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\begin{document}
\title{The Quantum Measurement Problem: State of Play}
\author{David Wallace\thanks{Balliol College, Oxford OX1 3BJ; \texttt{david.wallace@balliol.ox.ac.uk}}}
\date{July 2007}
\maketitle
\begin{abstract}
This is a preliminary version of an article to appear in the forthcoming \emph{Ashgate Companion to the New Philosophy of Physics}. In it, I aim to review, in a way accessible to foundationally interested physicists as well as physics-informed philosophers, just where we have got to in the quest for a solution to the measurement problem. I don't advocate any particular approach to the measurement problem (not here, at any rate!) but I do focus on the importance of decoherence theory to modern attempts to solve the measurement problem, and I am fairly sharply critical of some aspects of the ``traditional'' formulation of the problem.
\end{abstract}
\section*{Introduction}
By some measures, quantum mechanics (QM) is the great success story of modern physics: no other physical theory has come close to the range and accuracy of its predictions and explanations. By other measures, it is instead the great scandal of physics: despite these amazing successes, we have no satisfactory physical theory at all --- only an ill-defined heuristic which makes unacceptable reference to primitives such as ``measurement'', ``observer'' and even ``consciousness''.
This is the measurement problem, and it dominates philosophy of quantum mechanics. The great bulk of philosophical work on quantum theory over the last half-century has been concerned either with the strengths and weaknesses of particular interpretations of QM --- that is, of particular proposed solutions to the measurement problem --- or with general constraints on interpretations. Even questions which are notionally not connected to the measurement problem are hard to disentangle from it: one cannot long discuss the ontology of the wavefunction\footnote{Here and afterwards I follow the physicists' standard usage by using ``wavefunction'' to refer, as appropriate, either to the putatively \emph{physical} entity which evolves according to the Schr\"{o}dinger equation or to complex-valued function which represents it mathematically. I adopt a similar convention for ``state vector''.}, or the nature of locality in relativistic quantum physics, without having to make commitments which rule out one interpretation or another.
So I make no apologies that this review of ``the philosophy of quantum mechanics'' is focussed sharply on the measurement problem. Section \ref{DMWWproblem} sets up the problem from a modern perspective; section \ref{DMWWdecoherence} is a self-contained discussion of the phenomenon of decoherence, which has played a major part in physicists' recent writings on the measurement problem. In sections \ref{DMWW3candidates}--\ref{DMWWhidden} I discuss the main approaches to solving the measurement problem currently in vogue: modern versions of the Copenhagen interpretation; the Everett interpretation; dynamical collapse; and hidden variables. Finally, in section \ref{DMWWRQM} I generalise the discussion beyond non-relativistic physics. I give a self-contained, non-mathematical introduction to quantum field theory, discuss some of \emph{its} conceptual problems, and draw some conclusions for the measurement problem.
\section{The Measurement Problem: a modern approach}\label{DMWWproblem}
The goal of this section is a clean statement of what the measurement problem actually is. Roughly speaking, my statement will be that QM provides a very effective \emph{algorithm} to predict macroscopic phenomena (including the results of measurements which purportedly record microscopic phenomena) but that it does not provide a satisfactorily formulated physical \emph{theory} which explains the success of this algorithm. We begin by formulating this algorithm.
\subsection{QM: formalism and interpretation}\label{DMWWquantumalgorithm}
To specify a quantum system, we have to give three things:
\begin{enumerate}
\item A Hilbert space \mc{H}, whose normalised vectors represent the possible states of that system.\footnote{More accurately: whose \emph{rays} --- that is, equivalence classes of normalised vectors under phase transformations --- represent the possible states of the system. }
\item Some additional structure on \mc{H} (all Hilbert spaces of the same dimension are isomorphic, so we need additional structure in order to describe specific systems). The additional structure is given by one or both of
\begin{itemize}
\item Certain \emph{preferred operators} on Hilbert space (or, certain \emph{preferred sets of basis vectors}).
\item A \emph{preferred decomposition} of the system into subsystems.
\end{itemize}
\item A \emph{dynamics} on \mc{H}: a set of unitary transformations which take a state at one time to the state it evolves into at other times. (Normally the dynamics is specified by the \emph{Hamiltonian}, the self-adjoint operator which generates the unitary transformations).
\end{enumerate}
For instance:
\begin{enumerate}
\item Non-relativistic one-particle QM is usually specified by picking a particular triple of operators and designating them as the position operators, or equivalently by picking a particular representation of states as functions on $\mathbf{\mathrm{R}}^3$ and designating it as the configuration-space representation. More abstractly, it is sometimes specified by designating pairs of operators $\langle \op{Q}_i,\op{P}_i\rangle$ (required to satisfy the usual commutation relations) as being the position and momentum observables.
\item In quantum computation a certain decomposition of the global Hilbert space into 2-dimensional component spaces is designated as giving the Hilbert spaces of individual qubits (normally taken to have spatially definite locations); sometimes a particular basis for each qubit is also designated as the basis in which measurements are made.
\item In quantum field theory (described algebraically) a map is specified from spatial regions to subalgebras of the operator algebras of the space, so that the operators associated with region $R$ are designated as representing the observables localised in $R$. At least formally, this can be regarded as defining a component space comprising the degrees of freedom at $R$.
\end{enumerate}
(Note that, although the specification of a quantum system is a bit rough-and-ready, the quantum systems \emph{themselves} have perfectly precise mathematical formalisms).
I shall use the term \emph{bare quantum formalism} to refer to a quantum theory picked out in these terms, prior to any notion of probability, measurement etc. Most actual calculations done with quantum theory --- in particle physics, condensed matter physics, quantum chemistry, \etc --- can be characterised as calculations of certain mathematical properties of the bare quantum formalism (the expectation values of certain functions of the dynamical variables, in the majority of cases.)
Traditionally, we extract \emph{empirical content} from the bare formalism via some notion of \emph{measurement}. The standard ``textbook'' way to do this is to associate measurements with self-adjoint operators: if \ket{\psi} is the state of the system and \op{M} is the operator associated with some measurement, and if
\be
\op{M}=\sum_i m_i \op{P}_i
\ee
is \op{M}'s spectral decomposition into projectors, then the probability of getting result $m_i$ from the measurement is $\matel{\psi}{\op{P}_i}{\psi}$.
Now the really important thing here is the set of projectors $\op{P}_i$ and not $\op{M}$ itself: if we associate the measurement with $f(\op{M})$, which has spectral decomposition
\be
f(\op{M})=\sum_i f(m_i) \op{P}_i
\ee
then the only difference is that the different possible measurement outcomes are labeled differently. Hence it has become normal to call this sort of measurement a \emph{projection-valued measurement}, or PVM.
It has become widely accepted that PVMs are not adequate to represent all realistic sorts of measurement. In the more general \emph{Positive-operator-valued measurement} formalism (see, \egc, \citeN[282--9]{peres}, \citeN[pp.\,90--92]{nielsenchuang}) a measurement process is associated with a family of positive operators $\{\op{E}_1, \ldots \op{E}_n\}$.
Each possible outcome of the measurement is associated with one of the $\op{E}_i$, and the probability of getting result $i$ when a system in state \ket{\psi} is measured is \matel{\psi}{\op{E}_i}{\psi}. PVMs are a special case, obtained only when each of the $\op{E}_i$ is a projector. This framework has proved extremely powerful in analyzing actual measurements (see, for instance, the POVM account of the Stern-Gerlach experiment given by \citeN{buschmeasurement}).
How do we establish \emph{which} PVM or POVM should be associated with a particular measurement? There are a variety of more-or-less \emph{ad hoc} methods used in practice, e.g.
\begin{enumerate}
\item In non-relativistic particle mechanics we assume that the probability of finding the system in a given spatial region \mc{R} is given by the usual formula
\be
\Pr(x \in \mc{R})=\int_R |\psi|^2.
\ee
\item In high-energy particle physics, if the system is in a state of definite particle number and has momentum-space expansion
\be
\ket{\psi}=\int \dr{^3 k} \alpha(\vctr{k}) \ket{\vctr{k}}
\ee
then we assume that the probability of finding its momentum in the vicinity of some $\vctr{k}$ is proportional to $|\alpha(\vctr{k})|^2$.
\item Again in non-relativistic particle mechanics, if we are making a joint measurement of position and momentum then we take the probability of finding the system in the vicinity of some phase-space point $(\vctr{q},\vctr{p})$ is given by one of the various ``phase-space POVMs'' \cite{buschmeasurement}.
\end{enumerate}
But ``measurement'' is a physical process, not an unanalyzable primitive, and physicists routinely apply the formalism of quantum physics to the analysis of measurements themselves. Here we encounter a regress, though: if we have to construct a quantum-mechanical description of measurement, how do we extract empirical content from \emph{that description}? In actual physics, the answer is: the regress ends when the measurement process has been magnified up to have \emph{macroscopically large} consequences. That is: if we have some microscopic system in some superposed state then the empirical content of that state is in principle determined by careful analysis of the measurement process applied to it. If the superposition is between macroscopically different states, however, we may directly read empirical content from it: a system in state
\be
\alpha \ket{\mathrm{\mbox{Macroscopic state 1}}}+ \beta\ket{\mathrm{\mbox{Macroscopic state 2}}}
\ee
is interpreted, directly, as having probability $|\alpha|^2$ of being found in macroscopic state 1.
Let us get a little more precise about this.
\begin{enumerate}
\item We identify some of the system's dynamical variables (that is, some of its self-adjoint operators) somehow as being the positions $\op{Q}_i$ and momenta $\op{P}_i$ of some macroscopic degrees of freedom of the system . For instance, for a simple system such as a macroscopic pointer, the centre-of-mass position and conjugate momentum of the system will suffice. For something more complicated (such as a fluid) we normally take the macroscopic degrees of freedom to be the density of the fluid averaged over some spatial regions large compared to atomic scales but small compared to macroscopic ones.
\item We decompose the Hilbert space of the system into a component space $\mc{H}_{macro}$ described by these macroscopic variables, and a space $\mc{H}_{micro}$ for the remaining degrees of freedom:
\be \mc{H}=\mc{H}_{macro}\otimes \mc{H}_{micro}.\ee
\item We construct \emph{wave-packet states} \ket{q^i,p_i} in $\mc{H}_{macro}$ --- Gaussian states, fairly localised around particular values ($q^i,p_i$)of $\op{Q}_i$ and $\op{P}_i$. These are the states which physicists in practice regard as ``macroscopically definite'': that is, located at the phase-space point ($q^i,p_i$). (We leave aside the \emph{conceptual } problems with regarding them thus: for now, we are interested in explicating only the pragmatic method used to extract empirical content from QM.)
\item Next, we expand the state in terms of them:
\be \ket{\psi}=\int \dr{p_i} \dr{q^i} \alpha(q^i,p_i)\tpk{q^i,p_i}{\psi(q^i,p_i)}.\ee
\item We regard \ket{\psi}, expanded thus, as a probabilistic mixture. That is, we take the probability density of finding the system's macroscopic variables to be in the vicinity of ($q^i,p_i$) to be $|\alpha(q^i,p_i)|^2$. Or to be (slightly) more exact, we take the probability of finding the system's macroscopic variables to be in some reasonably large set $V$ to be
\be \ \int_V \dr{p_i} \dr{q^i} |\alpha(q^i,p_i)|^2.\ee
\end{enumerate}
We might call this the \emph{Quantum Algorithm}. Empirical results are extracted from the Bare Quantum Formalism by applying the Quantum Algorithm to it.
\subsection{The Measurement Problem}\label{DMWWinterpretation}
The Bare Quantum Formalism (for any given theory) is an elegant piece of mathematics; the Quantum Algorithm is an ill-defined and unattractive mess. And this is the Measurement Problem.
\begin{quote}
\textbf{The Measurement Problem:} Applying the Quantum Algorithm to the Bare Quantum Formalism produces extremely accurate predictions about macroscopic phenomena: from the results of measurement processes to the boiling points of liquids. But we have no satisfactorily formulated scientific theory which reproduces those predictions.
\end{quote}
A solution of the measurement problem, then, is a satisfactorily formulated scientific theory (``satisfactorily formulated'', that is, relative to your preferred philosophy of science) from which we can explain why the Quantum Algorithm appears to be correct. Most such solutions do so by providing theories from which we can prove that the Algorithm \emph{is} correct, at least in the vast majority of experimental situations. There is no requirement here that different solutions are empirically indistinguishable; two solutions may differ from one another, and from the predictions of the Algorithm, in some exotic and so-far-unexplored experimental regime.
(Why call it the \emph{measurement} problem? Because traditionally it has been the measurement process which has been taken as the source of macroscopic superpositions, and because only when we have such superpositions do we have any need to apply the Quantum Algorithm. But processes other than formal measurements --- the amplification of classical chaos into quantum-mechanical indeterminateness, in particular --- can also give rise to macroscopic superpositions.)
Solutions of the measurement problem are often called ``interpretations of QM'', the idea being that all such ``interpretations'' agree on the formalism and thus on the experimental predictions. But in fact, different proposed solutions of the measurement problem are often different physical theories with different formalism. Where possible, then, I avoid using ``interpretation'' in this way (though often tradition makes it unavoidable).
There is, however, a genuinely interesting distinction between those proposed solutions which do, and those which do not, modify the formalism. It will be helpful to make the following definition: a \emph{pure interpretation} is a (proposed) solution of the measurement problem which has \emph{no mathematical formalism} other than the Bare Quantum Formalism. Proposed solutions which are \emph{not} pure interpretations I call \emph{modificatory}: a modificatory solution either adds to the bare formalism, or modifies it (by changing the dynamics, for instance), or in principle eliminates it altogether.
\subsection{Against the traditional account of quantum mechanics}\label{DMWWagainst}
There is a more traditional way to formulate QM, which goes something like this:
\begin{enumerate}
\item A quantum system is represented by a vector \ket{\psi} in a Hilbert space \mc{H}.
\item Properties of the system are represented by projectors on \mc{H}.
\item If and only if \ket{\psi} is an eigenstate of some projector, the system possesses the property associated with that projector; otherwise the value is `indefinite' or `indeterminate' or somesuch. (The `eigenvalue-eigenvector link')
\item A measurement of some property associated with projector \op{P} will find that property to be possessed by the system with probability \matel{\psi}{\op{P}}{\psi}.
\end{enumerate}
From this perspective, the ``measurement problem'' is the problem of understanding what `indefinite' or `indeterminate' property possession means (or modifying the theory so as to remove it) and of reconciling the indefiniteness with the definite, probabilistically determined results of quantum measurements.
However, this ``traditional account'' is not an ``interpretation-neutral'' way of stating the basic assumptions of QM; it is a false friend. Primarily, this is because it fails to give a good account of how physicists in practice apply QM: it assumes that measurements can be treated as PVMs, whereas as we have seen, it is now generally accepted that many practical measurement processes are best understood via the more general POVM formalism.
