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\begin{document}
\title{Events and covariance in the interpretation of quantum field theory}
\author{Dennis Dieks\\ Institute for the History and Foundations of Science\\
Utrecht University, P.O.Box 80.000 \\ 3508 TA Utrecht, The
Netherlands\\Email: d.g.b.j.dieks@phys.uu.nl}
\date{January 2002}
\maketitle
\begin{abstract}
In relativistic quantum field theory the notion of a local {\em
operation} is regarded as basic: each open space-time region is
associated with an algebra of observables representing possible
measurements performed within this region. It is much more
difficult to accommodate the notions of \emph{events} taking place
in such regions or of localized objects. But how can the notion of
a local operation be basic in the theory if this same theory would
not be able to represent localized measuring devices and localized
events? After briefly reviewing these difficulties we discuss a
strategy for eliminating the tension, namely by interpreting
quantum theory in a realist way. To implement this strategy we use
the ideas of the modal interpretation of quantum mechanics. We
then consider the question of whether the resulting scheme can be
made Lorentz invariant.
\end{abstract}
\newpage
\sectiona{The problem}\label{sectproblem}
Relativistic quantum theory is notorious for the difficulties it
has with the notion of particle position. Relativistic quantum
mechanics does not accommodate `position' as a particle observable
in the same way as non-relativistic quantum mechanics does; the
Newton-Wigner position operator, which is the only observable that
comes into serious consideration, faces well-known difficulties in
connection with covariance. The standard response is that this
signals the inadequacy of the particle concept in relativistic
quantum theory, and that we should switch to quantum \emph{field}
theory. The most general form of this theory is `algebraic
relativistic quantum field theory' (ARQFT). At first sight, the
prospects for position as a central physical magnitude seem very
good in this theory. Indeed, ARQFT is formulated against the
background of Minkowski space-time, and regions of Minkowski
space-time figure prominently in the axioms of the theory.
Specifically, a $C^{\star}$-algebra of observables is associated
with each open region $O$ in Minkowski space-time. These operators
represent physical operations, measurements, that can be performed
within $O$ (\cite{haag}, p.\ 105). The localization of observables
is thus a fundamental notion in the theory. However, the
localization of an event or object are more troublesome notions in
ARQFT. As we already mentioned, the notion of a localized system
is problematic in relativistic quantum mechanics
(\cite{Hegerfeldt,Malament}), and it turns out that the same
problems persist in field theory.
Within the field theoretical framework the notorious
Reeh-Schlieder-theorem is responsible for additional concern about
locality. According to this theorem (\cite{haag}, p.\ 101), the
set of states obtained by applying the local operations associated
with any particular $O$ to a state of bounded energy is dense in
the Hilbert space of all states of the total field. That means
that any state can be approximated, arbitrarily closely, by
applying local operations to the vacuum (or any other state of
bounded energy). So, even states that could be thought of as
candidates for representing objects far from $O$ can be generated
by local operations within $O$. A corollary is that the local
$C^{\star}$-algebras do not contain observables corresponding to
tests of whether or not the vacuum state is the state of the total
field (\cite{haag}, p.\ 102). In other words, even in the vacuum
state local tests (corresponding to local observables) of whether
or not matter, energy or charge are present will with some
probability yield positive results---particle counters measuring
these local quantities will click from time to time. The vacuum
can therefore not be thought of as a `sum of local vacua'; it is
an inherently global concept. The same applies to $N$-particle
states: it is impossible to verify the presence of such a state by
local operations. The particles of the $N$-particle states are not
local objects.
The Reeh-Schlieder-theorem demonstrates that the vacuum, and all
other states of bounded energy, have long-distance correlations
built into them. It is therefore not surprising to find that
Bell-inequalities are violated in these states---a standard sign
of non-locality.
However, one should be careful not to lump all such non-local
features together. That there are non-local correlations does not
mean that it is impossible to speak about localized objects; to
the contrary, it only makes sense to discuss non-local
correlations if there are more or less localized things, far apart
from each other, that can be correlated with each other. And a
natural conclusion to draw from the Hegerfeldt and Malament
results is that relativistic quantum theory does not admit states
that assign non-zero probabilities to a bounded region only. But
that is not in {\em a priori} conflict with the idea that there
are localized events or localized objects; there could be very
many of them, (almost) everywhere.
In this paper we will concentrate on the question of whether it is
possible to interpret algebraic quantum field theory in such a way
that the theory is able to accommodate the concept of an event
localized in a small space-time region, and whether it is possible
to work---at least in some circumstances---with the concept of a
localized physical system. We will not address the ramifications
of violations of the Bell-inequalities and similar features of
non-locality relating to long-distance correlations.
The question of how to handle localized systems is vital for the
interpretation of relativistic quantum field theory. Without an
answer to it the theory is incomplete at best (and inconsistent at
worst). For how can the local operations whose existence the
theory assumes be performed if the theory itself does not allow
the description of localized devices? If the treatment of such
devices is outside the scope of the theory, the theory is
incomplete in its description of physical reality. Even more
worrying would be the alternative that the theory is complete and
still cannot handle localized systems. That would signal
inconsistency, given the fundamental importance of localized
operations in the theory.
A closely related issue concerns the meaning of the Minkowski
background that is assumed in the formulation of ARQFT. Of course,
we think of this four-dimensional manifold as the space-time arena
in which physical processes take place. However, without the
existence of at least approximately localized objects or events
the usual operational meaning and material realization of
space-time regions would no longer be available. It is not clear
that the four-dimensional manifold would in this case still have
its usual significance as space-time. Notions like causality
between events, and the space-time distance between events, which
figure in the axioms of ARQFT, would not have the physical
correlates they are commonly assumed to possess (e.g., in terms of
signals connecting the events). That would make the physical
motivation of these axioms (like the micro-causality axiom)
uncertain.
