\\\
&\\\
-\;\;(\frac{\hbar^{2}}{2m})\,\psi^{*}\,\\{\sum_{j<k}\,[(\mathbf{\nabla_{j}}^{2}\hat{\mathcal{V}}_{jk}\,+\,\mathbf{\nabla_{k}}^{2}\hat{\mathcal{V}}_{jk})\psi\;+\;2(\mathbf{\nabla_{j}}\hat{\mathcal{V}}_{jk}\,\cdot\,\mathbf{\nabla_{j}}\psi\,+\,\mathbf{\nabla_{k}}\hat{\mathcal{V}}_{jk}\,\cdot\,\mathbf{\nabla_{k}}\psi)]\\}\,d\xi&\\\
&\\\
+\;\;(\frac{\hbar^{2}}{2m})\psi^{*}\psi\,(\mathbf{\nabla}\hat{\mathcal{V}}\cdot\mathbf{\nabla}\hat{\mathcal{V}})\,dt.\end{array}$
(8.2)

The first line expresses the desired proportionality between
$d(\psi^{*}\,\mathbf{\hat{Q}}\,\psi)_{C}$ and $d(\psi^{*}\,\psi)_{C}$, and
mirrors the relationships for momentum and angular momentum derived in the
previous section. If it were not for the deviations shown on the second and
third line this would imply that strict energy conservation is maintained
within the nonrelativistic theory. If it can be shown that the problematic
terms displayed on the second and third lines are artifacts of the
nonrelativistic formulation, and that they do not represent real deviations
then the case will have been made that wave function collapse is consistent
with strict conservation of energy.131313As argued in Section 6, the common
presumption that the apparent narrowing of the wave function associated with
collapse must lead to violations of energy conservation stems from the failure
to recognize the entanglement of the measured system with systems with which
it has previously interacted and the fact that these systems are also involved
in the collapse.

Unless a very high level of precision is required relativistic corrections to
the nonrelativistic expressions become relevant only at fairly high energies.
The magnitudes of the inaccuracies in the predictions of the conventional
theory are well below the level of precision that is usually required. It will
be shown here that the apparent deviations implied by the collapse proposal
are of similar magnitudes. There are substantial differences between the
deviations represented by the terms on the second and third line of 8.2, and
so they will be dealt with separately.

Consider an interaction between system $j$ and system $k$, and focus on just
the effect on system $j$. Using the definition of $\hat{\mathcal{V}}_{jk}$
from Section 3 the relevant part of the second line can be expanded as:

$(-\frac{\hbar^{2}}{2m})\mathbf{\big{[}}\psi^{*}\,\psi(\mathbf{\nabla_{j}}^{2}\,\mathbf{\hat{V}_{jk}})\;+\;2\psi^{*}(\mathbf{\nabla_{j}}\mathbf{\hat{V}_{jk}}\,\cdot\,\mathbf{\nabla_{j}}\psi)\mathbf{\big{]}}(\frac{1}{(m_{j}+m_{k})c^{2})})\,(\sqrt{\gamma_{jk}}\,d\xi).$
(8.3)

When there is a change in kinetic energy some of that change is unaccounted
for in the conventional theory due to relativistic change in mass. (There is
also some radiation, although it is several orders of magnitude less than the
kinetic energy discrepancies.) So let us compare the deviations expressed in
8.3 to the rate at which potential energy is converted to kinetic energy
according to the Schrödinger equation:

$(\frac{i\hbar}{2m})\mathbf{\big{[}}\psi^{*}\,\psi(\mathbf{\nabla_{j}}^{2}\,\mathbf{\hat{V}_{jk}})\;+\;2\psi^{*}(\mathbf{\nabla_{j}}\mathbf{\hat{V}_{jk}}\,\cdot\,\mathbf{\nabla_{j}}\psi)\mathbf{\big{]}}.$
(8.4)

The expressions in the square brackets in 8.3 and 8.4 are identical. This
illustrates the fact that the deviations implied by 3.3 arise in exactly the
same situations in which the conventional theory fails to account for all of
the changes in kinetic energy associated with the lowest order relativistic
corrections.