This is particularly clear where continuous variables are concerned --- that is, where almost all the quantities measured in practice are concerned. Here, physicists will normally regard a system as ``localised'' at some particular value of some continuous variable --- position, usually --- if its wavefunction is strongly peaked around that value. The fact that the wavefunction strictly speaking vanishes nowhere does not seem to bother them. In particular, measurements frequently measure continuous variables, and frequently output the result using further continuous variables (such as a pointer position). The practical criterion for such measurements is that if the system being measured is localised in the vicinity of $x$, the pointer displaying the result of the measurement should end up localised near whatever pointer position is supposed to display ``$x$''. This is straightforwardly represented via a POVM, but there is no natural way to understand it in terms of projections and the properties which they are supposed to represent.
Independent of recent progress in physics, there are reasons internal to philosophy of QM to be skeptical about the traditional account. As we shall see, very few mainstream interpretations of QM fit this framework: mostly they either treat the wavefunction as a physical thing (whose ``properties'' are then any properties at all of that thing, not just the property of being an eigenstate of some particular operator); or they associate physical properties to some additional ``hidden variables''; or they deny that the system has observer-independent properties at all.
One of the recurring themes of this chapter will be that the traditional account, having been decisively rejected in the practice of physicists, should likewise be discarded by philosophers: it distorts the philosophy of QM, forcing interpretations into Procrustean beds and encouraging wild metaphysics.
\section{Decoherence theory}\label{DMWWdecoherence}
Quite apart from its conceptual weaknesses, it is \emph{prima facie} surprising that the Quantum Algorithm is well-defined enough to give any determinate predictions at all. For the division between `macroscopic' and `microscopic' degrees of freedom, essential to its statement, was defined with enormous vagueness. Over \emph{how large} a spatial region must we average to get macroscopic density? --- $10^{-5} \mathrm{m}$? $10^{-4} \mathrm{m}$? Fortunately, it is now fairly well understood how to think about this question, thanks to one of the most important quantum-foundational developments of recent years: decoherence theory.
\subsection{The concept of decoherence}\label{DMWWdecoherencegeneral}
Suppose we have some unitarily-evolving quantum system, with Hilbert space \mc{H}, and consider some decomposition of the system into component subsystems:
\be \mc{H}=\mc{H}_{sys}\otimes \mc{H}_{env},\ee
which we will refer to as the \emph{system} and the \emph{environment}.
Now suppose that $\{\ket{\alpha}\}$ is some (not-necessarily-orthogonal) basis of $\mc{H}_{sys}$ and that the dynamics of the joint system is such that, if we prepare it in a product state
\be
\tpk{\alpha}{\psi}
\ee
then it evolves rapidly into another pure state
\be
\tpk{\alpha}{\psi;\alpha}
\ee
with $\bk{\psi;\alpha}{\psi;\beta}\simeq \delta(\alpha-\beta)$. (Here, ``rapidly'' means rapidly relative to other relevant dynamical timescales). In other words, we suppose that the environment measures the system in the $\{\ket{\alpha}\}$ basis and records the result.
Suppose further that this ``recording'' is reasonably robust, so that subsequent system-environment interactions do not tend to erase it: that is, we don't get evolutions like
\be
\lambda_1 \tpk{\alpha_1}{\psi;\alpha_1}+\lambda_2 \tpk{\alpha_2}{\psi;\alpha_2}
\longrightarrow
\tpk{\phi}{\chi}.
\ee
In this (loosely-defined) situation, we say that the environment \emph{decoheres} the system, and that the basis $\{\ket{\alpha}\}$ is a \emph{preferred} basis or \emph{pointer} basis. The timescale on which the recording of the system's state occurs is called the \emph{decoherence timescale}.
Much follows from decoherence. The most obvious effects are \emph{synchronic} (or at least, have a consequence which may be expressed synchronically): the system cannot stably be prepared in superpositions of pointer-basis states. Such superpositions very rapidly become entangled with the environment. Conversely, if the system is prepared in a pointer-basis state, it will remain stably in that pointer-basis state (at least for times long compared to the decoherence timescale). Equivalently, the density operator of the system, when expressed in the pointer basis, will be diagonal or nearly so.
However, the more important consequence is diachronic. If the environment is keeping the density operator almost diagonal, then interference terms between elements of the pointer basis must be being very rapidly suppressed, and the evolution is effectively \emph{quasi-classical}.
To see this more clearly, suppose that the dynamics of the system is such that after time $t$, we have the evolution
\be \label{DMWWdeco1}\tpk{\alpha_1}{\psi_1} \longrightarrow \ket{\Lambda_1}=\lambda_{11}\tpk{\alpha_1}{\psi_{11}}+\lambda_{12}\tpk{\alpha_2}{\psi_{12}};
\ee
\be \label{DMWWdeco2}\tpk{\alpha_2}{\psi_2} \longrightarrow \ket{\Lambda_2}=\lambda_{21}\tpk{\alpha_1}{\psi_{21}}+\lambda_{22}\tpk{\alpha_2}{\psi_{22}}.
\ee
By linearity, the superposition
\be \label{DMWWdecosup}
\ket{\Psi}=\mu_1 \tpk{\alpha_1}{\psi_1}
+\mu_2 \tpk{\alpha_2}{\psi_2}
\ee
evolves in the same time to
\[
\mu_1 \left( \lambda_{11}\tpk{\alpha_1}{\psi_{11}}+\lambda_{12}\tpk{\alpha_2}{\psi_{12}}\right)
+
\mu_2 \left( \lambda_{21}\tpk{\alpha_1}{\psi_{21}}+\lambda_{22}\tpk{\alpha_2}{\psi_{22}}\right)
\]
\be \label{DMWWdecoresult}
= \ket{\alpha_1}\left( \mu_1 \lambda_{11}\ket{\psi_{11}} +\mu_2 \lambda_{21}\ket{\psi_{21}}\right)
+\ket{\alpha_2}\left( \mu_1 \lambda_{12}\ket{\psi_{12}} +\mu_2 \lambda_{22}\ket{\psi_{22}}\right).
\ee
Now, suppose that we want to interpret states \ket{\Psi}, \ket{\Lambda_1} and \ket{\Lambda_2} probabilistically with respect to the $\{\ket{\alpha}\}$ --- for example, in \ket{\Psi} we want to interpret $|\mu_1|^2$ as the probability of finding the system in state \ket{\alpha_1}. Generally speaking, interference makes this impossible: (\ref{DMWWdeco1}) and (\ref{DMWWdeco2}) would entail that if the joint system is initially in state \tpk{\alpha_i}{\psi_i}, after time $t$ there is probability $|\lambda_{i1}|^2$ of finding the system in state $\ket{\alpha_1}$. Applying the probabilistic interpretation to \ket{\Psi} tells us that the joint system initially has probability $|\mu_i|^2$ of indeed being initially in state \tpk{\alpha_i}{\psi_i}, and hence the system has probability
\be P= |\mu_1|^2 |\lambda_{11}|^2 + |\mu_2|^2 |\lambda_{21}|^2
\ee
of being found in $\ket{\alpha_1}$ after time $t$. But if we apply the probabilistic interpretation directly to (\ref{DMWWdecoresult}), we get a contradictory result:
\be
P' = |\mu_1|^2 |\lambda_{11}|^2 + |\mu_2|^2 |\lambda_{21}|^2 + 2\mathrm{\mbox{Re}}(\mu_1^*\lambda_{11}^*\mu_2\lambda_{21}\bk{\psi_{11}}{\psi_{21}}).
\ee
Crucially, though, the contradiction is eliminated and we get the same result in both cases (irrespective of the precise values of the coefficients) \emph{provided} that $\bk{\psi_{11}}{\psi_{21}}=0$. And this is exactly what decoherence, approximately speaking, guarantees: the states \ket{\psi_{11}} and \ket{\psi_{21}} are approximately-orthogonal records of the distinct states of the system in the original superposition.
So: we conclude that in the presence of decoherence, and provided that we are interested only in the state of the system and not of the environment, it is impossible to distinguish between a \emph{superposition} of states like $\tpk{\alpha}{\psi_\alpha}$ and a mere probabilistic \emph{mixture} of such states.
\subsection{Domains and rates of decoherence}\label{DMWWdecoherencerates}
When does decoherence actually occur? Some clear results have been established:
\begin{enumerate}
\item The macroscopic degrees of freedom of a system are decohered by the microscopic degrees of freedom.
\item The pointer basis picked out in $\mc{H}_{macro}$ is a basis of quasi-classical, Gaussian states.
\end{enumerate}
This should not be surprising. Decoherence occurs because the state of the system is recorded by the environment; and, because the dynamics of our universe are spatially local, the environment of a macroscopically large system in a given spatial position will inevitably record that position. (A single photon bouncing off the system will do it, for instance.) So superpositions of systems in macroscopically distinct positions will rapidly become entangled with the environment. And superpositions of states with macroscopically distinct \emph{momentums} will very rapidly evolve into states of macroscopically distinct positions. The only states which will be reasonably stable against the decoherence process will be wave-packets whose macroscopic degrees of freedom have reasonably definite position \emph{and} momentum.
Modelling of this decoherence process (both computationally and mathematically) shows that\footnote{These figures are derived from data presented in \citeN{joosetal}.}
\begin{enumerate}
\item The process is extremely rapid. For example:
\begin{enumerate}
\item A dust particle of size $\sim 10^{-3} \mathrm{cm}$ in a superposition of states $\sim 10^{-8} \mathrm{m}$ apart will become decohered by sunlight after $\sim 10^{-5}$ seconds, and by the Earth's atmosphere after $\sim 10^{-18} \mathrm{s}$; the same particle in a superposition of states $\sim 10^{-5} \mathrm{m}$ apart will become decohered by sunlight in $\sim 10^{-13} \mathrm{s}$ (and by the atmosphere in $10^{-18}\mathrm{s}$ again: once the separation is large compared to the wavelength of particles in the environment then the separation distance becomes irrelevant.)
\item A kitten in a superposition of states $10^{-10}\mathrm{m}$ apart is decohered by the atmosphere in $\sim 10^{-25}\mathrm{s}$ and by sunlight in $\sim 10^{-8}\mathrm{s}$; the same kitten in a superposition of states $10^{-5} \mathrm{m}$ apart is decohered by the atmosphere in $\sim 10^{-26}\mathrm{s}$, by sunlight in $\sim 10^{-21}\mathrm{s}$, and by the microwave background radiation in $\sim 10^{-15}\mathrm{s}$.
\end{enumerate}
\item In general there is no need for the ``environment'' to be in some sense \emph{external} to the system. In general, the macroscopic degrees of freedom of a system can be decohered by the residual degrees of freedom of that same system: in fluids, for instance, the `hydrodynamic' variables determined by averaging particle density and velocity over regions large compared to particle size are decohered by the remaining degrees of freedom of the fluid.
\item The dynamics of the macroscopic degrees of freedom seem, in general, to be `quasi-classical' not just in the abstract sense that they permit a probabilistic interpretation, but in the more concrete sense that they approximate classical equations of motion. To be more precise:
\begin{enumerate}
\item If the classical limit of the system's dynamics is classically regular (\iec, non-chaotic), as would be the case for a heavy particle moving inertially, then the pointer-basis states evolve, to a good approximation, like the classical states they are supposed to represent. That is, if the classical-limit dynamics would take the phase-space point $(\vctr{q},\vctr{p})$ to (\vctr{q}(t),\vctr{p}(t)), then the quantum dynamics are approximately
\be
\tpk{\vctr{q},\vctr{p}}{\psi}\longrightarrow \tpk{\vctr{q}(t),\vctr{p}(t)}{\psi(t)}.
\ee
\item If the classical limit of the system's dynamics is chaotic, then classically speaking a localised region in phase space will become highly fibrillated, spreading out over the energetically available part of phase-space (while retaining its volume). The quantum system is unable to follow this fibrillation: on timescales comparable to those on which the system becomes classically unpredictable, it spreads itself across the entire available phase space region:
\be \tpk{\vctr{q},\vctr{p}}{\psi}\longrightarrow \int_\Omega \dr{q}\dr{p}\tpk{\vctr{q},\vctr{p}}{\psi_{q,p}(t)}
\ee
(where $\Omega$ is the available region of phase space). In doing so, it still tracks the coarse-grained behaviour of the classical system, but fails to track the fine details: thus, classical unpredictability is transformed into quantum indeterminacy.
\end{enumerate}
\end{enumerate}
For our purposes, though, the most important point is this: decoherence gives a criterion for applicability of the Quantum Algorithm. For the `quasi-classical' dynamics that it entails for macroscopic degrees of freedom is a guarantee of the \emph{consistency} of that algorithm: provided `macroscopic' is interpreted as `decohered by the residual degrees of freedom beyond our ability to detect coherence', then the algorithm will give the same results regardless of exactly when, and at what scales, the algorithm is deployed to make a probabilistic interpretation of the quantum state.
\subsection{Sharpening decoherence: consistent histories}\label{DMWWhistoryframework}
The presentation of decoherence given in the previous section was somewhat loosely defined, and it will be useful to consider the most well-developed attempt at a cleaner definition: the \emph{consistent histories} formalism. To motivate this formalism, consider a decoherent system with pointer basis $\{\ket{\alpha}\}$, as above, and suppose (as is not in fact normally the case) that the pointer basis is discrete and orthonormal: $\bk{\alpha}{\beta}$=$\delta_{\alpha,\beta}$.
Suppose also that we consider the system only at discrete times $t_0,t_1,\ldots t_n$.
Now, decoherence as we defined it above is driven by the establishment of records of the state of the system (in the pointer basis) made by the environment. Since we are discretising time it will suffice to consider this record as made only at the discrete times (so the separation $(t_{n+1}-t_n)$ must be large compared with the decoherence timescale). Then if the system's state at time $t_0$ is
\be \ket{\Psi_0}=\sum_{i_0} \mu_{i_0} \tpk{\alpha_{i_0}}{\psi(i_0)}
\ee
it should evolve by time $t_1$ into some state like
\be \ket{\Psi_1}=\sum_{i_0,i_1}\mu_{i_0}\Lambda_1(i_0,i_1)\tpk{\alpha_{i_1}}{\psi(i_0,i_1)}
\ee
(for some transition coefficients $\Lambda_1(i_0,i_1)$), with the states \ket{\psi(i_0,i_1)} \emph{recording} the fact that the system (relative to that state) was in state $\ket{\alpha_{i_0}}$ and is now into $\ket{\alpha_{i_1}}$ (and thus being orthogonal to one another). Similarly, by time $t_2$ the system will be in state
\be \ket{\Psi_1}=\sum_{i_0,i_1,i_2}\mu_{i_0}\Lambda_1(i_0,i_1)\Lambda_2(i_1,i_2)\tpk{\alpha_{i_2}}{\psi(i_0,i_1,i_2)}
\ee
and (iterating)
by time $t_n$ will finish up in state
\be \ket{\Psi_n}=\sum_{i_0,i_1,\cdots i_n}\mu_{i_0}\Lambda_1(i_0,i_1)\cdots\Lambda_n(i_{n-1},i_n)\tpk{\alpha_{i_n}}{\psi(i_0,i_1,\ldots,i_n)}.
