\documentclass[12pt]{article}
\usepackage{graphicx,amsfonts,amsmath,amssymb,mathrsfs, comment}
\title{On the Common Structure of Bohmian Mechanics and the
Ghirardi--Rimini--Weber Theory}
\author{
Valia Allori\footnote{Department of Philosophy, Davison Hall,
Rutgers, The State University of New Jersey, 26 Nichol Avenue,
New Brunswick, NJ 08901-1411, USA.
E-mail: vallori@eden.rutgers.edu},
Sheldon Goldstein\footnote{Departments of Mathematics, Physics and
Philosophy, Hill Center, Rutgers, The State University of New
Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA.
E-mail: oldstein@math.rutgers.edu},\\
Roderich Tumulka\footnote{Mathematisches Institut,
Eberhard-Karls-Universit\"at, Auf der Morgenstelle 10, 72076
T\"ubingen, Germany. E-mail:
tumulka@everest.mathematik.uni-tuebingen.de},
and Nino Zangh\`\i\footnote{Dipartimento di Fisica dell'Universit\`a
di Genova and INFN sezione di Genova, Via Dodecaneso 33, 16146
Genova, Italy. E-mail: zanghi@ge.infn.it}
}
\date{June 17, 2006}
\addtolength{\textwidth}{2.0cm}
\addtolength{\hoffset}{-1.0cm}
\addtolength{\textheight}{3.0cm}
\addtolength{\voffset}{-1.5cm}
\newcommand{\Hilbert}{\mathscr{H}}
\newcommand{\conf}{\mathcal{Q}}
\newcommand{\Q}{\conf}
\renewcommand{\Re}{\mathrm{Re}}
\renewcommand{\Im}{\mathrm{Im}}
\newcommand{\EEE}{\mathbb{E}}
\newcommand{\III}{\mathbb{I}}
\newcommand{\PPP}{\mathbb{P}}
\newcommand{\RRR}{\mathbb{R}}
\newcommand{\CCC}{\mathbb{C}}
\newcommand{\QQQ}{\mathbb{Q}}
\newcommand{ \Z}{\mathbb{Z}}
\newcommand{\NNN}{\mathbb{N}}
\newcommand{\prob}{\mathrm{Prob}}
\renewcommand{\sp}[2]{\langle #1|#2 \rangle}
\newcommand{\Laplace}{\Delta}
\newcommand{\vx}{\boldsymbol{x}}
\newcommand{\vX}{\boldsymbol{X}}
\newcommand{\vQ}{\boldsymbol{Q}}
\newcommand{\vq}{\boldsymbol{q}}
\newcommand{\vv}{\boldsymbol{v}}
\newcommand{\macroX}{\mathscr{X}}
\newcommand{\BM}{Bohmian mechanics }
\newcommand{\wf}{wave function }
\newcommand{\po}{primitive ontology }
\renewcommand{\baselinestretch}{1.1}
\newcommand{\z}[1]{{#1}}
%\newcommand{\z}[1]{{#1}}
%\newcommand{\mw}[1]{{#1}}
\begin{document}
\maketitle
\begin{abstract}
Bohmian mechanics and the Ghirardi--Rimini--Weber theory provide opposite resolutions of the quantum measurement problem: the former postulates additional variables (the particle positions) besides the wave function, whereas the latter implements spontaneous collapses of the wave function by a nonlinear and stochastic modification of Schr\"odinger's equation. Still, both theories, when understood appropriately, share the following structure: They are ultimately not about wave functions but about ``matter'' moving in space, represented by either particle trajectories, fields on space-time, or a discrete set of space-time points. The role of the wave function then is to govern the motion of the matter.
\medskip
\noindent
PACS: 03.65.Ta.
Key words: quantum theory without observers; Bohmian mechanics;
Ghirardi--Rimini--Weber theory of spontaneous wave function collapse;
primitive ontology; local beables.
\end{abstract}
\begin{center}
\textit{Dedicated to GianCarlo Ghirardi on the occasion of his 70th birthday}
\end{center}
\newpage
\tableofcontents
\section{Introduction}
Bohmian mechanics ({\sf BM}) and the Ghirardi--Rimini--Weber ({\sf GRW})
theory are two quantum theories without observers, and thus provide two
possible solutions of the measurement problem of quantum
mechanics. However, they would seem to have little in common beyond
achieving the goal of describing a possible reality in which observers
would find, for the outcomes of their experiments, the probabilities
prescribed by the quantum formalism. They are two precise, unambiguous
fundamental physical theories that describe and explain the world around
us, but they appear to do this by employing opposite strategies. In Bohmian
mechanics \cite{Bohm52, Bell66, DGZ92, survey} the wave function evolves
according to the Schr\"odinger equation but is not the complete description
of the state at a given time; this description involves further variables,
traditionally called ``hidden variables,'' namely the particle
positions. In the {\sf GRW} theory \cite{Pe76, GRW86, Bell87, BG03}, in
contrast, the wave function $\psi$ describes the state of any physical
system completely, but $\psi$ collapses spontaneously, thus departing from
the Schr\"odinger evolution. That is, the two theories choose different
horns of the alternative that Bell formulated as his conclusion from the
measurement problem \cite{Bell87}: ``Either the wave function, as given by
the Schr\"odinger equation, is not everything, or it is not right.''
The two theories are always presented almost as dichotomical, as in the recent paper by Putnam \cite{putnam}.
Our suggestion here is instead that {\sf BM} and {\sf GRW} theory have much more in common than one would expect at first sight.
So much, indeed, that they should be regarded as being close to each other, rather than opposite. The differences are less profound than the similarities, provided that the {\sf GRW} theory is understood appropriately, as involving variables describing matter in space-time.
