\documentclass[onecolumn]{svjour2} % onecolumn
%\documentclass[twocolumn,fleqn]{svjour2} % twocolumn
%\documentclass[onecollarge]{svjour2} % onecolumn "king-size"
%
\smartqed % flush right qed marks, e.g. at end of proof
%
\usepackage{graphicx}% Include figure files
\usepackage{dcolumn}% Align table columns on decimal point
\usepackage{bm}% bold math
\usepackage{amssymb}%
\usepackage{mathptmx} % use Times fonts if available on your TeX system
%
\journalname{Foundations of Physics}
%
\begin{document}
\title{Objectivity in perspective: relationism in the interpretation of quantum mechanics}
\titlerunning{Objectivity in perspective}
\author{Dennis Dieks}
\institute{D. Dieks \at
Institute for the History and
Foundations of Science, Utrecht University\\ P.O.Box 80.010, 3508
TA Utrecht, The Netherlands \\
\email{d.dieks@uu.nl} }
\date{Received: date / Accepted: date}
% The correct dates will be entered by the editor
\maketitle
\begin{abstract}
Pekka Lahti is a prominent exponent of the renaissance of
foundational studies in quantum mechanics that has taken place
during the last few decades. Among other things, he and coworkers
have drawn renewed attention to, and have analyzed with fresh
mathematical rigor, the threat of inconsistency at the basis of
quantum theory: ordinary measurement interactions, described
within the mathematical formalism by Schr\"{o}dinger-type
equations of motion, seem to be unable to lead to the occurrence
of definite measurement outcomes, whereas the same formalism is
interpreted in terms of probabilities of precisely such definite
outcomes. Of course, it is essential here to be explicit about how
definite measurement results (or definite properties in general)
should be represented in the formalism. To this end Lahti et al.\
have introduced their \emph{objectification requirement} that says
that a system can be taken to possess a definite property if it is
\emph{certain} (in the sense of probability $1$) that this
property will be found upon measurement. As they have gone on to
demonstrate, this requirement entails that in general definite
outcomes cannot arise in unitary measuring processes.
In this paper we investigate whether it is possible to escape from
this deadlock. As we shall argue, there is a way out in which the
objectification requirement is fully maintained. The key idea is
to adapt the notion of objectivity itself, by introducing
\emph{relational} or \emph{perspectival} properties. It seems that
such a ``relational perspective'' offers prospects of overcoming
some of the long-standing problems in the interpretation of
quantum mechanics. \keywords{objectification \and perspectivalism
\and relational quantum mechanics \and locality \and realism}
\PACS{03.65+b}
\end{abstract}
\section{Objectification}
The last couple of decades have seen a renewed interest in
foundational problems in quantum mechanics; indeed, important
parts of present-day cutting edge research in physics owe their
existence to this renaissance of foundational studies. The brunt
of the revival may be situated in the 1980s and was stimulated by
a number of important and timely conferences. Foremost among these
were the Joensuu meetings on the Foundations of Modern Physics
(\cite{Joensuu1,Joensuu2,Joensuu3}) organized by Pekka Lahti.
They, together with the impressive monograph \emph{The Quantum
Theory of Measurement} by Busch, Lahti and Mittelstaedt
(\cite{Lahti}) defined the field and its problems to a
considerable extent and set the agenda for further research.
In \emph{The Quantum Theory of Measurement} the authors review the
measurement problem, and the interpretation of quantum mechanics
in general, on a new level of logical and mathematical rigor. A
central theme of their book is the problem of ``objectification'':
how does quantum theory deal with definite, objective, physical
states of affairs in the face of the ubiquity, on the theoretical
level, of entangled states of the Schr\"{o}dinger cat type? On the
basis of experience we require that successful measurements lead
to definite results, so that after the measurement a ``pointer
observable'' belonging to the measuring device possesses one
unique and objective value. Is the occurrence of such definite
results compatible with the fact that the state of object plus
device after the measurement is generally entangled? In order to
answer this question we obviously need to know how definite
properties are to be represented in the formalism. Here Busch,
Lahti and Mittelstaedt propose their \emph{objectification
requirement}: a property of a system is definite and objective if
and only if the system's quantum state is a mixture, \emph{in the
ignorance sense}, of eigenstates of the observable corresponding
to the property in question (\cite[p.\ 21]{Lahti}). In such a
mixture, in which the presence of different pure states as
components reflects our lack of knowledge about which one of these
states actually obtains, there can evidently never be any effect
of interference between the components.
Now consider a measurement with an arbitrary initial object state.
As already noted, after such an interaction (described by unitary
evolution) the final state of the apparatus and the object will be
entangled, which expresses the correlation brought about between
object and apparatus. Now, as Busch, Lahti and Mittelstaedt argue,
if there exists a specific pointer observable of the apparatus
such that no interference whatsoever can ever be found between
different pointer eigenstates, this pointer observable must be
\emph{classical}, i.e.\ it must commute with all other apparatus
observables. But then an inconsistency arises (\cite[pp.\
85-86]{Lahti}): If the pointer observable is classical, it will
also commute with the Hamiltonian generating the interaction, so
that it cannot change during this interaction. In other words,
such a classical ``pointer observable'' will not be able to
correlate to any property of the object system; it cannot be a
pointer observable at all!
This no-go result reinforces the general result that mixed states
derived from entangled states by the technique of partial tracing
(so-called \emph{improper mixtures}) are different from ignorance
mixtures over pure states (\emph{proper mixtures}). One could
attempt to argue nevertheless for an observational equivalence of
entangled states and such proper mixtures. This strategy would
have to be based on the existence of superselection rules
expressing classicality, but then the just-mentioned no-go result
implies a conflict with the core idea of measurement, namely the
possibility of establishing correlations between a measuring
device and an object.