\ee
Since we require (by definition) that record states are orthogonal or nearly so, we have
\be
\bk{\psi(i_0,\ldots i_n)}{\psi(j_0,\ldots j_n)}\simeq \delta_{i_0,j_0}\cdots \delta_{i_n,j_n}.
\ee
There is an elegant way to express this, originally due to \citeN{griffiths} and developed by \citeN{gellmannhartle} and others. For each
$\ket{\alpha_i}$, and each of our discrete times $t_0, \ldots t_n$, let $\op{P}_i$ be the projector
\be \op{P}_i(t_j)=\opad{U}(t_j,t_0)\left( \proj{\alpha_i}{\alpha_i}\otimes \id \right) \op{U}(t_j,t_0),
\ee
where $\op{U}(t_j,t_0)$ is the unitary evolution operator taking states at time $t_0$ to states at time $t_j$ (unless the Hamiltonian is time-dependent, $\op{U}(t_j,t_0)=\exp(-i(t_j-t_0)\op{H}/\hbar)$).
Then for any sequence $\vctr{i}=(i_0,i_1,\ldots i_n)$ of indices we may define the \emph{history operator} $\op{C}_\vctr{i}$
by
\be \label{DMWWhistdef}
\op{C}_\vctr{i}= \op{P}_{i_n}(t_n)\cdots \op{P}_{i_0}(t_0).
\ee
Now,
\[ \op{P}_{i_0}(t_0)\ket{\Psi_0}=\mu_{i_0}\tpk{\alpha_{i_0}}{\psi(i_0)};\]
\[\op{P}_{i_1}(t_1)\op{P}_{i_0}(t_0)\ket{\Psi_0}=\mu_{i_0}\op{P}_{i_1}(t_1)\tpk{\alpha_{i_0}}{\psi(i_0)}
=\mu_{i_0}\Lambda_{t_1}(i_0,i_1)\tpk{\alpha_{i_1}}{\psi(i_0,i_1)}
\]
\[\cdots\]
\be
\op{C}_\vctr{i}\ket{\Psi_0}=\mu_{i_0}\Lambda_1(i_0,i_1)\cdots\Lambda_n(i_{n-1},i_n)\tpk{\alpha_{i_n}}{\psi(i_0,i_1,\ldots,i_n)}.
\ee
This has an immediate corollary:
\be
\matel{\Psi_0}{\opad{C}_{\vctr{j}}\op{C}_{\vctr{i}}}{\Psi_0}\propto \bk{\alpha_{j_n}}{\alpha_{i_n}}\bk{\psi(j_0,j_1,\ldots,j_n)}{\psi(i_0,i_1,\ldots,i_n)}
\ee
and hence
\be \label{DMWWmeddec}
\matel{\Psi_0}{\opad{C}_{\vctr{j}}\op{C}_{\vctr{i}}}{\Psi_0}\simeq 0 \mathrm{\mbox{ unless }}\vctr{i}=\vctr{j}.
\ee
Furthermore, if we apply the Quantum Algorithm, it tells us that the probability of the system being found successively in states (corresponding to)
$\ket{\alpha_{i_0}}, \ldots \ket{\alpha_{i_n}}$ is given by $\matel{\Psi_0}{\opad{C}_{\vctr{i}}\op{C}_{\vctr{i}}}{\Psi_0}$ . The condition (\ref{DMWWmeddec}) then has a natural interpretation: it tells us that there is no interference between distinct histories, so that the Quantum Algorithm can be applied at successive times without fear of contradiction.
Now let us generalise. Given an \emph{arbitrary} complete set of projectors $\op{P}_i(t_j)$ for each time $t_j$ in our finite set
we can define histories $\op{C}_{\vctr{i}}$ via (\ref{DMWWhistdef}).
We say that these histories satisfy the \emph{medium decoherence condition} (\cite{gellmannhartle93}) with respect to some state \ket{\Psi} if $\matel{\Psi}{\opad{C}_{\vctr{j}}\op{C}_{\vctr{i}}}{\Psi}\simeq 0$ whenever $\vctr{i}\neq\vctr{j}$.
A set of histories satisfying medium decoherence has the following attractive properties:
\begin{enumerate}
\item If (as above) the quantities $\matel{\Psi}{\opad{C}_{\vctr{i}}\op{C}_{\vctr{i}}}{\Psi}$ are interpreted as probabilities of a given history being realised then medium decoherence guarantees that this can be done consistently, at least within the limits of what we can experimentally determine. In particular, it guarantees that if we define coarse-grained histories (by, \egc, leaving out some intermediate time $t_i$ or amalgamating some projectors into a single joint projector), the coarse-graining obeys the probability calculus:
\be\label{DMWWconsistent}\Pr(\sum_{\vctr{i}\in I}\op{C}_{\vctr{i}})\simeq \sum_{\vctr{i}\in I}\Pr(\op{C}_{\vctr{i}}).\ee
For
\[ \Pr(\sum_{\vctr{i}\in I}\op{C}_{\vctr{i}})\simeq \matel{\Psi}{(\sum_{\vctr{j}\in I}\opad{C}_{\vctr{j}})(\sum_{\vctr{i}\in I}\op{C}_{\vctr{i}})}{\Psi}
\]
\be
\simeq\sum_{\vctr{j}\in I}\sum_{\vctr{i}\in I}\matel{\Psi}{\opad{C}_{\vctr{j}}\op{C}_{\vctr{i}}}{\Psi}
\ee
which in the presence of medium decoherence is just equal to $\sum_{\vctr{i}\in I}\matel{\Psi}{\opad{C}_{\vctr{i}}\op{C}_{\vctr{i}}}{\Psi}$.
(Actually a weaker condition --- that the \emph{real part} of $\matel{\Psi}{\opad{C}_{\vctr{j}}\op{C}_{\vctr{i}}}{\Psi}=0$ --- is sufficient to deliver (\ref{DMWWconsistent}). This condition is called consistency; it does not seem to occur in natural situations other than those which also deliver medium decoherence.)
\item Medium decoherence guarantees the existence of records (in an abstract sense). The probabilistic interpretation tells us that at time $t_n$ the system should be thought of as having one of the states
\be \ket{\Psi(\vctr{i})}=\mc{N}\op{C}_\vctr{i}\ket{\Psi}\ee
(where \mc{N} is a normalising factor). These states are mutually orthogonal: as such, a single measurement (in the traditional sense) suffices, in principle, to determine the entire history and not just the current state.
\end{enumerate}
In light of its elegance, it is tempting to adopt the criterion of medium decoherence of a set of histories as the \emph{definition} of decoherence, with the decoherence of the previous section only a special case (and an ill-defined one at that). And in fact the resultant formalism (call it the `decoherent histories' formalism) has more than just elegance to recommend it. For one thing, it makes explicit the state-dependence of decoherence. This was in any case implicit in the previous section's analysis: for the `environment' to decohere the system, it must be in an appropriate state. (If the state of the `environment' is a quintillion-degree plasma, for instance, the system will certainly not undergo quasi-classical evolution!) For another, it allows for a system/environment division which is not imposed once and for all, but can vary from history to history.
It would be a mistake, however, to regard the decoherent histories formalism as \emph{conceptually} generalising the environment-induced decoherence discussed in section \ref{DMWWdecoherencegeneral}. In both cases, the mechanism of decoherence is the same: some subset of the degrees of freedom are recorded by the other degrees of freedom, with the local nature of interactions picking out a phase-space-local basis as the one which is measured; this recording process breaks the coherence of the macroscopic degrees of freedom, suppressing interference and leading to dynamics which are quasi-classical and admit of a probabilistic interpretation (at least approximately). And although the decoherent-histories formalism in theory has the power to incorporate history-dependent system/environment divisions, in practice even simple models where this actually occurs have proven elusive, and actual applications of the decoherent-histories formalism have in the main been restricted to the same sort of system/environment split considered in section \ref{DMWWdecoherencegeneral} (although the `environment' is often taken to be microscopic degrees of freedom of the same system).
Furthermore, there are some infelicities of the decoherent-histories formalism as applied to realistic cases of decoherence. In particular, the natural pointer basis for realistic systems seems to be non-orthonormal wave-packet states and the rate of decoherence of superpositions of such states depends smoothly on the spatial distance between them. This does not sit altogether easily with the decoherent-histories formalism's use of discrete times and an orthonormal pointer basis.
Perhaps most importantly, though, the consistency condition alone is insufficient to restore quasi-classical dynamics in the `concrete' sense of section \ref{DMWWdecoherencerates} --- that is, it is insufficient to provide approximately classical equations of motion. I return to this point in section \ref{DMWWsolutionthatisnt}.
In any case, for our purposes what is important is that (no matter how `decoherence' is actually defined) the basis of quasi-classical states of a macroscopic system is very rapidly decohered by its environment. This guarantees the consistency, for all practical purposes, of the Quantum Algorithm; whether it goes further and actually solves the measurement problem is a matter to which I will return in sections \ref{DMWW3candidates} and \ref{DMWWeverett}.
\subsection{Further Reading}\label{DMWWdecoherencefurtherreading}
Joos \emph{et al}~\citeyear{joosetal} and \citeN{zurek01review} provide detailed reviews of decoherence theory; \citeN{zurek91} is an accessible short introduction. \citeN{bacciagaluppiencyclopedia} reviews the philosophical implications of decoherence.
\section{Three candidates for orthodoxy}\label{DMWW3candidates}
In philosophy of QM, terminology is not the least source of confusion. Authors frequently discuss the ``orthodox'' interpretation of QM as if everyone knew what they meant, even though different authors ascribe different and indeed contradictory properties to their respective versions of orthodoxy. It does not help that physicists use the term ``Copenhagen interpretation'' almost interchangeably with ``orthodox interpretation'' or ``textbook interpretation'', while philosophers tend to reserve the term for Bohr's actual, historical position, and use a term like ``Dirac-von Neumann interpretation'' for the textbook version.
In this section --- which aims to present the ``orthodox interpretation'' --- I follow the sage advice of Humpty Dumpty, who reminded Alice that words mean what we want them to mean. There are at least three mainstream positions on the measurement problem which are often described as ``orthodoxy''. Two of them --- operationalism and the consistent-histories formalism --- are highly controversial pure interpretations of QM, which their proponents nonetheless often describe as the ``orthodox'' or indeed the only possible interpretation. (In their different ways, both are also claimed to capture the true spirit of Copenhagen). The third (which I call the ``new pragmatism'') is not actually regarded by anyone as a \emph{successful} solution to the measurement problem but, arguably, best captures the pragmatic quantum theory actually used by working physicists. It is best understood by considering, first, an attractive but failed solution.
\subsection{The solution that isn't: non-uniqueness of decoherent histories}\label{DMWWsolutionthatisnt}
Suppose that there was \emph{exactly one} finest-grained set of decoherent histories --- defined, say, by projectors $\op{P}_i(t_i)$ which satisfied the medium decoherence condition exactly; suppose also that this set of histories picked out a preferred basis reasonably close to the ``quasi-classical'' states used in the Quantum Algorithm, so that each $\op{P}_i(t_i)$ projected onto those states interpreted by the Quantum Algorithm as saying: the macroscopic degrees of freedom of the system will certainly be found to have some particular values $(q^i,p_i)$.
In this case, a solution of sorts to the measurement problem would be at hand. It would simply be a stochastic theory of the macroscopic degrees of freedom, specified as follows:
\begin{quote}
Given that:
\begin{enumerate}
\item the \emph{universal state} is \ket{\Psi};
\item the unique finest-grained decoherent-history space consistent with \ket{\Psi} is generated by projectors $\op{P}_i(t_j)$, associated with values $(q^i,p_i)$ for the macroscopic degrees of freedom at time $t_j$;
\item the macroscopic degrees of freedom at time $t_j$ have values $(q^i,p_i)$, corresponding to projector $\op{P}_i(t_j)$
\end{enumerate}
then the probability of the macroscopic degrees of freedom at time $t_{j'}$ having values $(q^{i'},p_{i'})$
is given by
\be
\Pr (q^{i'},p_{i'};t_{j'}| q^i,p_i;t_i)=\frac{\matel{\Psi}{\op{P}_{i'}(t_{j'})\op{P}_i(t_i)\op{P}_i(t_i)\op{P}_{i'}(t_{j'})}{\Psi}}{\matel{\Psi}{\op{P}_i(t_i)\op{P}_i(t_i)}{\Psi}}.
\ee
(It follows from this and the decoherence condition, of course, that the probability of a given \emph{history} $\op{C}_\vctr{i}$ is just \matel{\Psi}{\opad{C}_\vctr{i}\op{C}_\vctr{i}}{\Psi}.)
\end{quote}
How satisfactory is this as an interpretation of QM? It is not a \emph{pure} interpretation; on the other hand, since it is (ex hypothesi) a successful interpretation, it is unclear that this matters. It is not obviously compatible with relativity, since it makes explicit use of a preferred time; perhaps this could be avoided via a spacetime-local version of the preferred projectors, but it seems unlikely that genuinely pointlike degrees of freedom would decohere. The role of the `universal state' is pretty unclear --- in fact, the ontology as a whole is pretty unclear, and the state-dependent nature of the preferred set of histories is at least odd.
These questions are moot, though. For the basic assumption which grounds the interpretation --- that there exists a unique (finest-grained) exactly-decoherent history space --- is badly mistaken, as has been shown by \citeN{dowkerkent} and \citeN{kenthistory}. The problem does not appear to be \emph{existence}: as section \ref{DMWWdecoherence} showed, there are good reasons to expect the histories defined by macroscopic degrees of freedom of large systems to approximately decohere, and Dowker and Kent have provided plausibility arguments to show that in the close vicinity of any almost-decoherent family of histories we can find an exactly-decoherent one. It is \emph{uniqueness}, rather, that causes the difficulties: there are excellent reasons to believe that the set of exactly decoherent history spaces is huge, and contains (continuously) many history spaces which are not remotely classical. Indeed, given a family of decoherent histories defined up to some time $t$, there are continuously many distinct ways to continue that family. As such, the simple decoherence-based interpretation above becomes untenable.
The temptation, for those seeking to solve the measurement problem via decoherence, is to introduce some additional criterion stronger than medium decoherence --- some $X$ such that there is a unique finest-grained history space satisfying medium-decoherence-plus-$X$. And in fact there is a popular candidate in the literature: \emph{quasi-classicality} \cite{gellmannhartle93}. That is: the preferred history space not only decoheres: the decohering degrees of freedom obey approximately classical equations of motion.
