## 1. Introduction




The quantum realm involves curious correlations between distant
events. A well-known example is David Bohm's (1951) version of
the famous thought experiment that Einstein, Podolsky and Rosen
proposed in 1935 (henceforth, the EPR/B experiment). Pairs of particles
are emitted from a source in the so-called spin singlet state and rush
in opposite directions (see Fig. 1 below). When the particles are
widely separated from each other, they each encounter a measuring
apparatus that can be set to measure their spin components along
various directions. Although the measurement events are distant from
each other, so that no slower-than-light or light signal can travel
between them, the measurement outcomes are curiously
 correlated.[1]
 That is, while the outcome of each of the distant spin measurements
seems to be a matter of pure chance, they are correlated with each
other: The joint probability of the distant outcomes is different from
the product of their single probabilities. For example, the
probability that each of the particles will spin clockwise about the
*z*-axis in a *z*-spin measurement (i.e., a measurement
of the spin component along the *z* direction) appears to be
1/2. Yet, the outcomes of such measurements are perfectly
anti-correlated: If the left-hand-side (L-) particle happens to spin
clockwise (anti-clockwise) about the *z*-axis, the
right-hand-side (R-) particle will spin anti-clockwise (clockwise)
about that axis. And this is true even if the measurements are made
simultaneously.



> 
> ![figure1 - schematic diagram of EPR/B experiment](figure1.jpg)
> 
> **Figure 1**: A schematic illustration of the EPR/B
> experiment. Particle pairs in the spin singlet state are emitted in
> opposite directions and when they are distant from each other
> (i.e., space-like separated), they encounter measurement apparatuses
> that can be set to measure spin components along various
> directions.
> 
> 
> 




The curious EPR/B correlations strongly suggest the existence of
non-local influences between the two measurement events, and indeed
orthodox 'collapse' quantum mechanics supports this
suggestion. According to this theory, before the measurements the
particles do not have any definite spin. The particles come to possess
a definite spin only with the first spin measurement, and the outcome
of this measurement is a matter of chance. If, for example, the first
measurement is a *z*-spin measurement on the L-particle, the
L-particle will spin either clockwise or anti-clockwise about the
*z*-axis with equal chance. And the outcome of the
L-measurement causes an instantaneous change in the spin properties of
the distant R-particle. If the L-particle spins clockwise
(anti-clockwise) about the *z*-axis, the R-particle will
instantly spin anti-clockwise (clockwise) about the same axis. (It is
common to call spins in opposite directions 'spin up' and
'spin down,' where by convention a clockwise spinning may
be called 'spin up' and anti-clockwise spinning may be
called 'spin down.')




It may be argued that orthodox quantum mechanics is false, and that
the non-locality postulated by it does not reflect any non-locality in
the quantum realm. Alternatively, it may be argued that orthodox
quantum mechanics is a good instrument for predictions rather than a
fundamental theory of the physical nature of the universe. On this
instrumental interpretation, the predictions of quantum mechanics are
not an adequate basis for any conclusion about non-locality: This
theory is just an incredible oracle (or a crystal ball), which provides
a very successful algorithm for predicting measurement outcomes and
their probabilities, but it offers little information about ontological
matters, such as the nature of objects, properties and causation in the
quantum realm.




Einstein, Podolsky and Rosen (1935) thought that quantum mechanics is
incomplete and that the curious correlations between distant systems
do not amount to action at a distance between them. The apparent
instantaneous change in the R-particle's properties during the
L-measurement is not really a change of properties, but rather a
change of knowledge. (For more about the EPR argument, see the entry
on the EPR argument, Redhead 1987, chapter 3, and Albert 1992, chapter
3. For discussions of the EPR argument in the relativistic context,
see Ghirardi and Grassi 1994 and Redhead and La Riviere 1997.) On this
view, quantum states of systems do not always reflect their complete
state. Quantum states of systems generally provide information about
*some* of the properties that systems possess and information
about the probabilities of outcomes of measurements on them, and this
information does not generally reflect the complete state of the
systems. In particular, the information encoded in the spin singlet
state is about the probabilities of measurement outcomes of spin
properties in various directions, about the conditional probabilities
that the L- (R-) particle has a certain spin property given that the
R- (L-) particle has another spin property, and about the
anti-correlation between the spins that the particles may have in any
given direction (for more details, see section 5.1). Thus, the outcome
of a *z*-spin measurement on the L-particle and the spin
singlet state (interpreted as a state of knowledge) jointly provide
information about the *z*-spin property of the R-particle. For
example, if the outcome of the L-measurement is *z*-spin
'up,' we know that the R-particle has *z*-spin
'down'; and if we assume, as EPR did, that there is no
curious action at a distance between the distant wings (and that the
change of the quantum-mechanical state of the particle pair in the
L-measurement is only a change in state of knowledge), we could also
conclude that the R-particle had *z*-spin 'down'
even before the L-measurement occurs.




How could the L-outcome change our
knowledge/ignorance about the R-outcome if it has no influence on it?
The simplest and most straightforward reply is that the L- and the R-
outcome have a common cause that causes them to be correlated, so that
knowledge of one outcome provides knowledge about the
 other.[2]
 Yet, the question is whether the predictions of orthodox quantum
mechanics, which have been highly confirmed by various experiments,
are compatible with the quantum realm being local in the sense of
involving no influences between systems between which light and
slower-than-light signals cannot travel (i.e., space-like separated
systems). More particularly, the question is whether it is possible to
construct a local, common-cause model of the EPR/B experiment, i.e., a
model that postulates no influence between systems/events in the
distant wings of the experiment, and that the correlation between them
are due to the state of the particle pair at the source. In 1935,
Einstein, Podolsky and Rosen believed that this is possible. But, as
John Bell demonstrated in 1964, this belief is difficult to
uphold.


## 2. Bell's theorem and non-locality




In a famous theorem, John Bell (1964) demonstrated that granted some
plausible assumptions, any local model of the EPR/B experiment is
committed to certain inequalities about the probabilities of
measurement outcomes, 'the Bell inequalities,' which are
incompatible with the quantum-mechanical predictions. When Bell proved
his theorem, the EPR/B experiment was only a thought experiment. But
due to technological advances, various versions of this experiment have
been conducted since the 1970s, and their results have overwhelmingly
supported the quantum-mechanical predictions (for brief reviews of
these experiments and further references, see the entry on Bell's
theorem and Redhead 1987, chapter 4, section 4.3 and 'Notes and
References'). Thus, a wide consensus has it that the quantum
realm involves some type of non-locality.




The basic idea of Bell's theorem is as follows. A model of the EPR/B
experiment postulates that the state of the particle pair together
with the apparatus settings to measure (or not to measure) certain
spin properties determine the probabilities for single and joint
spin-measurement outcomes. A local Bell model of this experiment also
postulates that probabilities of joint outcomes factorize into the
single probabilities of the L- and the R- outcomes: The probability of
joint outcomes is equal to the product of the probabilities of the
single outcomes. More formally, let l denote the pair's state
before any measurement occurs. Let *l* denote the setting of
the L-measurement apparatus to measure spin along the *l*-axis
(i.e., the *l*-spin of the L-particle), and let *r*
denote the setting of the R-measurement apparatus to measure spin
along the *r*-axis (i.e., the *r*-spin of the
R-particle). Let *xl* be the outcome of a
*l*-spin measurement in the L-wing, and let
*yr* be the outcome of a *r*-spin measurement
in the R-wing; where *xl* is either the L-outcome
*l*-spin 'up' or the L-outcome *l*-spin
'down,' and *yr* is either the R-outcome
*r*-spin 'up' or the R-outcome *r*-spin
'down.' Let *P*l *l
r*(*xl* & *yr*) be
the joint probability of the L- and the R-outcome, and
*P*l
*l*(*xl*) and
*P*l
*r*(*yr*) be the single
probabilities of the L- and the R-outcome, respectively; where the
subscripts l, *l* and *r* denote the factors that
are relevant for the probabilities of the outcomes
*xl* and *yr*. Then, for any
l, *l*, *r*, *xl* and
 *yr*:[3]



> 
> **Factorizability**  
> 
> *P*l *l* *r*(*xl*
>  & *yr*) =
>  *P*l *l*(*xl*)
>  *
>  *P*l *r*(*yr*).
> 




(Here and henceforth, for simplicity's sake we shall denote events and
states, such as the measurement outcomes, and the propositions that
they occur by the same symbols.)