After recalling what Bohmian mechanics is in Section \ref{sec:bm}, we introduce
two concrete examples of {\sf GRW} theories in Section \ref{sec:GRW}. These examples involve rather different choices of crucial variables, describing matter in space-time, and give us a sense of the range of possibilities for such variables.
We discuss in Section \ref{sec:PO}
the notion of the {\it{primitive ontology} }(PO) of a theory
(a notion introduced in \cite{DGZ92})
and connect it to
Bell's notion of ``local beables'' \cite{Bell76}.
In Section \ref{sec:morePO} we relate the primitive ontology of a theory to the notion of physical equivalence between theories. We stress in Section \ref{sec:symmetry} the connection between the \po and symmetry properties, with particular concern for the generalization of such theories to a relativistically invariant quantum theory without observers. In Section \ref{sec:grw0}
we argue that a theory without a primitive ontology is at best profoundly problematical.
We proceed in Sections \ref{sec:diff} to an analysis of the differences between {\sf GRW} (with primitive ontology) and {\sf BM}, and in Section \ref{sec:differences} we discuss a variety of possible theories. We consider in Section \ref{sec:lgrwf} a ``no-collapse'' reformulation of one of the {\sf GRW} theories and in Section \ref{sec:BMC} a ``collapse'' interpretation of {\sf BM}. These formulations enable us to better appreciate the common structure of {\sf BM} and the {\sf GRW} theories, as well as the differences, as we discuss in Section \ref{sec:equiv}. We conclude in Section \ref{sec:essential} with a summary of this common structure.
\section{Bohmian Mechanics}
\label{sec:bm}
Bohmian mechanics is a theory of (non-relativistic) particles in motion. The motion of a system of $N$ particles is provided by their world lines $t \mapsto Q_i(t)$, $i=1, \ldots, N$, where $Q_i(t)$ denotes the position in $\RRR^3$ of the $i$-th particle at time $t$. These world lines are determined by Bohm's law of motion \cite{Bohm52, Bell66, DGZ92, survey},
\begin{equation}\label{Bohm}
\frac{dQ_i}{dt}=v_i^{\psi}(Q_1, \ldots, Q_N)=\frac{\hbar}{m_i}
\Im \frac{\psi^{*}\nabla_i \psi}{\psi^{*}\psi}(Q_1\ldots,Q_N),
\end{equation}
where $m_i$, $i=1, \ldots, N$, are the masses of the particles;
the wave function $\psi$ evolves according to Schr\"odinger's equation
\begin{equation}\label{Schr}
i\hbar\frac{\partial \psi}{\partial t} = H\psi \,,
\end{equation}
where
$H$ is the usual nonrelativistic Schr\"odinger Hamiltonian; for
spinless particles it is of the form
\begin{equation}
\label{eq:H}
H=-\sum_{k=1}^N\frac{\hbar^2}{2m_k}\nabla^2_k+V,
\end{equation}
containing as parameters the masses of the particles as
well as the potential energy function $V$ of the system.
In the usual yet unfortunate terminology, the actual positions $Q_1, ..., Q_N $ of the particles are the \emph{hidden variables} of the theory: the variables which, together with the wave function, provide a complete description of the system, the wave function alone providing only a partial, incomplete, description.
From the point of view of {\sf BM}, however, this is a strange terminology since it suggests that the main object of the theory is the wave function, with the additional information provided by the particles' positions playing a secondary role. The situation is rather much the opposite: {\sf BM} is a theory of particles; their positions are the primary variables, and the description in terms of them must be completed by specifying the wave function to define the dynamics \eqref{Bohm}.
As a consequence of Schr\"odinger's equation and of Bohm's law of motion,
the quantum equilibrium distribution $|\psi(q)|^2$ is equivariant. This means that if the configuration $Q(t) = (Q_1(t), \ldots, Q_N(t))$ of a system is random with distribution $|\psi_t|^2$ at some time $t$, then this will be true also for any other time $t$. Thus, the \emph{quantum equilibrium hypothesis}, which asserts that whenever a system has wave function $\psi_t$, its configuration $Q(t)$ is random with distribution $|\psi_t|^2$, can consistently be assumed. This hypothesis is not as hypothetical as its name may suggest: the quantum equilibrium hypothesis follows, in fact, by the law of large numbers from the assumption that the (initial) configuration of the universe is typical (i.e., not-too-special) for the $|\Psi|^2$ distribution, with $\Psi$ the (initial) wave function of the universe \cite{DGZ92}. The situation resembles the way Maxwell's distribution for velocities in a classical
gas follows from the assumption that the phase point of the gas is typical for the uniform distribution on the energy surface.
As a consequence of the quantum equilibrium hypothesis, a Bohmian universe, even if deterministic, appears random to its inhabitants. In fact, the probability distributions observed by the inhabitants agree exactly with those of the quantum formalism. To begin to understand why, note that any measurement apparatus must also consist of Bohmian particles. Calling $Q_{S}$ the configuration of the particles of the system to be measured and $Q_{A}$ the configuration of
the particles of the apparatus,
we can write for the configuration of the big Bohmian system relevant to the analysis of the measurement $Q=(Q_{S},Q_{A})$. Let us suppose that the initial wave function $\psi$ of the big system is a product state $\Psi(q)=\Psi(q_{S},q_{A})= \psi(q_{S}) \, \phi(q_{A})$.
During the measurement, this $\Psi$ evolves according to the Schr\"odinger equation, and in the case of an ideal measurement it evolves to $\Psi_t = \sum_\alpha \psi_\alpha \, \phi_\alpha$, where $\alpha$ runs through the eigenvalues of the observable measured, $\phi_\alpha$ is a state of the apparatus in which the pointer points to the value $\alpha$, and $\psi_\alpha$ is the projection of $\psi$ to the appropriate eigenspace of the observable. By the quantum equilibrium hypothesis, the probability for the random apparatus configuration $Q_A(t)$ to be such as to correspond to the pointer pointing to the value $\alpha$ is $\|\psi_\alpha\|^2$. For a more detailed discussion see \cite{DGZ92,DGZ04}.