It therefore follows that unitary measurement dynamics cannot lead
to objective pointer properties if this means that the final state
must be a mixture in the sense of the objectification requirement
(or a state equivalent to such a mixture). As Busch, Lahti and
Mittelstaedt discuss (\cite[chapter IV]{Lahti}), one may respond
to this situation in a variety of ways. One may decide to opt for
a hidden variable model, so that definiteness of properties of
physical systems is built in from the very start; this of course
means renouncing the idea that quantum mechanics may be a
fundamental and complete theory. Alternatively, one may assume
that the Schr\"{o}dinger dynamics should be modified, for example
by assuming the occurrence of collapses in addition to the unitary
evolution. Collapses destroy the coherence between the terms in a
superposition, so this option implies empirical differences, in
principle, with unitary quantum mechanics---a rather unpromising
prospect in view of the ever-increasing empirical evidence in
favor of the existence of even macroscopic entanglement (see,
e.g., \cite{cavalcanti} and the references contained therein to a
sample of recent experimental work).
A third way to go is to see whether it is possible to stay
completely within the formalism of unitary quantum mechanics and
nevertheless accommodate definite measurement results (and
definite properties in general). From the foregoing it is clear
that this strategy can only work if the objectification
requirement in the exact form mentioned above is dropped. It is
this response to the objectification problem that we shall now
explore in some more detail. As we shall argue, the
objectification requirement can consistently be maintained; but
the price to pay is that the definite properties satisfying this
requirement must be assumed to have a \emph{relational} character.
\section{A different type of objectivity}\label{gambling}
The core of the objectification requirement is the idea that a
system can be concluded to possesses a certain property if a
yes-no measurement of that property leads to a positive answer
with certainty, i.e.\ with probability $1$ (\cite[p.\ 21]{Lahti}).
Of course, we are speaking of ideal measurements here,
non-intrusive and not subject to inaccuracy. It is also assumed
that all possible properties correspond to quantum mechanical
observables. With these provisos the idea behind the
objectification requirement appears eminently reasonable. It might
perhaps be objected that there is an operationalist flavor to it;
but this would be off target, since there is no demand here that
the measurements in question are feasible in actual practice---the
requirement is about what the formalism predicts for the
expectation values of theoretically defined observables. Moreover,
the requirement is a \emph{criterion} for testing the presence of
objective properties rather than a verificationist
\emph{definition} of properties.
Another type of doubt might arise as follows. Consider the
following completely classical situation. I know that a die will
be cast in another room, and I also know that my friend, who is in
that room and who is a perfect observer, will be watching the
outcome closely. After some time I shall be sure that the
experiment is over and that my friend will have observed a
definite result. Still, I possess no certainty about the outcome:
if I know that the die is fair I can express this knowledge by
assigning a probability of $1/6$ to all possible situations
(consisting of a particular outcome plus my friend having noted
that outcome). So it seems that probability $1$ and objectivity of
possessed properties do not necessarily go hand in hand.
This argument does not yield an objection to the objectification
requirement, however---at least not in the context of classical
physics. It is precisely because of this type of situation that
the requirement was formulated in terms of \textit{ignorance
mixtures}. Indeed, because I stand outside the room in which the
experiment takes place, I happen to be blocked from direct access
to observables pertaining to what is going on inside, so that I
remain ignorant about the actual situation. My probability
assignments of $1/6$ represent this ignorance. But there exist
other observables than the ones I have direct access to that do
lead to a definite result with probability $1$, given the
condition of the die after the roll. These are precisely the
observables that my friend, inside the room, is measuring.
According to classical physics nothing will change in the state of
the room or the die when I make my way in cautiously enough and
take cognizance of the outcome---it is only my knowledge that then
changes. In other words, the objectification requirement does not
need to refer only to the information that is actually available
to the outside observer---one may also appeal to knowledge the
observer could obtain without disturbing the system.
Within the framework of classical physical ideas the conclusion
that both for me and my friend the same properties of the die are
definite is consequently perfectly reasonable. Still, this
conclusion depends on assumptions that cannot pretend to be valid
\emph{a priori}, but depend on the applicable physics. Even if it
is granted that if I enter the room and look at the die I shall
find a definite outcome, and that my friend will tell me that this
outcome was there all along, this does not compel me logically to
accept that this property is also objective for me now, standing
outside and not performing any measurement. At least by way of a
thought experiment I could contemplate the conceptual possibility
that \textit{for me} the situation in the room is really
undecided, objectively ``hovering between'' the different
outcomes; or as being in a state that should not be described in
terms of outcomes at all. Admittedly this suggestion seems rather
weird, in particular when we take into account the situation of my
friend who is inside the room. It should be acknowledged that my
friend at a certain point in time becomes aware of a definite
outcome, and is not undecidedly hovering between possibilities at
all. Is this not a direct refutation of the hypothesis according
to which things in the room are undecided?
No such refutation results if we complicate our thought experiment
further. Assume, at this point just for argument's sake, that we
are going to describe the situation in and outside the room not
with the usual physical properties that characterize objects as
they are in themselves, but rather in terms of \emph{relational}
properties. That is, instead of saying that the die has landed on
a particular face \emph{tout court}, we are thinking of the
possibility that this is the case \emph{with respect to} my friend
who has watched the roll of the die. And instead of simply saying
that the room and its contents do not possess a property
corresponding to a definite outcome, we are now going to say that
this is so with respect to me as outside observer who has not
interacted with what is inside. With this relativization manoeuver
logical contradiction can be avoided: ``there is a definite
outcome'' could be true for the inside observer, ``there is no
definite outcome'' could be true for the outside observer, whereas
a contradiction would require both the presence and absence of
definiteness for \emph{one and the same} observer. In this
scenario I as an outside observer could admit that \emph{for my
friend} the roll of the die has ended in some definite outcome,
while at the same time maintaining that \emph{for me} the state of
die plus friend is completely different, not describable as an
ignorance mixture of definite outcomes at all. The objectification
that has taken place for my friend is in this case not represented
by an ignorance mixture over states with respect to me, but by an
ignorance mixture of the \emph{states for my friend} that I
consider possible (of course, since I am shut off from the
experiment, I do not know what my friend has seen).