In 8.3 the term $(\sqrt{\gamma_{jk}}\,d\xi)$ integrates to a magnitude of
order, $1$, as decribed in Section 3. One can see that the size of the
deviation is determined by the ratio of the interaction energy to the total
relativistic energy. This deviation can be compared to the discrepancy
associated with the conventional theory. The full relativistic kinetic energy
can be represented as:
$KE_{rel}\,=\,\sqrt{m_{0}^{2}c^{4}+p^{2}c^{2}}\,-m_{0}c^{2}.$ Expanding this
expression about $\mathbf{p}^{2}\,=\,0$, and retaining the two lowest order
terms we get:

$KE_{rel}\,=\,m_{0}c^{2}\\{[1+\frac{1}{2}p^{2}/m_{0}^{2}c^{2}-\frac{1}{8}p^{4}/m_{0}^{4}c^{4}+\,...]\,-1\\}\,=\,\frac{1}{2}p^{2}/m_{0}-\frac{1}{8}p^{4}/m_{0}^{3}c^{2}\,+\,...$
(8.5)

Since the nonrelativistic formula for kinetic energy is
$KE_{nr}\,=\,p^{2}/(2m),$ the first-order correction can be rewritten as
$-KE_{nr}*(\frac{1}{2})(KE_{nr}/mc^{2}).$ The change in kinetic energy is
(approximately) equal to the change in potential energy. Thus, like the
deviation implied by 3.3, the unaccounted for change in kinetic energy,
$\sim(\Delta\mathbf{\hat{V}}/mc^{2})*KE_{nr}$, in the conventional
nonrelativistic theory is also proportional to the ratio of the interaction
energy to the total relativistic energy.

It is also worth noting that for interactions involved in transitions from
free states to free states the energy deviations that occur when the two
systems are receding from one another tend to cancel those that occurred when
they were approaching.

For interactions that lead to transitions to or from bound states the
conventional theory fails completely to account for energy changes. These
situations involve the emission or absorption of photons with energies
approximately equal to the change in kinetic energy. The standard theory is
useful for calculating the energy discrepancies, but cannot explain them.

The deviations listed on the third line of 8.2 involve two factors of the
ratio of interaction energy to total relativistic energy. Thus, for typical
nonrelativistic situations they are several orders of magnitude smaller than
the those listed on the second line. There are a couple of compelling reasons
for regarding these apparent deviations as artifacts of the nonrelativistic
formulation rather than as real effects. First, they are less than radiative
effects by a ratio of $(c/v)$. Second, in cases with much higher ratios of
interaction energy to total relativistic energy, creation and annihilation
processes would become relevant, and antiparticle content of the wave would
need to be considered. Since there is always some small amount of antiparticle
content in any localized wave function141414For a calculation of the
antiparticle content of localized wave packets see the discussion by Bjorken
and Drell in [40] (p.39). The ratio of the apparent deviation to radiated
energy is based on the classical Larmor formula which yields a ratio of
radiated power to energy change of
$(\frac{2}{3})\frac{c}{v}\mathbf{\big{[}}\frac{(e^{2}/r)}{(mc^{2})}\mathbf{\big{]}}^{2}.$
it is reasonable to view these small discrepancies as resulting from the
limitations of the nonrelativistic formulation.

To summarize, wave function collapse is not restricted to just the measured
subsystem and the measurement apparatus. It also involves the systems with
which the measured subsystem has previously interacted, even if the relevant
entanglement is very small. The change in energy of the measured subsystem
that is not attributable to interactions with the measurement apparatus is
compensated for by correlated changes in these entangled systems. The small
deviations that remain unaccounted for are attributable to the fact that the
nonrelativistic theory is not able to describe certain forms of energy.

The next section examines the possibility of extending the current collapse
proposal to account for relativistic effects.