\sectiona{What has to be done}\label{sectsteps}
What we would like to have, in order to achieve a consistent and
possibly complete theory, is a representation of the physical
systems performing the local operations mentioned in the axioms.
This would make it possible to consider the concepts of
measurement and operation not as fundamental but as derived: a
local measurement should be an interaction between an object and a
localized measuring device. The natural way to achieve this is to
describe measuring devices in terms of `beables', that is as
systems characterized by the values certain physical magnitudes
assume in them. Clearly, a prerequisite for the successful
implementation of this program is that we have an interpretation
of the formalism of ARQFT in terms of physical magnitudes and the
values that they take; as opposed to the usual interpretation,
which is only about measurement results. This is nothing else than
a variant of the general problem of the interpretation of quantum
theory.
Although the traditional interpretational problems are rarely
discussed in the context of quantum field theory, it is clear that they exist there in
analogous form and
are no less urgent than in non-relativistic quantum mechanics.
As in the non-relativistic theory,
the central issue is that it is not obvious that the theory is about objective physical
states of affairs, even
in circumstances in which no macroscopic measurements are being made. This is because the fields
in quantum field theory do not attach values
of physical magnitudes to space-time points. Rather, they are fields of operators, with
a standard interpretation
in terms of macroscopic measurement results.
As just pointed out, we
would like to give another meaning to the formalism, namely in terms of
physical systems that possess certain properties.
What we would like to
do is to provide
an interpretation in which not only operators, but also {\em
properties} are assigned to
space-time regions. That is, we would like at least some of the observables to have
definite values. This would lead
to a picture in which it is possible to speak of objective {\em
events} (if some physical
magnitude takes on a definite value in a certain spacetime region, this constitutes the
event that this magnitude has that value there and then).
As a second step, we need to show that it is possible to have
consistent joint probabilities of the properties assigned to
different space-time regions. Finally, it should be made clear
that it is possible, at least in principle, to assign properties
to space-time regions in such a way that (approximately) localized
systems can be represented. This does not mean that it is
necessary to establish that there exist localized systems in every
physical state allowed by the theory. The Minkowski manifold need
not have the space-time implementation that we are used to in
every possible world allowed by the theory. Rather, we have to
show that localized systems exist in some states; and in
particular, we expect localized systems to emerge in a classical
limiting situation.
Obviously, the above programme is ambitious and addresses
complicated and controversial issues. On several points we will
not be able to do much more than suggesting possibilities and
indicating open questions.
\sectiona{Modal property attribution schemes}\label{sect1}
In order to be able to speak about events and physical properties
of systems rather than merely about the results of operations and
measurements, we will make an attempt to apply modal
interpretational ideas to ARQFT. Modal interpretations of quantum
mechanics \cite{vanfra,dieks1,healey,dieks2} interpret the
mathematical formalism of quantum theory in terms of properties
possessed by physical systems, i.e.\ quantum mechanical
observables taking on definite values. Because of the Kochen and
Specker no-go theorem, not all observables pertaining to a system
can be definite-valued at the same time. Modal interpretations
therefore specify a \emph{subset} of all observables, such that
only the observables in this subset are definite-valued. It is
characteristic of the modal approach in the version that we shall
use that this is done in a state-dependent way: the quantum
mechanical state of the system contains all information needed to
determine the set of definite-valued observables.
In non-relativistic quantum mechanics, and in the most common
version of the modal interpretation \cite{verm}, the precise
prescription for finding this set makes use of the spectral
decomposition of the density operator of the system, in the
following way. Let $\A$ be our system and let $\B$ represent its
total environment (the rest of the universe). Let $\A\&\B$ be
represented by $\ket{\psi^{\A\B}} \in {\cal H}^{\A} \otimes {\cal
H}^{\B}$. The bi-orthonormal decomposition of $\ket{\psi^{\A\B}}$,
\be \ket{\psi^{\A\B}} & = & \sum_{i} c_{i} \: \ket{\psi^{\A}_{i}}
\: \ket{\psi^{\B}_{i}} \label{eq1} \;\;\; , \ee with
$\inpr{\psi^{\A}_{i}}{\psi^{\A}_{j}} =
\inpr{\psi^{\B}_{i}}{\psi^{\B}_{j}} = \delta_{ij}$, generates a
set of projectors operating on ${\cal H}^{\A}$:
$\{\proj{\psi^{\A}_{i}} \}_{i}$. If there is no degeneracy among
the numbers $\{ | c_{i} |^{2} \}$, this is a uniquely determined
set of one-dimensional projectors. If there is degeneracy, the
projectors belonging to one value of $\{ | c_{i} |^{2} \}$ can be
added to form a multi-dimensional projector; the thus generated
new set of projectors, including multi-dimensional ones, is again
uniquely determined. These projectors are the ones occurring in
the spectral decomposition of the reduced density operator of
$\A$.
The modal interpretation of non-relativistic quantum mechanics
assigns definite values to the subset of all physical magnitudes
that are generated by these projectors; i.e., the subset obtained
by starting with these projectors, and then including their
continuous functions, real linear combinations, symmetric and
antisymmetric products, and finally closing the set \cite{clifton}
(the thus defined subset of all observables constitutes the set of
`well-defined' or `applicable' physical magnitudes, in Bohrian
parlance). {\em Which} value among the possible values of a
definite magnitude is actually realized is not fixed by the
interpretation. For each possible value a probability is
specified: the probability that the magnitude represented by
$\proj{\psi^{\A}_{i}}$ has the value $1$ is given by
$|c_{i}|^{2}$. In the case of degeneracy it is stipulated that the
magnitude represented by $\sum_{i\in I_{l}} \proj{\psi^{\A}_{i}}$
has value $1$ with probability $\sum_{i\in I_{l}} |c_{i}|^2$
($I_{l}$ is the index-set containing indices $j$, $k$ such that
$|c_{j}|^{2} = |c_{k}|^{2}$).