In the context of classical physics this proposal would boil down
to accepting the objectification requirement in an implausibly
strong form, namely that if I cannot \emph{actually} make a
prediction with probability $1$ about the presence of a property,
this property is not there. We have already pointed out that this
version of the requirement is unduly strict, and the formulation
of the requirement in \cite{Lahti} indeed allowed that such an
impossibility of actually making certain predictions could be due
to mere ignorance, i.e.\ an ontologically insignificant lack of
knowledge about the real situation.
Going this way in the case of our example would lead to a much
more complicated description than the usual one. Although it is
true that such a perspectival description could be consistently
maintained even within the context of classical physics, there is
no physical reason to actually do so---as already emphasized,
classical physical theory says that it makes no difference for the
room if I enter and make (perfectly gentle, ideal) observations.
The description available to my friend can therefore be considered
valid for me too. It would complicate matters enormously if all
physical laws in such cases had to be reformulated in terms of
relational properties, without there being any compensation in the
way of better or more predictions. The only motivation to engage
in such a move would be philosophical, for example a desire to
indulge in skeptical or verificationist predilections.
This is not to say that relational properties have no place at all
in classical physics. The obvious counterexample are the length
and time determinations of special relativity, which vary with the
inertial frame of reference from which they are made. In fact,
awareness that these and similar instances of relational
properties already occur in classical physics may help to make our
later introduction of perspectivalism in quantum theory more
palatable. Still, there is a difference between these well-known
cases of perspectivalism and the one considered in our discussion
of the gambling experiment. In the latter case the perspective
decided whether or not the die had landed on a definite face
\emph{at all}; whereas the different descriptions given from
different frames in special relativity pertain to different
numerical values characterizing a situation that is definite as
judged from all frames. In the case of the Lorentz contraction,
for example, all observers agree that a moving rod possesses a
definite length; but they differ on the \emph{value} to be
ascribed to this length. Likewise, in relativity observers may
disagree about the sizes of the faces of our die and its distance
to other objects, but not about whether the faces have sizes at
all or about whether the die has landed---or will land---on a
definite face at all. The perspectivalism considered above, in the
discussion of what is going on inside the room, is therefore
really thoroughgoing, of a more dramatic kind than classical
perspectivalism. No wonder that the philosophical and
epistemological motivations that we mentioned do not suffice to
justify the introduction of this kind of perspectivalism in
classical physical theory.
But in quantum mechanics the situation becomes different. Think
back of the von Neumann measurement scheme, applied to a situation
like the above die experiment. The quantum equations of motion
tell us that after the conclusion of the measurement interaction
the state of my friend plus die will be given by a linear
superposition of terms each of which will be a product of a die
state with a definite face-up and a friend state representing
awareness of that same definite face. Also experience seems to
support the ascription of this superposed state: as we have
pointed out before, experiments on so-called Schr\"{o}dinger cat
states definitely indicate that superpositions are needed to do
justice to the experimental facts (\cite{cavalcanti}).
Accordingly, if I, standing outside the room, wish to make
predictions about the results of measurements on the room I had
better use the full superposition. In particular, if I am going to
measure the projection operator $|\Psi\rangle \langle \Psi |$
(where $|\Psi\rangle $ stands for the superposed state of the room
and its contents), the formalism tells me that I shall find the
result ``$1$'' with certainty. The objectification requirement
thus leads me to consider the corresponding physical property as
objectively belonging to the room. Now, experience also tells me
that during the experiment my friend becomes aware of exactly one
result; for him the process certainly ends with a definite die
property. Furthermore, my friend can predict with certainty, after
having noted the face the die has landed on, what subsequent looks
at the die will show. Therefore, application of the
objectification requirement in his case will lead to the
attribution of a definite-face state to the die. I, on the other
hand, can only derive a mixed state for the die from my
superposition for the total room, and well-known arguments forbid
me to think that this mixture is an ontologically insignificant
ignorance mixture (indeed, if the die actually were in one of the
component states of the mixture, the total system would have to be
a mixture as well, which conflicts with the assumption that the
total state is a superposition).
In other words, quantum mechanics makes it physically plausible to
ascribe more than one state to the same physical system. The
contradiction that looms can be avoided if we drop the implicit
assumption that only \emph{one} state can belong to any object; in
other words, if we decide to assign relational or perspectival
states, i.e.\ states of a physical system $A$ \emph{from the
perspective} of a physical system $B$. This manoeuvre creates room
for the possibility that the state and physical properties of a
system $A$ are different from different perspectives.
Consequently, a reconciliation between the unitary evolution that
takes place during a quantum measurement and the occurrence of
definite outcomes is perhaps no longer out of the question: the
properties associated with the superposition and the definite
outcomes, respectively, could relate to two different
perspectives.
The significant difference between the classical and the quantum
cases is that in the latter there are \textit{physical},
\textit{empirical} reasons for investigating the viability of
working with relational properties of this drastic kind. By
contrast, in the classical case physics favors the simpler picture
of monadic (i.e.\ non-relational) states and properties.
\section{A quantum scheme for attributing perspectival
states}\label{schmidt}
The foregoing sections offered a motivation for introducing
perspectival states, but this should evidently be supplemented by
a more precise account that makes it clear what these new states
are and in what way they relate to the standard formalism of
quantum mechanics. As it turns out, proposals are already
available in the literature; in particular, a recent relational
version of the modal interpretation (\cite{benedieks}) captures
the above intuitions very well. There are also other---though
similar---proposals in the literature
(\cite{Rovelli,Rovelli2,kochen}, see also
\cite{Bitbol,VanFraassen}), with which we should compare. But let
us first reformulate the ideas from \cite{benedieks} for our
present purposes.
Our point of departure is the standard Hilbert space formalism of
quantum mechanics with only unitary time evolution (as governed by
equations of motion of the Schr\"{o}dinger-type). We assume the
universal validity of quantum mechanics, so there is no division
of the world in a classical and a quantum part. Within this
framework it is not problematic, in principle, to speak about the
quantum state of the whole universe; we assume this state to be a
pure vector state $|\Psi\rangle$.