\section{Ghirardi, Rimini, and Weber}\label{sec:GRW}
The theory proposed by Ghirardi, Rimini and Weber \cite{GRW86} is in agreement with the predictions of nonrelativistic quantum mechanics as far as
all present
experiments are concerned \cite{BG03}; for a discussion of future experiments that may
distinguish this theory from quantum mechanics, see
Section~V of \cite{BG03}. According to the way in which this theory is usually presented, the evolution of the wave function follows, instead of Schr\"odinger's equation, a stochastic jump process in Hilbert space.
We shall succinctly summarize this process as follows.
Consider a quantum system described (in the standard language) by
an $N$-``particle''\footnote{We wish to emphasize here that there are no particles in this theory: the word ``particle'' is used only for convenience in order to be able to use the standard notation and terminology.} wave function $\psi = \psi(q_1,...,q_N)$, ${q}_i\in \RRR^3$, $i=1,\dots, N$; for any point $x$ in $\mathbb{R}^3$ (the ``center'' of the collapse that will be defined next), define on the Hilbert space of the system the \emph{collapse operator}
\begin{equation}
\Lambda_i (x) =\frac{1}{(2\pi \sigma^2)^{3/2}}\, e^{-\frac{( \widehat{Q}_i-x)^2}{2\sigma^2}}\,,
\label{eq:collapseoperator}
\end{equation}
where $\widehat{Q}_i$ is the position operator of ``particle'' $i$. Here $\sigma$ is a new constant of nature of order of $10^{-7}$m.
Let $\psi_{t_0}$ be the initial wave function, i.e., the normalized wave function at some time $t_0$ arbitrarily chosen as initial time. Then $\psi$ evolves in the following way:
\begin{enumerate}
\item It evolves unitarily, according to Schr\"odinger's equation, until a random time $T_1= t_0 + \Delta T_1$, so that
\begin{equation}
\psi_{T_1}= U_{\Delta T_1} \psi_{t_0},
\end{equation}
where $U_t$ is the unitary operator $U_t=e^{-\frac{i}{\hbar}Ht}$ corresponding to the standard Hamiltonian $H$ governing the system, e.g., given by (\ref{eq:H}) for $N$ spinless particles,
and $\Delta T_1$ is a random time distributed according to the exponential distribution with rate $N\lambda$ (where the quantity $\lambda$ is another constant of nature of the theory,\footnote{Pearle and Squires \cite{PS94} have argued that $\lambda$ should be chosen differently for every ``particle,'' with $\lambda_i$ proportional to the mass $m_i$.} of order of $10^{-15}$ s$^{-1}$).
\item At time $T_1$ it undergoes an instantaneous collapse with random center
$X_1$ and random label $I_1$ according to
\begin{equation}
\psi_{T_1} \mapsto\psi_{T_1+}= \frac{\Lambda_{I_1} (X_{1})^{1/2}\psi_{T_1}}{\| \Lambda_{I_1} (X_{1})^{1/2} \psi_{T_1} \|}.
\end{equation}
$I_1$ is chosen at random in the set $\{1, \ldots, N\}$ with uniform distribution. The center of the collapse
$X_1$ is chosen randomly with probability distribution\footnote{Hereafter, when no ambiguity could arise, we use the standard notations of probability theory, according to which a capital letter, such as $X$, is used to denote a random variable, while the the values taken by it are denoted by small letters; $ X\in dx$ is a shorthand for $X\in [x, x+dx]$, etc. }
\begin{equation}\label{p}
\mathbb{P}(X_1\in dx_{1}| \psi_{T_1}, I_1=i_1) \left\langle \psi_{T_1}|\Lambda_{i_1}(x_1)\psi_{T_1}\right\rangle dx_{1}\|\Lambda_{i_1} (x_1)^{1/2} \psi_{T_1}\|^2 dx_{1}.
\end{equation}
\item Then the algorithm is iterated: $\psi_{T_1+}$ evolves unitarily until a random time $T_2 = T_1 + \Delta T_2$, where $\Delta T_2$ is a random time (independent of $\Delta T_1$) distributed according to the exponential distribution with rate $N\lambda$, and so on.
\end{enumerate}
In other words, the evolution of the wave function is the Schr\"odinger evolution interrupted by collapses. When the wave function is
$\psi$ a collapse with center $x$ and label $i$ occurs at rate
\begin{equation}\label{rate}
r(x,i|\psi)=\lambda\left\langle\psi\,|\,\Lambda_{i}(x)\psi\right\rangle
\end{equation}
and when this happens, the wave function changes to $
{\Lambda_{i} (x)^{1/2}\psi}/{\| \Lambda_{i} (x)^{1/2} \psi \|}$.
Thus, if between time $t_0$ and any time $t>t_0$, $n$ collapses have occurred at the times $t_0< T_1 < T_2 < \ldots < T_n < t $, with centers $X_1, \ldots, X_n$ and labels $I_1, \ldots, I_n$, the wave function at time $t$ will be
\begin{equation}\label{eq:psit}
\psi_t = \frac{L^{F_n}_{t, t_0} \psi_{t_0}
}{\| L^{F_n}_{t, t_0} \psi_{t_0}
\|}\,
\end{equation}
where $F_n = \{(X_1,T_1,I_1), \ldots, (X_n,T_n,I_n)\}$ and
\begin{equation}\label{eq:long}
L^{F_n}_{t, t_0}
= U_{t-T_n} \Lambda_{I_n}(X_n)^{1/2} \,U_{T_n-T_{n-1}} \Lambda_{I_{n-1}}(X_{n-1})^{1/2}
\,U_{T_{n-1}-T_{n-2}} \cdots \Lambda_{I_1}(X_1)^{1/2} \, U_{T_1-t_0}.