##  9 Possible Relativistic Extensions

Although the formulation of the collapse equation presented here is
essentially nonrelativistic, as mentioned earlier the idea that wave function
collapse is induced by the interactions that establish correlations between
systems was originally motivated by the desire to reconcile relativity with
the nonlocal aspects of quantum theory. The focus on these interactions
stemmed from the central role that they play in the transmission of
information. It is, therefore, reasonable to ask what are the prospects for
extending this proposal to achieve full consistency with relativity.

As argued in previous works[15, 16] full consistency requires a
reconsideration of the foundations of both quantum theory and relativity.
Recalling Einstein’s repeated insistence that any inferences about the metric
properties of space and time must be based on the observation of physical
objects and processes[41, 42], one must ask whether those properties are
underdetermined by the probabilistic character of some physical processes. To
what extent could such underdetermination hide some features of spacetime
structure?

Consider that in quantum field theory Lorentz invariance is not guaranteed
simply by the limiting speed of light. It is necessary to add the assumption
that spacelike-separated interactions commute.[43] Given the demonstration by
Bell[44] that quantum correlations imply some sort of nonlocal effects, what
is it that prevents the assignment of an unambiguous sequencing to spacelike-
separated interactions involving entangled systems? It is, precisely, the
probabilistic character of nonlocal quantum effects. The reason that the
relativistic description of spacetime is not disrupted by nonlocal quantum
effects is that these effects are inherently probabilistic.

These considerations raise the possibility that there are spacetime features
such as an evolving, spacelike hypersurface, along which the nonlocal quantum
effects propagate. This structure remains hidden because of the
nondeterministic nature of these effects. From this perspective the preferred
reference frame used here to construct a low energy collapse theory can be
viewed as just a simplified special case of such an evolving surface.

In order to extend the current proposal to a relativistic version the most
straightforward approach would be to simply replace the preferred frame by a
randomly evolving spacelike surface. Although any such surface does imply a
sequencing of spacelike-separated interactions, since it evolves in a purely
random fashion it remains impossible, in principle, to associate the specific
sequencing with any observable physical effects. This approach also has the
advantage that it maintains the equivalence of all reference frames, which is
one of the essential features of relativity. Since the evolving surface would
be unobservable in principle, the usual relativistic description of spacetime
should emerge from the proposed extension of 3.3.

Such an extension would need to characterize interactions in a fully
relativistic manner. So a stochastic operator based on interactions would
involve more than scalar potentials. The account would need to incorporate all
of the features of quantum field theory - massless particles, antiparticles,
particle creation and annihilation, and other high energy effects. Such a
development is, obviously, not trivial. But I believe that the challenges it
would face are of a technical, rather than a fundamental conceptual nature.

##  10 Summary

The hypothesis that wave function collapse is induced by the entangling
interactions that generate decoherent branches leads to a stochastic collapse
equation with several attractive properties. It reduces to the Schrödinger
equation for stationary states and freely evolving systems, and deviates from
it by a very small amount in situations that involve a few interacting
elementary particles. It insures collapse to a definite outcome for systems of
mesoscopic size with the correct probability and on a time scale that is
consistent with our macroscopic experience. Because the strength and duration
of the collapse effects are determined by the ratio of interaction energy to
total relativistic energy it does not require the introduction of any new
physical constants.

Furthermore, it is consistent with exact conservation of momentum and orbital
angular momentum in individual experiments, and it is consistent with energy
conservation in those circumstances in which conventional nonrelativistic
quantum theory correctly predicts such conservation. This consistency is based
on the recognition that all physical systems are quantum systems, and that
they are almost always entangled to some extent. Since conserved quantities
are shared across entangled branches the apparent nonconservation of a
quantity in an elementary target system is compensated for by corresponding
changes in (usually larger) systems with which it has interacted in the past.

Finally, compatibility with relativity can be achieved by replacing the fixed
rest frame with an evolving spacelike surface. Nonlocal quantum effects
propagate along the surface, and their nondeterministic nature prevents
superluminal information transmission. This surface could be taken to evolve
in a purely random fashion. This possibility respects the equivalence of all
inertial frames, one of the central principles of relativity. With this
approach one might reasonably hope to develop an account that explains
measurement outcomes in terms of fundamental processes, while preserving the
essential features of contemporary theory.

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