It is plausible (though to my knowledge unproven) that this condition is essentially unique; it is \emph{highly} plausible that there are not continuously many essentially different ways to vary a quasi-classical decoherent history space. But as a candidate for $X$, quasi-classicality is pretty unsatisfactory. For one thing, it is essentially vague: while we have good theoretical reasons to expect exactly-decoherent histories in the vicinity of approximately decoherent ones, we have no reason at all to expect exactly classical histories in the vicinity of quasi-classical ones. For another, it is a high-level notion all-but-impossible to define in microphysical terms. It is as if we were to write a theory of atomic decay which included ``density of multicellular organisms'' as a term in its equations.
As such, it seems that no satisfactory decoherent-history-based interpretation can be developed along the lines suggested here.
\subsection{The new pragmatism}
However, an interpretation need not be \emph{satisfactory} to be \emph{coherent} (so to speak). No-one who took the measurement problem seriously regarded the Dirac-von Neumann formulation of QM, with its objective collapse of the wavefunction at the moment of measurement, as a \emph{satisfactory} physical theory; yet it was widely discussed and used in physics when one wanted a reasonably clear statement of the theory being applied (and never mind its foundational problems). The quasi-classical condition discussed in the previous section lets us improve on the Dirac-von Neumann interpretation by making (somewhat) more precise and objective its essential appeal to `measurement' and `observation'. The resultant theory has been called the `unknown set' interpretation by \citeN{kentbohmhistory}; I prefer to call it the \emph{New Pragmatism}, to emphasise that no-one really regards it as acceptable. It is, nonetheless, one of our three ``candidates for orthodoxy''; though it has not been explicitly stated in quite the form which I shall use, it seems to conform quite closely to the theory that is in practice appealed to by working physicists.
\begin{quote}
\textbf{The New Pragmatism (decoherent-histories version):}
The state of the Universe at time $t$ is given by specifying some state vector \ket{\Psi(t)}, which evolves unitarily, and some
\emph{particular} quasi-classical, approximately decoherent consistent-history space, generated by
the projectors $\op{P}_i(t_j)$
The state \ket{\Psi(t)} is to be interpreted as a probabilistic mixture of eigenstates of the quasi-classical projectors: that is, expanding it as
\be
\ket{\Psi(t)}=\sum_{i}\op{P}_i(t)\ket{\Psi(t)}\matel{\Psi(t)}{\op{P}_i(t)}{\Psi(t)},
\ee
the probability that the state of the Universe is (up to normalisation) $\op{P}_i(t)\ket{\Psi(t)}$ is
$|\matel{\Psi(t)}{\op{P}_i(t)}{\Psi(t)}|^2$. Because the history space is approximately decoherent, any interference-generated inconsistencies caused by this probabilistic reading of the state will be undetectable; if that is felt to be unsatisfactory, just require that the history space is exactly decoherent (some such will be in the vicinity of any given approximately-decoherent history space).
\end{quote}
According to the New Pragmatism, then, the quantum state vector is physical --- is, indeed, the complete physical description of the system. It evolves in some mixture of unitary steps and stochastic jumps, and at any given time it assigns approximately-definite values of position and momentum to the macroscopic degrees of freedom of the system. We do not know the actual decoherent-history space used (hence `unknown set), but we know it well enough to predict all probabilities to any reasonably-experimentally-accessible accuracy.
The New Pragmatism, it will be apparent, is a pretty minimal step beyond the Quantum Algorithm itself: if we were to ask for the most simple-minded way to embed the Algorithm into a theory, without any concern for precision or elegance, we would get something rather like the New Pragmatism. This is even more obvious if we reformulate it from the language of decoherent histories to the environment-induced decoherence of section \ref{DMWWdecoherence}:
\begin{quote}
\textbf{The New Pragmatism (wave-packet version):}
Fix some \emph{particular} division of Hilbert space into macroscopic and microscopic degrees of freedom: $\mc{H}=\mc{H}_{macro}\otimes \mc{H}_{micro}$; and fix some particular basis $\ket{\vctr{q},\vctr{p}}$ of wave-packet states for $\mc{H}_{macro}$. Then the state vector \ket{\Psi(t)} of the Universe always evolves unitarily, but is to be understood as a probabilistic mixture of approximately-macroscopically-definite states: if the universal state is the superposition
\be
\ket{\Psi(t)}=\int\dr{\vctr{p}}\dr{\vctr{q}}\alpha(\vctr{q},\vctr{p})\tpk{\vctr{q},\vctr{p}}{\psi(\vctr{q},\vctr{p};t)}
\ee
then the actual state is one of the components of this superposition, and has probability
$|\alpha(\vctr{q},\vctr{p})|^2$ of being $\tpk{\vctr{q},\vctr{p}}{\psi(\vctr{q},\vctr{p};t)}$.
(And of course this state in turn is somehow to be understood as having macroscopic phase-space location $(\vctr{q},\vctr{p})$.)
\end{quote}
It is an interesting philosophy-of-science question to pin down exactly what is unacceptable about the New Pragmatism. And it is not obvious at all that it \emph{is} unacceptable from some anti-realist standpoints (from the standpoint of \citeN{vanfraassenscientificimage}, for instance). Nonetheless, it is accepted as unsatisfactory. Unlike our other two candidates for orthodoxy, and despite the frequency with which it is in fact used, no-one really takes it seriously.
\subsection{The consistent-histories interpretation}\label{DMWWhistoryinterpretation}
A more `serious' interpretation of QM, still based on the decoherent-histories formalism, has been advanced by Griffiths \citeyear{griffiths,griffithsbook} and Omnes \citeyear{omnes,omnesbook}: it might be called the `consistent histories' interpretation,\footnote{Terminology is very confused here. Some of those who advocate `consistent-histories' interpretations --- notably Gell-Mann and Hartle --- appear to mean something very different from Griffiths and Omnes, and much closer in spirit to the Everett interpretation.} and its adherents claim that it incorporates the essential insights of Bohr's complementarity, and should be viewed as the natural modern successor to the Copenhagen interpretation.
The positions of Griffiths and Omnes are subtle, and differ in the details. However, I think that it is possible to give a general framework which fits reasonably well to both of them. We begin, as with the impossible single-decoherent-history-space theory, with some universal state \ket{\Psi}. Now, however, we consider \emph{all} of the maximally-fine-grained consistent history spaces associated with \ket{\Psi}. (Recall that a history space is \emph{consistent} iff the \emph{real} part of \matel{\Psi}{\opad{C}_\vctr{i}\op{C}_\vctr{j}}{\Psi} vanishes for $\vctr{i}\neq \vctr{j}$; it is a mildly weaker condition than decoherence, necessary if the probabilities of histories are to obey the probability calculus.)
Now in fact, these ``maximally fine-grained'' history spaces are actually constructed from \emph{one-dimensional} projectors. For any exactly-consistent history which does not so consist can always be fine grained, as follows: let it be constructed as usual from projectors $\op{P}_{i_j}(t_j)$, and define the state \ket{i_k, \ldots i_0} by
\be
\ket{i_k, \ldots i_0}= \mc{N}\op{P}_{i_k}(t_k)\cdots \op{P}_{i_0}(t_0)\ket{\Psi}
\ee
(where \mc{N} is just a normalising factor.) Then define a fine-graining $\op{P}^m_{i_k}(t_k)$ as follows:
\be \op{P}^0_{i_k}(t_k)=\proj{i_k, \ldots i_0}{i_k, \ldots i_0};\ee
the other $\op{P}^m_{i_k}(t_k)$ are arbitrary one-dimensional projectors chosen to satisfy
\be
\sum_m \op{P}^m_{i_k}(t_k)=\op{P}_{i_k}(t_k).
\ee
It is easy to see that
\[
\op{P}^{m_n}_{i_n}(t_n)\cdots \op{P}^{m_0}_{i_0}(t_0)\ket{\Psi}=0\mathrm {\mbox{ whenever any }}m_k \neq 0
\]
\be
\op{P}^{0}_{i_n}(t_n)\cdots \op{P}^{0}_{i_0}(t_0)\ket{\Psi}=\op{P}_{i_n}(t_n)\cdots \op{P}_{i_0}(t_0)\ket{\Psi},
\ee
and hence the fine-graining also satisfies the consistency condition. Notice (this is why I give the proof explicitly, in fact) how sensitive this fine-graining process is to the universal state \ket{\Psi} (by contrast, when we are dealing with the coarse-grained approximately-decoherent histories given by dynamical decoherence, the history space is fairly insensitive to all but broad details of \ket{\Psi}).
Griffiths and Omnes now regard each consistent history space as providing some valid \emph{description} of the quantum system under study. And under a given description $\{\op{C}_\vctr{i}\}$, they take the probability of the system's actual history being $\op{C}_\vctr{i}$ to be given by the usual formula $\matel{\Psi}{\opad{C}_\vctr{i}\op{C}_\vctr{i}}{\Psi}$.
Were there in fact only one consistent history space, this would reduce to the `impossible' interpretation which I discussed in section \ref{DMWWsolutionthatisnt} . But of course this is not the case, so that a great deal of conceptual work must be done by the phrase `under a given description'.
It is very unclear how this work is in fact to be done. There are of course multiple descriptions of even classical systems, but these descriptions can in all cases be understood as coarse-grainings of a single exhaustive description (\citeN{griffithsbook} dubs this the \emph{principle of unicity}). By contrast, in the consistent-histories form of QM this is not the case:
\begin{quote}
The principle of unicity does not hold: there is not a unique exhaustive description of a physical system or a physical process. Instead, reality is such that it can be described in various alternative, incompatible ways,using descriptions which cannot be combined or compared. \cite[p.\,365]{griffithsbook}
\end{quote}
There is a close analogy between this `failure of unicity' and Bohrian complementarity, as proponents of the consistent-histories interpretation recognise. The analogy becomes sharper in the concrete context of measurement: which histories are `consistent' in a given measurement process depends sensitively on what the measurement device is constructed to measure. If, for instance, we choose to measure a spin-half particle's spin in the $x$ direction, then schematically the process looks something like
\[
(\alpha\ket{+_x}+\beta \ket{-_x})\otimes\ket{\mathrm{\mbox{untriggered device}}}
\]
\be
\longrightarrow
\alpha \tpk{+_x}{\mathrm{\mbox{device reads `up'}}}
+
\beta
\tpk{-_x}{\mathrm{\mbox{device reads `down'}}}.
\ee
A consistent-history space for this process might include histories containing the projectors
\[
\proj{\pm_x}{\pm_x}\otimes\proj{\mathrm{\mbox{untriggered device}}}{\mathrm{\mbox{untriggered device}}},
\]
\[
\proj{\pm_x}{\pm_x}\otimes\proj{\mathrm{\mbox{device reads `up'}}}{\mathrm{\mbox{device reads `up'}}},
\]
and
\[
\proj{\pm_x}{\pm_x}\otimes\proj{\mathrm{\mbox{device reads `down'}}}{\mathrm{\mbox{device reads `down'}}},
\]
But if the experimenter instead chooses to measure the $z$ component of spin, then this set will no longer be consistent
and we will instead need to use a set containing projectors like
\[
\proj{\pm_z}{\pm_z}\otimes\proj{\mathrm{\mbox{device reads `down'}}}{\mathrm{\mbox{device reads `down'}}},
\]
So while for Bohr the classical context of \emph{measurement} was crucial, for the consistent-histories interpretation this just emerges as a special case of the consistency requirement, applied to the measurement process. (Note that it is, in particular, perfectly possible to construct situations where the consistent histories at time $t$ are fixed by the experimenter's choices at times far later than $t$ --- cf \citeN[p.\,255]{griffithsbook}, \citeN[p.\,54--5]{dicksonbook} --- in keeping with Bohr's response to the EPR paradox.)
But serious conceptual problems remain for the consistent-histories interpretation:
\begin{enumerate}
\item What is the ontological status of the universal state vector \ket{\Psi}? It plays an absolutely crucial role in the theory in determining which histories are consistent: as we have seen, when we try to fine-grain histories down below the coarse-grained level set by dynamical decoherence then the details of which histories are consistent becomes extremely sensitively dependent on \ket{\Psi}. Perhaps it can be interpreted as somehow `lawlike'; perhaps not. It is certainly difficult to see how it can be treated as physical without letting the consistent-histories interpretation collapse into something more like the Everett interpretation.
\item Does the theory actually have predictive power? The criticisms of Kent and Dowker continue to apply, and indeed can be placed into a sharp form here: they prove that a given consistent history can be embedded into two different history spaces identical up to a given time and divergent afterwards, such that the probabilities assigned to the history vary sharply from one space to the other. In practice, accounts of the consistent-history interpretation seem to get around this objection by foreswearing cosmology and falling back on some externally-imposed context to fix the correct history; shades of Bohr, again.
\item Most severely, is Griffith's `failure of unicity' really coherent? It is hard to make sense of it; no wonder that many commentators on the consistent-history formalism (\egc, \citeN[p.788]{penroseroadtoreality}) find that they can make sense of it only by regarding every history in every history space as actualised: an ontology that puts Everett to shame.
\end{enumerate}
\subsection{Operationalism}\label{DMWWoperationalist}
The consistent-histories interpretation can make a reasonable case for being the natural home for Bohr's complementarity. But there is another reading of the Copenhagen interpretation which arguably has had more influence on physicists' attitude to the measurement problem: the operationalist doctrine that physics is concerned not with an objective `reality' but only with the result of experiments. This position has been sharpened in recent years into a relatively well-defined interpretation (stated in particularly clear form by \citeN{peres}; see also \citeN{fuchsperes}): the \emph{operationalist interpretation} that is our third candidate for orthodoxy.
Following Peres, we can state operationalism as follows:
\begin{quote}
\textbf{The operationalist interpretation:}
Possible measurements performable on a quantum system are represented by the POVMs of that system's Hilbert space.
All physics tells us is the probability, for each measurement, of a given outcome: specifically, it tells us that the probability of the outcome corresponding
to positive operator \op{A} obtaining is $\tr(\denop \op{A})$ (or $\matel{\psi}{\op{A}}{\psi}$ in the special case where a pure state may be used). As such, the state of the system is not a physical thing at all, but simply a shorthand way of recording the probabilities of various outcomes of measurements; and the evolution rule
\be \ket{\psi(t)}=\exp(-it \op{H}/\hbar)\ket{\psi}\ee
is just a shorthand way of recording how the various probabilities change over time for an isolated system.
\end{quote}
In fact, we do not even need to \emph{postulate} the rule $\Pr(\op{A})=\tr(\denop \op{A})$. It is enough to require \emph{non-contextuality}: that is, to require that the probability of obtaining the result associated with \op{A} is independent of which POVM \op{A} is embedded into. Suppose $\Pr$ is any non-contextual probability measure on the positive operators: that is, suppose it is a function from the positive operators to $[0,1]$ satisfying
\be
\sum_i\op{A}_i=\id \longrightarrow \sum_i\\Pr(\op{A}_i)=1.