Our task is now to assign states to smaller systems $S$,
components of the whole universe $U$. These states will in general
be density operators, i.e., positive semi-definite and unit-trace
Hermitian operators acting on the Hilbert space of $S$:
$\rho^S_R$. The upper and lower indices attached to $\rho$
anticipate that in our approach the state of a physical system $S$
needs the specification of another system, the ``reference
system'' $R$, with respect to which the state is defined:
$\rho^S_R$ is the state of $S$ with respect to $R$. As explained
in the previous section, we wish to create room for the
possibility that one and the same system, at one and the same
instant of time, has different states with respect to different
reference systems.
An important special case is the one in which $R$ coincides with
$S$: the state of $S$ with respect to itself. This state we define
as one of the projectors occurring in the spectral decomposition
of the reduced density operator of $S$ in the standard formalism,
i.e.\ the density operator that is obtained for $S$ by partial
tracing from $|\Psi\rangle\langle\Psi|$. In other words,
$\rho^S_S$ follows from $|\Psi\rangle\langle\Psi|$ by ``tracing
out'' over all degrees of freedom not pertaining to $S$. If there
is no degeneracy, this state will be a one-dimensional projector
\begin{eqnarray}
\rho^S_S=|\psi_S\rangle\langle\psi_S|,\label{g3}
\end{eqnarray}
or equivalently a vector state $|\psi_S\rangle$. In accordance
with the ideas of modal interpretations
(\cite{dieksx,dieks&vermaas,vermaas&dieks,bub,dieksy}), we posit
that \emph{which} projector from the spectral decomposition of
$S$'s reduced density operator is $\rho^S_S$ is not fixed by
quantum mechanics; the theory only specifies probabilities
(namely, the usual Born probabilities) for the various
possibilities. Also in accordance with modal ideas is that we
assume this state $\rho^S_S$, the ``state of $S$ with respect to
itself'', to codify the physical properties $S$ actually has
(i.e.\ the quantities that possess definite values): all operators
of which $|\psi_S\rangle$ is an eigenvector have the corresponding
definite value or, put differently, the observable $|\psi_S\rangle
\langle \psi_S | = \rho^S_S$ possesses the definite value $1$.
These properties, since they are derived from the state of $S$
\emph{with respect to itself}, are interpreted as properties
possessed by $S$ ``on its own'', without reference to anything
external.
So far, there is nothing explicitly relational or perspectival
going on; $|\psi_S\rangle$ is just the ``physical state'' assigned
to $S$ in earlier versions of the modal interpretation of quantum
mechanics (\cite{dieks&vermaas}). The relational aspect enters
when we consider states $\rho^S_R$ for arbitrary $S$ and $R$.
We are interested in situations in which there is a reference
system, $R$, outside $S$, such that $A \equiv {U\setminus R}$
contains $S$. By virtue of the Schmidt decomposition of
$|\Psi\rangle\langle\Psi|$ (the state of the universe), there is a
unique state $\rho^A_A$ that is coupled to $\rho^R_R$ (in the
sense of being perfectly correlated to it via $|\Psi\rangle$'s
Schmidt decomposition). Again in accordance with earlier modal
ideas, we posit that this correlated state $\rho^A_A$ is $A$'s
state with respect to itself \emph{given} $R$'s state $\rho^R_R$.
Now, since system $S$ is contained in $A$ (remember: $A =
{U\setminus R}$), the state $\rho^S_R$ can be defined as the
density operator that follows from this $\rho^A_A$ by taking the
partial trace over the degrees of freedom in $A$ that do not
pertain to $S$:
\begin{eqnarray}
\rho^S_R={\rm Tr}_{A\setminus S}\; \rho^A_A. \label{g4}
\end{eqnarray}
Any relational state of a system with respect to another system,
outside of it, can be determined by means of Eq.\ (\ref{g4}).
So for an arbitrary system $S$ contained in the universe $U$,
$\rho^S_S$ is one of the projectors occurring in the spectral
resolution of ${\rm Tr}_{U\setminus S}\;
|\Psi\rangle\langle\Psi|$. If there is no degeneracy among the
eigenvalues of this density operator these projectors are
one-dimensional and the state can be represented by a vector
$|\psi_S\rangle$, see Eq.\ (\ref{g3}). In the case of degeneracy
this generalizes: now, the state of the system with respect to
itself becomes a multi-dimensional projector (\cite{dieksy}). For
simplicity we shall in the following focus on the non-degenerate
case. One-to-one coupled with $\rho^S_S$, via the correlation
represented in the Schmidt decomposition, is a state $\rho^R_R$ of
the rest of the universe $U$. The state attribution rule of the
foregoing paragraph says that the relational state of a component
$C$ of $S$, with respect to $R$, $\rho^C_R$, is found from
$\rho^S_S$ by tracing out the degrees of freedom not belonging to
$C$.
The state $|\Psi\rangle$ evolves unitarily in time. Because there
is no collapse of the wave function in our approach, this unitary
evolution of $|\Psi\rangle$ is the main dynamical principle of the
theory. Furthermore, we assume that the state assigned to a
\emph{closed} system $S$ undergoes a unitary time evolution
\begin{eqnarray}
i\hbar\frac{\partial}{\partial t}\rho^S_S=\left[H_S,\;
\rho^S_S\right].\label{g5}
\end{eqnarray}
For more about the dynamics, (joint) probabilities and other
details, see \cite{benedieks,dieksy}. Here we intend to focus on
the relational aspects of the just-introduced scheme of state
attribution. These can be illustrated by looking at how this
scheme works for the case of the gambling experiment from the
previous section.