\end{equation}
Since $T_i$, $X_i$, $I_i$ and $n$ are random, $\psi_t$ is also random.
It should be observed that---unless $t_0$ is the initial time of the universe---also $\psi_{t_0}$ should be regarded as random, being determined by the collapses that occurred at times earlier that $t_0$. However, \emph{given} $\psi_{t_0}$, the statistics of the future evolution of the wave function is completely determined; for example, the joint distribution of the first $n$ collapses after $t_0$, with particle labels $I_1, \ldots, I_n \in \{1,\ldots,N\}$, is
\begin{multline}\label{nflashdist}
\PPP\bigl( X_1\in d x_1, T_1 \in d t_1, I_1 = i_1, \ldots,
X_n \in dx_n, T_n \in d t_n, I_n = i_n | \psi_{t_0} \bigr) =\\
\lambda^n e^{-N\lambda (t_n-t_0)} \| L^{f_n}_{t_n, t_0} \psi_{t_0}
\|^2 \, dx_{1}dt_1 \cdots dx_{n}dt_n \,,
\end{multline}
with $f_n = \{(x_1,t_1,i_1), \ldots, (x_n,t_n,i_n)\}$ and $L^{f_n}_{t_n, t_0}$ given, \emph{mutatis mutandis}, by \eqref{eq:long}.
\bigskip
This is, {more or less}, all there is to say about the formulation of the {\sf GRW} theory according to most theorists.
In contrast, GianCarlo Ghirardi believes that the description provided above is not the whole story, and we agree with him. We believe that, depending on the choice of what we call the \emph{primitive ontology} (PO) of the theory, there are correspondingly
different versions of the theory. We will discuss the notion of primitive
ontology in detail in Section \ref{sec:PO}.
In the subsections below we present two versions of the {\sf GRW} theory, based on two different choices of the PO, namely the \emph{matter density ontology} (in Section~\ref{sec:GRWm}) and the \emph{flash ontology} (in Section~\ref{sec:GRWf}).
\subsection{GRWm}
\label{sec:GRWm}
In the first version of the {\sf GRW} theory, denoted by {\sf GRWm}, the PO is given by a field: We have a variable $m(x,t)$ for every point $x \in \RRR^3$ in space and every time $t$, defined by
\begin{equation}\label{mdef}
m(x,t) = \sum_{i=1}^N m_i \int\limits_{\RRR^{3N}} dq_1 \cdots dq_N \, \delta(q_i-x) \, \bigl|\psi(q_1, \ldots, q_N,t)\bigr|^2 \,.
\end{equation}
In words, one starts with the $|\psi|^2$--distribution in configuration
space $\RRR^{3N}$, then obtains the marginal distribution of
the $i$-th degree of freedom $q_i\in \RRR^3$
by integrating out
all other variables $q_j$, $j \neq i$, multiplies by the mass associated with $q_i$, and sums over $i$.
{\sf GRWm} was essentially proposed by Ghirardi and co--workers
in \cite{Ghi}.\footnote{They first proposed (for a model slightly more complicated than the one considered here)
that the matter density be given by an expression similar to \eqref{mdef} but this difference is not relevant for our purposes.}
The field $m(\cdot,t)$ is supposed to be understood as the
density of matter in space at time $t$. Since these variables are
functionals of the wave function $\psi$, they are not ``hidden
variables'' since, unlike the positions in {\sf BM}, they need not be specified in
addition to the wave function, but rather are determined by it. Nonetheless, they are additional elements of the {\sf GRW} theory that need to be posited in order to have a complete description of the world in the framework of that theory.
{\sf GRWm} is a theory about the behavior of a field $m(\cdot,t)$ on three-dimensional space. The microscopic description of reality provided by the matter density field $m(\cdot,t)$ is not particle-like but instead continuous, in contrast to the particle ontology of {\sf BM}.
This is reminiscent of Schr\"odinger's early view of the wave function as representing a continuous matter field. But while Schr\"odinger was obliged to abandon his early view because of the tendency of the wave function to spread,
the spontaneous wave function collapses built into the {\sf GRW} theory tend to localize the wave function, thus counteracting this tendency and overcoming the problem.
A parallel with {\sf BM} begins to emerge:
they both essentially involve more than the wave function.
In one the matter is spread out continuously, while in the other it is
concentrated in finitely many particles; however, both theories are concerned with matter in three-dimensional space, and in some regions of space there is more than in others.
You may find {\sf GRWm} a surprising proposal. You may ask, was it not the
point of {\sf GRW} --- perhaps even its main advantage over {\sf BM} ---
that it can do without objects beyond the wave function, such as particle
trajectories or matter density? Is not the dualism present in {\sf GRWm}
unnecessary? That is, what is wrong with the version of the {\sf GRW}
theory, which we call {\sf GRW0}, which involves just the wave function and
nothing else? We will return to these questions in Section
\ref{sec:grw0}. To be sure, it seems that if there was nothing wrong with
{\sf GRW0}, then, by simplicity, it should be preferable to {\sf GRWm}. We
stress, however, that Ghirardi must regard {\sf GRW0} as seriously
deficient; otherwise he would not have proposed anything like {\sf GRWm}.
We will indicate in Section \ref{sec:grw0} why we think Ghirardi is
correct. To establish the inadequacy of {\sf GRW0} is not, however, the
main point of this paper.
\subsection{GRWf}
\label{sec:GRWf}
According to another version of the {\sf GRW} theory, which was
first suggested by Bell \cite{Bell87,Bell89}, then adopted in \cite{kent,Tum04,Mau05,AZ05,Tum05},
and here denoted {\sf GRWf}, the PO is given by ``events'' in space-time called flashes, mathematically
described by points in space-time.