\ee
Then it is fairly simple (Caves \emph{et al}~\citeyearNP{cavesetalgleason}) to prove that $\Pr$ must be represented by a density operator: $\Pr(\op{A})=\tr(\denop \op{A})$ for some \denop.
Modifications of the operationalist interpretation are available. The probabilities may be taken to be subjective \cite{cavesetalprobability}, as referring to an ensemble of systems (\citeNP{ballentine}, \citeNP{taylorghost}), or as irreducible single-case chances \cite{fuchsperes}. The `possible measurements' may be taken to be given by the PVMs alone rather than the POVMs (in which case Gleason's theorem must be invoked in place of the simple proof above to justify the use of density operators). But the essential content of the interpretation remains: the `quantum state' is just a way of expressing the probabilities of various measurement outcomes, and --- more generally --- quantum theory itself is not in the business of supplying us with an objective picture of the world. Fuchs and Peres put this with admirable clarity:
\begin{quote}
We have learned something new when we can distill from the accumulated data a compact description of all that was seen and an indication of which further experiments will corroborate that description. This is what science is about. If, from such a description, we can \emph{further} distill a model of a free-standing ``reality'' independent of our interventions, then so much the better. Classical physics is the ultimate example of such a model. However, there is no logical necessity for a realistic worldview to always be obtainable. If the world is such that we can never identify a reality independent of our experimental activity, then we must be prepared for that, too. \ldots [Q]uantum theory does \emph{not} describe physical reality. What it does is provide an algorithm for computing \emph{probabilities} for the macroscopic events (``detector clicks'') that are the consequences of our experimental interventions. This strict definition of the scope of quantum theory is the only interpretation ever needed, whether by experimenters or theorists. \cite{fuchsperes}
\ldots
Todd Brun and Robert Griffiths point out [in \citeN{fuchsperescomments}] that ``physical theories have always had as much to do with providing a coherent picture of reality as they have with predicting the results of experiment.'' Indeed, have always had. This statement was true in the past, but it is untenable in the present (and likely to be untenable in the future). Some people may deplore this situation, but we were not led to reject a freestanding reality in the quantum world out of a predilection for positivism. We were led there because this is the overwhelming message quantum theory is trying to tell us.
\cite{fuchsperesreply}
\end{quote}
Whether or not Fuchs and Peres were led to their position `out of a predilection for positivism', the operationalist interpretation is nonetheless positivist in spirit, and is subject to many of the same criticisms. However, in one place it differs sharply. Where the positivists were committed to a once-and-for-all division between observable and unobservable, a quantum operationalist sees no difficulty in principle with applying QM to the measurement process itself. In a measurement of spin, for instance, the state
\be
\alpha\ket{+_x}+\beta \ket{-_x}
\ee
may just be a way of expressing that (among other regularities) the probability of getting result `up' on measuring spin in the $x$ direction is $|\alpha|^2$. But the measurement process may itself be modeled in QM in the usual way ---
\[
(\alpha\ket{+_x}+\beta \ket{-_x})\otimes\ket{\mathrm{\mbox{untriggered device}}}
\]
\be
\longrightarrow
\alpha \tpk{+_x}{\mathrm{\mbox{device reads `up'}}}
+
\beta
\tpk{-_x}{\mathrm{\mbox{device reads `down'}}}.
\ee
--- \emph{provided} that it is understood that this state is itself just a shorthand way of saying (among other regularities) that the probability of finding the measurement device to be in state ``reads up' '' is $|\alpha|^2$. It is not intended to describe a physical superposition any more than $\alpha\ket{+_x}+\beta \ket{-_x}$ is.
In principle, this can be carried right up to the observer:
\be
\alpha\ket{\mathrm{\mbox{Observer sees `up' result}}}
+
\beta\ket{\mathrm{\mbox{Observer sees `down' result}}}
\ee
is just a shorthand expression of the claim that if the `observer' is themselves observed, they will be found to have seen `up' with probability $|\alpha|^2$.
Of course, if analysis of any given measurement process only gives dispositions for certain results in subsequent measurement processes, then
there is a threat of infinite regress. The operationalist interpretation responds to this problem by adopting an aspect of the Copenhagen interpretation essentially lost in the consistent-histories interpretation: the need for a separate classical language to describe measurement devices, and the resultant ambiguity (\citeN[p.\,373]{peres} calls it \emph{ambivalence}) as to which language is appropriate when.
To spell this out (here I follow \citeN[pp.\,376--7]{peres}) a measurement device, or any other macroscopic system, may be described either via a density operator \denop\ on Hilbert space
(a quantum state, which gives only probabilities of certain results on measurement) or a probability distribution $W(\vctr{q},\vctr{p})$ over phase-space points (each of which gives an actual classical state of the system). These two descriptions then give different formulae for the probability of finding the system to have given position and momentum:
\begin{itemize}
\item The quantum description just \emph{is} a shorthand for the probabilities of getting various different results on measurement. In particular, there will exist some POVM $\op{A}_{q,p}$ such that the probability density of getting results $(q,p)$ on a joint measurement of position and momentum is $\tr(\denop \op{A}_{q,p})$.
\item According to the classical description, the system actually has some particular values of $q$ and $p$, and the probability density for any given values is just $W(q,p)$
\end{itemize}
If the two descriptions are not to give contradictory predictions for the result of experiment, then we require that $W(q,p)\simeq\tr(\denop \op{A}_{q,p})$; or, to be more precise, we require that the integrals of $W(q,p)$ and $\tr(\denop \op{A}_{q,p})$ over sufficiently large regions of phase space are equal to within the limits of experimental error. This gives us a recipe to construct the classical description from the quantum: just set $W(q,p)$ equal to $\tr(\denop \op{A}_{q,p})$. If this is done at a given time $t_0$, then at subsequent times $t>t_0$ the classical dynamics applied to $W$ and the quantum dynamics applied to \denop\ will break the equality:
\be
\tr(\denop(t) \op{A}_{q,p})\neq W(q,p;t)
\ee
(where $W(q,p;t)$ is the distribution obtained by time-evolving $W(q,p)$ using Hamilton's equations.) But if the system is sufficiently large, decoherence guarantees that the equality continues to hold approximately when $W(q,p;t)$ and $\tr(\denop(t) \op{A}_{q,p})$ are averaged over sufficiently large phase-space volumes.
The `operationalism' of this interpretation is apparent here. There is no \emph{exact} translation between classical and quantum descriptions, only one whose imprecisions are too small to be detected empirically.\footnote{A further ambiguity in the translation formula is the POVM $A(q,p)$: in fact, no unique POVM for phase-space measurement exists. Rather, there are many equivalently-good candidates which essentially agree with one another provided that their predictions are averaged over phase-space volumes large compared to $\hbar^n$, where $n$ is the number of degrees of freedom of the system.}
But if QM --- if science generally --- is merely a tool to predict results of experiments, it is unclear at best that we should be concerned about ambiguities which are empirically undetectable in practice. Whether this indeed a valid conception of science --- and whether the operationalist interpretation really succeeds in overcoming the old objections to logical positivism --- I leave to the reader's judgment.
\subsection{Further Reading}
Standard references for consistent histories are \citeN{griffithsbook} and \citeN{omnesbook}; for critical discussion, see \citeN[pp.\,52--57]{dicksonbook}, \citeN[212--236]{bubbook} and \citeN{ghirardiconsistent}. The best detailed presentation of operationalism is \citeN{peres}; for a briefer account see \citeN{fuchsperes}. For two rather different reappraisals of the original Copenhagen interpretation, see \citeN{cushingcopenhagen} and \citeN{saunderscopenhagen}.
Recently, operationalist approaches have taken on an ``information-theoretic'' flavour, inspired by quantum information. See Chris Timpson's contribution to this volume for more details.
Though they cannot really be called ``orthodox'', the family of interpretations that go under the name of ``quantum logic'' are also pure interpretations which attempt to solve the measurement problem by revising part of our pre-quantum philosophical picture of the world. In this case, though, the part to be revised is classical logic. Quantum logic is not especially popular at present, and so for reasons of space I have omitted it, but for a recent review see \citeN{dicksonlogic}.
\section{The Everett interpretation}\label{DMWWeverett}
Of our three `candidates for orthodoxy', only two are pure interpretations in the sense of section \ref{DMWWinterpretation}, and neither of these are `realist' in the conventional sense of the world. The consistent-histories interpretation purports to describe an objective reality, but that reality is unavoidably perspectival, making sense only when described from one of indefinitely many contradictory perspectives; whether or not this is coherent, it is not how scientific realism is conventionally understood! The operationalist interpretation, more straightforwardly, simply denies explicitly that it describes an independent reality. And although the new pragmatism does describe such a reality, it does it in a way universally agreed to be ad hoc and unacceptable.
There is, however, one pure interpretation which purports to be realist in a completely conventional sense: the Everett interpretation. Unlike the three interpretations we have considered so far, its adherents make no claim that it is any sort of orthodoxy; yet among physicists if not philosophers it seems to tie with operationalism and consistent histories for popularity. Its correct formulation, and its weaknesses, are the subject of this section.
\subsection{Multiplicity from indefiniteness?}
At first sight, applying straightforward realism to QM without modifying the formalism seems absurd. Undeniably, unitary QM produces superpositions of macroscopically distinct quasi-classical states; whether or not such macroscopic superpositions even make sense, their existence seems in flat contradiction with the fact that we actually seem to observe macroscopic objects only in definite states.
The central insight in the Everett interpretation is this: \emph{superpositions} of macroscopically distinct states are somehow to be understood in terms of \emph{multiplicity}. For instance (to take the time-worn example)
\be \label{DMWWschrodingercat}\alpha\ket{\mathrm{\mbox{Live cat}}}+\beta\ket{\mathrm{\mbox{Dead cat}}}\ee
is to be understood (somehow) as representing not a single cat in an indefinite state, but rather a multiplicity of cats, one (or more) of which is alive, one (or more) of which is dead. Given the propensity of macroscopic superpositions to become entangled with their environment, this `many cats' interpretation becomes in practice a `many worlds' interpretation: quantum measurement continually causes the macroscopic world to branch into countless copies.
The problems in cashing out this insight are traditionally broken in two:
\begin{enumerate}
\item The `preferred basis problem': \emph{how} can the superposition justifiably be understood as some kind of multiplicity?
\item The `probability problem': how is probability to be incorporated into a theory which treats wavefunction collapse as some kind of branching process?
\end{enumerate}
\subsection{The preferred-basis problem: solution by modification}\label{DMWWmodification}
If the preferred basis problem is a question (``how can quantum superpositions be understood as multiplicities?'') then there is a traditional answer, more or less explicit in much criticism of the Everett interpretation \cite{barrettbook,kent,butterfieldeverett}: they cannot. That is: it is no good just \emph{stating} that a state like (\ref{DMWWschrodingercat}) describes multiple worlds: the formalism must be explicitly \emph{modified} to incorporate them. This position dominated discussion of the Everett interpretation in the 1980s and early 1990s: even advocates like \citeN{deutsch85} accepted the criticism and rose to the challenge of providing such a modification.
Modificatory strategies can be divided into two categories.
\emph{Many-exact-worlds theories} augment the quantum formalism by adding an ensemble of `worlds' to the state vector. The `worlds' are each represented by an element in some particular choice of `world basis' $\ket{\psi_i(t)}$ at each time $t$: the proportion of worlds in state $\ket{\psi_i(t)}$ at time $t$ is $|\bk{\Psi(t)}{\psi_i(t)}$, where \ket{\Psi(t)} is the (unitarily-evolving) universal state. Our own world is just one element of this ensemble.
Examples of many-exact-worlds theories are the early Deutsch (\citeyearNP{deutsch85,deutschghost}), who tried to use the tensor-product structure of Hilbert space to define the world basis\footnote{A move criticised on technical grounds by \citeN{brownondeutsch}.}, and Barbour(\citeyearNP{barbour2,barbour99}), who chooses the position basis.
In \emph{Many-minds theories}, by contrast, the multiplicity is to be understood as illusory. A state like (\ref{DMWWschrodingercat}) really is indefinite, and when an observer looks at the cat and thus enters an entangled state like
\be\alpha\tpk{\mathrm{\mbox{Live cat}}}{\mathrm{\mbox{Observer sees live cat}}}+\beta\tpk{\mathrm{\mbox{Dead cat}}}{\mathrm{\mbox{Observer sees dead cat}}}\ee
then the observer too has an indefinite state. However: to each physical observer is associated not one mental state, but an ensemble of them: each mental state has a definite experience, and the proportion of mental states where the observer sees the cat alive is $|\alpha|^2$. Effectively, this means that in place of a global `world-defining basis' (as in the many-exact-worlds theories) we have a `consciousness basis' for each observer.\footnote{Given that an `observer' is represented in the quantum theory by some Hilbert space many of whose states are not conscious at all, and that conversely almost any sufficiently-large agglomeration of matter can be formed into a human being, it would be more accurate to say that we have a consciousness basis for all \emph{systems}, but one with many elements which correspond to no conscious experience at all.} When an observer's state is an element of the consciousness basis, all the minds associated with that observer have the same experience and so we might as well say that the observer is having that experience. But in all realistic situations the observer will be in some superposition of consciousness-basis states, and the ensemble of minds associated with that observer will be having a wide variety of distinct experiences. Examples of many-minds theories are \citeN{albertloewermm}, Lockwood (\citeyearNP{lockwoodbook,lockwoodbjps1}), \citeN{pagesensible} and Donald(\citeyearNP{donald90,donald92,donald02}).
It has increasingly become recognised, by supporters and detractors alike that there are severe problems with either of these approaches to developing the Everett interpretation. Firstly, and most philosophically, both the many-exact-worlds and the many-minds theories are committed to a very strong (and arguably very anti-scientific) position in philosophy of mind: the rejection of \emph{functionalism}, the view that mental properties should be ascribed to a system in accordance with the functional role of that system (see \egc, \citeN{armstrongmind}, \citeN{lewisradical}, \citeN{hofstadterdennett}, \citeN{levinencyclopedia} for various explications of functionalism). This is particularly obvious in the case of the Many-Minds theories, where some rule associating conscious states to physical systems is simply postulated in the same way that the other laws of physics are postulated. If it is just a \emph{fundamental law} that consciousness is associated with some given basis, clearly there is no hope of a functional \emph{explanation} of how consciousness emerges from basic physics (and hence much, perhaps all, of modern AI, cognitive science and neuroscience is a waste of time). And in fact many adherents of Many-Minds theories (\egc, Lockwood and Donald) embrace this conclusion, having been led to reject functionalism on independent grounds.
It is perhaps less obvious that the many-exact-worlds theories are equally committed to the rejection of functionalism. But
if the `many worlds' of these theories are supposed to include \emph{our world}, it follows that conscious observers are found within each world. This is only possible compatible with functionalism if the worlds are capable of containing independent complex structures which can instantiate the `functions' that subserve consciousness. This in turn requires that the world basis is decoherent (else the structure would be washed away by interference effects) and --- as we have seen --- the decoherence basis is not effectively specifiable in any precise microphysical way. (See \citeN{wallaceworlds} for further discussion of the difficulty of localising conscious agents within `worlds' defined in this sense.)