The observation that the definite-valued projections occur in the
spectral decomposition of $\A$'s density operator gives rise to a
generalization of the above scheme that is also applicable to the
case in which the total system $\A\&\B$ is not represented by a
pure state: find $\A$'s density operator by partial tracing from
the total density operator, determine its spectral resolution and
construct the set of definite-valued observables from the
projection operators in this spectral resolution \cite{verm}.
The above recipe for assigning properties is meant to apply to each physical system in
a non-overlapping collection of
systems that together make up the total universe \cite{baccia,dieks3}. It is easy to
write down a satisfactory {\em joint}
probability distribution for the properties of such a collection (or a subset of it):
\begin{equation}
Prob(P^{\A}_{i}, P^{\B}_{j},...., P^{\G}_{k},...., P^{\X}_{l})
= \langle \Psi | P^{\A}_{i}.
P^{\B}_{j}.....P^{\G}_{k}.....P^{\X}_{l} | \Psi
\rangle ,\label{prob}
\end{equation}
where the left-hand side represents the joint probability for the
projectors occurring in the argument of taking the value $1$, and
where $\Psi$ is the state of the total system consisting of $\A$,
$\B$, $\G$, etc. \cite{verm}. It is important for the consistency
of this probability ascription that the projection operators
occurring in the formula all commute (which they do, since they
operate in different Hilbert spaces).
\sectiona{A perspectival version of the modal
interpretation}\label{perspective} In the usual version of the
modal interpretation, as it was just explained, physical
properties are represented by \emph{monadic} predicates. Such
properties belong to a system without reference to other systems.
We will now briefly describe an alternative, according to which
physical properties have a \emph{relational} character \cite{bene
and dieks}. These relational properties need \emph{two} systems
for their definition: the physical system $S$ and a `reference
system' $R$ that defines the `perspective' from which $S$ is
considered. This reference system is a larger system, of which
$S$ is a part. We will allow that one and the same system, at one
and the same instant of time, can have different states with
respect to different reference systems. However, the system will
have one single state with respect to any given reference system.
This state of $S$ with respect to $R$ is a density matrix denoted
by $\rho^S_R$. In the special case in which $R$ coincides with
$S$, we have the `state of $S$ with respect to itself', which we
take to be the same as the state of $S$ assigned by the modal
scheme explained in the previous section; i.e.\ it is one of the
projectors occurring in the spectral decomposition of the reduced
density operator of $S$.
The rules for determining all states, for arbitrary $S$ and $R$,
are as follows. If $U$ is the whole universe, then $\rho^U_U$ is
taken as the quantum state assigned to $U$ by standard quantum
theory. If system $S$ is contained in system $A$, the state
$\rho^S_A$ is defined as the density operator that can be derived
from $\rho^A_A$ by taking the partial trace over the degrees of
freedom in $A$ that do not pertain to $S$:
\begin{eqnarray}
\rho^S_A={\rm Tr}_{A\setminus S}\;\rho^A_A\label{g4}
\end{eqnarray}
Any relational state of a system with respect to a bigger system
containing it can be derived by means of Eq.(\ref{g4}).
As in the ordinary modal scheme, the state $\rho^U_U$ evolves
unitarily in time. Because there is no collapse of the wave
function in the modal interpretation, this unitary evolution of
the total quantum state is the main dynamical principle of the
theory. Furthermore, it is assumed that the state assigned to a
\emph{closed} system $S$ undergoes a unitary time evolution
\begin{eqnarray}
i\hbar\frac{\partial}{\partial t}\hat \rho^S_S=\left[ H_S,\;
\rho^S_S\right]\label{g5}
\end{eqnarray}
As always in the modal interpretation, the idea is that the theory
should specify only the \emph{probabilities} of the various
possibilities. For a collection of pair-wise disjoint systems,
with respect to one reference system, one could postulate that the
joint probability of the states of the various systems is given
by the usual formula, Eq.\ (\ref{prob}) (\cite{bene and dieks}
uses a different probability postulate). A significant point is
that joint probabilities should not always be expected to exist
within the perspectival approach because states that are defined
with respect to different quantum reference systems need not be
commensurable.
If a system $A$ and its complement $U\setminus A$ are concerned, a
simple calculation based on the Schmidt representation of the
total state shows that the states, with respect to themselves, of
these two systems are one-to-one correlated. Therefore, knowledge
of the state of $U\setminus A$, plus the total state, makes it
possible to infer the state of $A$. This suggests that one may
consider the state of $S$ with respect to the reference system
$A$, $\rho^S_A$, alternatively as being defined {\it from the
perspective} $U\setminus A$ (here $A$ is an arbitrary quantum
reference system, while $U$ is the whole universe). Sometimes the
concept of a `perspective' is intuitively more appealing than the
concept of a quantum reference system (cf.\ \cite{Rovelli}).
However, this alternative point of view has its limitations.
First, if $A$ itself is the whole universe, the concept of an
external perspective cannot be applied. Moreover, the state of the
system $U\setminus A$ in itself does not contain sufficient
information to determine the state of system $A$; one also needs
the additional information provided by $|\psi_U>$ in order to
compute $|\psi_A>$. But $|\psi_A>$ does contain all the
information needed to calculate $\rho^S_A$ (cf. Eq.(\ref{g4})). We
will therefore relativize the states of $S$ to reference systems
that contain $S$, although we shall sometimes---in cases in which
this is equivalent--- also speak about the state of $S$ {\em from
the perspective} of the complement of the reference system.