\section{The quantum world of perspectives}
The essential points are not affected if we simplify by taking the
universe $U$ to consist of only three systems: I (or a measuring
device, initially not partaking in the interaction), the die, and
``my friend'' (a measuring device recording the outcome of the die
roll), denoted by $I$, $D$ and $F$, respectively. Let us take the
initial state $|\Psi\rangle$ as a product state: $|\Psi\rangle =
|I_0\rangle \otimes |D_0\rangle \otimes |F_0\rangle$, with
$|D_0\rangle = \sum c_i |D_i\rangle$, where the states
$|D_i\rangle$ are eigenstates of the observable corresponding to
``face $i$ coming up''. Using the familiar von Neumann measurement
scheme we can represent the first stage of the experiment (in
which the die is rolled and my friend looks at the outcome) as
follows:
\begin{eqnarray}
|I_0\rangle \otimes |D_0\rangle \otimes |F_0\rangle
\longrightarrow |I_0\rangle \otimes \sum c_i |D_i\rangle \otimes
|F_i\rangle .\label{g6}
\end{eqnarray}
This may be followed by a second stage, in which I receive
information about the outcome. In the same von Neumann style
(which is obviously highly idealized and simplistic; but we are
here interested in general characteristics of the situation) this
is represented as:
\begin{eqnarray}
|I_0\rangle \otimes \sum c_i |D_i\rangle \otimes |F_i\rangle
\longrightarrow \sum c_i |I_i\rangle \otimes |D_i\rangle \otimes
|F_i\rangle;\label{g7}
\end{eqnarray}
with obvious notation both in Eq.\ (\ref{g6}) and Eq.\
(\ref{g7})---the states with different values of $i$ are mutually
orthogonal.
We can now apply the rules of the previous section to the states
occurring in Eq.\ (\ref{g6}) and Eq.\ (\ref{g7}) in order to
determine what the properties of the various component systems
are. Looking at the final state in (\ref{g6}), we see that $F$ by
itself has registered a definite result $i$ and also that $D$ by
itself has landed with a definite face up (possibility $i=k$ being
realized with probability $|c_k|^2$). The state of $I$ is, as
expected, as it was before the experiment started, since $I$ has
not been involved in the process. The state of $D$ plus $F$
\emph{for $I$} is given by $\sum c_i |D_i\rangle \otimes
|F_i\rangle$, from which it follows that the state of $F$
\emph{for $I$} is the improper mixture $\sum |c_i|^2 |F_i\rangle
\langle F_i |$; and analogously for the state of $D$ with respect
to $I$. So clearly, the states of $D$ and $F$ vary depending on
the reference system with respect to which they are defined: they
are different when taken with respect to the systems themselves
from when taken with respect to $I$. The result reproduces what
was discussed in the previous section: after the die has been cast
there is a clear-cut outcome of the experiment \emph{for my
friend}, whereas \emph{for me} the situation is still objectively
undecided. The latter statement should not be interpreted in the
sense that I do not know what the outcome has been, but rather as
expressing that with respect to me the die \emph{does not possess}
a well-defined position showing a definite face. Indeed, with
respect to me $\sum |c_i|^2 |D_i \rangle \langle D_i |$ is the
state of the die. Taking our clue from the ``objectification
requirement'' in answering the question to which definite physical
property this state corresponds (in other words, looking for a
projector $P$ such that ${\rm Tr}\rho P = \sum |c_i|^2 \langle D_i
| P | D_i \rangle = 1)$, we readily find that the projector $\sum
|D_i \rangle \langle D_i |$ is definite but that the individual
projectors $|D_i \rangle$ are not.
But this becomes different when I enter into interaction with
either the die, my friend, or both. During this new interaction
the total state changes, with the right-hand side of (\ref{g7}) as
the final result. At the end of this process I will have recorded
a definite result: my state with respect to myself has become one
of the states $|I_i\rangle$. With respect to me, the state of my
friend plus the die will now be given by the corresponding
$|D_i\rangle \otimes |F_i\rangle$, from which it follows that with
respect to me the die has acquired a definite face-up position.
The state attribution scheme under discussion thus yields the type
of relational states that a philosophical skeptic or
verificationist would perhaps already be inclined to assign even
in the context of classical physics. But again, the salient point
is that in the context of quantum theory this way of attributing
states has empirical consequences and corresponding empirical
support: we may take it as supported by experiment that
measurements performed by $I$ on the composite $D$ plus $F$
system, after the die has been cast but before any further
interaction with $I$ has taken place (so that the state is given
by (\ref{g6})), can generally be successfully described only if
the entangled state $\sum c_i |D_i\rangle \otimes |F_i\rangle$ is
used instead of one of the ``definite result states'' $|D_i\rangle
\otimes |F_i\rangle$.
We thus find that a relational description of the world, in the
sense that the state and physical properties of any physical
system depend on a reference system with respect to which they are
defined, is implicitly present in the formalism of quantum
mechanics. There is certainly some affinity here with the
many-worlds or relative-state interpretation of quantum mechanics.
One important point of difference is that we have not been
assuming that \emph{all} possible outcomes of experiments are
equally realized; we have taken the modal point of view that only
one result becomes actual and that quantum theory only contains
probabilistic information about which result that will be. In this
and other respects our proposals are akin to those made by Rovelli
(\cite{Rovelli,Rovelli2}). However, before going into a comparison
with his ``relational quantum mechanics'', let us look at some
more characteristics of the relational perspective developed
above.
\section{Holism, locality and EPR}\label{epr}
One of central themes in discussions about the interpretation of
quantum mechanics is that of holism: it seems generally accepted
today that quantum mechanics contains holistic features. Various
types of holism have been distinguished in the literature, but the
most important one relates directly to the existence of entangled
states. Indeed, in a state like $\sum c_i |D_i\rangle \otimes
|F_i\rangle$ the whole system, comprising $D$ and $F$, physical
properties appear to be instantiated that cannot be understood as
being ``built up'' from properties of the individual component
systems. A paradigmatic example is that of definite and fixed
correlations between component systems, represented by
well-defined fixed values of global quantities, in spite of the
fact that the component systems do not possess fixed values for
the correlated quantities. In the notorious case of the singlet
spin state there is a perfect anti-correlation between the spins,
following from the fact that the total spin has the definite value
$0$, whereas there are no definite individual spin values. In
situations in which the component systems can be thought of as
being at a large spatial distance from each other, this holistic
feature manifests itself in the guise of non-locality.