The state l is typically thought of as the pair's state at the
emission time, and it is assumed that this state does not change in
any relevant sense between the emission and the first measurement. It
is (generally) a different state from the quantum-mechanical pair's
state ps. ps is assumed to be an incomplete state of the pair,
whereas l is supposed to be a (more) complete state of the
pair. Accordingly, pairs with the same state ps may have different
states l which give rise to different probabilities of outcomes
for the same type of measurements. Also, the states l may be
unknown, hidden, inaccessible or uncontrollable.




Factorizability is commonly motivated as a locality condition. In
non-local models of the EPR/B experiment, the correlations between the
distant outcomes are accounted for by non-local influences between the
distant measurement events. For example, in orthodox quantum mechanics
the first spin measurement on, say, the L-particle causes an immediate
change in the spin properties of the R-particle and in the
probabilities of future outcomes of spin measurements on this particle.
By contrast, in local models of this experiment the correlations are
supposed to be accounted for by a common cause--the pair's
state l (see Fig. 2 below): The pair's state and the
L-setting determine the probability of the L-outcome; the pair's
state and the R-setting determine the probability of the
R-outcome; and the pair's state and the L- and the
R-setting determine the probability of joint outcomes, which (as
mentioned above) is simply the product of these single probabilities.
The idea is that the probability of each of the outcomes is determined
by 'local events,' i.e., events that are confined to its
backward light-cone, and which can only exert subluminal or luminal
influences on it (see Figure 3 below); and the distant outcomes are
fundamentally independent of each other, and thus their joint
probability factorizes. (For more about this reasoning, see sections 6
and 8-9.)



> 
> ![figure2 - common-cause model of the EPR/B experiment](figure2.jpg)
> 
> **Figure 2**: A schematic common-cause model of the EPR/B
> experiment. Arrows denote causal connections.  
>   
> 
> 
> 
> 



> 
> ![figure3 - space-time diagram of a local model of the EPR/B experiment](figure3.jpg)
> 
> **Figure 3**: A space-time diagram of a local model of
> the EPR/B experiment. The circles represent the measurement events,
> and the cones represent their backward light cones, i.e., the
> boundaries of all the subluminal and luminal influences on them. The
> dotted lines denote the propagation of the influences of the pair's
> state at the emission and of the settings of the measurement
> apparatuses on the measurement outcomes.
> 
> 
> 




A Bell model of the EPR/B experiment also postulates that for each
quantum-mechanical state ps there is a distribution r over all
the possible pair states l, which is
independent of the settings of the apparatuses. That is, the
distribution of the ('complete') states l depends on
the ('incomplete') state ps, and this distribution
is independent of the particular choice of measurements in the L- and
R-wing (including the choice not to measure any quantity). Or
formally, for any quantum-mechanical state ps, L-settings
*l* and *l*', and R-settings *r* and
*r*':



> 
> **l-independence**   
> 
>  rps *l* *r*(l) =
>  rps *l*' *r*(l) =
>  rps *l* *r*'(l) =
>  rps *l*' *r*'(l) =
>  rps(l)
> 




where the subscripts denote the factors that are potentially
relevant for the distribution of the states l.




Although the model probabilities (i.e., the probabilities of outcomes
prescribed by the states l) are different from the
corresponding quantum-mechanical probabilities of outcomes (i.e., the
probabilities prescribed by the quantum-mechanical states ps),
the quantum mechanical probabilities (which have been systematically
confirmed) are recovered by averaging over the model probabilities.
That is, it is supposed that the quantum-mechanical probabilities
*P*ps *l
r*(*xl* & *yr*),
*P*ps *l*(*xl*)
and
*P*ps *r*(*yr*)
are obtained by averaging over the model probabilities
*P*l *l
r*(*xl* & *yr*),
*P*l *l*
(*xl*) and
*P*l *r*(*yr*),
respectively: For any ps, *l*, *r*, *xl* and *yr*,



> 
> **Empirical Adequacy**   
> 
> *P*ps *l* *r*(*x**l*
>  & *yr*) = 
>  [?]l *P*l *l* *r*(*xl* 
>  & *yr*) *
>  rps *l* *r*(l)  
> 
> *P*ps *l*(*x**l*) 
>  =
>  [?]l *P*l *l*(*x**l* )
>  *
>  rps *l*(l)  
> 
> *P*ps *r*(*y**r*) 
>  =
>  [?]l *P*l *r*(*y**r*)
>  *
>  rps *r*(l).[4]
> 




The assumption of l-independence is very plausible. It
postulates that (complete) pair states at the source are uncorrelated
with the settings of the measurement apparatuses. And independently of
one's philosophical view about free will, this assumption is strongly
suggested by our experience, according to which it seems possible to
prepare the state of particle pairs at the source independently of the
set up of the measurement apparatuses.




There are two ways to try to explain a failure of
l-independence. One possible explanation is that
pairs' states and apparatus settings share a common cause, which
always correlates certain types of pairs' states l with
certain types of L- and R-setting. Such a causal hypothesis will be
difficult to reconcile with the common belief that apparatus settings
are controllable at experimenters' will, and thus could be set
independently of the pair's state at the source. Furthermore,
thinking of all the different ways one can measure spin properties and
the variety of ways in which apparatus settings can be chosen, the
postulation of such common cause explanation for settings and
pairs' states would seem highly *ad hoc* and its existence
conspiratorial.




Another possible explanation for the failure of l-independence
is that the apparatus settings influence the pair's state at the
source, and accordingly the distribution of the possible pairs' states
l is dependent upon the settings. Since the settings can be
made after the emission of the particle pair from the source, this
kind of violation of l-independence would require backward
causation. (For advocates of this way out of non-locality, see Costa
de Beauregard 1977, 1979, 1985, Sutherland 1983, 1998, 2006 and Price
1984, 1994, 1996, chapters 3, 8 and 9.) On some readings of John
Cramer's (1980, 1986) transactional interpretation of quantum
mechanics (see Maudlin 1994, pp. 197-199), such violation of
l-independence is postulated. According to this interpretation,
the source sends 'offer' waves forward to the measurement
apparatuses, and the apparatuses send 'confirmation' waves
(from the space-time regions of the measurement events) backward to
the source, thus affecting the states of emitted pairs according to
the settings of the apparatuses. The question of whether such a
theory can reproduce the predictions of quantum mechanics is a
controversial matter (see Maudlin 1994, pp. 197-199, Berkovitz 2002,
section 5, and Kastner 2006). It is noteworthy, however, that while
the violation of l-independence is sufficient for circumventing
Bell's theorem, the failure of this condition *per se* does not
substantiate locality. The challenge of providing a local model of the
EPR/B experiment also applies to models that violate
l-independence. (For more about these issues, see sections 9
and 10.3.)