Of course, we must address the question of the physical meaning of
the states $\rho^S_A$. In the perspectival approach it is a
fundamental assumption that basic descriptions of the physical
world have a relational character, and therefore we cannot explain
the relational states by appealing to a definition in terms of
more basic, and more familiar, non-relational states. But we
should at the very least explain how these relational states
connect to actual experience. Minimally, the theory has to give an
account of what observers observe. We postulate that experience in
this sense is represented by the state of a part of the observer's
perceptual apparatus (the part characterized by a relevant
indicator variable, like the display of a measuring device) {\em
with respect to itself}. More generally, the states of systems
with respect to themselves correspond to the (monadic)properties
assigned by the earlier, non-perspectival, version of the modal
interpretation. The empirical meaning of many other states can be
understood and explained - by using the rules of the
interpretation - through their relation to these states of
observers, measuring devices, and other systems, with respect to
themselves. For more details concerning these ideas see \cite{bene
and dieks}.
\sectiona{Application to ARQFT}\label{arqft}
The modal interpretations that were discussed in the previous
sections were devised for the case of quantum mechanics, in which
each physical system is represented within its own Hilbert space
and in which the total Hilbert space is the tensor product of the
Hilbert spaces of the individual component systems. The possible
physical properties of the systems correspond to observables
defined as operators on the Hilbert spaces of these systems. In
axiomatic quantum field theory algebras of observables are
associated with open space-time regions (`local algebras'). It is
therefore natural to think of these observables as representative
of possible local physical characteristics and to regard the
space-time regions as the analogues of the physical systems to
which we applied the modal scheme before. If the open space-time
regions would correspond to subspaces of the total Hilbert space,
an immediate application of the modal scheme would be possible and
would lead to the selection of definite-valued observables from
the local algebras. Unfortunately, things are not that simple. The
local algebras should generally be expected to be of type III
(algebra's with only infinite-dimensional projectors, without
minimal projectors). This implies that they cannot be represented
as algebras of bounded observables on a Hilbert space (such
algebras are of type I). In other words, the local algebras cannot
be thought of as algebras of observables belonging to a physical
system described within its own Hilbert space, and the total
Hilbert space is not a tensor product of Hilbert spaces of such
local subsystems.
There are, however, other ways to introduce the notion of a
localized subsystem. One possibility is to use the algebras of
type I that `lie between two local algebras'. That such type-I
algebras exist is assumed in the postulate of the `split property'
(\cite{haag}, Ch.\ V.5). If this postulate is accepted one can
consider the algebras of type I lying between the
$C^{\star}$-algebras associated with concentric standard `diamond'
regions with radii $r$ and $r+\epsilon$, respectively, with $r$
and $\epsilon$ very small numbers. In this way we approximate the
notion of a space-time point as a physical system, represented in
a subspace of the total Hilbert space \cite{diekspla}. The
advantage of this approach is that all systems that we consider
have their own Hilbert spaces, and that we can therefore use the
same techniques as in non-relativistic quantum mechanics. However,
a disadvantage is the arbitrariness in fixing the values of $r$
and $\epsilon$, and in choosing one type-I algebra from the
infinity of such algebras lying between the two type-III algebras
associated with the two diamond regions \cite{cliftonpla}.
Another way of applying the modal ideas to ARQFT was suggested by
Clifton \cite{cliftonpla}; in this proposal there is much less
arbitrariness. Clifton considers an arbitrary von Neumann algebra
$\mathcal{R}$. A state $\rho $ on $\mathcal{R}$ defines the
`centralizer subalgebra'
\begin{equation}
\mathcal{C}_{\rho ,\mathcal{R}}\equiv \{A\in \mathcal{R}:\rho
([A,B])=0 \; \mbox{for all} \; B\in \mathcal{R}\}.
\end{equation}
Further, let $\mathcal{Z}(\mathcal{C}_{\rho ,\mathcal{R}})$ be the
\emph{center algebra} of $\mathcal{C}_{\rho ,\mathcal{R}}$, i.e.\
the elements of $\mathcal{C}_{\rho ,\mathcal{R}}$ that commute
with all elements of $\mathcal{C}_{\rho ,\mathcal{R}}$. Clifton
proves the following theorem:
\mathstrut
\textit{Let }$\mathcal{R}$\textit{\ be a von Neumann
algebra and }$\rho $\textit{\ a faithful normal state of }$\mathcal{R}$%
\textit{\ with centralizer }$\mathcal{C}_{\rho ,\mathcal{R}}\;\mathcal{%
\subseteq R}$\textit{. Then }$\mathcal{Z}(\mathcal{C}_{\rho ,\mathcal{R}})$%
\textit{, the center of }$\mathcal{C}_{\rho
,\mathcal{R}}$\textit{, is the unique subalgebra
}$\mathcal{S\subseteq R}$\textit{\ such that:}
\begin{enumerate}
\item \textit{The restriction of }$\rho $\textit{\ to }$\mathcal{S}$\textit{
\ \ is a mixture of dispersion-free states.}
\item $\mathcal{S}$\textit{\ is definable solely in terms of }$\rho $%
\textit{\ and the algebraic structure of }$\mathcal{R}$\textit{.}
\item $\mathcal{S}$\textit{\ is maximal with respect to properties 1. and 2.%
}
\end{enumerate}
Moreover, for faithful states projections from
$\mathcal{Z}(\mathcal{C}_{\rho ,\mathcal{R}})$ are strictly
correlated with projections from $\mathcal{Z}(\mathcal{C}_{\rho
,\mathcal{R}^\prime})$, where $\mathcal{R}^\prime$ is the
commutant of $\mathcal{R}$. This generalizes the strict
correlation between projectors occurring in the bi-orthonormal
decomposition in quantum mechanics \cite{cliftonpla}.
