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\pagestyle{myheadings} \markboth{Wayne C. Myrvold}{Einstein's
Untimely Burial}
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\begin{document}
\title{Einstein's Untimely Burial}
\author{Wayne C. Myrvold \\ Department of Philosophy \\
University of Western Ontario \\ London, ON Canada N6A 3K7 \\
e-mail: wmyrvold@julian.uwo.ca} \maketitle
\begin{center}\emph{Presented at PSA \\ November 4, 2000 \\ Vancouver, BC}\end{center}
\begin{abstract}
There seems to be a growing consensus that any interpretation of
quantum mechanics other than an instrumentalist interpretation
will have to abandon the requirement of Lorentz invariance, at
least at the fundamental level, preserving at best Lorentz
invariance of phenomena. In particular, it is often said that the
collapse postulate is incompatible with the demands of relativity.
It is the purpose of this paper to argue that such a conclusion is
premature, and that a covariant account of collapse can be given
according to which the state histories yielded by different
reference frames are the Lorentz transforms of each other.
\end{abstract}
%\doublespacing
\section{Introduction}
There seems to be a growing consensus that any interpretation of
quantum mechanics other than a instrumentalist one---that is, any
interpretation that purports to provide a description of events
and processes between measurements---will have to sacrifice the
requirement of Lorentz invariance at the fundamental
level,\footnote{The term ``fundamental Lorentz invariance'' was
introduced by Dickson and Clifton (1998).} preserving at best
phenomenal Lorentz invariance, or invariance regarding empirical
predictions (see, \emph{e.g.}, Cushing 1994, Maudlin 1994, 1996).
In particular, it has often been said that the collapse postulate
is incompatible with relativity and hence that interpretations
that take state-vector reduction to be a real physical process
must abandon relativity, perhaps by introducing a preferred notion
of distant simultaneity. It is the purpose of this paper to argue
that such a conclusion is premature at best, and that there is
reason to believe that an account of quantum-mechanical state
reduction can be given that does not introduce a preferred
foliation.
It is not enough to formulate the theory in a manifestly covariant
form. If there are differences in the accounts given with respect
to different reference frames, then one must be able to argue, as
Einstein did in formulating the special theory of relativity, that
these differences are nothing more than different descriptions of
the same reality, using different coordinates.
%\begin{itemize} \item The notion of a state of an extended
%system, be it a classical or quantum state,
% is, in a relativistic context, a foliation-relative notion.
% \item Nevertheless, we can regard quantum states as evolving
% relativistically, as long as the global evolution is a product of
% local evolutions.
% \item This remains the case even when collapse evolution is
% taken into account.
%\end{itemize}
\section{The Relativity of Entanglement}
The notion of an instantaneous state of a spatially extended
object, or an extended system of objects, is a foliation-relative
notion. The state of the system is its state along some temporal
slice, or, in other words, its state along some spacelike
hyperplane or other spacelike hypersurface of simultaneity. This
is vividly illustrated by the well-known example of the pole and
the barn. With respect to a foliation of hyperplanes orthogonal
to the barn's worldline, there is a temporal slice of the pole
that lies entirely within the temporal slice of the barn along the
same hyperplane. With respect to a foliation of hyperplanes
orthogonal to the pole's worldline, there is no temporal slice of
the pole that lies entirely within the temporal slice of the barn
along the same hyperplane.
A comment is in order to forestall potential misunderstandings.
Although, in special relativity, one associates a particular
foliation of spacetime into spacelike hyperplanes with each state
of inertial motion, which could be the state of motion of some
observer, to say that something is foliation-relative is not to
say that it is relative to an observer or that it is subjective in
any way. No one is obliged to use his or her rest frame as a
preferred frame of reference; quite the contrary, it is the lesson
of relativity that it is quite immaterial which inertial reference
frame one uses. Those who hold that an observer must always refer
motion to that observer's rest frame, and hence must always regard
him or herself as being at rest, are invited to consider whether
it makes sense, while driving in a car, to look at the speedometer
and say, ``I'm moving at 50 miles per hour (with respect to the
road)'' rather than ``The road is moving at 50 miles per hour
(with respect to me).'' A choice of reference frame is a choice
of coordinate system; a choice of foliation is a choice of a
global time coordinate, nothing more.
Let us now consider the transformation of quantum states. Consider
two spin-$\half$ particles, at rest with respect to some reference
frame $K$. We will assume that the particles are localized in two
regions, located at $\mathbf{x}_1$ and $\mathbf{x}_2$,
respectively, that are small enough, compared to the distances
between them, that the regions may be regarded as points, but
large enough compared to the Compton wavelength of the systems
that the systems can indeed be regarded as localized within the
regions. Let $P$ be a point on $S_1$'s worldline, and let $Q$ and
$R$ be two points on $S_2$'s worldline, with $R$ in the past of
$Q$ (see Figure 1).\begin{figure}[htb]
\setlength{\unitlength}{1in} \centering
\includegraphics[width=2.5in]{Figure_1.eps}
\center{\textbf{Figure 1.}}
\end{figure} Let $s$ be a spacelike
hypersurface that intersects $S_1$'s worldline at $P$ and $S_2$'s
worldline at $R$, and let $s'$ be a spacelike hypersurface that
also intersects $S_1$'s worldline at $P$, and intersects $S_2$'s
worldline at $Q$.