In any case, as Bell's theorem demonstrates, factorizability,
l-independence and empirical adequacy jointly imply the Bell
inequalities, which are violated by the predictions of orthodox quantum
mechanics (Bell 1964, 1966, 1971, 1975a,b). Granted the systematic
confirmation of the predictions of orthodox quantum mechanics and the
plausibility of l-independence, Bell inferred that
factorizability fails in the EPR/B experiment. Thus, interpreting
factorizability as a locality condition, he concluded that the quantum
realm is non-local. (For further discussions of Bell's theorem,
the Bell inequalities and non-locality, see Bell 1966, 1971, 1975a,b,
1981, Clauser *et al* 1969, Clauser and Horne 1974, Shimony
1993, chapter 8, Fine 1982a,b, Redhead 1987, chapter 4, Butterfield
1989, 1992a, Pitowsky 1989, Greenberger, Horne and Zeilinger 1989,
Greenberger, Horne, Shimony and Zeilinger 1990, Mermin 1990, and the
entry on Bell's theorem.)


## 3. The analysis of factorizability




Following Bell's work, a broad consensus has it that the
quantum realm involves some type of non-locality (for examples, see
Clauser and Horne 1974, Jarrett 1984,1989, Shimony 1984, Redhead 1987,
Butterfield 1989, 1992a,b, 1994, Howard 1989, Healey 1991, 1992, 1994,
Teller 1989, Clifton, Butterfield and Redhead 1990, Clifton 1991,
Maudlin 1994, Berkovitz 1995a,b, 1998a,b, and references
 therein).[5]
 But there is an ongoing controversy as to its exact nature and its
compatibility with relativity theory. One aspect of this controversy
is over whether the analysis of factorizability and the different ways
it could be violated may shed light on these issues. Factorizability
is equivalent to the conjunction of two conditions (Jarrett 1984,
1989, Shimony
 1984):[6]



> 
> **Parameter independence**. The probability of a distant
> measurement outcome in the EPR/B experiment is independent of the
> setting of the nearby measurement apparatus. Or formally, for any
> pair's state l, L-setting *l*, R-setting *r*,
> L-outcome *xl* and R-outcome
> *yr*:
> 
> 
> > 
> > **PI**
> > 
> > 
> > |  |  |  |
> > | --- | --- | --- |
> > | *P*l *l
> > r*(*xl*) =
> > *P*l *l*(*xl*) |   and    | *P*l *l
> > r*(*yr*) =
> > *P*l *r*(*yr*). |
> > 
> > 
> > 
> 
> 
> 



> 
> **Outcome independence**. The probability of a distant
> measurement outcome in the EPR/B experiment is independent of the
> nearby measurement outcome. Or formally, for any pair's state
> l, L-setting *l*, R-setting *r*, L-outcome
> *xl* and R-outcome *yr*:
> 
> 
> > 
> > 
> > 
> > |  |  |  |
> > | --- | --- | --- |
> > | *P*l *l
> > r*(*xl* / *yr*) =
> > *P*l *l
> > r*(*xl*) |    and    | *P*l *l
> > r*(*yr* / *xl*) =
> > *P*l *l
> > r*(*yr*) |
> > |  |
> > | *P*l *l
> > r*(*yr*) > 0 |  | *P*l *l
> > r*(*xl*) > 0, |
> > 
> > 
> > 
> 
> 
> 



> 
>  or more generally,
> 
>  
> > 
> > **OI**  
> > 
> > *P*l *l
> > r*(*xl* & *yr*) =
> > *P*l *l
> > r*(*xl*) *
> > *P*l *l
> > r*(*yr*).
> >  
> 
> 
> 




Assuming l-independence (see section 2), any empirically
adequate theory will have to violate OI or PI. A common view has it
that violations of PI involve a different type of non-locality than
violations of OI: Violations of PI involve some type of
action-at-a-distance that is impossible to reconcile with relativity
(Shimony 1984, Redhead 1987, p. 108), whereas violations of OI involve
some type of holism, non-separability and/or passion-at-a-distance that
may be possible to reconcile with relativity (Shimony 1984, Readhead
1987, pp. 107, 168-169, Howard 1989, Teller 1989).




On the other hand, there is the view that the analysis above (as
well as other similar analyses of
 factorizability[7])
 is immaterial for studying quantum non-locality (Butterfield 1992a,
pp. 63-64, Jones and Clifton 1993, Maudlin 1994, pp. 96 and 149) and
even misleading (Maudlin 1994, pp. 94-95 and 97-98). On this
alternative view, the way to examine the nature of quantum
non-locality is to study the ontology postulated by the various
interpretations of quantum mechanics and alternative quantum
 theories.[8]
 In sections 4-7, we shall follow this methodology and discuss the
nature of non-locality postulated by several quantum theories. The
discussion in these sections will furnish the ground for evaluating
the above controversy in section 8.


## 4. Action at a distance, holism and non-separability


### 4.1 Action at a distance




In orthodox quantum mechanics as well as in any other current quantum
theory that postulates non-locality (i.e., influences between distant,
space-like separated systems), the influences between the distant
measurement events in the EPR/B experiment do not propagate
continuously in space-time. They seem to involve action at a
distance. Yet, a common view has it that these influences are due to
some type of holism and/or non-separability of states of composite
systems, which are characteristic of systems in entangled states (like
the spin singlet state), and which exclude the very possibility of
action at a distance. The paradigm case of action at a distance is the
Newtonian gravitational force. This force acts between distinct
objects that are separated by some (non-vanishing) spatial distance,
its influence is symmetric (in that any two massive objects influence
each other), instantaneous and does not propagate continuously in
space. And it is frequently claimed or presupposed that such action at
a distance could only exist between systems with separate states in
non-holistic universes (i.e., universes in which the states of
composite systems are determined by, or supervene upon the states of
their subsystems and the spacetime relations between them), which are
commonly taken to characterize the classical
 realm.[9]




In sections 4.2 and 4.3, we shall briefly review the relevant
notions of holism and non-separability (for a more comprehensive
review, see the entry on holism and nonseparability in physics and
Healey 1991). In section 5, we shall discuss the nature of holism and
non-separability in the quantum realm as depicted by various quantum
theories. Based on this discussion, we shall consider whether the
non-local influences in the EPR/B experiment constitute action at a
distance.


### 4.2 Holism




In the literature, there are various characterizations of holism.
Discussions of quantum non-locality frequently focus on property
holism, where certain physical properties of objects are not determined
by the physical properties of their parts. The intuitive idea is that
some intrinsic properties of wholes (e.g. physical systems) are not
determined by the intrinsic properties of their parts and the
spatiotemporal relations that obtain between these parts. This idea can
be expressed in terms of supervenience relations.



> 
> **Property Holism**. Some objects have intrinsic
> qualitative properties and/or relations that do not supervene upon the
> intrinsic qualitative properties and relations of their parts and the
> spatiotemporal relations between these parts.




It is difficult to give a general precise specification of the terms
'intrinsic qualitative property' and
'supervenience.' Intuitively, a property of an object is
intrinsic just in case that object has this property in and for itself
and independently of the existence or the state of any other object. A
property is qualitative (as opposed to individual) if it does not
depend on the existence of any particular object. And the intrinsic
qualitative properties of an object *O* supervene upon the
intrinsic qualitative properties and relations of its parts and the
spatiotemporal relations between them just in case there is no change
in the properties and relations of *O* without a change in the
properties and relations of its parts and/or the spatiotemporal
relations between them. (For attempts to analyze the term
'intrinsic property,' see for example Langton and Lewis
1998 and the entry on intrinsic vs. extrinsic properties. For a review
of different types of supervenience, see for example Kim 1978,
McLaughlin 1994 and the entry on supervenience.)




Paul Teller (1989, p. 213) proposes a related notion of holism,
'relational holism,' which is characterized as the
violation of the following condition:



> 
> **Particularism**. The world is composed of
> individuals. All individuals have non-relational properties and all
> relations supervene on the non-relational properties of the
> relata.