These results make it natural to take
$\mathcal{Z}(\mathcal{C}_{\rho ,\mathcal{R}(\diamondsuit_{r})})$
as the subalgebra of definite-valued observables in
$\mathcal{R}(\diamondsuit_{r})$, the algebra associated with a
diamond region $\diamondsuit_{r}$, if $\rho $ is a pure state of
the field that induces a faithful state on
$\mathcal{R}(\diamondsuit_{r})$ (for example, $\rho $ could be the
vacuum or any other state with bounded energy). In this way it
becomes possible to assign definite physical properties to regions
of space-time.
As in the case of non-relativistic quantum mechanics, we will take
the projectors in the (Abelian) algebra of definite-valued
observables as a `base set' of definite-valued quantities. The
complete collection of definite-valued observables can be
constructed from this base set by closing the set under the
operations of taking continuous functions, real linear
combinations, and symmetric and anti-symmetric products
\cite{clifton}. The probability of projector $P_{l}$ having the
value $1$ is $\langle \Psi | P_{l} | \Psi \rangle$, with $|\Psi>$
the state of the total field. Subdividing Minkowski space-time
into a collection of non-overlapping point-like regions, and
applying the above prescription to the associated algebras, we
achieve the picture aimed at: to each space-time region belong
definite values of some physical magnitude, and this constitutes
an event localized in that region.
It is important, though, to realize that the resulting picture is
not classical. A typical quantum feature is that there is no
guarantee that the definite-valued quantities associated with a
space-time region will also be definite-valued quantities of
larger space-time regions that contain the original one. This is
analogous to what is found in non-relativistic quantum mechanics:
in general, the properties of systems do not follow from the
properties of their components. That such a simple relation
between wholes and parts nevertheless does obtain in classically
describable situations needs to be explained by some physical
mechanism; decoherence is the prime candidate. Indeed, it is easy
to see that the principle of `property composition' (asserting
that the properties of composite systems are built up from the
properties of their components) holds according to the modal
interpretation of non-relativistic quantum mechanics if the
environment `decoheres' all component systems separately. For a
composite system $\A\&\B$, with environments $E^\A$ and $E^\B$ of
$\A$ and $\B$, respectively, we have in this case the state \be
\ket{{\psi}} & = & \sum_{ij} c_{ij} \: \ket{\psi^{\A}_{i}} \:
\ket{E^\A_i} \: \ket{\psi^{\B}_{j}} \: \ket{E^\B_j}
\label{propcomp} \; , \ee with
$\inpr{\psi^{\A}_{i}}{\psi^{\A}_{j}} =
\inpr{\psi^{\B}_{i}}{\psi^{\B}_{j}} =
\inpr{E^{\A}_{i}}{E^{\A}_{j}} = \inpr{E^{\B}_{i}}{E^{\B}_{j}} =
\delta_{ij}$. The definite-valued projectors for $\A\&\B$ are
therefore the products of the definite-valued projectors for $\A$
and $\B$ separately.
In the field-theoretic case decoherence similarly tends to mask
typical quantum effects. Above, we discussed two possible ways of
implementing modal ideas in the field-theoretic context. If the
split property is used, physical subsystems are represented by
Hilbert spaces, and the same reasoning can be employed as in the
non-relativistic case. If the algebras of definite-valued
observables as defined by Clifton are taken as a starting point,
the situation does not become very different. These algebras are
generated by projection operators $P$ that are strictly correlated
to projections $\overline{P}$ associated with the environment:
$<\Psi| P \overline{P} |\Psi> = <\Psi| P |\Psi> =
<\Psi|\overline{P} |\Psi>$. Now suppose that decoherence
mechanisms have been effective, in the sense that information has
been carried away to distant regions, and that (semi-)permanent
memories have been formed of the definite-valued observables in
the space-time region $O$. If there are thus `copy' projectors
$\overline{P}$, strictly correlated to $P$, that belong to
far-away regions, we have that $\rho ([P,B]) = \rho
([\overline{P},B]= 0$, for all observables $B$ associated with a
region $O^\prime$ that contains $O$ but is not too big (so that
$\overline{P}$ belongs to an algebra that is distant enough to be
in the commutant of the observables $B$). Therefore, the
centralizer subalgebra of the von Neumann algebra associated with
$O$ will be contained in the centralizer subalgebra pertaining to
$O^\prime$. It follows that the definite-valued projectors
associated with region $O$ commute with those associated with
$O^\prime$ (the latter by definition commute with all elements of
the centralizer subalgebra of $O^\prime$, and therefore with all
elements of the centralizer subalgebra of $O$). The properties of
$O$ and $O^\prime$ are therefore compatible.
\sectiona{Histories of modal properties}\label{history}
Let us return to the non-relativistic case to consider the problem
of correlations in time. There is a natural analogue of expression
(\ref{prob}) for the case of Heisenberg projection operators
pertaining to different instants of time:
\begin{eqnarray}
\lefteqn{Prob(P_{i}(t_{1}), P_{j}(t_{2}),.... P_{l}(t_{n})) =}
\nonumber \\ & & \langle \Psi | P_{i}(t_{1}).
P_{j}(t_{2}).....P_{l}(t_{n}) . P_{l}(t_{n}) . .... P_{j}(t_{2}) .