Suppose, first, that the initial state of the combined system is
\begin{equation}
\ket{\psi_1(0)} = \biket{z+}{z-},
\end{equation}
and suppose that a measurement of spin-$x$ is performed on $S_2$
at the spacetime point $M$, between $R$ and $Q$. The result of
such a measurement will be either $+1$ or $-1$, and, by the
projection postulate, the state of $S_2$ after the measurement
will be either $\ket{x+}$ or $\ket{x-}$. Suppose that the outcome
is $+1$. Then the state of the combined system along $s$ is
\begin{equation}
\ket{\psi_1(t_s)} = \biket{z+}{z-}
\end{equation}
whereas the state of the system along $s'$ is
\begin{equation}
\ket{\psi'_1(t'_{s'})} = \biket{z'+}{x'+}.
\end{equation}
There is nothing paradoxical in this state of affairs; the
difference in the two states is merely a consequence of a local
change pertaining to $S_2$, plus the familiar relativity of
simultaneity; the two hyperplanes link up two different moments in
the evolution of $S_2$ with the same moment in the evolution of
$S_1$ to form their respective instantaneous temporal slices of
the combined system.
Now consider a scenario in which the initial state of the combined
system is
\begin{equation}
\ket{\psi_2(0)} = \biket{z-}{z+}.
\end{equation}
Suppose, again, that spin-$x$ is measured at $M$, and that again
the result is +1. Then, along $s$, the state of the system is
\begin{equation}
\ket{\psi_2(t_s)} = \biket{z-}{z+}
\end{equation}
while the state of the combined system along $s'$ is
\begin{equation}
\ket{\psi'_2(t'_{s'})} = \biket{z'-}{x'+}.
\end{equation}
Now let the initial state of the combined system be a linear
superposition of the states in the first two scenarios:
\begin{equation}
\ket{\psi_3(0)} = C_1 \: \biket{z+}{z-} + C_2 \: \biket{z-}{z+}
\end{equation}
We will suppose once again that a measurement of spin-$x$ is
performed at $M$, and that the outcome is $+1$. Then, in this
scenario, the state at any time, and along any hyperplane, is
simply a superposition of the states in the first two scenarios.
Along $s$ we have,
\begin{equation}\label{tangle}
\ket{\psi_3(t_s)} = C_1 \: \biket{z+}{z-} + C_2 \: \biket{z-}{z+}
\end{equation}
Along $s'$, the state is:\footnote{In discussing collapse, it will
be more convenient not to require the state vector to be
normalized at all times.}
\begin{eqnarray}\label{product}
\nonumber \ket{\psi'_3(t'_{s'})} &=& C_1 \: \ket{z'+}_1 \otimes
\left(_2\bkt{x'+}{z'-}_2 \right)\ket{x'+}_2 + C_2 \: \ket{z'-}_1
\otimes \left(_2\bkt{x'+}{z'-}_2 \right) \ket{x'+}
\\ &=& \frac{1}{\sqrt{2}} \left( C_1 \ket{z'+}_1 + C_2 \ket{z'-}_1 \right) \otimes
\ket{x'+}_2.
\end{eqnarray}
Here we have a superposition of two states that undergo local
changes in the transition from $s$ to $s'$, pertaining only to
$S_2$. This change in the state vector ought also to count as a
local change pertaining to $S_2$. Here again the two hyperplanes
merely link up two different moments in the evolution of $S_2$
with the same moment in the evolution of $S_1$ to form
instantaneous temporal slices of the combined system; the only
difference is that we have here to do with a superposition of such
spliced states.
There is, of course, an important difference between the third
scenario and the first two. In the first two scenarios, the state
of the system is a factorizable state along both hyperplanes. In
the third, the state of the combined system is an entangled state
along $s$ and a factorizable state along $s'$. This seems odd, as
we think of the factorizable state (\ref{product}) as attributing
a definite spin state to $S_1$, whereas in the entangled state
(\ref{tangle}), $S_1$ has no spin state of its own at all. An
analogy might help: it is as if a marriage could be dissolved
unilaterally by the declaration of one spouse. If a husband and
wife are some distance apart at the moment that the wife declares
the marriage dissolved, then the question of whether, at a given
point on his worldline that is spacelike separated from the
declaration, the husband is married or divorced, requires a choice
of hypersurface of simultaneity.