Here, by a non-relational property Teller means an intrinsic property
(1986a, p. 72); and by 'the supervenience of a relational
property on the non-relational properties of the relata,' he
means that 'if two objects, 1 and 2, bear a relation *R*
to each other, then, *necessarily*, if two further objects,
1' and 2' have the same non-relational properties, then
1' and 2' will also bear the same relation *R* to
each other' (1989, p. 213). Teller (1986b, pp. 425-7) believes
that spatiotemporal relations between objects supervene upon the
objects' intrinsic physical properties. Thus, he does not
include the spatiotemporal relations in the supervenience basis. This
view is controversial, however, as many believe that spatiotemporal
relations between objects are neither intrinsic nor supervene upon the
intrinsic qualitative properties of these objects. But, if such
supervenience does not obtain, particularism will also be violated in
classical physics, and accordingly relational holism will fail to mark
the essential distinction between the classical and the quantum
realms. Yet, one may slightly revise Teller's definition of
particularism as follows:



> 
> **Particularism\*.** The world is composed of
> individuals. All individuals have non-relational properties and all
> relations supervene upon the non-relational properties of the relata
> and the spatiotemporal relations between them.




In what follows in this entry, by relational holism we shall mean a
violation of particularism\*.


### 4.3 Non-separability




Like holism, there are various notions of non-separability on offer.
The most common notion in the literature is state non-separability,
i.e., the violation of the following condition:



> 
> **State separability.** Each system possesses a
> separate state that determines its qualitative intrinsic properties,
> and the state of any composite system is wholly determined by the
> separate states of its subsystems.




The term 'wholly determined' is vague. But, as before, one
may spell it out in terms of supervenience relations: State
separability obtains just in case each system possesses a separate
state that determines its qualitative intrinsic properties and
relations, and the state of any composite system is supervenient upon
the separate states of its subsystems.




Another notion of non-separability is spatiotemporal non-separability.
Inspired by Einstein (1948), Howard (1989, pp. 225-6) characterizes
spatiotemporal non-separability as the violation of the following
separability condition:



> 
> **Spatiotemporal separability**. The contents of any
> two regions of space-time separated by a non-vanishing spatiotemporal
> interval constitute two separate physical systems. Each separated
> space-time region possesses its own, distinct state and the joint state
> of any two separated space-time regions is wholly determined by the
> separated states of these regions.




A different notion of spatiotemporal non-separability, proposed by
Healey (see the entry on holism and nonseparability in physics), is
process non-separability. It is the violation of the following
condition:



> 
> **Process separability**. Any physical process
> occupying a spacetime region *R* supervenes upon an assignment
> of qualitative intrinsic physical properties at spacetime points in
> *R*.


## 5. Holism, non-separability and action at a distance in quantum mechanics




The quantum realm as depicted by all the quantum theories that
postulate non-locality, i.e., influences between distant (space-like
separated) systems, involves some type of non-separability or
holism. In what follows in this section, we shall consider the nature
of the non-separability and holism manifested by various
interpretations of quantum mechanics. On the basis of this
consideration, we shall address the question of whether these
interpretations predicate the existence of action at a distance. We
start with the so-called 'collapse theories.'


### 5.1 Collapse theories


#### 5.1.1 Orthodox quantum mechanics




In orthodox quantum mechanics, normalized vectors in Hilbert spaces
represent states of physical systems. When the Hilbert space is of
infinite dimension, state vectors can be represented by functions, the
so-called 'wave functions.' In any given basis, there is a
unique wave function that corresponds to the state vector in that
basis. (For an entry level review of the highlights of the
mathematical formalism and the basic principles of quantum mechanics,
see the entry on quantum mechanics, Albert 1992, Hughes 1989, Part I,
and references therein; for more advanced reviews, see Bohm 1951 and
Redhead 1987, chapters 1-2 and the mathematical appendix.)




For example, the state of the L-particle having *z*-spin
'up' (i.e., spinning 'up' about the
*z*-axis) can be represented by the vector |*z*-up>
in the Hilbert space associated with the L-particle, and the state of
the L-particle having *z*-spin 'down' (i.e.,
spinning 'down' about the *z*-axis) can be
represented by the orthogonal vector, |*z*-down>. Particle
pairs may be in a state in which the L-particle and the R-particle
have opposite spins, for instance either a state
|ps1> in which the L-particle has *z*-spin
'up' and the R-particle has *z*-spin
'down,' or a state |ps2> in which the
L-particle has *z*-spin 'down' and the R-particle
has *z*-spin 'up.' Each of these states is
represented by a tensor product of vectors in the Hilbert space of the
particle pair:
 |ps1>
 = |*z*-up>*L*
|*z*-down>*R* and |ps2> =
|*z*-down>*L*
|*z*-up>*R*; where the subscripts L and R
refer to the Hilbert spaces associated with the L- and the R-particle,
respectively. But particle pairs may also be in a superposition of
these states, i.e., a state that is a linear sum of the states
|ps1> and |ps2>, e.g. the state
represented by



> 
> 
> 
> |  |  |  |
> | --- | --- | --- |
> | |ps3> |  =  | 1/[?]2 (|ps1>
> [?] |ps2>) |
> |  |  =  | 1/[?]2
> (|*z*-up>*L*
> |*z*-down>*R*
> [?] |*z*-down>*L*
> |*z*-up>*R*). |
> 
> 
> 




In fact, this is exactly the case in the spin singlet state. In this
state, the particles are entangled in a non-separable state (i.e., a
state that cannot be decomposed into a product of separate states of
the L- and the R-particle), in which (according to the
property-assignment rules of orthodox quantum mechanics) the particles
do not possess any definite *z*-spin (or definite spin in any
other direction). Thus, the condition of state separability fails: The
state of the particle pair (which determines its intrinsic qualitative
properties) is not wholly determined by the separate states of the
particles (which determine their intrinsic qualitative properties). Or
more precisely, the pair's state is not supervenient upon the
separable states of the particles. In particular, the superposition
state of the particle pair assigns a 'correlational'
property that dictates that the outcomes of (ideal) *z*-spin
measurements on both the L- and the R-particle will be
anti-correlated, and this correlational property is not supervenient
upon properties assigned by any separable states of the particles (for
more details, see Healey 1992, 1994). For similar reasons, the spin
singlet state also involves property and relational holism; for the
above correlational property of the particle pair also fails to
supervene upon the intrinsic qualitative properties of the particles
and the spatiotemporal relations between them. Furthermore, the
process that leads to each of the measurement outcomes is also
non-separable, i.e., process separability fails (see Healey 1994 and
the entry on holism and nonseparability in physics).




This correlational property is also 'responsible' for the
action at a distance that the orthodox theory seems to postulate
between the distant wings in the EPR/B experiment. Recall (section 1)
that Einstein, Podolsky and Rosen thought that this curious action at
a distance reflects the incompleteness of this theory rather than a
state of nature. The EPR argument for the incompleteness of the
orthodox theory is controversial. But the orthodox theory seems to be
incomplete for a different reason. This theory postulates that in
non-measurement interactions, the evolution of states obeys a linear
and unitary equation of motion, the so-called Schrodinger
equation (see the entry on quantum mechanics), according to which the
particle pair in the EPR/B experiment remains in an entangled
state. This equation of motion also dictates that in a spin
measurement, the pointers of the measurement apparatuses get entangled
with the particle pair in a non-separable state in which (according to
the theory's property assignment, see below) the indefiniteness of
particles' spins is 'transmitted' to the pointer's
position: In this entangled state of the particle pair and the
pointer, the pointer lacks any definite position, in contradiction to
our experience of perceiving it pointing to either 'up' or
'down.'