We can use the results of the previous section to discuss this
issue for the case of our gambling experiment. After the initial
stage of the experiment the combined system of $D$ and $F$ is for
$I$ in an entangled pure state, whereas the states, again for $I$,
of the component systems are improper mixtures. The associated
physical properties are represented by the projector on the
entangled state in the case of the total system, and by the rather
uninformative projectors of the form $\sum |D_i \rangle \langle
D_i |$ in the case of the component states. So from the
perspective of $I$ the total system indeed has properties that
cannot be understood as constructed from the properties of the
components: a manifestation of holism. From the point of view of
$F$, however, there is nothing holistic: the die has landed on a
definite side, this has been recorded by $F$, and the total system
is characterized by the conjunction of these properties. So
whether or not a system displays holistic features becomes itself
a perspectival matter in this approach.
An important question is whether the holism that is present gives
rise to non-locality in cases in which the component systems are
spatially separated. In one sense of the question the answer is
evident: if the properties of a spatially extended system do not
supervene on the properties of the individual localized parts,
there is non-locality in exactly this meaning of the term. But we
are more interested in another question, namely whether in
EPR-like situations it can be said that there are
instantaneous---or faster-than-light---influences going from one
part of the total system to another. It is this meaning of
non-locality that is usually discussed in the context of debates
concerning the implications of violations of Bell inequalities.
The essential features of the EPR case are already present in our
gambling thought experiment. After the rolling of the die the
state of $D$ plus $F$ is entangled, with the consequence that for
$I$ there is no definite face that has come up. However, if $I$
looks at the die (and thus enters in interaction with it) the die
will acquire such a definite face-up state for $I$, as is clear
from Eq.\ (\ref{g7}). This offers no problem for locality, of
course: the interaction between $I$ and $D$ can be assumed to be
purely local. However, as a result of this same interaction the
state of $F$ for $I$ also changes---namely into a state telling us
that $F$ had already observed the same outcome as $I$ is observing
now. This change of $F$'s state with respect to $I$ takes place
without any causal contact between $I$ and $F$. Appearances
therefore are that there is some strange influence on $F$ arising
from the interaction between $I$ and $D$; it is exactly this
effect that translates into a non-local influence in the case of
spatially separated systems.
But a more careful consideration of the relational character of
the states takes away the impression of mystery here. It is true
that there is no physical interaction between $I$ and $F$, and
that in this sense $I$'s observation of $D$ cannot have an effect
on $F$'s state. However, $I$'s \emph{own} state is modified during
the (purely local) interaction with $D$, and since we are
considering the state of $F$ \emph{with respect to $I$}, it need
not cause bewilderment that this relational state also becomes
different from what it was. As a result of $I$'s looking at the
die, $I$'s state by itself becomes aligned with $D$'s state by
itself (cf. (\ref{g7}); it is therefore no wonder that $F$'s state
with respect to $I$ becomes the same as it already was with
respect to $D$. We do not need to invoke any non-local influences
here: the changes in the various states can be understood on the
basis of the combination of local physical interactions and the
relational character of the states.
This line of reasoning carries over directly to the case of the
EPR experiment---see \cite{benedieks}. Einstein, Podolsky and
Rosen famously proposed that {\em if, without in any way
disturbing a system, we can predict with certainty (i.e., with
probability equal to unity) the value of a physical quantity, then
there exists an element of physical reality corresponding to this
physical quantity} \cite{EPR}. We can accept this criterion within
our relational framework---in its essence, it is the
objectification requirement discussed before---but here its
application does not lead to the conclusion that \emph{for $I$}
the properties of $F$ were already before the experiment what they
are after $I$'s interaction with $D$. The crux of the matter is
that in spite of the absence of physical disturbances going from
$I$ to $F$, there \emph{does} exist an influence of the
observation of $D$. This influence comes about via the relational,
perspectival, character of the state we are considering: it is a
state of $F$ \emph{with respect to} $I$, which makes the local
change in $I$ relevant. The premiss that this relational state of
$F$ is not ``in any way disturbed'' during the experiment is thus
not fulfilled.
More in general, the perspectival character of the states is
responsible for the failure of counterfactual reasoning in the way
we are used to it: from the fact that no physical disturbance has
affected an object, it cannot always be concluded that the object
state is the same as it would have been if no interactions at all
had been present. One should also look at the reference system,
with respect to which the state is defined, and see whether
anything has changed \emph{there} that is relevant.
These considerations show the extent to which the very concept of
reality is modified in this perspectival approach to the
interpretation of quantum mechanics. However, to a point the
quantum formalism itself hides these relational features from
view. Indeed, when different observers compare their findings,
they will agree on what they have seen, as is illustrated by the
agreement between $I$ and $F$ about the outcome of rolling the
die. This is a general feature of the formalism \cite{benedieks}.
Furthermore, there are the omnipresent effects of decoherence that
will blur the observability of entanglement. Yet, in principle the
existence of superpositions (as in the case of the state of $D$
and $F$ before $I$ has taken a look at the $D$) can be found out
experimentally, even in the macroscopic domain. In fact, the
violation of Bell inequalities can be interpreted as empirical
support for the thesis that the traditional notion of reality
(monadic properties combined with locality) is inappropriate.
Let us for clarity's sake look explicitly at the EPR case of
distant correlated particles. We find that the state of the second
particle that becomes known after a measurement on the first
particle is the state of this second particle from the perspective
of the measuring device that has interacted with particle $1$.
However, it cannot be concluded that this state was already
present before the measurement, because the state of the measuring
device with respect to itself changes during the measurement. If
one writes down the states explicitly, applying the given rules to
the situations before and after the measurement, one easily
establishes that the relational state of particle $2$ with respect
to the measuring device at the position of particle $1$ indeed
changes as a result of the measurement, in spite of the fact that
there was no mechanical disturbance propagating between $1$ and
$2$. It is important to note that, by contrast, the state of
particle $2$ \emph{with respect to itself} does not change---this
is a direct consequence of the no-signalling theorem.
The modification of the reality concept that is inherent in the
introduction of relational (perspectival) states thus appears to
make the introduction of `quantum nonlocality' unnecessary. The
change in the relational state of particle $2$ can be understood
as a consequence of the local change in the reference system,
brought about by the measurement interaction. One could express it
like this: the local measurement interaction is responsible for
the creation of a new perspective (namely the perspective
connected with the new state of the measuring device), and from
this new perspective even far-away systems look differently.