P_{i}(t_{1}) | \Psi \rangle .\label{sevtimeprob}
\end{eqnarray}
This expression is in accordance with the standard prescription
for calculating the joint probability of outcomes of consecutive
measurements. It agrees also with the joint distribution assigned
to `consistent histories' in the consistent histories approach to
the interpretation of quantum mechanics \cite{grif}. However, it
should be noted that the projection operators in
(\ref{sevtimeprob}), pertaining to different times as they do,
need not commute. As a result, (\ref{sevtimeprob}) does not
automatically yield a consistent probability distribution. For
this reason it is an essential part of the consistent histories
approach to impose the following decoherence condition, in order
to guarantee that expression (\ref{sevtimeprob}) is an ordinary
Kolmogorov probability:
\begin{eqnarray}
\lefteqn{\langle \Psi | P_{i}(t_{1}).
P_{j}(t_{2}).....P_{l}(t_{n}) . P_{l^{\prime}}(t_{n}) ......
P_{j^{\prime}}(t_{2}) . P_{i^{\prime}}(t_{1}) | \Psi \rangle =0}
\nonumber \\ \mbox{if} & i \neq i^{\prime} \vee j \neq j^{\prime}
\vee ....\vee l \neq l^{\prime}.\label{consist}
\end{eqnarray}
In the consistent histories approach the only sequences of
properties which are considered are those satisfying the
decoherence condition (\ref{consist}). It has been argued in the
literature (\cite{kent}) that the projection operators singled out
by the modal interpretation will in general not satisfy this
decoherence condition. That argument is not valid, however
(\cite{diekspla}). On the contrary, it is natural to introduce the
idea of decoherence in the modal scheme in such a way that
condition (\ref{consist}) is satisfied. Eq.\ (\ref{sevtimeprob})
yields a consistent joint multi-times probability distribution for
modal properties if this decoherence condition is fulfilled.
The notion of decoherence to be used is the following. It is a
general feature of the modal interpretation that if a system
acquires a certain property, this happens by virtue of its
interaction with the environment, as expressed in Eq.\
(\ref{eq1}). As can be seen from this equation, in this process
the system's property becomes correlated with a property of the
environment. Decoherence is now defined to imply the
irreversibility of this process of correlation formation: the rest
of the universe retains a trace of the system's property, also at
later times when the properties of the system itself may have
changed. In other words, the rest of the universe acts as a memory
of the properties the system has had; decoherence guarantees that
this memory remains intact. For the state $\ket{\Psi}$ this means
that in the Schr\"{o}dinger picture it can be written in the
following form: \be \ket{\Psi(t_{n})} & = & \sum_{i,j,...,l}
c_{i,j,...,l} \: \ket{\psi_{i,j,...,l}} \: \ket{\Phi_{i,j,...,l}}
\label{eqdeco} \;\;\; , \ee where $\ket{\psi_{i,j,...,l}}$ is
defined in the Hilbert space of the system,
$\ket{\Phi_{i,j,...,l}}$ in the Hilbert space of the rest of the
universe, and where $\langle \Phi_{i,j,...,l} |
\Phi_{i^{\prime},j^{\prime},...,l^{\prime}} \rangle = \delta_{i
i^{\prime} j j^{\prime}...l l^{\prime}}$. In (\ref{eqdeco}) $l$
refers to the properties $P_{l}(t_{n})$, $j$ to the properties
$P_{j}(t_{2})$, $i$ to the properties $P_{i}(t_{1})$, and so on.
The physical picture that motivates a $\ket{\Psi}$ of this form is
that the final state results from consecutive measurement-like
interactions, each of which is responsible for generating new
properties. Suppose that in the first interaction with the
environment the properties $ \proj{\A_{i}}$ become definite: then
the state obtains the form $\sum_{i} c_{i} \ket{\A_{i}}
\ket{E_{i}}$, with $\ket{E_{i}}$ mutually orthogonal states of the
environment. In a subsequent interaction, in which the properties
$ \proj{\B_{j}}$ become definite, and in which the environment
`remembers' the presence of the $\ket{\A_{i}}$, the state is
transformed into $\sum_{i,j} c_{i} \inpr{\B_{j}}{\A_{i}}
\ket{\B_{j}} \ket{E_{i,j}}$, with mutually orthogonal environment
states $\ket{E_{i,j}}$. Continuation of this series of
interactions eventually leads to Eq.\ (\ref{eqdeco}), with in this
case $ \ket{\psi_{i,j,...,l}} = \ket{\psi_{l}}$.
If this picture of consecutive measurement-like interactions
applies, it follows that in the Heisenberg picture we have
$P_{l}(t_{n}) ...... P_{j}(t_{2}) . P_{i}(t_{1}) \ket{\Psi} =
c_{i,j,...,l} \ket{\psi_{i,j,...,l}} \ket{\Phi_{i,j,...,l}}$.
Substituting this in the expression at the left-hand side of Eq.\
(\ref{consist}), and making use of the orthogonality properties of
the states $\ket{\Phi_{i,j,...l}}$, we find immediately that the
consistent histories decoherence condition (\ref{consist}) is
satisfied. As a result, expression (\ref{sevtimeprob}) yields a
classical Kolmogorov probability distribution of the modal
properties at several times.
\sectiona{Joint probabilities of events}\label{joint}
In order to complete the space-time picture that we discussed in
section \ref{arqft}, we should specify the joint probability of
events taking place in different space-time regions. It is natural
to consider, for this purpose, a generalization of expression
(\ref{sevtimeprob}). The first problem encountered in generalizing
this expression to the relativistic context is that we no longer
have absolute time available to order the sequence $P_{i}(t_{1})$,
$P_{j}(t_{2})$, ...., $P_{l}(t_{n})$. In Minkovski space-time we
only have the partial ordering $y < x$ (i.e., $y$ is in the causal
past of $x$) as an objective relation between space-time points.