This circumstance, that, at a spacetime point $P$ a system may be
part of an entangled state along one hypersurface passing through
$P$, and part of a factorizable state along another hypersurface
passing through $P$, may be called the \emph{relativity of
entanglement}. It is a consequence jointly of the relativity of
simultaneity and of modelling collapse as a local change in the
state vector.
%\newpage
\section{Transformations between foliations}
The instantaneous state of an extended system, classical or
quantum, is defined along a spacelike hypersurface. Within a
foliation, the transition from one hypersurface to another is
given by the dynamical evolution of the system. How are states
defined hypersurfaces belonging to different foliations related to
each other?
Consider two systems,\footnote{The generalization to any finite
number of systems is straightforward, and to continuous fields by
a limiting process. See Tomonaga (1946).} $S_1$ and $S_2$, which
at time $t$ are located, with respect to some reference frame
$\Sigma$, at positions $\mathbf{x}_1(t)$ and $\mathbf{x}_2(t)$,
respectively. Let $\Sigma'$ be a reference frame moving with
velocity $v$ in the positive $x$-direction relative to $\Sigma$,
and let the transformation from $\Sigma$ to $\Sigma'$ be given by
the Lorentz boost,
\begin{eqnarray}\label{boost}
x' &=& \gamma\left(x - v \: t\right) \nonumber \\ y' &=& y \qquad
\qquad z' = z \nonumber
\\ t^{\prime}&=& \gamma\left(t- {v x}/{c^2}\right),
\end{eqnarray}
where $\gamma = 1 / \sqrt{1- v^2/c^2}$. Let $\mathcal{H}_1$ and
$\mathcal{H}_2$ be the Hilbert spaces associated with $S_1$ and
$S_2$, respectively, and let the states of the two systems at time
$t$ be given, with respect to $\Sigma$, by $\ket{u(t)}_1$ and
$\ket{v(t)}_2$.
By Wigner's theorem (see Weinberg 1995, 91--96), there is a
unitary transformation operator $\Lambda_1$ that takes vectors in
$\mathcal{H}_1$ into their transforms under the Lorentz boost
(\ref{boost}). Similarly, there is a unitary Lorentz boost
operator $\Lambda_2$ on $\mathcal{H}_2$. As Dickson and Clifton
(1998, 15) have shown, the transformation operator on
$\mathcal{H}_1 \otimes \mathcal{H}_2$ is simply $\Lambda_1 \otimes
\Lambda_2$. Therefore, for any time $t$, the Lorentz transform of
$\left|\psi(t)\right\rangle_{12}$ is
\begin{eqnarray}
\left|\psi^{\prime}\right\rangle_{12} & = & \Lambda_1
\left|u(t)\right\rangle_1 \otimes \Lambda_2
\left|v(t)\right\rangle_2 \nonumber \\ &=&
\left|u^{\prime}(t^{\prime}_1(t))\right\rangle_1 \otimes
\left|v^{\prime}(t^{\prime}_2(t))\right\rangle_2,
\end{eqnarray}
where
\begin{equation}
t'_i(t) = \gamma \left( t - v \: x_i(t)/{c^2} \right).
\end{equation}
These are not instantaneous temporal slices of the combined system
with respect to the $\Sigma'$'s hyperplanes of simultaneity.
Rather, the transform of a state $\biket{u(t)}{v(t)}$ at a given
$\Sigma$-time $t$ is a description of the states of the two
component systems at two different times, as measured by
$\Sigma'$'s time coordinate. In order to get instantaneous states
with respect to $\Sigma'$ from instantaneous states with respect
to $\Sigma$, we need to know something of the dynamical evolution
of the combined system.
Assume that the Hamiltonian of the combined system is simply the
sum of the component Hamiltonians, with no interaction term,
\begin{equation}
H_{12} = H_1 \otimes I_2 + I_1 \otimes H_2.
\end{equation}
Then the unitary evolution operator $U(t;t_0) = e^{{H (t-t_0)}/{i
\hbar}}$ factors:
\begin{equation}
U(t;t_0) = U_1(t; t_0) \otimes U_2(t; t_0).
\end{equation}
Once we have a factorizable evolution operator, we can obtain
states along the hyperplanes of constant $t'$ from state defined
on hyperplanes of constant $t$.
\begin{equation}\label{FTransform}
\biket{u'(t')}{v'(t')} = \Lambda_1 \: U_1(t_1(t');t) \ket{u(t)}
\otimes \Lambda_2 \: U_2(t_2(t');t) \ket{v(t)},
\end{equation}
where
\begin{equation}
t_i(t') = \gamma \left( t' + v \: x'_i(t') / c^2 \right).