\section{A brief survey of relational ideas}
The idea that quantum theory points into the direction of
perspectivalism is certainly not new. Some of the basic thoughts
can already be recognized in Bohr's writings, in particular in his
complementarity doctrine. Bohr emphasized that quantum objects
cannot be characterized by a fixed set of physical magnitudes that
are definite-valued at all times; instead, which properties can be
attributed to quantum objects depends on the experimental set-up
of which the object is a part. According to Bohr this experimental
context should not be interpreted in terms of the presence of
conscious observers; it is rather the presence of a measuring
device measuring an observable $A$ that makes it possible to speak
about an object in terms of $A$-values. So it is the nature of the
physical interaction between the object and the system with which
it is in interaction and with which it establishes correlations
that determines the validity of property ascriptions---for
example, whether momentum or position values can be properly
ascribed\footnote{Bohr stressed that measuring devices are
macroscopic, and this has sometimes created the misunderstanding
that Bohr thought that these devices were themselves not subject
to quantum mechanics. Bohr has mentioned in several places,
however, that these macroscopic objects should be treated by
quantum mechanics if we want to make predictions about
measurements performed \emph{on them}---see, e.g., \cite{Bohr}.
The invocation of the macroscopicity of the devices serves the
purpose of making contact with experience and ordinary language,
by referring to situations in which the concepts of classical
physics, like position and momentum, are applicable.}. There is
clearly a similarity here to some of the features of the approach
explained in the foregoing sections here. However, one should be
very careful in attributing to Bohr (or any other author) ideas
that are not formulated explicitly by himself; and although it
seems plausible to interpret Bohr's statements in terms of
relational property ascriptions, he has never mentioned the
possibility of different property ascriptions to one system
depending on the perspective that is taken.
Kochen in \cite{kochen} has proposed an interpretation of quantum
mechanics that he dubs the ``witnessing'' interpretation, which
strongly suggests that it is about properties of systems as
``witnessed'' or viewed from other systems---this seems an example
of the explicit introduction of perspectival properties. However,
in Kochen's only published paper on the subject there is only a
discussion of a physical system consisting of two components, in
which case the Schmidt decomposition of the total state (cf.\
section \ref{schmidt} above) is used to assign properties to these
components. This makes the witnessing interpretation identical,
for this particular case, to a version of the modal interpretation
\cite{dieksx,dieks&vermaas,bub}. Since there is neither mention of
other systems in Kochen's proposal nor discussion of its
supposedly perspectival nature, it is difficult to be certain
about what the meaning and the relevance of the ``witnessing
relation'' is.
This is very different in the work of Rovelli and coworkers, who
propose an explicit \emph{Relational Quantum Mechanics} (RQM) and
emphasize the possibility of different descriptions of a physical
system depending on the perspective
\cite{Rovelli,Rovelli2,Laudisa}. The spirit of RQM, in particular
the central idea that quantum systems have states and properties
that are defined relatively to reference systems, is very similar
to what was discussed in the foregoing sections. It appears that
there are also a number of differences, although it remains to be
seen how important these are.
In Relational Quantum Mechanics the concept of measurement
interactions, and definite outcomes of measurements, is primary;
the state $|\psi\rangle$ is introduced as a derivative quantity, a
bookkeeping device that takes into account the information about
previous interactions with a system $S$ that has been stored in a
system $A$ and can be used for making predictions. As Smerlak and
Rovelli write \cite{Rovelli2}: ``The state $\psi$ that we
associate with a system $S$ is therefore, first of all, just a
coding of the outcome of these previous interactions with $S$.
Since these are actual only with respect to $A$, the state $\psi$
is only relative to $A$: $\psi$ is the coding of the information
that $A$ has about $S$. Because of this irreducible epistemic
character, $\psi$ is but a relative state, which cannot be taken
to be an objective property of the single system $S$, independent
from $A$. Every state of quantum theory is a relative state''.
Smerlak and Rovelli append a footnote to this, saying ``From this
perspective, probability needs clearly to be interpreted
subjectively''. The basic idea of perspectivalism in RQM seems
thus to be motivated by epistemological or operationalistic
considerations (there are indeed references to operational
definitions and operationally justified procedures elsewhere in
the same paper). But it would be a mistake to read too much into
this, since the papers \cite{Rovelli,Rovelli2} make it also clear
that ``measurements'' are here meant not in the sense of human
acts, but rather as physical interactions that give rise to
correlations between physical systems. Moreover, all physical
systems are treated as quantum systems: quantum theory is taken to
be universally applicable, both to microscopic and macroscopic
systems. In these latter points there is complete agreement with
our proposals from the foregoing sections. However, the first
point---the primary role played by ``measurement results'' and the
derivative role of $\psi$---gives rise to differences with modal
approaches.
Since the occurrence of definite events (as registered by some
physical system) is taken as primary in RQM, and since the
bookkeeping device $\psi$ has to be \emph{updated} every time such
an event occurs, $\psi$ changes discontinuously with every new
event. As the authors of \cite{Rovelli2} say, ``the state $\psi$
is a tool that can be used by $A$ to predict future outcomes of
interactions between $S$ and $A$. In general these predictions
depend on the time $t$ at which the interaction will take place.
In the Schr\"{o}dinger picture this time dependence is coded into
a time evolution of the state $\psi$ itself. In this picture,
there are therefore two distinct manners in which $\psi$ can
evolve: (i) in a discrete way, when $S$ and $A$ interact, in order
for the information to be adjusted, and (ii) in a continuous way,
to reflect the time dependence of the probabilistic relation
between past and future events''. By contrast, according to modal
interpretational ideas unitary evolution is the main dynamical
principle, also when systems interact. Whether or not definite
events occur, and what their characteristics are, is in the modal
interpretation (which was on the background of our perspectival
proposals in the foregoing) derived from the form of
$\Psi$---instead of the other way around as in RQM. It may be
true, however, that on the level of the physical properties that
become realized over time (which process is treated in general as
stochastic in the modal interpretation) continuity cannot always
be guaranteed even in the modal scheme. The extent and the
significance of the differences here therefore remain a question
for further investigation.