This is, admittedly, an unusual PO, but it is a possible
one nonetheless. In {\sf GRWf} matter is neither made of particles following world
lines, such as in classical or Bohmian mechanics, nor of a continuous
distribution of matter such as in {\sf GRWm}, but rather of discrete points
in space--time, in fact finitely many points in every bounded space-time
region, see Figure~\ref{flashes}.
\begin{figure}[ht]\begin{center}\includegraphics[width=.4 \textwidth]{flashes.eps}\end{center}\caption{A typical pattern of flashes in space-time, and thus a possible
world according to the {\sf GRWf} theory}\label{flashes}\end{figure}
In the {\sf GRWf} theory, the space-time locations of the flashes can be read
off from the history of the wave function {given by \eqref{eq:psit} and \eqref{eq:long}}: every flash corresponds to one of the spontaneous collapses of the wave function, and its space-time location is just the space-time location of that
collapse.
Accordingly, equation \eqref{nflashdist} gives the joint distribution of the first $n$ flashes, after some initial time $t_0$.
The flashes form the set
\[
F=\{(X_{1},T_{1}), \ldots, (X_{k},T_{k}), \ldots\}
\]
(with $T_10, $ be the restriction mapping $\Omega_0 \to
\Omega_t$, and $T_{\tau}$ the time translation mapping $\Omega_t \to
\Omega_{t+\tau}$; then $S_t = T_{-t} \circ R_t : \Omega_0 \to \Omega_0$ is
the time shift. Consider an association $\psi \mapsto \PPP^{\psi}$ where
$\PPP^{\psi}$ is a probability measure on $\Omega_0$ that is compatible
with the dynamics of the theory. We say that this association is
\textit{equivariant} relative to a deterministic evolution $\psi \mapsto
\psi_t$ if $S_t^{\star} \PPP^{\psi} = \PPP^{\psi_t}$, where ${\star}$
denotes the action of the mapping on measures. More generally, for an
evolution that may be stochastic, we say that the association is
\textit{equivariant} relative to the evolution if
\begin{equation}S_t^{\star} \PPP^{\psi} = \EEE \PPP^{\psi_t},\end{equation}
where $\EEE$ denotes the average over the random $\psi_t$.
With this definition, {\sf BM} is equivariant relative to the Schr\"odinger evolution, and {\sf GRWf} and {\sf GRWm} are equivariant relative to the {\sf GRW} evolution.
\section{A Plethora of Theories}
\label{sec:differences}
One may wonder whether some primitive ontologies (flashes and
continuous matter density) work only with {\sf GRW}-type theories while others
(particle trajectories) work only with Bohm-type theories. This is
not the case, as we shall explain in this section.
\subsection{Particles, Fields, and Flashes}
Let us analyze, with the aid of Table 1, several possibilities:
\begin{table}[t!]
\begin{center}
\begin{tabular}{|c||c|c|c|}\hline & Particles & Fields & Flashes \\\hline\hline Deterministic & {\sf BM} & {\sf BQFT}, {\sf Sm} & \\\hline Indeterministic & {\sf SM}, {\sf BTQFT}, {\sf BMW}, {\sf GRWp} &
{\sf GRWm} & {\sf GRWf}, {\sf Sf}, {\sf Sf$'$} \\\hline \end{tabular} \caption{\small{Different possibilities for the PO of a theory are presented: particles, fields and flashes. These different primitive ontologies can evolve according to either deterministic or stochastic laws.
Corresponding to these possibilities we have a variety of physical theories: Bohmian mechanics ({\sf BM}), Bohmian quantum field theory ({\sf BQFT}), a mass density field theory with Schr\"odinger evolving wave function ({\sf Sm}), stochastic mechanics ({\sf SM}), Bell-type quantum field theory ({\sf BTQFT}), Bell's version of many-worlds ({\sf BMW}), a particle {\sf GRW} theory ({\sf GRWp}),
{\sf GRW} theory with mass density ({\sf GRWm}), {\sf GRW} theory with flashes ({\sf GRWf}),
and two theories with flashes governed by Schr\"odinger (or Dirac) wave functions ({\sf Sf} and {\sf Sf$'$}). For a detailed description of these theories, see the text.}}
\end{center}
\label{defaulttable}
\end{table}
there can be at least three different kinds of primitive ontologies for a fundamental physical theory, namely particles, fields, and flashes. Those primitive ontologies can evolve either according to a deterministic or to a stochastic law and this law can be implemented with the aid of a wave function evolving either stochastically or deterministically.
{\sf BM} is the prototype of a theory in which we have a particle ontology that evolves deterministically according to a law specified by a wave function that also evolves deterministically. The natural analog for a theory with particle ontology with indeterministic evolution is stochastic mechanics ({\sf SM}), in which the law of evolution of the particles is given by a diffusion process while the evolution of the wave function, the usual Schr\"odin\-ger evolution, remains deterministic
(see \cite{stochmech1,stochmech2} for details).
Another example involving stochastically evolving particles with a deterministically evolving wave function is provided by a Bell-type quantum field theory ({\sf BTQFT}) in which, despite the name, the PO is given by particles evolving indeterministically to allow for creation and annihilation (for a description, see \cite{crlet,crea2B,Bell86}).
Another possibility for a stochastic theory of particles is a theory {\sf GRWp} in which the particle motion is governed by \eqref{Bohm} but with a wave function that obeys a {\sf GRW}-like evolution in which the collapses occur exactly as in {\sf GRW} except that, once the time and label for the collapse has been chosen, the collapse is centered at the actual position of the particle with the chosen label, rather than at random according to equation \eqref{p}. (A garbled formulation of this theory is presented in \cite[p. 346]{BohmHiley}.)
What in Table 1 we call a Bohmian quantum field theory ({\sf BQFT}) involves only fields, evolving deterministically \cite{Bohm52}, \cite{westman}.