However, we can still impose a linear ordering on the space-time
points in any region in space-time by considering equivalence
classes of points that all have space-like separation with respect
to each other. The standard simultaneity hyperplanes provide
examples of such classes. Of course, there are infinitely many
ways of subdividing the region into such space-like collections of
points. It will have to be shown that the joint probability
distribution that we are going to construct is independent of the
particular subdivision that is chosen.
Take one particular linear time ordering of the points in a closed
region of Minkowski space-time, for instance one generated by a
set of simultaneity hyperplanes (i.e.\ hyperplanes that are
Minkowski-orthogonal to a given time-like worldline). Let the time
parameter $t$ label thin slices of space-time (approximating
hyperplanes) in which small space-time regions---`points'---with
mutual space-like separation, are located. We can now write down a
joint probability distribution for the properties on the various
`hyperplanes', in exactly the same form as in Eq.\
(\ref{sevtimeprob}):
\begin{eqnarray}
\lefteqn{Prob(P^{\ast}_{i}(t_{1}), P^{\ast}_{j}(t_{2}),....
P^{\ast}_{l}(t_{n}))=}
\nonumber \\ & & \langle \Psi | P^{\ast}_{i}(t_{1}).
P^{\ast}_{j}(t_{2}).....P^{\ast}_{l}(t_{n}) . P^{\ast}_{l}(t_{n})
. .... P^{\ast}_{j}(t_{2}) . P^{\ast}_{i}(t_{1}) | \Psi \rangle.
\label{sevtimeprob1}
\end{eqnarray}
In this formula the projector $P^{\ast}_{m}(t_{l})$ represents the
properties of the space-time `points' on the `hyperplane' labelled
by $t_{l}$. That is:
\begin{equation}
P^{\ast}_{m}(t_{l}) = \Pi_{i} P_{m_{i}}(x_{i}, t_{l}),
\label{product}
\end{equation}
with $\{ x_{i} \}$ the central positions of the point-like regions
considered on the hyperplane. The index $m$ is symbolic for the
set of indices $\{m_{i}\}$. Because all the considered point-like
regions on the hyperplane $t_{l}$ are space-like separated from
each other, the associated projectors commute (the principle of
micro-causality). This important feature of local quantum physics
guarantees that the product operator of Eq.\ (\ref{product}) is
again a projection operator, so that expression
(\ref{sevtimeprob1}) can be treated in the same way as Eq.\
(\ref{sevtimeprob}). In particular, we will need an additional
condition to ensure that (\ref{sevtimeprob1}) will yield a
Kolmogorovian probability.
The decoherence condition that we propose to use is the same as
the one discussed in sections \ref{arqft} and \ref{history}.
Suppose that at space-time point $(x,t)$ the magnitude represented
by the set of projector operators $\{P_{k}\}$ is definite-valued;
$P_{l}$ has value $1$, say. In physical terms the notion of
decoherence that we invoke is that in the course of the further
evolution there subsists a trace of this property in the future
lightcone of $(x,t)$. That is, decoherence implies that on each
space-like hyperplane intersecting the future lightcone of $(x,t)$
there are (perhaps very many) local projectors that are strictly
correlated to the earlier property $P_{l}$.
If this decoherence condition is fulfilled, we have because of the
assumed permanence of the traces just as in section \ref{history}:
\begin{eqnarray}
\lefteqn{\langle \Psi | P^{\ast}_{i}(t_{1}).
P^{\ast}_{j}(t_{2}).....P^{\ast}_{l}(t_{n}) .
P^{\ast}_{l^{\prime}}(t_{n}) ......
P^{\ast}_{j^{\prime}}(t_{2}) . P^{\ast}_{i^{\prime}}(t_{1}) | \Psi \rangle
=0}
\nonumber \\ & & i \neq
i^{\prime} \vee j \neq j^{\prime} \vee
....\vee l \neq l^{\prime}.\label{consist1}
\end{eqnarray}
This makes (\ref{sevtimeprob1}) a consistent Kolmogorovian
joint probability for the joint occurrence of the events represented by
$P^{\ast}_{i}(t_{1})$, $P^{\ast}_{j}(t_{2})$, ....$P^{\ast}_{l}(t_{n})$.
The projectors $P^{\ast}_{i}(t_{1})$, $P^{\ast}_{j}(t_{2})$,
....$P^{\ast}_{l}(t_{n})$ depend for their definition on the
chosen set of hyperplanes, labelled by $t$. Therefore
(\ref{sevtimeprob1}) is not manifestly Lorentz invariant. However,
the projectors $P^{\ast}(t)$ are products of projectors pertaining
to the individual space-time points lying on the $t$-hyperplanes,
so (\ref{sevtimeprob1}) can alternatively be written in terms of
these latter projectors. The specification of the joint
probability of the values of a field at all considered `points' in
a given space-time region requires (\ref{sevtimeprob1}) with
projectors for all those points appearing in it. Depending on the
way in which the space-time region has been subdivided in
space-like hyperplanes in the definition of $P^{\ast}_{i}(t_{1})$,
$P^{\ast}_{j}(t_{2})$, ....$P^{\ast}_{l}(t_{n})$, the projectors
occur in different orders in this complete probability
specification. However, there is a lot of conventionality in this
ordering. All operators attached to point-like regions with
space-like separation commute, so that their ordering can be
arbitrarily changed. The only characteristic of the ordering that
is invariant under all these allowed permutations is that if $y <
x$ (i.e.\ $y$ is in the causal past of $x$), $P(y)$ should appear
before $P(x)$ in the expression for the joint probability. But
this is exactly the characteristic that is common to all
expressions that follow from writing out (\ref{sevtimeprob1}),
starting from all different ways of ordering events with a time
parameter $t$. All these expressions can therefore be transformed
into each other by permutations of projectors belonging to
space-time points with space-like separation. The joint
probability thus depends only on how the events in the space-time
region are ordered with respect to the Lorentz-invariant relation
$<$; it is therefore Lorentz-invariant itself.