\end{equation}
Because of the linearity of the Lorentz boost operators and of the
evolution operators, the transformation (\ref{FTransform}) applies
to any state of the combined system, and not only to factorizable
states:
\begin{equation}
\ket{\psi'(t')} = \Lambda_1 \: U_1(t_1(t');t) \otimes \Lambda_2 \:
U_2(t_2(t');t) \ket{\psi(t)}.
\end{equation}
Thus, the state history of the combined system with respect to
$\Sigma'$ can be obtained from a state history given with respect
to $\Sigma$, together with knowledge of the dynamics of the
system.
Let us now add collapse to the picture,\footnote{Though the notion
of foliation-relative state evolution was present from the early
days work on relativistic quantum theories (see Dirac 1933,
Tomonaga 1946, Schwinger 1951), the application of this notion to
state-vector collapse was perhaps first made by Aharonov and
Albert (1984). Fleming (1986, 1989, 1996) has perhaps been the
most prominent exponent of foliation-relative collapse.} and
assume the existence of generalized evolution operators
$E_1(t;t_0)$, $E_2(t;t_0)$ that approximate unitary evolution most
of the time but produce a collapse whenever conditions are ripe
(whatever that may turn out to be; such events ought not be
confined to the laboratory). We can assume these operators to be
linear if we don't require that they preserve the norm of the
state vector. After a measurement of, say, spin-$x$ on $S_1$, the
operator $E_1(t;t_0)$ becomes either $P^{\ket{x+}_1}$ or
$P^{\ket{x-}_1}$, with the probability for each transition given
by the usual quantum-mechanical rules for computing probabilities
(note that these involve the \emph{entire} global state vector;
the probabilities are non-local quantities). Collapse evolution
will \emph{not} be assumed to be a reversible process, and so in
general $E_i(t;t_0)$ will be undefined for $t < t_0$. Because of
this, we will not always be able to obtain a state of the system
along any given hyperplane of constant $t'$ from any state of
constant $t$; in order to obtain a $\Sigma'$-state at time $t'$,
we must start with a $\Sigma$-state at a time $t$ such that $t
\leq t_1(t')$ and $t \leq t_2(t')$. With this proviso, we have:
\begin{equation}\label{CollapseTransform}
\ket{\psi'(t')} = \Lambda_1 \: E_1(t_1(t');t) \otimes \Lambda_2 \:
E_2(t_2(t');t) \ket{\psi(t)}.
\end{equation}
for $t \leq \mbox{Min}[t_1(t'), t_2(t')]$. It thus remains true
that a complete state history given with respect to $\Sigma$
uniquely determines the state history given with respect to
$\Sigma'$.
\section{The EPR-Bohm experiment}
Let us now apply this picture of the collapse of
foliation-relative states to the familiar EPR-Bohm experiment.
Let $S_1$ and $S_2$ be spin-$\half$ particles, initially in the
singlet spin state,
\begin{eqnarray}
\nonumber \ket{\psi(0)} &=& \biket{z+}{z-} - \biket{z-}{z+} \\
&=& \biket{x+}{x-} - \biket{x-}{x+}.
\end{eqnarray}
Suppose that measurements of spin-$x$ and spin-$z$ are performed
on $S_1$ and $S_2$, respectively, at spacelike separation, and
suppose that the outcomes of the two experiments are both $+1$.
Let $\Sigma$ be a reference frame with respect to which the
measurement on $S_1$ is performed first, and let $\Sigma'$ be a
reference frame with respect to which the order of the
measurements is reversed. Let $t_a$ be a time, with respect to
$\Sigma$, prior to the measurement on $S_1$, let $t_b$ be after
the measurement on $S_1$ but before the measurement on $S_2$, and
let $t_c$ be a time after both measurements are completed (see
Figure 2).
\begin{figure}[htb] \setlength{\unitlength}{1in}
\centering
\includegraphics[width=3in]{Figure_2.eps}
\center{\textbf{Figure 2. Spacetime diagram of the EPR-Bohm
experiment.}}
\end{figure}
Define the `collapse evolution operators,'
\begin{eqnarray}\label{CollapseOp}
E_1(t) = \left\{
\begin{array}{r@{\quad}l}
I_1, & \mbox{before the measurement on }S_1. \\
P^{\left|x+\right\rangle_1}, & \mbox{after the measurement on
}S_1.
\end{array}
\right. \nonumber \\
\\
E_2(t) = \left\{
\begin{array}{r@{\quad}l}
I_2, & \mbox{before the measurement on }S_2. \\
P^{\left|z+\right\rangle_2}, & \mbox{after the measurement on
}S_2.
\end{array}
\right. \nonumber
\end{eqnarray}
The state history of the two-particle system, with respect to
$\Sigma$, is given by
\begin{equation}
\ket{\psi(t)} = E_1(t) \otimes E_2(t) \ket{\psi}_{singlet}.