There is a more straightforwardly physical problem with these approaches to the Everett interpretation. Suppose that a wholly satisfactory Many-Exact-Worlds or Many-Minds theory were to be developed, specifying an exact `preferred basis' and an exact transition rule defining identity for worlds or minds. Nothing would then stop us from taking that theory, discarding all but one of the worlds/minds\footnote{It would actually be a case of discarding all but one \emph{set} of minds --- one for each observer.} and obtaining an equally empirically effective theory without any of the ontological excess which makes Everett-type interpretations so unappealing. Put another way: an Everett-type theory developed along the lines that I have sketched would really just be a hidden-variables theory with the additional assumption that continuum many non-interacting sets of hidden variables exist, each defining a different classical world. (This point is made with some clarity by \citeN{bellqmforcosmologists} in his classic attack on the Everett interpretation.)
In the light of these sorts of criticisms, these modify-the-formalism approaches to the Everett interpretation have largely fallen from favour. Almost no advocate of ``the Many-Worlds Interpretation'' actually advocates anything like the Many-Exact-Worlds approach\footnote{\citeN{barbour99} may be an exception.} (Deutsch, for instance, clearly abandoned it some years ago) and Many-Minds strategies which elevate consciousness to a preferred role continue to find favour mostly in the small group of philosophers of physics strongly committed for independent reasons to a non-functionalist philosophy of mind. Advocates of the Everett interpretation among physicists (almost exclusively) and philosophers (for the most part) have returned to Everett's original conception of the Everett interpretation as a pure interpretation: something which emerges simply from a realist attitude to the unitarily-evolving quantum state.
\subsection{The Bare Theory: how not to think about the wave function}\label{DMWWbaretheory}
One way of understanding the Everett interpretation as pure interpretation --- the so-called `Bare Theory' --- was suggested by \citeN{albertqmbook}. It has been surprisingly influential among philosophers of physics --- not as a \emph{plausible interpretation of QM}, but as the correct reading of the Everett interpretation.
\citeN[p.\,94]{barrettbook} describes the Bare Theory as follows:
\begin{quote}
The bare theory is simply the standard von Neumann-Dirac formulation of QM with the standard interpretation of states (the eigenvalue-eigenstate link) but stripped of the collapse postulate --- hence, \emph{bare}.
\end{quote}
From this perspective, a state like (\ref{DMWWschrodingercat}) is not an eigenstate of the `cat-is-alive' operator (that is, the projector which projects onto all states where the cat is alive); hence, given the eigenstate-eigenvalue link the cat is in an indefinite state of aliveness. Nor is it an eigenstate of the `agent-sees-cat-as-alive' operator, so the agent's mental state is indefinite between seeing the cat alive and seeing it dead. But it \emph{is} an eigenstate of the `agent-sees-cat-as-alive-\textbf{or}-agent-sees-cat-as-dead' operator: the states
\be
\tpk{\mathrm{\mbox{Live cat}}}{\mathrm{\mbox{Observer sees live cat}}}
\ee
and
\be
\tpk{\mathrm{\mbox{Dead cat}}}{\mathrm{\mbox{Observer sees dead cat}}}
\ee
are both eigenstates of that operator with eigenvalue one, so their superposition is also an eigenstate of that operator. Hence if we \emph{ask} the agent, `did you see the cat as either alive or dead' they will answer `yes'.
That is: the bare theory --- without any flaky claims of `multiplicity' or `branching' --- undermines the claim that macroscopic superpositions contradict our experience. It predicts that we will \emph{think}, and \emph{claim}, that we do not observe superpositions at all, even when our own states are highly indefinite, and that we are simply mistaken in the belief that we see a particular outcome or other. That is, it preserves unitary QM --- at the expense of a scepticism that ``makes Descartes's demon and other brain-in-the-vat stories look like wildly optimistic appraisals of our epistemic situation'' \cite[p.\,94]{barrettbook}. As Albert puts it:
\begin{quote}
[M]aybe \ldots the linear dynamical laws are nonetheless the complete laws of the evolution of the \emph{entire world}, and maybe all the appearances to the contrary (like the appearance that experiments have outcomes, and the localised that the world doesn't evolve deterministically) turn out to be just the sorts of \emph{delusions} which \emph{those laws themselves} can be shown to \emph{bring on!}
\end{quote}
A quite extensive literature has developed trying to point out exactly what is wrong with the Bare Theory (see, \egc, \citeN[pp.\,117--125]{albertqmbook}, \citeN{barrettbare}, \citeN[pp.\,92--120]{barrettbook}, \citeN{bubcliftonmonton}, \citeN[pp.\,45--47]{dicksonbook}). The consensus seems to be that:
\begin{enumerate}
\item If we take a `minimalist', pure-interpretation reading of Everett, we are led to the Bare Theory; and
\item The bare theory has some extremely suggestive features; but
\item It is not ultimately satisfactory as an interpretation of QM because it fails to account for probability/is empirically self-undermining/smuggles in a preferred basis (delete as applicable); and so
\item Any attempt to solve the measurement problem along Everettian lines cannot be `bare' but must add additional assumptions.
\end{enumerate}
From the perspective of this review, however, this line of argument is badly mistaken. It relies essentially on the assumption that the eigenstate-eigenvalue link is part of the basic formalism of QM, whereas --- as I argued in section \ref{DMWWagainst} --- it plays no part in modern practice and is flatly in contradiction with most interpretative strategies. It will be instructive, however, to revisit this point in the context of the state-vector realism that is essential to the Everett interpretation.
If the state vector is to be taken as physically real, the eigenstate-eigenvalue link becomes a claim about the properties of that state vector. Specifically:
\begin{quote}All properties of the state vector are represented by projectors, and the state vector \ket{\psi} has the property represented by \op{P} if it is inside the subspace onto which \op{P} projects: that is, if $\op{P}\ket{\psi}=\ket{\psi}$. If $\op{P}\ket{\psi}=0$ then \ket{\psi} certainly lacks the property represented by \op{P}; if neither is true then either \ket{\psi} definitely lacks the property or it is indefinite whether \ket{\psi} has the property.
\end{quote}
As is well known, it follows that the logic of state-vector properties is highly non-classical: it is perfectly possible, indeed typical, for a system to have a definite value of the property $(p \vee q)$ without definitely having either $p$ or $q$. The quantum logic program developed this into a mathematically highly elegant formalism; see \citeN{bubbook} for an clear presentation.
What is less clear is why we should take this account of properties at all seriously. We have seen that it fails to do justice to modern analyses of quantum measurement; furthermore, from the perspective of state-vector realism it seems to leave out all manner of perfectly ordinary properties of the state. Why not ask:
\begin{itemize}
\item Is the system's state an eigenstate of energy?
\item Is its expectation value with respect to the Hamiltonian greater than $100$ joules?
\item Is its wavefunction on momentum space negative in any open set?
\item Does its wavefunction on configuration space tend towards any Gaussian?
\end{itemize}
If the state vector is physical, these all seem perfectly reasonable questions to ask about it. Certainly, each has a determinate true/false answer for any given state vector. Yet none correspond to any projector (it is, for instance, obvious that there is no projector that projects onto all and only eigenstates of some operator!)
Put more systematically: if the state vector is physical, then the set \mc{S} of normalised vectors in Hilbert space is (or at least represents) the set of possible states in which an (isolated, non-entangled) system can be found. If we assume standard logic, then a property is defined (at least for the purposes of physics) once we have specified the states in \mc{S} for which the property holds. That is: properties in quantum physics correspond to subsets of the state space, just as properties in classical physics correspond to subsets of the phase space.
If we assume \emph{non}-standard logic, of course, we doubtless get some different account of properties; if we assume a particular non-standard logic, very probably we get the eigenstate-eigenvalue account of properties. The fact remains that if we wish to assume state-vector realism and standard logic (as did Everett) we do not get the Bare Theory.
(There is, to be fair, an important question about what we \emph{do} get. That is, how can we think about the state vector (construed as real) other than through the eigenstate-eigenvalue link? This question has seen a certain amount of attention in recent years. The most common answer seems to be `wavefunction realism': if the state vector is a physical thing at all it should be thought of as a field on $3N$-dimensional space. Bell proposed this (in the context of the de Broglie-Bohm theory):
\begin{quote}
\emph{No one can understand this theory until he is willing to think of $\psi$ as a real physical field rather than a `probability amplitude'. Even though it propagates not in 3-space but in 3N-space.} (\citeNP{bellqmforcosmologists}; emphasis his)
\end{quote}
\citeN{albertmetaphysics} proposes it as the correct reading of the state vector in any state-vector-realist theory;\footnote{This would seem to imply that Albert would concur with my criticism of the Bare Theory, but I am not aware of any actual comment of his to this effect.} \citeN{lewisconfiguration} and \citeN{montonconfiguration} concur. (Monton argues that wavefunction realism is unacceptable, but he does so in order to argue against state-vector realism altogether rather than to advocate an alternative).
There are alternatives, however. Chris Timpson and myself \citeyear{wallacetimpsonshort} suggest a more spatio-temporal ontology, in which each spacetime region has a (possibly impure) quantum state but in which, due to entanglement the state of region $A \cup B$ is not determined simply by the states of regions $A$ and $B$ separately (a form of nonlocality which we claim is closely analogous to what is found in the Aharonov-Bohm effect). \citeN{deutschhayden} argue for an ontology based on the Heisenberg interpretation which appears straightforwardly local (but see \citeN{wallacetimpsonshort} for an argument that this locality is more apparent than real). \citeN{saundersmetaphysics} argues for a thoroughly relational ontology reminiscent of Leibniz's monadology. To what extent these represent real metaphysical alternatives rather than just different ways of describing the quantum state's structure is a question for wider philosophy of science and metaphysics.)
\subsection{Decoherence and the preferred basis}\label{DMWWeverettdecoherence}
In any case, once we have understood the ontological fallacy on which the Bare Theory rests, it remains to consider whether multiplicity does indeed emerge from a realist reading of the quantum state, and if so how. The 1990s saw an emerging consensus on this issue, developed by Zurek, Gell-Mann and Hartle, Zeh, and many others\footnote{See, \egc, Zurek~\citeyear{zurek91,zurekroughguide}, Gell-Mann and Hartle~ \citeyear{gellmannhartle,gellmannhartle93}), \citeN{zeh93}.} and explored from a philosophical perspective by Saunders~\citeyear{saundersevolution,saundersmetaphysics,saundersdecoherence}: the multiplicity is a consequence of decoherence. That is, the structure of ``branching worlds'' suggested by the Everett interpretation is to be identified with the branching structure induced by the decoherence process. And since the decoherence-defined branching structure is comprised of quasi-classical histories, it would follow that Everett branches too are quasi-classical.
It is important to be clear on the nature of this ``identification''. It cannot be taken as an additional axiom (else we would be back to the Many-Exact-Worlds theory); rather, it must somehow be forced on us by a realist interpretation of the quantum state. \citeN{gellmannhartle93} made the first sustained attempt to defend why this is so, with their concept of an IGUS: an ``information-gathering-and-utilising-system'' (similar proposals were made by \citeN{saundersevolution} and \citeN{zurekroughguide}) An IGUS, argue Gell-Mann and Hartle, can only function if the information it gathers and utilises is information about particular decoherent histories. If it attempts to store information about superpositions of such histories, then that information will be washed out almost instantly by the decoherence process. As such, for an IGUS to function it must perceive the world in terms of decoherent histories: proto-IGUSes which do not will fail to function. Natural selection then ensures that if the world contains IGUSes at all --- and in particular if it contains intelligent life --- those IGUSes will perceive the decoherence-selected branches as separate realities.
The IGUS approach is committed, implicitly, to functionalism: it assumes that intelligent, conscious beings just are information-processing systems, and it furthermore assumes that these systems are instantiated in certain structures within the quantum state. (Recall that in the ontology I have defended, the quantum state is a highly structured object, with its structure being describable in terms of the expectation values of whatever the structurally preferred observables are in whichever bare quantum formalism we are considering.) In \citeN{wallacestructure} I argued that this should be made explicit, and extended to a general functionalism about higher-level ontology: quite generally (and independent of the Everett interpretation) we should regard macroscopic objects like tables, chairs, tigers, planets and the like as structures instantiated in a lower-level theory. A tiger, for instance, is a pattern instantiated in certain collections of molecules; an economy is a pattern instantiated in certain collections of agents.
\citeN{dennettrealpatterns} proposed a particular formulation of this functionalist ontology: those formally-describable structures which deserve the name `real' are those which are predictively and explanatorily necessary to our account of a system (in endorsing this view in \citeN{wallacestructure}, I dubbed it `Dennett's criterion'). So for instance (and to borrow an example from \citeN{wallacestructure}) what makes a tiger-structure ``real'' is the phenomenal gain in our understanding of systems involving tigers, and the phenomenal predictive improvements that result, if we choose to describe the system using tiger-language rather than restricting ourselves to talking about the molecular constituents of the tigers. A variant of Dennett's approach has been developed by \citeN{rossinformation} and \citeN{ladymanbook}; ignoring the fine details of how it is to be cashed out, let us call this general approach to higher-level ontology simply \emph{functionalism}, eliding the distinction between the general position and the restricted version which considers only the philosophy of mind.
Functionalism in this sense is not an uncontroversial position. \citeN{kim98}, in particular, criticises it and develops a rival framework based on mereology (this framework is in turn criticised in \citeN{rossspurrett}; see also \citeN{wallacebbs} and other commentaries following \citeN{rossspurrett}, and also the general comments on this sort of approach to metaphysics in chapter 1 of \citeN{ladymanbook}). Personally I find it difficult to see how any account of higher-level ontology that is not functionalist in nature can possibly do justice to science as we find it; as \citeN[p.\,17]{dennettsweetdreams} puts it, ``functionalism in this broadest sense is so ubiquitous in science that it is tantamount to a reigning presumption of all science''; but in any event, the validity or otherwise of functionalism is a very general debate, not to be settled in the narrow context of the measurement problem.
The reason for discussing functionalism here is that (as I argued in \citeN{wallacestructure}) it entails that the decohering branches really should be treated --- really \emph{are} approximately independent quasi-classical worlds. Consider: if a system happens to be in a quasi-classical state
$\tpk{\vctr{q}(t),\vctr{p}(t)}{\psi(t)}$ (as defined in section \ref{DMWWquantumalgorithm} and made more precise in section \ref{DMWWdecoherence}) then\footnote{I ignore the possibility of chaos; if this is included, then the quantum system would be better described as instantiating an ensemble of classical worlds.} its evolution will very accurately track the phase-space point $(\vctr{q}(t),\vctr{p}(t))$ in its classical evolution, and so instantiates the same structures. As such, insofar as that phase-space point actually represents a macroscopic system, and insofar is what it \emph{is} to represent a macroscopic system is to instantiate certain structures, it follows that $\tpk{\vctr{q}(t),\vctr{p}(t)}{\psi}$ represents that same macroscopic system. The fact that these `certain structures' are instantiated in the expectation values\footnote{Note that here `expectation value' of an operator $\op{P}$ simply denotes \matel{\psi}{\op{P}}{\psi}; no probabilistic interpretation is intended. } of some phase-space POVM rather than in the location of a collection of classical particles is, from a functionalist perspective, quite beside the point.