The above problem, commonly called 'the measurement
problem,' arises in orthodox no-collapse quantum mechanics from
two features that account very successfully for the behavior of
microscopic systems: The linear dynamics of quantum states as
described by the Schrodinger equation and the property assignment
rule called 'eigenstate-eigenvalue link.' According to the
eigenstate-eigenvalue link, a physical observable, i.e., a physical
quantity, of a system has definite value (one of its eigenvalues) just
in case the system is in the corresponding eigenstate of that
observable (see the entry on quantum mechanics, section
4). Microscopic systems may be in a superposition state of spin
components, energies, positions, momenta as well as other physical
observables. Accordingly, microscopic systems may be in a state of
indefinite *z*-spin, energy, position, momentum and various
other quantities. The problem is that given the linear and unitary
Schrodinger dynamics, these indefinite quantities are also
endemic in the macroscopic realm. For example, in a *z*-spin
measurement on a particle in a superposition state of *z*-spin
'up' and *z*-spin 'down,' the position
of the apparatus's pointer gets entangled with the indefinite
*z*-spin of the particle, thus transforming the pointer into a
state of indefinite position, i.e., a superposition of pointing
'up' and pointing 'down' (see Albert 1992,
chapter 4, and the entry on collapse theories, section 3). In
particular, in the EPR/B experiment the L-measurement causes the
L-apparatus pointer to get entangled with the particle pair,
transforming it into a state of indefinite position:




> 
>  |ps4> = 1/[?]2
> (|*z*-up>*L* 
> |*z*-down>*R*
> |up>*LA* [?] 
> |*z*-down>*L* 
> |*z*-up>*R* 
> |down>*LA*)






where |up>*LA* and
|down>*LA* are the states of the L-apparatus
pointer displaying the outcomes *z*-spin 'up' and
*z*-spin 'down,' respectively. Since the above type
of indefiniteness is generic in orthodox no-collapse quantum
mechanics, in this theory measurements typically have no definite
outcomes, in contradiction to our experience.




In order to solve this problem, the orthodox theory postulates that in
measurement interactions, entangled states of measured systems and the
corresponding measurement apparatuses do not evolve according to the
Schrodinger equation. Rather, they undergo a
'collapse' into product (non-entangled) states, where the
systems involved have the relevant definite properties. For example,
the entangled state of the particle pair and the L-apparatus in the
EPR/B experiment may collapse into a product state in which the
L-particle comes to possess *z*-spin 'up,' the
R-particle comes to possess *z*-spin 'down' and the
L-apparatus pointer displaying the outcome *z*-spin
'up':



> 
>  |ps5> = 
> |*z*-up>*L* 
> |*z*-down>*R*
> |up>*LA*.




The problem is that in the orthodox theory, the notions of
measurement and the time, duration and nature of state collapses remain
completely unspecified. As John Bell (1987b, p. 205) remarks, the
collapse postulate in this theory, i.e., the postulate that dictates
that in measurement interactions the entangled states of the relevant
systems do not follow the Schrodinger equation but rather undergo
a collapse, is no more than 'supplementary, imprecise, verbal,
prescriptions.'




This problem of accounting for our experience of perceiving definite
measurement outcomes in orthodox quantum mechanics, is an aspect of the
more general problem of accounting for the classical-like behavior of
macroscopic systems in this theory.


#### 5.1.2 Dynamical models for state vector reduction




The dynamical models for state-vector reduction were developed to
account for state collapses as real physical processes (for a review
of the collapse models and a detailed reference list, see the entry on
collapse theories). The origin of the collapse models may be dated to
Bohm and Bub's (1966) hidden variable theory and Pearle's (1976)
spontaneous localization approach, but the program has received its
crucial impetus with the more sophisticated models developed by
Ghirardi, Rimini and Weber in 1986 (see also Bell 1987a and Albert
1992) and their consequent development by Pearle (1989) (see also
Ghirardi, Pearle and Rimini 1990, and Butterfield et al. 1993).
Similarly to orthodox collapse quantum mechanics, in the GRW models
the quantum-mechanical state of systems (whether it is expressed by a
vector or a wave function) provides a complete specification of their
intrinsic properties and relations. The state of systems follows the
Schrodinger equation, except that it has a probability for
spontaneous collapse, independently of whether or not the systems are
measured. The chance of collapse depends on the 'size' of
the entangled systems--in the earlier models the 'size'
of systems is predicated on the number of the elementary particles,
whereas in later models it is measured in terms of mass densities. In
any case, in microscopic systems, such as the particle pairs in the
EPR/B experiment, the chance of collapse is very small and
negligible--the chance of spontaneous state collapse in such
systems is cooked up so that it will occur, on average, every hundred
million years or so. This means that the chance that the entangled
state of the particle pair in the EPR/B experiment will collapse to a
product state between the emission from the source and the first
measurement is virtually zero. In an earlier L-measurement, the state
of the particle pair gets entangled with the state of the
L-measurement apparatus. Thus, the state of the pointer of the
L-apparatus evolves from being 'ready' to measure a
certain spin property to an indefinite outcome. For instance, in a
*z*-spin measurement the L-apparatus gets entangled with the
particle pair in a superposition state of pointing to 'up'
and pointing to 'down' (corresponding to the states of the
L-particle having *z*-spin 'up' and having
*z*-spin 'down'), and the R-apparatus remains
un-entangled with these systems in the state of being ready to measure
*z*-spin. Or formally:




> 
>  |ps6> = 1/[?]2
> (|*z*-up>*L* 
> |up>*AL* 
> |*z*-down>*R* [?] 
> |*z*-down>*L* 
> |down>*AL* 
> |*z*-up>*R*) 
> |ready>*AR*






where, as before, |up>*AL* and
|down>*AL* denote the states of the L-apparatus
displaying the outcomes *z*-spin 'up' and
'down' respectively, and |ready>*AR*
denotes the state of the R-apparatus being ready to measure
*z*-spin. In this state, a gigantic number of particles of the
L-apparatus pointer are entangled together in the superposition state
of being in the position (corresponding to pointing to)
'up' and the position (corresponding to pointing to)
'down.' For assuming, for simplicity of presentation, that
the position of all particles of the L-apparatus pointer in the state
of pointing to 'up' ('down') is the same, the
state |ps6> can be rewritten as:




> 
>  |ps7> = 1/[?]2
> (|*z*-up>*L* 
> |up>*p1* 
> |up>*p2* 
> |up>*p3 ...* 
> |*z*-down>*R* [?]
>    
> 
>         
>         
>  |*z*-down>*L* 
> |down>*p1* 
> |down>*p2* 
> |down>*p3 ...* 
> |*z*-up>*R*) |ready
> >*AR*






where *pi* denotes the *i*-particle of the
L-apparatus pointer, and |up>*pi*
(|down>*pi*) is the state of the
*i*-particle being in the position corresponding to the outcome
*z*-spin 'up'
 ('down').[10]
 The chance that at least one of the vast number of the pointer's
particles will endure a spontaneous localization toward
being in the position corresponding to either the outcome
*z*-spin 'up' or the outcome *z*-spin
'down' within a very short time (a split of a micro
second) is very high. And since all the particles of the pointer and
the particle pair are entangled with each other, such a collapse will
carry with it a collapse of the entangled state of the pointer of the
L-apparatus and the particle pair toward either



> 
>  |*z*-up>*L* |up>*p1* |up>*p2*|up>*p3 ...* 
> |*z*-down>*R*
> 



 or



> 
>  |*z*-down>*L* |down>*p1* |down>*p2*|down>*p3 ...* 
> |*z*-up>*R*.



 Thus, the pointer will very quickly move in the direction of pointing
to either the outcome *z*-spin 'up' or the outcome
*z*-spin 'down.'