Within the just-discussed interpretational scheme, it is possible
to speak of values of physical magnitudes attached to small
space-time regions. The notion of an event can therefore now be
accommodated. This also gives us the conceptual tools needed to
work with the notion of an object. An object can be treated as a
particular distribution of field values. In particular, an
(approximately) localized object can be regarded as a distribution
of field values that vary continuously and fill a narrow
world-tube in Minkowski space-time. The notion of state
localization to be used here implies that all observables have
their vacuum expectation values in regions within the causal
complement of the world-tube (\cite{haag}, sect.V.5.3). Note that
this does not mean that the probabilities of local observables
are zero: the probabilities are the same as in the vacuum state.
As we have noted in sect.\ref{sectproblem}, such local vacuum
probabilities do not vanish.
\sectiona{A possible alternative: perspectivalism}
The approach sketched in the previous sections relied on the
presence of decoherence mechanisms; only because we assumed that
decoherence conditions were fulfilled could we obtain joint
probabilities of events and Lorentz-invariance. Although this is
perhaps enough from a pragmatic point of view, fundamentally
speaking it seems unlikely that basic points like the existence
of a joint probability distribution and Lorentz-invariance could
depend on the satisfaction of contingent, fact-like conditions.
It seems therefore worth-while to consider the question of
whether a more fundamental approach is possible that offers
prospects of interpreting quantum field theory in a consistent
realist way that is Lorentz-invariant. After all, that problems
with joint probabilities and Lorentz-invariance can be made to
disappear when decoherence mechanisms enter the stage is not
really surprising. Decoherence can be regarded as a measurement
performed by the environment, and it has often been observed in
the literature that problems with joint probabilities and
fundamental Lorentz-invariance do not show up in measurement
results. By contrast, these topics cause considerable difficulty
in the case of systems on which no measurements are made---see
\cite{dickson} for an explanation of the problems in the context
of modal interpretations.
The core of the problems identified in \cite{dickson} is that
different Lorentz observers cannot---on the pain of
inconsistency---use the same joint probability expression
(\ref{prob}) for the simultaneous properties of two (more or less
localized) systems in an EPR-type situation (see also
\cite{dieks4}). By exploiting the fact that in some Lorentz
frames a measurement is made on system 1 before a measurement on
system 2 takes place, whereas this order is reversed in other
frames, Dickson and Clifton in essence show that the transitions
undergone by the two systems during the measurements are locally
determined. Indeed, in the frames in which the measurements are
not simultaneous there is no other measurement to take into
account. But this result conflicts with the treatment given in a
Lorentz frame in which the two measurements take place
simultaneously, and in which such a local account is notoriously
impossible.
One possible way out (see also \cite{dieks4}) is to think of the
properties assigned to the systems not as monadic predicates but
as relations, in the manner explained in section
\ref{perspective}. If the properties of system 1, say, thus need
the specification of a perspective, it becomes natural to do
without simultaneity as an important ingredient in the expression
for the joint probabilities. Instead, it becomes essential to
specify from which point of view the properties of system 1 are
defined: from the point of view of system 2 before it has
undergone a measurement, from the point of view of system 2
during the measurement, or from a still later perspective.
Applying the rules of section \ref{perspective}, we find that
different states are found for system 1, at one and the same
space-time point, from these different perspectives. If the
properties of system 1 are regarded as non-relational, a
contradiction results immediately because a system cannot
simultaneously possess different states (reflecting its physical
properties) at one space-time point. Essentially, this is the
background of the contradiction derived by Dickson and Clifton.
The contradiction disappears, however, if properties are
relational. There is no logical difficulty involved in assuming
that one and the same system, at one particular stage in its
evolution, has different properties in relation to different
reference systems.
If the systems have definite spatio-temporal positions, as in the
above example, the object-perspective relation is associated with
a relation between two or more space-time regions, which itself
is Lorentz-invariant. If, moreover, all probability
considerations are also defined from a perspective, no problems
with Lorentz-invariance can arise. It remains to be seen,
however, whether this idea works also in more general situations.
\sectiona{Conclusion}
We have discussed the possibility of interpreting quantum field
theory in terms of a space-time picture involving localized
events, by means of the application of ideas from the modal
interpretation of quantum mechanics. We have argued that the usual
modal ideas, together with the fulfillment of a decoherence
condition, ensure the existence of a simple, natural and
Lorentz-invariant joint probability expression for the values of
definite-valued observables at several space-time locations.
Because of the uncountable infinity of degrees of freedom in the
quantum field, the occurrence of decoherence, involving
irreversibility, is something very natural to assume
\cite{schroer} in the context of quantum field theory. Still, it
would probably be more satisfactory to have a scheme in which the
existence of a consistent joint probability distribution and
Lorentz-invariance would not depend on such fact-like, contingent,
circumstances. We have therefore devoted a brief and tentative
discussion to the idea of applying a new form of the modal
interpretation, according to which properties have a relational
character. This perspectival modal interpretation may offer
prospects of obtaining a Lorentz-invariant picture even if no
decoherence mechanisms operate.
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\end{document}