Now if we consider instead a superposition
\be \label{DMWWtwostructures}
\alpha\tpk{\vctr{q}_1(t),\vctr{p}_1(t)}{\psi_1(t)}+\beta\tpk{\vctr{q}_2(t),\vctr{p}(t)}{\psi_2(t)}
\ee
then not one but two structures are instantiated in the expectation values of that same phase-space POVM: one corresponding to the classical history $(\vctr{q}_1(t),\vctr{p}_1(t))$, one to $(\vctr{q}_2(t),\vctr{p}_2(t))$, with decoherence ensuring that the structures do not interfere and cancel each other out but continue to evolve independently, each in its own region of phase space.
Generalising to arbitrary such superpositions, we deduce that functionalism applied to the unitarily-evolving, realistically-interpreted quantum state yields the result that decoherence-defined branches are classical worlds. Not worlds in the sense of universes, precisely defined and dynamically completely isolated, but worlds in the sense of planets --- very accurately defined but with a little inexactness, and not quite dynamically isolated, but with a self-interaction amongst constituents of a world which completely dwarfs interactions between worlds.
This functionalist account of multiplicity is not in conflict with the IGUS strategy, but rather contains it. For not only could IGUSes not process information not restricted to a single branch, they could not even \emph{exist} across branches. The structures in which they are instantiated will be robust against decoherence only if they lie within a single branch.
\subsection{Probability: the Incoherence Problem}
The decoherence solution to the preferred-basis problem tells us that the quantum state is really a constantly-branching structure of quasi-classical worlds. It is much less clear how notions of probability fit into this account: if an agent knows for certain that he is about to branch into many copies of himself --- some of which see a live cat, some a dead cat --- then how can this be reconciled with the Quantum Algorithm's requirement that he should expect with a certain probability to see a live cat?
It is useful to split this problem in two:
\begin{description}
\item[The Incoherence Problem:] In a deterministic theory where we can have perfect knowledge of the details of the branching process, how can it even make sense to
assign probabilities to outcomes?
\item[The Quantitative Problem:]Even if it does make sense to assign
probabilities to outcomes, why should they be the probabilities given by the
Born rule?
\end{description}
The incoherence problem rests on problems with personal identity. In branching, one person is replaced by a multitude of (initially near-identical) copies of that person, and it might be thought that this one-to-many relation of past to future selves renders any talk of personal identity simply incoherent in the face of branching (see, \egc, \citeN{albertloewermm} for a defence of this point).
However (as pointed out by \citeN{saundersprobability}) this charge of incoherence fails to take account of what grounds ordinary personal identity: namely (unless we believe in Cartesian egos) it is grounded by the causal and structural relations between past and future selves. These relations exist no less strongly between past and future selves when there exist \emph{additional} such future selves; as such, if it is rational to care about one's unique future self (as we must assume if personal identity in \emph{non}-branching universes is to be made sense of) then it seems no less rational to care about one's multiple future selves in the case of branching. This point was first made --- entirely independently of QM --- by Parfit; see his \citeyear{parfitbook}.
This still leaves the question of how \emph{probability} fits in, and at this point there are two strategies available: the \emph{Fission Program} and the \emph{Subjective Uncertainty Program} \cite{wallaceepist}. The Fission Program works by considering situations where the interests of future selves are in \emph{conflict}. For instance, suppose the agent, about to observe Schr\"{o}dinger's Cat and thus to undergo branching, is offered an each-way bet on the cat being left alive. If he takes the bet, those future versions of himself who exist in live-cat branches will benefit and those who live in dead-cat branches will lose out. In deciding whether to take the bet, then, the agent will have to weigh the interests of some of his successors against those of others. Assigning a (formal) probability to each set of successors and choosing that action which benefits the highest-probability subset of successors is at least \emph{one way} of carrying out this weighing of interests.
This strategy is implicit in \citeN{deutschprob} and has been explicitly defended by Greaves (\citeyearNP{greaves}).
It has the advantage of conforming unproblematically to our intuition that ``I can feel uncertain over $P$ only if I think that there is a fact of the matter regarding $P$ of which I am ignorant'' \cite{greaves}; it has the disadvantage of doing violence to our intuitions that uncertainty about the future is generally justified; it is open to question what epistemic weight these intuitions should bear.\footnote{See \citeN{wallacebranching}, especially section 6, for more discussion of this point.} There is, however, a more serious problem with the Fission Program: it is at best uncertain whether it solves the measurement problem. For recall: in the framework of this review, `to solve the measurement problem' is to construct a theory which entails the truth (exact or approximate) of the Quantum Algorithm, and that Algorithm dictates that we should regard macroscopic superpositions as probabilistic, and hence that an agent expecting branching should be in a state of uncertainty. The challenge for fission-program advocates is to find an alternative account of our epistemic situation according to which the Everett interpretation is nonetheless explanatory of our evidence. See \citeN{greavesepistemic} for Greaves' proposed account, which draws heavily on Bayesian epistemology.
The Subjective Uncertainty Program aims to establish that probability really, literally, makes sense in the Everett universe: that is, that an agent who knows for certain that he is about to undergo branching is nonetheless justified in being \emph{uncertain} about what to expect. (This form of uncertainty cannot depend on ignorance of some facts describable from a God's-eye perspective, since the relevant features of the universal state are \emph{ex hypothesi} perfectly knowable by the agent --- hence, \emph{subjective} uncertainty).
Subjective uncertainty was first defended by \citeN{saundersprobability}, who asks: suppose that you are about to be split into multiple copies, then \emph{what} should you expect to happen? He argues that, given that each of your multiple successors has the same structural/causal connections to you as would have been the case in the absence of splitting, the only coherent possibility is \emph{uncertainty}: I should expect to be one of my future selves but I cannot know which.
I presented an alternative strategy for justifying subjective uncertainty in \citeN{wallacebranching} (and more briefly in \citeN{wallaceepist}). My proposal is that we are led to subjective uncertainty by considerations in the philosophy of language: namely, if we ask how we would analyse the semantics of a community of language-users in a constantly branching universe, we conclude that claims like ``$X$ might happen'' come out true if $X$ happens in some but not all branches.
If the Subjective Uncertainty program can be made to work, it avoids the epistemological problem of the Fission Program, for it aims to recover the quantum algorithm itself (and not just to account for its empirical success.) It remains controversial, however, whether subjective uncertainty really makes sense. For further discussion of subjective uncertainty and identity across branching, see \citeN{greaves}, \citeN{saunderswallace}, \citeN{wallaceepist} and \citeN{lewissu}.
\subsection{Probability: the Quantitative Problem}\label{DMWWprobabilityquantitative}
The Quantitative Problem of probability in the Everett interpretation is often posed as a paradox: the \emph{number} of branches has nothing to do with the \emph{weight} (\iec, modulus-squared of the amplitude) of each branch, and the only reasonable choice of probability is that each branch is equiprobable, so the probabilities in the Everett interpretation can have nothing to do with the Born rule.
This sort of criticism has sometimes driven advocates of the Everett interpretation back to the strategy of modifying the formalism, adding a continuous infinity of worlds \cite{deutsch85} or minds \cite{albertloewermm,lockwoodbook} in proportion to the weight of the corresponding branch. But this is unnecessary, for the criticism was mistaken in the first place: it relies on the idea that there is some sort of remotely well-defined branch number, whereas there is no such thing.
This can most easily be seen using the decoherent-histories formalism. Recall that the `branches' are decoherent histories in which quasi-classical dynamics apply, but recall too that the criteria of decoherence and quasi-classicality are approximate rather than exact. We can always fine-grain a given history space at the cost of slightly less complete decoherence, or coarse-grain it to ensure more complete decoherence; we can always replace the projectors in a history space by ever-so-slightly-different projectors and obtain an equally decoherent, equally quasi-classical space. These transformations do not affect the \emph{structures} which can be identified in the decoherent histories (for those structures are themselves only approximately defined) but they wildly affect the \emph{number} of branches with a given macroscopic property.
The point is also apparent using the formalism of quasi-classical states discussed in section \ref{DMWWquantumalgorithm}. Recall that in that framework, a macroscopic superposition is written
\be
\int \dr{\vctr{q}}\dr{\vctr{p}}\alpha(\vctr{q},\vctr{p})\tpk{\vctr{q},\vctr{p}}{\psi_{q,p}}.
\ee
If the states $\tpk{\vctr{q},\vctr{p}}{\psi_{q,p}}$ are to be taken as each defining a branch, there are continuum many of them, but if they are too close to one another then they will not be effectively decohered. So we will have to define branches via some coarse-graining of phase space into cells $\mc{Q}_n$, in terms of which we can define states
\be \ket{n}=\int_{\mc{Q}_n}\dr{\vctr{q}}\dr{\vctr{p}}\alpha(\vctr{q},\vctr{p})\tpk{\vctr{q},\vctr{p}}{\psi_{q,p}}.
\ee
The coarse-graining must be chosen such that the states \ket{n} are effectively decohered, but there will be no precisely-determined `best choice' (and in any case no precisely-determined division of Hilbert space into macroscopic and microscopic degrees of freedom in the first place.)
As such, the `count-the-branches' method for assigning probabilities is ill-defined.\footnote{\citeN{wallace3branch} presents an argument that the count-the-branches rule is incoherent even if the branch number \emph{were} to be exactly definable.} But if this dispels the \emph{paradox} of objective probability, still a \emph{puzzle} remains: why use the Born rule rather than any other probability rule?
Broadly speaking, three strategies have been proposed to address this problem without modifying the formalism. The oldest strategy is to appeal to relative frequencies of experiments. It has long been known (\citeNP{everett}) that if many copies of a system are prepared and measured in some fixed basis, the total weight of those branches where the relative frequency of any result differs appreciably from the weight of that result tends to zero as the number of copies tends to infinity. But it has been recognised for almost as long that this account of probability courts circularity: the claim that a branch has \emph{very small weight} cannot be equated with the claim that it is \emph{improbable}, unless we assume that which we are trying to prove, namely that weight=probability.
It is perhaps worth noting, though, that precisely equivalent objections can be made against the frequentist definition of probability. Frequentists equate probability with long-run relative frequency, but again they run into a potential circularity. For we cannot prove that relative frequencies converge on probabilities, only that they \emph{probably} do: that is, that the probability of the relative frequencies differing appreciably from the probabilities tends to zero as the number of repetitions of an experiment tends to infinity (the maths is formally almost identical in the classical and Everettian cases). As such, it is at least arguable that anyone who is happy with frequentism \emph{in general} as an account of probability should have no additional worries in the case of the Everett interpretation.\footnote{\citeN{fgg} try to evade the circularity by direct consideration of infinitely many measurements, rather than just by taking limits; their work has recently criticised by \citeN{cavesschackfrequentism}. }
The second strategy might be called \emph{primitivism}: simply postulate that weight=probability. This strategy is explicitly defended by \citeN{saundersprobability}; it is implicit in Vaidman's ``Behaviour Principle'' \cite{vaidmanencyclopedia}. It is open to the criticism of being unmotivated and even incoherent: effectively, to make the postulate is simply to stipulate that it is rationally compelling to care about one's successors in proportion to their weight (or to expect to be a given successor in proportion to his weight, in subjective-uncertainty terms), and it is unclear that we have any right to \emph{postulate} any such rationality principle, as if it were a law of nature. But again, it can be argued that classical probability theory is no better off here --- what is a ``propensity'', really, other than a primitively postulated rationality principle? (This is David Lewis's ``big bad bug'' objection to Humean supervenience; see \citeN[pp.\,xiv-xvii]{lewispapers2} and \citeN{lewis94} for further discussion of it). \citeN{papineaubjps} extends this objection to a general claim about probability in Everett: namely, although we do not understand it at all, we do not understand classical probability any better! --- so it is unfair to reject the Everett interpretation simply on the grounds that it has an inadequate account of probability.
The third, and most recent, strategy has no real classical analogue (though it has some connections with the `classical' program in philosophy of probability, which aims to derive probability from symmetry). This third strategy aims to derive the principle that weight=probability from considering the constraints upon rational action of agents living in an Everettian universe.\footnote{Given this, it is tempting to consider the Deutsch program as a form of subjectivism about probability, but --- as I argue more extensively in \citeN{wallaceepist} --- this is not the case. There was always a conceptual connection between objective probability and the actions of rational agents (as recognised in Lewis's Principal Principle \cite{lewischance}) --- what makes a probability `objective' is that all rational agents are constrained by it in the same way, and this is what Deutsch's proofs (purport to) establish for the quantum weight. In other words, there are objective probabilities --- and they have turned out to be the quantum weights.} It was initially proposed by \citeN{deutschprob}, who presented what he claimed to be a proof of the Born rule from decision-theoretic assumptions; this proof was criticised by Barnum \emph{et al}~\citeyear{barnumetal}, and defended by \citeN{decshort}. Subsequently, I have presented various expansions and developments on the proof (Wallace \citeyearNP{wallaceprobdec},\citeyearNP{wallace3branch}), and Zurek \citeyear{zurekenvariance03,zurekenvariance05} has presented another variant of it. It remains a subject of controversy whether or not these `proofs' indeed prove what they set out to prove.
\subsection{Further Reading}
\citeN{barrettbook} is an extended discussion of Everett-type interpretations (from a perspective markedly different from mine); \citeN{vaidmanencyclopedia} is a short (and fairly opinionated) survey.
\citeN{kent} is a classic criticism of ``old-style'' many-worlds theories; \citeN{baker}, \citeN{lewissu} and \citeN{hemmopitowsky07} criticise various aspects of the Everett interpretation as presented in this chapter.
\section{Dynamical-collapse theories}\label{DMWWdynamicalcollapse}
In this section and the next, we move away from pure \emph{interpretations} of the bare quantum formalism, and begin to consider substantive \emph{modifications} to it. There are essentially two ways to do this:
\begin{quote}
Either the wavefunction, as given by the Schr\"{o}dinger equation, is not everything, or it is not right (\citeNP{bellbook}, p.\,201)
\end{quote}
That is, if unitary QM predicts that the quantum state is in a macroscopic superposition, then either
\begin{enumerate}
\item the macroscopic world does not supervene on the quantum state alone but also (or instead) on so-called ``hidden variables'', which pick out one term in the superposition as corresponding to the macroscopic world; or
\item the predictions of unitary QM are false: unitary evolution is an approximation, valid at the microscopic level but violated at the macroscopic, so that macroscopic superpositions do not in fact come into existence.