If (as portrayed above) the spontaneous localization of particles were
to a precise position, i.e., to the position corresponding to the
outcome 'up' or the outcome 'down,' the GRW
collapse models would successfully resolve the measurement
problem. Technically speaking, a precise localization is achieved by
multiplying |ps7> by a delta function centered on the
position corresponding to either the outcome 'up' or the
outcome 'down' (see the entry on collapse theories,
section 5 and Albert 1992, chapter 5); where the probability of each
of these mutually exhaustive possibilities is 1/2. The problem is
that it follows from the uncertainty principle (see the entry on the
uncertainty principle) that in such localizations the momenta and the
energies of the localized particles would be totally uncertain, so
that gases may spontaneously heat up and electrons may be knocked out
of their orbits, in contradiction to our experience. To avoid this
kind of problems, GRW postulated that spontaneous localizations are
characterized by multiplications by Gaussians that are centered around
certain positions, e.g. the position corresponding to either the
outcome 'up' or the outcome 'down' in the
state |ps7>. This may be problematic, because in
either case the state of the L-apparatus pointer at (what we
characteristically conceive as) the end of the L-measurement would be
a superposition of the positions 'up' and
'down.' For although this superposition
'concentrates' on either the outcome 'up' or
the outcome 'down' (i.e., the peak of the wave function
that corresponds to this state concentrates on one of these
positions), it also has 'tails' that go everywhere: The
state of the L-apparatus is a superposition of an infinite number of
different positions. Thus, it follows from the eigenstate-eigenvalue
link that the position observable of the L-apparatus has no definite
value at the end of the measurement. But if the position observable
having a definite value is indeed required in order for the
L-apparatus to have a definite location, then the pointer will point
to neither 'up' nor 'down,' and the GRW
collapse models will fail to reproduce the classical-like behavior of
such
 systems.[11]




In later models, GRW proposed to interpret the quantum state as a
density of mass and they postulated that if almost all the density of
mass of a system is concentrated in a certain region, then the system
is located in that region. Accordingly, pointers of
measurement apparatuses do have definite positions at the end of
measurement interactions. Yet, this solution has also given rise to a
debate (see Albert and Loewer 1995, Lewis 1997, 2003a, 2004, Ghirardi
and Bassi 1999, Bassi and Ghirardi 1999, 2001, Clifton and Monton 1999,
2000, Frigg 2003, and Parker 2003).




The exact details of the collapse mechanism and its characteristics in
the GRW/Pearle models have no significant implications for the type of
non-separability and holism they postulate--all these models
basically postulate the same kinds of non-separability and holism as
orthodox quantum mechanics (see section 5.1.1). And action at a
distance between the L- and the R-wing will occur if the L-measurement
interaction, a supposedly local event in the L-wing, causes some local
events in the R-wing, such as the event of the pointer of the
measurement apparatus coming to possess a definite measurement outcome
during the R-measurement. That is, action at a distance will occur if
the L-measurement causes the R-particle to come to possess a definite
*z*-spin and this in turn causes the pointer of the R-apparatus
to come to possess the corresponding measurement outcome in the
R-measurement. Furthermore, if the L-measurement causes the R-particle
to come to possess (momentarily) a definite position in the R-wing,
then the action at a distance between the L- and the R-wing will occur
independently of whether the R-particle undergoes a spin
measurement.




The above discussion is based on an intuitive notion of action at a
distance and it presupposes that action at a distance is compatible
with non-separability and holism. In the next section we shall provide
more precise characterizations of action at a distance and in light of
these characterizations reconsider the question of the nature of
action at a distance in the GRW/Pearle collapse models.


### 5.2 Can action-at-a-distance co-exist with non-separability and holism?




The action at a distance in the GRW/Pearle models is different from
the Newtonian action at a distance in various respects. First, in
contrast to Newtonian action at a distance, this action is independent
of the distance between the measurement events. Second, while
Newtonian action is symmetric, the action in the GRW/Pearle models is
(generally) asymmetric: Either the L-measurement influences the
properties of the R-particle or the R-measurement influences the
properties of the L-particle, depending on which measurement occurs
first (the action will be symmetric when both measurements occur
simultaneously). Third (and more important to our consideration), in
contrast to Newtonian action at a distance, before the end of the
L-measurement the state of the L-apparatus and the R-particle is not
separable and accordingly it is not clear that the influence is
between separate existences, as the case is supposed to be in
Newtonian gravity.




This non-separability of the states of the particle pair and the
L-measurement apparatus, and more generally the fact that the
non-locality in collapse theories is due to state non-separability, has
led a number of philosophers and physicists to think that wave
collapses do not involve action at a distance. Yet, the question of
whether there is an action at a distance in the GRW/Pearle models (and
various other quantum theories) depends on how we interpret the term
'action at a distance.' And, as I will suggest below, on a
natural reading of Isaac Newton's and Samuel Clarke's
comments concerning action at a distance, there may be a peaceful
coexistence between action at a distance and non-separability and
holism.




Newton famously struggled to find out the cause of
 gravity.[12]
 In a letter to Bentley, dated January 17 1692/3, he said:



> 
> 
> You sometimes speak of Gravity as essential and inherent to Matter.
> Pray do not ascribe that Notion to me, for the Cause of Gravity is what
> I do not pretend to know, and therefore would take more Time to
> consider it. (Cohen 1978, p. 298)




In a subsequent letter to Bentley, dated February 25, 1692/3, he
added:



> 
> 
> It is inconceivable that inanimate Matter should, without the
> Mediation of something else, which is not material, operate upon, and
> affect other matter without mutual Contact...That Gravity should
> be innate, inherent and essential to Matter, so that one body may act
> upon another at a distance thro' a Vacuum, without the Mediation
> of any thing else, by and through which their Action and Force may be
> conveyed from one to another, is to me so great an Absurdity that I
> believe no Man who has in philosophical Matters a competent Faculty of
> thinking can ever fall into it. Gravity must be caused by an Agent
> acting constantly according to certain laws; but whether this Agent be
> material or immaterial, I have left to the Consideration of my
> readers. (Cohen 1978, pp. 302-3)




Samuel Clarke, Newton's follower, similarly struggled with the
question of the cause of gravitational phenomenon. In his famous
controversy with Leibniz, he
 said:[13]



> 
> 
> That one body attracts another without any intermediate means, is
> indeed not a miracle but a contradiction; for 'tis supposing
> something to act where it is not. But the means by which two bodies
> attract each other, may be invisible and intangible and of a different
> nature from mechanism ...




And he added:



> 
> 
> That this phenomenon is not produced *sans moyen*, that is
> without a cause capable of producing such an effect, is undoubtedly
> true. Philosophers therefore can search after and discover that cause,
> if they can; be it mechanical or not. But if they cannot discover the
> cause, is therefore the effect itself, the phenomenon, or the matter of
> fact discovered by experience ... ever the less true?




Newton's and Clarke's comments suggest that for them
gravity was a law-governed phenomenon, i.e., a phenomenon in which
objects influence each other at a distance according to the Newtonian
law of gravity, and that this influence is due to some means which may
be invisible and intangible and of a different nature from mechanism.
On this conception of action at a distance, there seems to be no reason
to exclude the possibility of action at a distance in the quantum realm
even if that realm is holistic or the state of the relevant systems is
non-separable. That is, action at a distance may be characterized as
follows:



> 
> **Action at a distance** is a phenomenon in which a
> change in intrinsic properties of one system induces a change in the
> intrinsic properties of a distant system, *independently* of the
> influence of any other systems on the distant system, and without
> there being a process that carries this influence contiguously in space
> and time.
>  




We may alternatively characterize action at a distance in a more
liberal way:



> 
> **Action\* at a distance** is a phenomenon in which a
> change in intrinsic properties of one system induces a change in the
> intrinsic properties of a distant system without there being a process
> that carries this influence contiguously in space and time.
>  




And while Newton and Clarke did not have an explanation for the action
at a distance involved in Newtonian gravity, on the above
characterizations action at a distance in the quantum realm would be
explained by the holistic nature of the quantum realm and/or
non-separability of the states of the systems involved. In particular,
if in the EPR/B experiment the L-apparatus pointer has a definite
position before the L-measurement and the R-particle temporarily comes
to possess definite position during the L-measurement, then the
GRW/Pearle models involve action at a distance and thus also action\*
at a distance. On the other hand, if the R-particle never comes to
possess a definite position during the L-measurement, then the
GRW/Pearle models only involve action\* at a distance.