\end{equation}
Along the hyperplane $t = t_a$, the state is
\begin{eqnarray}
\nonumber \ket{\psi(t_a)} &=& E_1(t_a) \otimes E_2(t_a)
\ket{\psi}_{singlet}
\\
&=& I_1 \otimes I_2 \ket{\psi}_{singlet} = \ket{\psi}_{singlet}.
\end{eqnarray}
Along the hyperplane $t = t_b$, we have,
\begin{eqnarray}
\nonumber \ket{\psi(t_b)} &=& E_1(t_b) \otimes E_2(t_b)
\ket{\psi}_{singlet}
\\ \nonumber
&=& P^{\ket{x+}_1} \otimes I_2 \ket{\psi}_{singlet}
\\ \nonumber
&=& P^{\ket{x+}_1} \ket{x+}_1 \otimes \ket{x-}_2 - P^{\ket{x-}_1} \ket{x+}_1 \otimes \ket{x+}_2
\\ &=& \biket{x+}{x-}.
\end{eqnarray}
Along the hyperplane $t = t_c$, the state is,
\begin{eqnarray}
\nonumber \ket{\psi(t_c)} &=& E_1(t_c) \otimes E_2(t_c)
\ket{\psi}_{singlet} \\ \nonumber &=& P^{\ket{x+}_1} \otimes
P^{\ket{z+}_2} \ket{\psi}_{singlet} \\ \nonumber &=&
P^{\ket{x+}_1} \ket{x+}_1 \otimes P^{\ket{z+}_2} \ket{x-}_2
\\ &=& - \frac{1}{\sqrt{2}} \: \biket{x+}{z+}.
\end{eqnarray}
To describe the evolution of the state as given by $\Sigma'$, we
define,
\begin{eqnarray}\label{CollapseOp'}
E^{\prime}_1(t^{\prime}) = \left\{
\begin{array}{r@{\quad}l}
I_1, & \mbox{before the measurement on }S_1. \\
P^{\left|x^{\prime}+\right\rangle_1}, & \mbox{after the
measurement on }S_1.
\end{array}
\right. \nonumber \\
\\
E^{\prime}_2(t^{\prime}) = \left\{
\begin{array}{r@{\quad}l}
I_2, & \mbox{before the measurement on }S_2. \\
P^{\left|z^{\prime}+\right\rangle_2}, & \mbox{after the
measurement on }S_2.
\end{array}
\right. \nonumber
\end{eqnarray}
These operators are simply the Lorentz transforms of the operators
(\ref{CollapseOp}).
Along the hyperplane $t' = t'_A$, the state is
\begin{eqnarray}
\nonumber \ket{\psi'(t'_A)} &=& E'_1(t'_A) \otimes E'_2(t'_A)
\ket{\psi'}_{singlet}
\\
&=& I_1 \otimes I_2 \ket{\psi'}_{singlet} = \ket{\psi'}_{singlet}.
\end{eqnarray}
Along the hyperplane $t' = t'_B$, we have,
\begin{eqnarray}
\nonumber \ket{\psi'(t'_B)} &=& E'_1(t'_B) \otimes E'_2(t'_B)
\ket{\psi}_{singlet}
\\ \nonumber
&=& I_1 \otimes P^{\ket{z'+}_2} \ket{\psi'}_{singlet}
\\ \nonumber
&=& \ket{z'+}_1 \otimes P^{\ket{z'+}_2} \ket{z'-}_2 - \ket{z'-}_1
\otimes P^{\ket{z'+}_2} \ket{z'+}_2 \\ \nonumber
&=& - \biket{z'-}{z'+}.
\end{eqnarray}
Along the hyperplane $t' = t'_C$, the state is
\begin{eqnarray}
\nonumber \ket{\psi'(t'_C)} &=& E'_1(t'_C) \otimes E'_2(t'_C)
\ket{\psi'}_{singlet} \\ \nonumber &=& P^{\ket{x'+}_1} \otimes
P^{\ket{z'+}_2} \ket{\psi'}_{singlet} \\ \nonumber &=& -
P^{\ket{x'+}_1} \ket{z'+}_1 \otimes P^{\ket{z'+}_2} \ket{z'+}_2
\\ &=& - \frac{1}{\sqrt{2}} \: \biket{x'+}{z'+}.