Another example is provided by the theory {\sf Sm} in which the PO is given by the mass density field \eqref{mdef} but evolving with a Schr\"odinger wave function --- always evolving according to Schr\"odinger's equation, with no collapses.
{\sf GRWm} provides an example of a theory of fields that evolve stochastically.
Concerning theories with flashes, these are inevitably stochastic, and {\sf GRWf}, in which the flashes track the collapses of the wave function, is the prototype. However, there are also theories with flashes in which the wave function never collapses. Such theories are thus arguably closer
to {\sf BM} than to {\sf GRWf}. We consider two examples.
In the first example, denoted by {\sf Sf},\footnote{Here {\sf S} stands for Schr\"odinger (evolution). Using this notation we have that {\sf BM} = {\sf Sp}.} the PO consists of flashes with their distribution determined by a Schr\"odinger wave function $\psi =\psi (q_1, \ldots, q_N)$, that evolves always unitarily, as in {\sf BM}, according to the $N$--``particle'' Schr\"odinger evolution \eqref{Schr}.
The flashes are generated by the wave function exactly as in {\sf GRWf}. Thus, the algorithm, whose output is the flashes, is the same as the one described in Section \ref{sec:GRW}, with steps 1., 2. and 3., with the following difference: the first sentence in step 2. is dropped, since no collapse takes place.
In other words, in {\sf Sf} flashes occur with rate \eqref{rate} but are accompanied by no changes in the wave function.\footnote{Accordingly, equation \eqref{nflashdist} is replaced by
\begin{multline}
\PPP\bigl( X_1\in d x_1, T_1 \in d t_1, I_1 = i_1, \ldots,
X_n \in dx_n, T_n \in d t_n, I_n = i_n | \psi_{t_0} \bigr) \\
=\lambda^n e^{-N\lambda (t_n-t_0)} \prod_{k=1}^{n}
\langle \psi_{t_{k}} | \Lambda_{i_k}(x_{k})\psi_{t_{k}}
\rangle \, dx_{1}dt_1 \cdots dx_{n}dt_n \,, \nonumber
%\label{eq:BMWfmany}
\end{multline}
where $\Lambda_{i}(x)$ is the collapse operator given by
\eqref{eq:collapseoperator}.} \z{(This flash process defines, in fact, a
Poisson process in space-time---more precisely, a Poisson system of points in
$\RRR^4\times\{1,\dots,,N\}$---with intensity measure
$r((x,t),i)=r(x,i|\psi_t)$ given by \eqref{rate}.)} Note that, in contrast
to the case of {\sf GRWf}, one obtains a well defined theory by taking the
limit $\sigma\rightarrow 0$ in \eqref{eq:collapseoperator}, that is by
replacing $\Lambda_{i}(x)$ in \eqref{rate} with $\tilde{\Lambda}_i(x)$
given by $ \tilde{\Lambda}_i(x)=\delta(\widehat{Q}_i-x),
\label{eq:opsa}
$
where $\widehat{Q}_i$ is the position operator of the $i$-th ``particle.''
Our last example ({\sf Sf$'$}) is the following.
Consider a nonrelativistic system of $N$ noninteracting quantum
particles with wave function satisfying the Schr\"odinger equation
\begin{equation}\label{schr}
i\hbar \frac{\partial \psi}{\partial t} = -\sum_{i=1}^N
\frac{\hbar^2}{2m_i} \nabla_i^2 \psi + \sum_{i=1}^N V_i(q_i)
\, \psi\,,
\end{equation}
and suppose that, as in {\sf GRWf}, each of the flashes is associated with
one of the particle labels $1, \ldots, N$.
Given the flashes up to the present, the next flash occurs with
rate $N\lambda$, and has a label $I \in \{1, \ldots, N\}$ that is randomly chosen with uniform distribution. If this flash occurs at time $T_I$,
its location $X$ is random with probability distribution
\begin{equation}\label{flashdistr}
\mathbb{P}(X\in d X_I| I, T_I, \{X_k,T_k\}_{k\neq I}) = \mathcal{N} \, \bigl|\psi(X_1, T_1, \ldots,
X_N,T_N) \bigr|^2 \, d X_I \,,
\end{equation}
where $\mathcal{N}$ is a normalizing factor, $\psi= \psi(q_1,t_1,
\ldots, q_N,t_N)$ is a multi-time wave function evolving according
to the set of $N$ equations
\begin{equation}\label{multitime}
i\hbar \frac{\partial \psi}{\partial t_i} = -\frac{\hbar^2}{2m_i}
\nabla_i^2 \psi + V_i(q_i) \, \psi
\end{equation}
for every $i \in \{1,\ldots, N\}$, and $T_k$ and $X_k$ are, for $k \neq I$,
the time and location of the last flash with label $k$. \z{The reason that
this model is assumed to be noninteracting is precisely to guarantee the
existence of the multi-time wave function in \eqref{flashdistr}. {\sf Sf$'$}
is an example of a theory with a flash ontology that arguably is
empirically equivalent to {\sf OQT} (unlike {\sf GRWf})---at least, it
would be if it were extended to incorporate interactions between
particles---and avoids the many-worlds character of {\sf Sf} (see Section
\ref{subsec:swamw} below).}
A provisional moral that emerges is that relativistic invariance might be
connected with a flash ontology, since {\sf GRWf} is the only theory in
Table 1 \z{(except for {\sf Sm} and {\sf Sf}, which have a rather
extraordinary character that we discuss in Section \ref{subsec:swamw}
below)} of which we know how it can be made relativistically invariant
without postulating a preferred foliation of space-time (or any other
equivalent additional structure). Finally, note that all the theories in
Table 1 are empirically equivalent (suitably understood) to {\sf OQT}
except {\sf GRWm}, {\sf GRWf}, and {\sf GRWp}.