### 5.3 No-collapse theories


#### 5.3.1 Bohm's theory




In 1952, David Bohm proposed a deterministic, 'hidden
variables' quantum theory that reproduces all the observable
predictions of orthodox quantum mechanics (see Bohm 1952, Bohm,
Schiller and Tiomno 1955, Bell 1982, Dewdney, Holland and Kyprianidis
1987, Durr, Goldstein and Zanghi 1992a, 1997, Albert 1992,
Valentini 1992, Bohm and Hiley 1993, Holland 1993, Cushing 1994, and
Cushing, Fine and Goldstein 1996; for an entry level review, see the
entry on Bohmian mechanics and Albert 1992, chapter 5).




In contrast to orthodox quantum mechanics and the GRW/Pearle
collapse models, in Bohm's theory wave functions always evolve
according to the Schrodinger equation, and thus they never
collapse. Wave functions do not represent the states of systems.
Rather, they are states of a 'quantum field (on configuration
space)' that influences the states of
 systems.[14]
 Also, particles always have definite positions, and the positions of
the particles and their wave function at a certain time jointly
determine the trajectories of the particles at all future times. Thus,
particles' positions and their wave function determine the
outcomes of any measurements (so long as these outcomes are recorded
in the positions of some physical systems, as in any practical
measurements).




There are various versions of Bohm's theory. In the
'minimal' Bohm theory, formulated by Bell
 (1982),[15]
 the wave function is interpreted as a 'guiding' field
(which has no source or any dependence on the particles) that
deterministically governs the trajectories of the particles according
to the so-called 'guiding equation' (which expresses the
velocities of the particles in terms of the wave
 function).[16]
 The states of systems are separable (the state of any composite
system is completely determined by the state of its subsystems), and
they are completely specified by the particles'
positions. Spins, and any other properties which are not directly
derived from positions, are not intrinsic properties of
systems. Rather, they are relational properties that are determined by
the systems' positions and the guiding field. In particular,
each of the particles in the EPR/B experiment has dispositions to
'spin' in various directions, and these dispositions are
relational properties of the particles-- they are
(generally) determined by the guiding field and the positions of the
particles relative to the measurement apparatuses and to each
other.



> 
> ![figure4 - EPR/B experiment with Stern-Gerlach measurement devices](figure4.jpg)
>   
> 
> **Figure 4**. The EPR/B experiment with Stern-Gerlach
> measurement devices. Stern-Gerlach 1 is on, set up to measure the
> *z*-spin of the L-particle, and Stern-Gerlach 2 is off. The
> horizontal lines in the left-hand-side denote the trajectories of six
> L-particles in the spin singlet state after an (impulsive)
> *z*-spin measurement on the L-particle, and the horizontal
> lines in the right-hand-side denote the trajectories of the
> corresponding R-particles. The center plane is aligned orthogonally to
> the *z*-axis, so that particles that emerge above this plane
> correspond to *z*-spin 'up' outcome and particles
> that emerge below this plane correspond to *z*-spin
> 'down' outcome. The little arrows denote the
> *z*-spin components of the particles in the
> 'non-minimal' Bohm theory (where spins are intrinsic
> properties of particles), and are irrelevant for the
> 'minimal' Bohm theory (where spins are not intrinsic
> properties of particles).
> 
> 




To see the nature of non-locality postulated by the minimal Bohm
theory, consider again the EPR/B experiment and suppose that the
measurement apparatuses are Stern-Gerlach (S-G) magnets which are
prepared to measure *z*-spin. In any run of the experiment, the
measurement outcomes will depend on the initial positions of the
particles and the order of the measurements. Here is why. In the
minimal Bohm theory, the spin singlet state denotes the relevant state
of the guiding field rather than the intrinsic properties of the
particle pair. If the L-measurement occurs before the R-measurement,
the guiding field and the position of the L-particle at the emission
time jointly determine the disposition of the L-particle to emerge
from the S-G device either above or below a plane aligned in the
*z*-direction; where emerging above (below) the plane means
that the L-particle *z*-spins 'up'
('down') about the *z*-axis and the L-apparatus
'pointer' points to 'up' ('down')
(see Fig. 4 above). All the L-particles that are emitted above the
center plane aligned orthogonally to the *z*-direction, like
the L-particles 1-3, will be disposed to spin 'up'; and
all the particles that are emitted below this plane, like the
L-particles 4-6, will be disposed to spin 'down.'
Similarly, if the R-measurement occurs before the L-measurement, the
guiding field and the position of the R-particle at the emission time
jointly determine the disposition of the R-particle to emerge either
above the *z*-axis (i.e., to *z*-spin 'up')
or below the *z*-axis (i.e., to *z*-spin
'down') according to whether it is above or below the
center plane, independently of the position of the L-particle along
the *z*-axis.




But the *z*-spin disposition of the R-particle changes
immediately after an (earlier) *z*-spin measurement on the
L-particle: The R-particles 1-3 (see Fig. 4), which were previously
disposed to *z*-spin 'up,' will now be disposed to
*z*-spin 'down,' i.e., to emerge below the center
plane aligned orthogonally to the *z*-axis; and the R-particles
4-6, which were previously disposed to *z*-spin
'down,' will now be disposed to *z*-spin
'up,' i.e., to emerge above this center plane. Yet, the
L-measurement *per se* does not have any immediate influence on
the state of the R-particle: The L-measurement does not influence the
position of the R-particle or any other property that is directly
derived from this position. It only changes the guiding field, and
thus grounds new spin dispositions for the R-particle. But these
dispositions are not intrinsic properties of the R-particle. Rather,
they are relational properties of the R-particle, which are grounded
in the positions of both particles and the state of the guiding
 field.[17] 
(Note that in the particular case in which the L-particle is emitted
above the center plane aligned orthogonally to the *z*-axis and
the R-particle is emitted below that plane, an earlier *z*-spin
on the L-particle will have no influence on the outcome of a
*z*-spin on the R-particle.)




While there is no contiguous process to carry the influence of the
L-measurement outcome on events in the R-wing, the question of whether
this influence amounts to action at a distance depends on the exact
characterization of this term. In contrast to the GRW/Pearle collapse
models, the influence of the L-measurement outcome on the intrinsic
properties of the R-particle is dependent on the R-measurement: Before
this measurement occurs, there are no changes in the R-particle's
intrinsic properties. Yet, the influence of the L-measurement on the
R-particle is at a distance. Thus, the EPR/B experiment as depicted by
the minimal Bohm theory involves action\* at a distance but not action
at a distance. 




Bohm's theory portrays the quantum realm as deterministic. Thus, the
single-case objective probabilities, i.e., the chances, it assigns to
individual spin-measurement outcomes in the EPR/B experiment are
different from the corresponding quantum-mechanical probabilities. In
particular, while in quantum mechanics the chances of the outcomes
'up' and 'down' in an earlier L- (R-) spin
measurement are both 1/2, in Bohm's theory these chances are
either one or zero. Yet, Bohm's theory postulates a certain
distribution, the so-called 'quantum-equilibrium
distribution,' over all the possible positions of pairs with the
same guiding field. This distribution is computed from the
quantum-mechanical wave function, and it is typically interpreted as
ignorance over the actual position of the pair; an ignorance that may
be motivated by dynamical considerations and statistical patterns
exhibited by ensembles of pairs with the same wave function (for more
details, see the entry on bohmian mechanics, section 9). And the
sum-average (or more generally the integration) over this distribution
reproduces all the quantum-mechanical observable predictions.