\end{eqnarray}
The difference in the two state histories can clearly be seen to
arise solely from the difference in foliations used to define the
instantaneous states of the evolving system, and therefore, the
two state histories are, in a straightforward way, the Lorentz
transformations of each other. This in spite of the fact that,
along $t_b$, the state of the system is $\biket{x+}{x-}$, even
though $S_2$ is \emph{never} in the state $\ket{x-}_2$ on the
state history according to $\Sigma'$, and, along $t'_B$, the state
of the system is $-\biket{z-}{z+}$, even though, on the state
history according to $\Sigma$, $S_1$ is never in the state
$\ket{z'-}$.\footnote{It should be pointed out that, whenever the
states along two hypersurfaces passing through a point $P$ on
$S_1$'s worldline both assign a definite spin state to $S_1$, they
will agree on what spin state it is.}
Since there is a collapse between the hyperplane $t = t_b$ and the
hyperplane $t'= t'_B$ in both directions, we cannot apply
(\ref{CollapseTransform}) directly to one of these states to
obtain the other. This doesn't mean that the two states are
unrelated, however, as they both can be obtained from the state
along $t = t_a$:\footnote{This answers an objection raised by
Maudlin (1996, 302) to foliation-relative collapse.}
\begin{equation}
\ket{\psi(t_b)} = E_1(t_b) \otimes E_2(t_b) \ket{\psi(t_a)}
\end{equation}
\begin{equation}
\ket{\psi(t'_B)} = \Lambda_1 \otimes \Lambda_2 E_2(t_c)
\ket{\psi(t_a)}
\end{equation}
For convenience, the hyperplanes $t = t_a$ and $t' = t'_B$ have
been taken to intersect $S_1$'s worldline at the same point, and
the hyperplanes $t = t_c$ and $t' = t'_B$ have been taken to
intersect $S_2$'s worldline at the same point.
\section{Probabilities, causality, and passion-at-a-distance}
The quantum-mechanical rule for assigning probabilities to
outcomes of measurements is easily formulated in terms of
foliation-relative states. Let $F$ be any foliation, and let
$\sigma$ be the member of $F$ passing through the measurement
event (or, perhaps, immediately to the past of the measurement
event). Then the expectation value of a measurement of an
observable $A$ is given by
\begin{equation}\label{ProbRule}
\expval{A}_\sigma = \frac{\bra{\psi(\sigma)} A
\ket{\psi(\sigma)}}{\bkt{\psi(\sigma)}{\psi(\sigma)}}.
\end{equation}
Probabilities, therefore, are foliation-relative. In spite of
this, the rule (\ref{ProbRule}) does not pick out a
\emph{preferred} foliation, as long as the Hamiltonian contains no
nonlocal interaction terms. Although a statistical test of quantum
mechanics requires calculation of probabilities with respect to
some foliation-relative state evolution, it doesn't matter which
foliation is chosen.\footnote{This suggests that there ought to be
a way to formulate the rule in a manifestly covariant way, without
mention of any non-intrinsic structures; so far, however, this has
not been done. Unless and until such a formulation is available,
we must refer probability assignments to foliation-relative
states.}
The foliation-relativity of probabilities has the odd consequence
that a collapse event, even though it is a local event, will
sometimes be assigned different probabilities by states along
different hyperplanes passing through a single point. There is no
limit on \emph{how} different these probabilities can be. Suppose,
for example, that the initial state of our two-particle system is,
instead of the singlet state, the state,
\begin{equation}
\epsilon \biket{z+}{z-} + \sqrt{1 - \abs{\epsilon}^2} \:
\biket{z-}{z+}
\end{equation}
with $\abs{\epsilon} << 1$. Suppose that measurements of spin-$z$
are performed on both particles at spacelike separation, and that
the outcome of the measurement on $S_2$ is $-1$. Then, a
hypersurface passing through the measurement on $S_1$ and crossing
$S_2$'s worldline to the past of the measurement on $S_2$ will
assign a very small probability $\abs{\epsilon}^2$ to the outcome
$+1$ of the measurement on $S_1$, whereas a hypersurface passing
through the measurement on $S_1$ and crossing $S_2$'s worldline to
the future of the measurement on $S_2$ will assign a probability
$1$ to the same outcome.
The fact that probability assignments do not factor into
independent local probabilities (and, by Bell's theorem, cannot be
regarded as supervening on independent local probabilities), is
seen by some as a form of superluminal causal influence
(\emph{e.g.} Maudlin 1994). There is, however, a marked difference
between the Bell-inequality violating dependence between systems
in entangled states and an interaction of the sort that would be
modelled by an interaction term in the Hamiltonian of the combined
system. For spatially separated systems, such a term could not
result in Lorentz-invariant state evolution, or even
Lorentz-invariant statistics, but would on the contrary require a
preferred notion of distant simultaneity. Indeed, any such
interaction term will permit superluminal signalling via suitable
measurements on suitable states.
It is not a matter of great moment whether we call the
quantum-mechanical failure of probabilistic independence of
spacelike separated events a form of causal influence, or invent a
new term for it, such as ``passion-at-a-distance'' (Shimony 1984).