\subsection{Schr\"odinger Wave Functions and Many-Worlds}
\label{subsec:swamw}
A rather peculiar theory representing the world as if it were, at any given
time, a collection of particles with classical configuration
$Q=(Q_1,\ldots ,Q_N)$ is Bell's version of many-worlds ({\sf BMW})
\cite{BellMW}. In {\sf BMW} the wave function $\psi$ evolves according to
Schr\"odinger's equation and \cite{BellMW}
\begin{quote}
instantaneous classical configurations \ldots{} are
supposed to exist, and to be distributed \ldots{} with probability $|\psi|^2$.
But no pairing of configurations at different times, as would be
effected by the existence of trajectories, is supposed.
\end{quote}
This can be understood as suggesting that the configurations at different times are not connected by any law. It could also be regarded as suggesting that configurations at different times are (statistically) independent, and that is how we shall understand it here.
The world described by {\sf BMW} is so radically different from what we are accustomed to that it is hard to take {\sf BMW} seriously. In fact, for example, at some time during the past second, according to {\sf BMW}, there were on the earth dinosaurs instead of humans, because of the independence and the fact that, in any no-collapse version of quantum theory, there are parts of the wave function of the universe in which the dinosaurs have never become extinct. In this theory, the actual past will typically entirely disagree with what is suggested by our memories, by history books, by photographs and by other records of (what we call) the past.
Also {\sf Sf} and {\sf Sm}, though they are simple mathematical modifications of {\sf GRWf} and {\sf GRWm} respectively, provide very different pictures of reality, so different indeed from
what we usually believe reality should be like that it would seem hard to take these theories seriously. In {\sf Sf} and {\sf Sm}, apparatus pointers never
point in a specific direction (except when a certain direction in {\sf OQT} would
have probability more or less one), but rather all directions are, so to speak, realized at once. As a consequence, one is led to conclude that their predictions don't agree with those of the quantum formalism.
Still, it can be argued that these theories do not predict any \emph{observable} deviation from the quantum
formalism: there is, arguably, no conceivable experiment that could
help us decide whether our world is governed by {\sf Sf} or {\sf Sm} on the one hand or by the quantum formalism on the other. The reason for this surprising claim is that {\sf Sf} and {\sf Sm} can be regarded as many-worlds formulations of quantum mechanics. Let us explain.
At first glance, in an {\sf Sf} or {\sf Sm} world, the after-measurement state of the
apparatus seems only to suggest that matter is very spread out.
However, if one considers the flashes, governed by the rate \eqref{rate}, or the mass density \eqref{mdef}, that correspond to macroscopic superpositions, one sees that they form independent families
of correlated flashes or mass density associated with the terms of the superposition, with no interaction between the families. The families can indeed be regarded as comprising many worlds, superimposed on a single space-time. Metaphorically speaking, the universe according to {\sf Sf} or {\sf Sm} resembles the situation of a TV set that is not correctly tuned, so that one always sees a mixture of two channels. In principle, one might watch two movies at the same time in this way, with each movie conveying its own story composed of temporally and spatially correlated events.
Thus {\sf Sf} and {\sf Sm} are analogous to Everett's many-worlds ({\sf EMW}) formulation of quantum mechanics \cite{everett}, but with the ``worlds'' realized in the same space-time.
Since the different worlds do not interact among themselves---they are, so to speak, reciprocally transparent---this difference should not be regarded as crucial. Thus, to the extent that one is willing to grant that {\sf EMW} entails no observable deviation from the quantum formalism, the same should be granted to {\sf Sf} and {\sf Sm}.
Moreover, contrarily to {\sf EMW}, but similarly to {\sf BMW}, {\sf Sf} and
{\sf Sm} have a clear PO upon which the existence and behavior of the macroscopic counterparts of our experience can be
grounded.
This ontological clarity notwithstanding, in {\sf Sf} and {\sf Sm} reality is of course very different from what we usually believe it to be like. It is populated with ghosts we do not perceive, or rather, with what are like ghosts from our perspective, because the ghosts are as real as we are, and from their perspective we are the ghosts. \z{We plan to give a more complete discussion of {\sf Sf} and {\sf Sm} in a future work.}
We note that the theory {\sf Sm} is closely related to---if not precisely the same as---the version of quantum mechanics proposed by Schr\"odinger in 1926 \cite{sch1}. After all, Schr\"odinger originally regarded his theory as describing a continuous distribution of matter (or charge) spread out in {\em physical space} in accord with the wave function on {\em configuration space} \cite{sch1}. He soon rejected this theory because he thought that it rather clearly conflicted with experiment. Schr\"odinger's rejection of this theory was perhaps a bit hasty. Be that as it may, according to what we have said above, Schr\"odinger did in fact create the first many-worlds theory, though he probably was not aware that he had done so. (We wonder whether he would have been pleased if he had been).\footnote{However, Schr\"odinger did write that \cite[p. 120]{sch}``$\psi\bar{\psi}$ is a
kind of weight-function in the system's configuration space. The
wave-mechanical configuration of the system is a superposition of
many, strictly speaking of all, point-mechanical configurations kinematically possible. Thus, each point-mechanical configuration contributes to the true wave-mechanical configuration with a certain
weight, which is given precisely by $\psi\bar{\psi}$. If we like paradoxes, we may
say that the system exists, as it were, simultaneously in all the
positions kinematically imaginable, but not `equally strongly' in all.''}
\section{The Flexible Wave Function}
\label{sec:tfwf}
In this section we elaborate on the notion of physical equivalence by
considering physically equivalent formulations of {\sf{GRWf}} and {\sf{BM}}
for which the laws of evolution of the wave function are very different from
the standard ones. We conclude with some remarks on the notion of empirical
equivalence.