In RQM \emph{all} states are relative to some \emph{other} system,
there is no mention of states of systems with respect to
themselves. \emph{A fortiori}, RQM does not operate with the
notion of a state of the whole universe $U$. Nevertheless, it
seems that in cases like the gambling experiment from section
\ref{gambling}, RQM leads to the same relational state
attributions as derived in section \ref{schmidt}: the
``information'' mentioned in RQM is both in RQM and in the modal
scheme represented by the correlations between the systems, as
encoded in the entangled states. It is less clear whether the
state assignments agree in cases in which there is space-like
separation between systems $S$ and $A$, and in which no
information about $S$ has arrived at $A$ from the past. It seems
that in such cases RQM says that \emph{it makes no sense} to speak
of the state of $B$ with respect to $A$: $A$ cannot ``know''
anything about $B$ \cite{Rovelli2}. In our proposal, in which we
started from the assumption that there exists a well-defined state
of the total universe $\Psi$, such a state of $S$ with respect to
$A$ \emph{is} well-defined. More generally, it appears that RQM
defines less properties and states than are assigned in our
proposal: according to RQM the attribution of a state of $S$
relative to $A$ becomes \emph{meaningless} as soon as no actual
information transfer has taken place in any way between $S$ and
$A$.
Obviously, there is a lot more to say about this and other topics
(like the way EPR is dealt with in the different approaches). But
for the purpose of this paper it is enough to conclude that the
idea of perspectival properties makes it possible to give a new
twist to many an old debate in the foundations and interpretation
of quantum mechanics, and that a new field of detailed questions
is opened up by this development.
\section{Conclusion}
Interpreting quantum mechanics via perspectival (relational)
properties seems a promising way to go. One important result is
that the relational perspective sheds new light on the
long-standing locality debate: within a relational framework
locality in quantum mechanics does not need to be jettisoned. The
price to pay is that the nature of reality becomes different from
what we were used to: all properties need a perspective for their
definition. This is admittedly strange and far-fetched if judged
from a classical point of view. But it is not a gratuitous
philosophical suggestion: the very structure of the quantum
formalism points in this direction. If reality is conceived of in
this perspectival way, local realism seems to become a live option
again.
\begin{thebibliography}{99}
\bibitem{Joensuu1}
Lahti, P., Mittelstaedt, P. (eds.): Symposium on the Foundations
of Modern Physics 1985. World Scientific, Singapore (1985)
\bibitem{Joensuu2} Lahti, P., Mittelstaedt, P. (eds.): Symposium on the Foundations of Modern Physics 1987. World
Scientific, Singapore (1987)
\bibitem{Joensuu3} Lahti, P., Mittelstaedt, P. (eds.): Symposium on the Foundations of Modern
Physics 1990. World Scientific, Singapore (1990)
\bibitem{Lahti} Busch, P., Lahti, P., Mittelstaedt, P.: The Quantum Theory of
Measurement. Springer Verlag, Heidelberg (1991, 1996)
\bibitem{cavalcanti} Cavalcanti, E.G., Reid, M.D.: Criteria for Generalized Macroscopic and Mesoscopic Quantum
Coherence. Physical Review A 77, 062108 (2008)
\bibitem{benedieks} Bene, G., Dieks, D.: A Perspectival Version of the Modal Interpretation of Quantum Mechanics and the Origin of Macroscopic
Behavior. Foundations of Physics 32, 645-671 (2002)
\bibitem{Rovelli} Rovelli, C.: Relational Quantum Mechanics. International Journal of Theoretical Physics
35, 1637-1678 (1996); Rovelli, C.: Quantum Gravity, sec.\ 5.6.
Cambridge University Press, Cambridge (2004)
\bibitem{Rovelli2} Smerlak, M., Rovelli, C.: Relational EPR. Foundations of Physics 37, 427-445 (2007)
\bibitem{kochen}
Kochen, S.: A New Interpretation of Quantum Mechanics. In: Lahti,
P., Mittelstaedt, P. (eds.) Symposium on the Foundations of Modern
Physics 1985. World Scientific, Singapore (1985)
\bibitem{Bitbol} Bitbol, M.: Physical Relations or Functional
Relations? http://philsci-archive.pitt.edu/archive/00003506
\bibitem{VanFraassen} Van Fraassen, B.: Rovelli's World. Foundations of Science,
forthcoming
\bibitem{dieksx} Dieks, D.: Resolution of the measurement problem through decoherence of the quantum state.
Physics Letters A 142, 439-446 (1989)
\bibitem{dieks&vermaas} Dieks, D., Vermaas, P.E.: The Modal Interpretation of Quantum Mechanics.
Kluwer Academic Publishers, Dordrecht (1998)
\bibitem{vermaas&dieks} Vermaas, P.E., Dieks, D.: The modal interpretation of quantum mechanics and
its generalization to density operators. Foundations of Physics
25, 145-158 (1995)
\bibitem{bub} Bub, J.: Interpreting the Quantum World. Cambridge University Press, Cambridge (1997)
\bibitem{dieksy} Dieks, D.: Probability in modal interpretation of quantum mechanics.
Studies in History and Philosophy of Modern Physics 38, 292-310
(2007)
\bibitem{EPR} Einstein, A., Rosen, N., Podolsky, B.: Can quantum-mechanical description of physical reality be
considered complete? Physical Review 47, 777-780 (1935)
\bibitem{Bohr}
Bohr, N.: Discussion with Einstein on Epistemological Problems of
Atomic Physics. In: Schilpp, P.A. (ed.) Albert Einstein:
Philosopher-Scientist. Open Court, La Salle (1949)
\bibitem{Laudisa}
Laudisa, F., Rovelli, C.: Relational Quantum Mechanics. In: Zalta,
E.N. (ed.) The Stanford Encyclopedia of Philosophy, Fall 2008
Edition.
http://plato.stanford.edu/archives/fall2008/entries/qm-relational
\end{thebibliography}
\end{document}