Brian Skyrms (1984) is correct when he says that our notion of
causality is an ``amiably confused jumble'' which unravels when
applied to the quantum domain. What does matter is that we not
ignore the fact that there is a difference between the way in
which quantum mechanics treats this sort of influence and its
treatment of causal interactions modelled by interaction terms in
the Hamiltonian; if our physical theories are a guide for forming
causal notions and applying them to the world, this suggests that
at the very least we are dealing with something other than
familiar causal interactions. Moreover, there seems to a physical
basis for this distinction, as exhibited by the fact that
interactions given by interaction terms in the Hamiltonian can be
manipulated for signalling, whereas the lack of probabilistic
independence that arises from quantum entanglement cannot, in the
absence of nonlocal interaction terms in the Hamiltonian, be used
to send superluminal signals.\footnote{This point has, of course,
been made many times before, and is often couched in terms of Jon
Jarrett's factorization of the Bell locality condition into a
conjunction of the condition that Jarrett (1984) calls
``Locality'' and Shimony (1986) calls ``Parameter Independence,''
on the one hand, and the condition that Jarrett calls
``Completeness'' and Shimony calls ``Outcome Independence,'' on
the other.}
The distinction is important because, relativity theory does not,
by itself, prohibit superluminal causation. If the special theory
of relativity gives us reason to disbelieve in superluminal
causation, the reason lies in our belief that the causal relation
requires a unique temporal ordering of cause and effect. What is
ruled out by relativity is any relation between spacelike
separated events that requires a unique temporal ordering. The
relation of signal transmission to reception is such a relation.
It appears that the relation that holds between quantum systems in
an entangled state is not such a relation. If this is so, then
there is reason to hope for peaceful coexistence between quantum
mechanics and special relativity after all.
\section{Objections}
The picture presented here is of stochastic evolution of
foliation-relative states, with no foliation preferred. A number
of objections have appeared in the literature to such a notion.
There is not space here to address these objections in full, but
it is possible to give the outlines of a response to the chief
objections.
One objection is that foliation-relative state evolution, and the
associated notion of foliation-relative becoming, makes such
evolution a subjective matter and results in a breakdown of
intersubjectivity (Dorato 1995, 593). A preliminary answer to
this objection is to point out that the objection presupposes that
a foliation is the extended present of some actual or possible
observer; the special theory of relativity does not commit us to
this. A choice of foliation is a matter of choice of coordinates,
rather than a matter of the state of motion of the observer. A
fuller answer to this question would have to address the larger
issue of which it is a part, which is the issue of whether the
foliation-relative notion of becoming that goes along with
foliation-relative state evolution suffices for a realist,
probabilistic interpretation of quantum mechanics (see also
Maxwell 1985, Saunders 1996). I believe that it does, but I
cannot do justice to the issue within the scope of this paper.
A second objection, due to Maudlin, stems from a supposed
ontological independence of state histories defined along
different foliations. Maudlin, reacting primarily to Fleming
(1989), wrote,
\begin{quote}
The wave function on a hyperplane is, as it were, ontologically
atomic. Wave functions defined on hyperplanes in the same family
can be related by a dynamics which uses the family as a substitute
for absolute time, but relations among wave functions from
different families are obscure at best (1996, 302).
\end{quote}
If this were correct, then a dilemma would ensue. Either the
state histories along each foliation would agree as to the
macroscopic outcomes of experiments, or they wouldn't. If all
foliations agreed on the outcome of experiments, this would be a
coincidence inexplicable on the basis of the dynamics of the
theory, which only connect states \emph{within} a foliation
(Maudlin 1996, 301). If, on the other hand, the state histories
did not agree on such outcomes, ``[t]he wave functions on each
family of hyperplanes would then completely decouple, yielding an
independent world for each foliation''(302).
The alleged ontological independence of state histories along
differing foliations does not exist, however.\footnote{That is, it
does not exist on the interpretation advocated here. It is not my
intention to examine whether Maudlin has correctly interpreted
Fleming. The issue at hand is whether the objection succeeds
against the interpretation advocated in this paper.} Different
foliations are merely different ways of dividing spacetime into
3-dimensional spaces of simultaneity, and accounts given with
respect to different foliations are merely different accounts of
the same processes and events. Although one cannot always
transform directly from the state on one hypersurface to a state
on another, a state history given with respect to one foliation
determines the state history with respect to any other foliation.
The relations among state vectors from different families are
given by our equation (\ref{CollapseTransform}) and its
generalizations.
A third objection, also due to Maudlin, is that the relativity of
entanglement ``shocks intuitions which are formed by acquaintance
both with Relativity and with non-relativistic quantum mechanics''
(1994, 209). Although, when a system is entangled with another,
this is clearly a relation between the two systems (and hence it
would not, perhaps, be surprising if the nature of this
entanglement were a foliation-relative affair), when a system is
\emph{not} entangled with another, it seems plausible to assume
that this is a matter purely of the intrinsic state of it and
hence ought to be an invariant fact about the system. Acceptance
of collapse as part of foliation-relative state evolution requires
regarding this intuition, formed, as Maudlin points out, by
exposure to relativity and \emph{non-relativistic} quantum
mechanics, as mistaken. To go back to our anthropocentric
analogy: if marriage is a relation, so too is divorce! It would
interesting to explore whether the manner in which the relativity
of simultaneity shocks our intuitions shares any important
similarities with the counterintuitive features of relativistic
quantum theory without collapse.