\end{enumerate}
The first possibility leads us towards hidden variable theories, the topic of section \ref{DMWWhidden}. This section is concerned with ``dynamical collapse'' theories, which modify the dynamics to avoid macroscopic superpositions.
\subsection{The GRW theory as a paradigm of dynamical-collapse theories}\label{DMWWGRW}
How, exactly, should we modify the dynamics? Qualitatively it is fairly straightforward to see what is required. Firstly, given the enormous empirical success of QM at the microscopic level we would be well advised to leave the Schr\"{o}dinger equation alone at that level. At the other extreme, the Quantum Algorithm dictates that states like
\be \label{DMWWcatagain}\alpha\ket{\mathrm{\mbox{dead cat}}}+\beta \ket{\mathrm{\mbox{live cat}}}\ee
must be interpretable probabilistically, which means that our modification must ``collapse'' the wavefunction rapidly into either \ket{\mathrm{\mbox{dead cat}}} or \ket{\mathrm{\mbox{live cat}}} --- and furthermore, they must do it stochastically, so that the wavefunction collapses into \ket{\mathrm{\mbox{dead cat}}} with probability $|\alpha|^2$ and \ket{\mathrm{\mbox{live cat}}} with probability $|\beta|^2$.
Decoherence theory offers a way to make these qualitative remarks somewhat more precise. We know that even in unitary QM, probabilistic mixtures of pointer-basis states are effectively indistinguishable from coherent superpositions of those states. So we can be confident that our dynamical-collapse theory will not be in contradiction with the observed successes of quantum theory provided that coherent superpositions are decohered by the environment before they undergo dynamical collapse --- or, equivalently, provided that superpositions which are robust against decoherence generally \emph{do not} undergo dynamical collapse. Furthermore, dynamical collapse should leave the system in (or close to) a pointer-basis state --- this is in any case desirable, since the pointer-basis states are quasi-classical states, approximately localized in phase space.
The other constraint --- that macroscopic superpositions should collapse quickly --- is harder to quantify. \emph{How} quickly should they collapse? Proponents of dynamical-collapse theories --- such as \cite{bassighirardireview} --- generally require that the speed of collapse should be chosen so as to prevent ``the embarrassing occurrence of linear superpositions of appreciably different locations of a macroscopic object''. But it is unclear exactly when a given superposition counts as ``embarrassing''. One natural criterion is that the superpositions should collapse before humans have a chance to observe them. But the motivation for this is open to question. For suppose that a human observer looks at the state (\ref{DMWWcatagain}). If collapse is quick, the state rapidly collapses into
\be \ket{\mathrm{\mbox{dead cat}}} \,\,\,\,\,\,\,\mathrm{\mbox{or}}\,\,\,\,\,\, \ket{\mathrm{\mbox{live cat}}},
\ee
and observation puts the cat-observer system into the state
\be
\tpk{\mathrm{\mbox{dead cat}}}{\mathrm{\mbox{observer sees dead cat}}}\,\,\,\,\,\,\,\mathrm{\mbox{or}}\,\,\,\,\,\,
\tpk{\mathrm{\mbox{live cat}}}{\mathrm{\mbox{observer sees live cat}}}.
\ee
Given the stochastic nature of the collapse, the probability of the observer being in a state where he remembers seeing a dead cat is $|\alpha|^2$.
Now suppose that the collapse is much slower, taking several seconds to occur. Then the cat-observer system enters the superposition
\be
\alpha \tpk{\mathrm{\mbox{dead cat}}}{\mathrm{\mbox{observer sees dead cat}}}+ \beta
\tpk{\mathrm{\mbox{live cat}}}{\mathrm{\mbox{observer sees live cat}}}.
\ee
Who knows what it is like to be in such a state?\footnote{According to the functionalist analysis of section \ref{DMWWeverettdecoherence} ``it is like'' there being two people, one alive and one dead; but we shall not assume this here.} But no matter: in a few seconds the state collapses to
\be
\tpk{\mathrm{\mbox{dead cat}}}{\mathrm{\mbox{observer sees dead cat}}}\,\,\,\,\,\,\,\mathrm{\mbox{or}}\,\,\,\,\,\,
\tpk{\mathrm{\mbox{live cat}}}{\mathrm{\mbox{observer sees live cat}}}.
\ee
Once again, the agent is in a state where he remembers seeing either a live or dead cat, and the probability is $|\alpha|^2$ that he remembers seeing a dead cat --- since his memories are encoded in his physical state, he will have no memory of the superposition. So the fast and slow collapses appear indistinguishable empirically.
However, let us leave this point to one side. The basic constraints on a collapse theory remain: it must cause superpositions of pointer-basis states to collapse to pointer-basis states, and it must do so quickly enough to suppress ``embarrassing superpositions''; however, it must not have any appreciable affect on states which do not undergo decoherence.
Here we see again the difficulties caused by the approximate and ill-defined nature of decoherence. If decoherence were an exactly and uniquely defined process, we could just stipulate that collapse automatically occurs when states enter superpositions of pointer-basis states. Such a theory, in fact, would be exactly our `solution that isn't' from section \ref{DMWWsolutionthatisnt}. But since decoherence is not at all like this, we cannot use it directly to define a dynamical-collapse theory.
The requirement on a dynamical collapse theory is then: find a modification to the Schr\"{o}dinger equation that is cleanly defined in microphysical terms, and yet which closely approximates collapse to the decoherence-preferred basis. And such theories can in fact be found. The classic example is the ``GRW theory'' of \citeN{grw}. The GRW theory postulates that every particle in the Universe has some small spontaneous chance per unit time of collapsing into a localised Gaussian wave-packet:
\be\label{DMWWgrw}
\psi(x)\longrightarrow \mc{N}\exp(-(x-x_0)^2/2L^2) \psi(x)
\ee
where $L$ is a new fundamental constant (and \mc{N} is just a normalisation factor). The probability of collapse defines another new constant: $\tau$, the mean time between collapses. Crucially, the `collapse centre' $x_0$ is determined stochastically: the probability that $\psi$ collapses to a Gaussian with collapse centre in the vicinity of $x_0 $ is proportional to $|\psi(x_0)|^2$. If the particle is highly localised (that is, localised within a region small compared with $L$) then the collapse will have negligible effect on it; if it is in a superposition of such states, it will be left in just one of them, with the probability of collapse to a given state being equal to its mod-squared amplitude.
Now, $\tau$ is chosen to be extremely small, so that the chance of an isolated particle collapsing in a reasonable period of time is quite negligible. But things are otherwise if the particle is part of a macroscopic object. (The generalisation of (\ref{DMWWgrw}) to $N$-particle systems is just
\be\label{DMWWgrw2}
\psi(x_1,\ldots x_m, \ldots x_N)\longrightarrow \mc{N}\exp(-(x_m-x_0)^2/2L^2) \psi(x_1,\ldots x_m, \ldots x_N)
\ee
where the collapse occurs on the $m$th particle.) For suppose that that macroscopic object is in a superposition: something like (schematically)
\be
\alpha \ket{\mathrm{\mbox{at }}X}\otimes \cdots \otimes \ket{\mathrm{\mbox{at }}X}
+
\beta \ket{\mathrm{\mbox{at }}Y}\otimes \cdots \otimes \ket{\mathrm{\mbox{at }}Y}.
\ee
If $N \gg 1/\tau$, then within a small fraction of a second one of these particles will undergo collapse. Then the collapse will kick that particle (roughly speaking) into either \ket{\mathrm{\mbox{at }}X} (with probability $|\alpha|^2$) or \ket{\mathrm{\mbox{at }}Y} (with probability $|\beta|^2$). For convenience, suppose it in fact collapses to $X$. Then because of the entanglement, so do all of the other particles - the system as a whole collapses to a state very close to
\be
\ket{\mathrm{\mbox{at }}X}\otimes \cdots \otimes \ket{\mathrm{\mbox{at }}X}.
\ee
(Taking more mathematical care: if $\psi(x_1, \ldots x_N)$ is the wavefunction of a macroscopic $N$-particle body approximately localised at $x=0$, then
\be
\alpha \psi(x_1-X, \ldots x_N-X)
+ \beta
\psi(x_1-Y, \ldots x_N-Y).
\ee
If the first particle undergoes collapse, then its collapse centre has a probability $\simeq |\alpha|^2$ to be in the vicinity of $X$. Assuming this is so, the post-collapse wavefunction is approximately proportional to
\be
\alpha
\psi (x_1-X, \ldots x_N-X)
+
\beta \exp(-|X-Y|^2/L^2)\psi(x_1-Y, \ldots x_N-Y).
\ee
On the assumption that $|X-Y| \gg L$, the second term in the superposition is hugely suppressed compared with the first.)
So: the GRW theory causes superpositions of $N$ particles to collapse into localised states in a time $\sim \tau/N$, which will be very short if $\tau$ is chosen appropriately; but it has almost no detectable effect on small numbers of particles. From the perspective in which I have presented dynamical collapse, GRW incorporates two key observations:
\begin{enumerate}
\item Although the decoherence process is approximately defined and highly emergent, the actual pointer-basis states are fairly simple: they are Gaussians, approximately localised at a particular point in phase space. As such, it is sufficient to define collapse as suppressing superpositions of position states.
\item Similarly, although the \emph{definition} of `macroscopic system' given by decoherence is highly emergent, \emph{in practice} such systems can be picked out simply by the fact that they are compounds of a great many particles. So a collapse mechanism defined for single particles is sufficient to cause rapid collapse of macroscopic systems.
\end{enumerate}
The actual choice of GRW parameters is determined by the sorts of considerations discussed above. Typical choices are $L=10^{-5} \mathrm{cm}$, $\tau=10^{16} \mathrm{s}$, ensuring that an individual particle undergoes collapse only after $\sim 10^8$ years, but a grain of dust $\sim 10^{-2}$ cm across will undergo collapse within a hundredth of a second, and Schr\"{o}dinger's cat will undergo it after $\sim 10^{-11}$ seconds. (In fact, if the GRW theory holds then the cat never has the chance to get into the alive-dead superposition in the first place: dynamical collapse will occur in the cat-killing apparatus long before it begins its dread work.)
The GRW theory is not the ``last word'' on dynamical-collapse theories. Even in the non-relativistic domain it is not fully satisfactory: manifestly, the collapse mechanism does not preserve the symmetries of the wavefunction, and so it is not compatible with the existence of identical particles. These and other considerations led \citeN{pearle} to develop ``continuous state localisation'' (or CSL), a variant on GRW where the collapse mechanism preserves the symmetry of the wavefunction, and most advocates of dynamical collapse now support CSL rather than GRW. (See \citeN[section 8]{bassighirardireview} for a review of CSL.)
However, there seems to be a consensus that foundational issues with CSL can be equally well understood in the mathematically simpler context of GRW. As such, conceptual and philosophical work on dynamical collapse is predominantly concerned with GRW, in the reasonable expectation that lessons learned there will generalise to CSL and perhaps beyond.
\subsection{The problem of tails and the Fuzzy Link}\label{DMWWtails}
The main locus of purely \emph{philosophical} work on the GRW theory in the past decade has been the so-called ``problem of tails''.
As I shall argue (following \citeN{cordero} to some extent) there are actually two ``problems of tails'', only one of which is a particular problem of dynamical-collapse theories, but both are concerned with the stubborn resistance of the wavefunction to remain decently confined in a finite-volume region of space.
The original ``problem of tails'' introduced by \citeN{albertloewer1996} works as follows. Suppose we have a particle in a superposition of two fairly localised states \ket{\mathrm{\mbox{here}}} and \ket{\mathrm{\mbox{there}}}:
\be \ket{\psi}=\alpha\ket{\mathrm{\mbox{here}}}+\beta\ket{\mathrm{\mbox{there}}}.\ee
Dynamical collapse will rapidly occur, propelling the system into something like
\be \ket{\psi'}=\sqrt{1-\epsilon^2}\ket{\mathrm{\mbox{here}}}+\epsilon\ket{\mathrm{\mbox{there}}}.\ee
But (no matter how small $\epsilon$ may be) this is not the same state as
\be\ket{\psi''}=\ket{\mathrm{\mbox{here}}}.\ee
Why should the continued presence of the `there' term in the superposition --- the continued indefiniteness of the system between `here' and `there' --- be ameliorated in any way at all just because the `there' term has low amplitude?
Call this the \emph{problem of structured tails} (the reason for the name will become apparent). It is specific to dynamical collapse theories; it is a consequence of the GRW collapse mechanism, which represents collapse by multiplication by a Gaussian and so fails to annihilate terms in a superposition no matter how far they are from the collapse centre.
It is interesting, though, that most of the recent `tails' literature has dealt with a rather different problem which we might call the \emph{problem of bare tails}. Namely: even if we ignore the `there' state, the wave function of \ket{\mathrm{\mbox{here}}} is itself spatially highly delocalised. Its centre-of-mass wavefunction is no doubt a Gaussian, and Gaussians are completely delocalised in space, for all that they may be concentrated in one region or another. So how can a delocalised wave-packet possibly count as a localised particle?
\emph{This} problem has little or nothing to do with the GRW theory. Rather, it is an unavoidable consequence of using wave-packets to stand in for localised particles. For \emph{no} wave-packet evolving unitarily will remain in any finite spatial region for more than an instant (consider that infinite potentials would be required to prevent it tunneling to freedom.)
Apparent force is added to this objection by applying the eigenvector-eigenvalue link. The latter gives a perfectly clear criterion for when a particle is localised in any spatial region $R$: it must be an eigenstate of the operator
\be\op{P}_R=\int_R \dr{x}\proj{x}{x}.\ee
That is, it must have support within $R$; hence, no physically realisable state is every localised in a finite region.
One might be inclined to respond: so much the worse for the eigenvector-eigenvalue link, at least in the context of continuous observables. As we have seen in section \ref{DMWWagainst}, its motivation in modern QM is tenuous at best. But that simply transfers the problem: if the eigenvector-eigenvalue link is not to be the arbiter for which physical states count as localised, what is?
Albert and Loewer propose a solution: a natural extension of the eigenvector-eigenvalue link which they call the \emph{fuzzy link}. Recall that the eigenvector-eigenvalue link associates (at least a subset of) properties 1:1 with projectors, and regards a state \ket{\psi} as possessing (the property associated with) projector \op{P} iff $\op{P}\ket{\psi}=\ket{\psi}$; that is, if
\be|\op{P}\ket{\psi}-\ket{\psi}|=0.\ee
The fuzzy link is a relaxation of this condition: the properties remain in one-to-one correspondence with the projectors, but now \ket{\psi} has property (associated with) \op{P} if, for some fixed small $p$,
\be\label{DMWWfuzzylink}|\op{P}\ket{\psi}-\ket{\psi}|