\subsection{GRWf Without Collapse}
\label{sec:lgrwf}
As a consequence of the view that the {\sf GRW} theory is ultimately not about wave functions but about either flashes or matter density, the
process $\psi_t$ in Hilbert space (representing the collapsing wave function) should no longer be regarded as playing the central role in
the {\sf GRW} theory. Instead, the central role is played by the random set $F$ of flashes for {\sf GRWf}, respectively by the random matter density function $m(\cdot,t)$ for {\sf GRWm}.
From this understanding of {\sf GRWf} as being fundamentally about flashes, we obtain a lot of flexibility as to how we should regard the wave function and prescribe its behavior. As we point out in this section, it is not necessary to regard the wave function in {\sf GRWf} as undergoing collapse; instead, one can formulate {\sf GRWf} in such a way that it involves a wave function $\psi$ that evolves linearly (i.e., following the usual Schr\"odinger evolution).
Suppose the wave function at time $t$ is $\psi_t$. Then according to equation \eqref{rate}, for {\sf{GRWf}} the rate for the next flash is given by
\begin{equation}
r(x,i|\psi_t)=\lambda
\|\Lambda_i (x)^{1/2}\psi_{t}\|^2 .
\label{ps}
\end{equation}
Observe that $\psi_t$, given by equation \eqref{eq:psit}, is determined by $\psi_{t_0}$ and the flashes $(X_k,T_k)$ that occur between the times $t_0$ and $t$; it
can be rewritten as follows:
\begin{equation}\label{eq:newpsi}
\psi_t = %\frac{\tilde{\psi}_t }{\| \tilde{\psi}_t \|}=
\frac{ \Lambda_{I_n}(X_n,T_n;t)^{1/2} \; \cdots\;
\Lambda_{I_1}(X_1,T_1;t)^{1/2} \psi_{t}^L}
{\| \Lambda_{I_n}(X_n,T_n;t)^{1/2} \; \cdots\;
\Lambda_{I_1}(X_1,T_1;t)^{1/2} \psi_{t}^L\|}
\end{equation}
where we have introduced the Heisenberg-evolved operators (with respect to time $t$)
\begin{equation}\Lambda_{I_k}(X_k,T_k;t)^{1/2} =
U_{t-T_k} \Lambda_{I_k}(X_k)^{1/2} U_{T_k-t} =
U_{t-T_k} \Lambda_{I_k}(X_k)^{1/2} U_{t-T_k}^{-1}
\end{equation}
and the linearly evolved wave function
\begin{equation}
\psi_{t}^L= U_{t-t_0}\psi_{t_0}\, ,
\end{equation}
where $t_0$ is the initial (universal) time.
By inserting $\psi_t$ given by equation \eqref{eq:newpsi} in \eqref{ps} one obtains that
\begin{equation}
r(x,i|\psi_t)=\lambda \frac{ \| \Lambda_i (x)^{1/2}\Lambda_{I_n}(X_n,T_n;t)^{1/2}
\cdots\; \Lambda_{I_1}(X_1,T_1;t)^{1/2} \psi_{t}^L
\|^2}{\| \Lambda_{I_n}(X_n,T_n;t)^{1/2}
\cdots\; \Lambda_{I_1}(X_1,T_1;t)^{1/2} \psi_{t}^L
\|^2}.
\label{eq:newpsip}
\end{equation}
Suppose that the initial wave function is $\psi_{t_0}$, i.e., that the linearly evolved wave function at time $t$ is $\psi_t^L$.
Then the right hand side of equation \eqref{eq:newpsip} defines the conditional rate for the next flash after time $t$, given the flashes in the past of $t$.
Note that this conditional rate thus defines precisely the same flash process as {\sf GRWf}. In particular, we have that
\begin{equation}
\mathbb{P}_{\psi_t^L}( \mbox{future flashes} | \mbox{past flashes}) = \mathbb{P}(\mbox{future flashes}
| \psi_t).
\label{ppp}
\end{equation}
The collapsed wave function $\psi_t$ provides precisely the same information as the linearly evolving wave function $\psi_t^L$ together with all the flashes.
Thus, one arrives at the surprising conclusion that the Schr\"odinger wave function
can be regarded as governing the evolution of the space--time point process of {\sf GRWf},
so that {\sf GRWf} can indeed be regarded as a \textit{no-collapse} theory involving flashes. We say ``no-collapse'' to underline that the dynamics of the PO is then governed by a wave function evolving according to the standard, linear Schr\"odinger equation \eqref{Schr}.
However, while the probability distribution of the future flashes, given the collapsing wave function $\psi_t$, does not depend on the past flashes, given only $\psi_t^L$ it does.
The two versions of {\sf GRWf}, one using the collapsing \wf $\psi_t$ and the other using the non--collapsing \wf $\psi_t^L$, should be regarded not as two different theories but rather as two formulations of the same theory, {\sf GRWf}, because they lead to the same distribution of the flashes and thus are physically equivalent.
We conclude from this discussion that what many have considered to be the crucial, irreducible difference between {\sf BM} and {\sf GRWf}, namely that the wave function collapses in {\sf GRWf} but does not in {\sf BM}, is not in fact an objective difference at all, but rather a matter of how {\sf GRWf} is presented.
\bigskip
%\section{Markov Property}
\label{sec:markov}
We close this section with a remark. A notable difference between the two presentations of {\sf GRWf} is that while the {\sf GRW} collapse process $\psi_t$ is a Markov process,\footnote{This means that
$\PPP\bigl(\text{future}\big|\text{past } \& \text{ present}\bigr) = \PPP\bigl(\text{future}\big|\text{present}\bigr)$. In
more detail, the distribution of the $\psi_t$ for all $t>t_0$ conditional on the $\psi_t$ for all $t \leq t_0$ coincides with the distribution of the future conditional on $\psi_{t_0}$.}
the point-process $F$ of flashes is generically non-Markovian. In more detail, we regard a point process in space-time
as Markovian if for all $t_1