%\begin{enumerate}
%\item{Foliation-relative states are subjective (Dorato).}
%\item{State histories defined along different
% foliations are kinematically and dynamically independent of each other (Maudlin, Berkovitz).}
%\item{The relativity of entanglement is counterintuitive (Maudlin).}
%\item{The probability of an event cannot be a foliation-relative
%quantity (Maudlin).}
%\item{Ontological probabilism requires an invariant distinction
%between past and future (Maxwell, Saunders).}
%\end{enumerate}
%\subsection{Subjectivity of foliation-relative states}. This
%objection presupposes that a foliation is the extended present of
%some observer; this has already been rejected. A foliation of
%spacetime is a geometric object which is definable by the
%equal-time hypersurfaces of any choice of time coordinate.
%Although, in Minkowski space, one can associated with each
%inertial observer the family of hyperplanes orthogonal to that
%observer's worldline, it would be misleading to regard these
%hyperplanes as the observer's `extended present'; it is not clear
%that such an infinitely extended present can meaningfully be
%attributed to observers, and, if it could, it hard to see why
%hypersurfaces.
%\subsection{Independence of states between foliations}.
%Maudlin, in response to Fleming (1989), wrote,
%\subsection{The relativity of simultaneity is counterintuitive.}
%According to Maudlin, the relativity of entanglement not only
%shocks common-sense intuitions, but also ``shocks intuitions which
%are formed by acquaintance both with Relativity and with
%non-relativistic quantum mechanics'' (1994, 209). ``Indeed, one
%has the strong intuition that whether or not a photon is polarized
%should be a matter of the intrinsic state of it, independent of
%any considerations about hyperplanes'' (210). This last objection
%is phrased in a misleading way. A system in an entangled state is
%not ascribed its own state vector at all. In this context, to
%say that a photon is \emph{polarized} means that it is not
%entangled, at least with regard to polarization, with any other
%system, and hence that it is has its own state vector. To say
%that it is \emph{not polarized} means, in this context, not that
%the photon has its own state vector which is not a state of
%definite polarization, but that the photon is in an entangled
%state with some other system, and hence is not ascribed its own
%state vector. I do not believe that many have a strong intuition
%that whether or not a photon is entangled with another system is a
%matter of the intrinsic state of it, independent of its relations
%to other objects.
%However, we do, I believe, have a strong intuition that, when a
%system is \emph{not} entangled with another object, \emph{this
%circumstance} is purely a matter of the intrinsic state of it. An
%acceptance collapse evolution in a relativistic context requires
%one to regard this intuition, formed, as Maudlin points out, by
%acquaintance with relativity and \emph{non-relativistic} quantum
%mechanics, as mistaken.
%The relativity of entanglement does, indeed, shock intuitions
%trained by acquaintance with non-relativistic quantum mechanics
%and relativity. It does so in much the same way that relativistic
%quantum field theory without collapse shocks these intuitions.
%[Explain]
%\subsection{Foliation-relativity of probabilities}
%As noted above, it does not seem possible to formulate the rule
%for calculating probabilities from states without invoking some
%foliation. The Lorentz invariance in the rule consists in the
%fact that it makes no difference to the expected statistics which
%foliation is used. This suggests that there ought to be a way to
%formulate the rule in a manifestly covariant way, without mention
%of any non-intrinsic structures; so far, however, nobody knows how
%to do this. Unless and until such a formulation is available, we
%must refer probability assignments to foliation-relative states.
%This has the odd consequence that a collapse event, local though
%it may be, will sometimes be assigned different probabilities by
%states along different hyperplanes passing through a single point.
%There is no limit on \emph{how} different these probabilities can
%be. Suppose, for example, that the initial state of our
%two-particle system is, instead of the singlet state, the state,
%\begin{equation}
%\epsilon \biket{z+}{z-} + \sqrt{1 - \abs{\epsilon}^2} \:
%\biket{z-}{z+}
%\end{equation}
%with $\abs{\epsilon} << 1$. Suppose that measurements of spin-$z$
%are performed on both particles at spacelike separation, and that
%the outcome of the measurement on $S_2$ is $-1$. Then, a
%hypersurface passing through the measurement on $S_1$ and crossing
%$S_2$'s worldline to the past of the measurement on $S_2$ will
%assign a very small probability $\abs{\epsilon}^2$ to the outcome
%$+1$ of the measurement on $S_1$, whereas a hypersurface passing
%through the measurement on $S_1$ and crossing $S_2$'s worldline to
%the future of the measurement on $S_2$ will assign a probability
%$1$ to the same outcome.
\newpage
\begin{center}
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\end{document}