What is the status of this probability postulate? Is it a law of
nature or a contingent fact (if it is a fact at all)? The answers to
these questions vary (see Section 7.2.1, Bohm 1953, Valentini 1991a,b,
1992, 1996, 2002, Valentini and Westman 2004, Durr, Goldstein and
Zanghi 1992a,b, 1996, fn. 15, and Callender 2006).




Turning to the question of non-separability, the minimal Bohm theory
does not involve state non-separability. For recall that in this
theory the state of a system does not consist in its wave function,
but rather in the system's position, and the position of a composite
system always factorizes into the positions of its subsystems. Here,
the non-separability of the wave function reflects the state of the
guiding field. This state propagates not in ordinary three-space but
in configuration space, where each point specifies the configuration
of both particles. The guiding field of the particle pair cannot be
factorized into the guiding field that governs the trajectory of the
L-particle and the guiding field that governs the trajectory of the
R-particle. The evolution of the particles' trajectories,
properties and dispositions is non-separable, and accordingly the
particles' trajectories, properties and dispositions are
correlated even when the particles are far away from each other and do
not interact with each other. Thus, process separability fails.




In the non-minimal Bohm
 theory[18],
 the behavior of an *N*-particle system is determined by its
wave function and the intrinsic properties of the particles. But, in
contrast to the minimal theory, in the non-minimal theory spins are
intrinsic properties of particles. The wave function always evolves
according to the Schrodinger equation, and it is interpreted as a
'quantum field' (which has no sources or any dependence on
the particles). The quantum field guides the particles via the
'quantum potential,' an entity which is determined from
the quantum field, and the evolution of properties is fully
 deterministic.[19]




Like in the minimal Bohm theory, the non-separability of the wave
function in the EPR/B experiment dictates that the evolution of the
particles' trajectories, properties and dispositions is
non-separable, but the behavior of the particles is somewhat different.
In the earlier *z*-spin measurement on the L-particle, the
quantum potential continuously changes, and this change induces an
immediate change in the *z*-spin of the R-particle. If the
L-particle starts to spin 'up' ('down') in the
*z*-direction, the R-particle will start to spin
'down' ('up') in the same direction (see the
little arrows in Fig.
 4).[20]
 Accordingly, the L-measurement induces instantaneous action at a
distance between the L- and the R-wing. Yet, similarly to the minimal
Bohm theory, while the disposition of the R-particle to emerge above
or below the center plane aligned orthogonally to the
*z*-direction in a *z*-spin measurement may change
instantaneously, the actual trajectory of the R-particle along the
*z*-direction does not change before the measurement of the
R-particle's *z*-spin occurs. Only during the R-measurement,
the spin and the position of the R-particle get correlated and the
R-particle's trajectory along the *z*-direction is dictated by
the value of its (intrinsic) *z*-spin.




Various objections have been raised against Bohm's theory (for a
detailed list and replies, see the entry on Bohmian mechanics, section
15). One main objection is that in Bohmian mechanics, the guiding
field influences the particles, but the particles do not influence the
guiding field. Another common objection is that the theory is involved
with a radical type of non-locality, and that this type of
non-locality is incompatible with relativity. While it may be very
difficult, or even impossible, to reconcile Bohm's theory with
relativity, as is not difficult to see from the above discussion, the
type of non-locality that the minimal Bohm theory postulates in the
EPR/B experiment does not seem more radical than the non-locality
postulated by the orthodox interpretation and the GRW/Pearle collapse
models. 


#### 5.3.2 Modal interpretations




Modal interpretations of quantum mechanics were designed to solve the
measurement problem and to reconcile quantum mechanics with
relativity. They are no-collapse, (typically) indeterministic
hidden-variables theories. Quantum-mechanical states of systems
(which may be construed as denoting their states or information about
these states) always evolve according to unitary and linear dynamical
equations (the Schrodinger equation in the non-relativistic
case). And the orthodox quantum-mechanical state description of
systems is supplemented by a set of properties, which depends on the
quantum-mechanical state and which is supposed to be rich enough to
account for the occurrence of definite macroscopic events and their
classical-like behavior, but sufficiently restricted to escape all the
known no-hidden-variables theorems. (For modal interpretations, see
van Fraassen 1973, 1981, 1991, chapter 9, Kochen 1985, Krips 1987,
Dieks 1988, 1989, Healey 1989, Bub 1992, 1994, 1997, Vermaas and Dieks
1995, Clifton 1995, Bacciagaluppi 1996, Bacciagaluppi and Hemmo 1996,
Bub and Clifton 1996, Hemmo 1996b, Bacciagaluppi and Dickson 1999,
Clifton 2000, Spekkens and Sipe 2001a,b, Bene and Dieks 2002, and
Berkovitz and Hemmo 2006a,b. For an entry-level review, see the entry
on modal interpretations of quantum theory. For comprehensive reviews
and analyses of modal interpretations, see Bacciagaluppi 1996, Hemmo
1996a, chapters 1-3, Dieks and Vermaas 1998, Vermaas 1999, and the
entry on modal interpretations of quantum theory. For the
no-hidden-variables theorems, see Kochen and Specker 1967,
Greenberger, Horne and Zeilinger 1989, Mermin 1990 and the entry on
the Kochen-Specker 
 theorem.)[21]




Modal interpretations vary in their property assignment. For
simplicity, we shall focus on modal interpretations in which the
property assignment is based on the so-called Schmidt
biorthogonal-decomposition theorem (see Kochen 1985, Dieks 1989, and
Healey 1989). Let *S*1 and *S*2 be
systems associated with the Hilbert spaces
*HS*1 and
*HS*2, respectively. There exist bases
{|a*i*>} and
{|b*i*>} for
*HS*1 and
*HS*2 respectively such that the state of
*S*1+*S*2 can be expressed as a
linear combination of the following form of vectors from these
bases:



> 
>  |ps*8*
> >*S*1+*S*2 =
> [?]*i ci*
> |a*i*>*S*1 
> |b*i*>*S*2.




When the absolute values of the coefficients *ci*
are all unequal, the bases
{|a*i*>} and
{|b*i*>} and the above
decomposition of |ps8
>*S*1+*S*2 are
unique. In that case, it is postulated that *S*1 has
a determinate value for each observable associated with
*HS*1 with the basis
{|a*i*>} and
*S*2 has a determinate value for each observable
associated with *HS*2 with the basis
{|b*i*>}, and |*ci*|2
provide the (ignorance)
probabilities of the possible values that these observables may
 have.[22]
 For example, suppose that the state of the L- and the R-particle in
the EPR/B experiment before the measurements is:




> 
>  |ps9> = (1/[?]2+e) 
> |*z*-up>*L*|*z*-down>*R* [?] (1/[?]2-e') 
> |*z*-down>*L*|*z*-up>*R* 






where 1/[?]2 >> e,e',
(1/[?]2+e)2+(1/[?]2-e')2
= 1, and (as before) |*z*-up>*L*
(|*z*-up>*R*) and |*z*-down>*L* (|*z*-down>*R*) denote the states of the L- (R-)
particle having *z*-spin 'up' and *z*-spin 'down',
 respectively.[23]
 Then, either the L-particle spins 'up' and the R-particle
spins 'down' in the *z*-direction, or the
L-particle spins 'down' and the R-particle spins
'up' in the *z*-direction. Thus, in contrast to the
orthodox interpretation and the GRW/Pearle collapse models, in modal
interpretations the particles in the EPR/B experiment may have
definite spin properties even before any measurement occurs.




To see how the modal interpretation accounts for the curious
correlations in EPR/B-type experiments, let us suppose that the state
of the particle pair and the measurement apparatuses at the emission
time is:




> 
> |ps10> = ((1/[?]2+e)
> |*z*-up>*L*
> |*z*-down>*R* [?]
> (1/[?]2[?]e')
> |*z*-down>*L*
> |*z*-up>*R*)
> |ready>*AL*|ready>*AR*
> 






where |ready>*AL*
(|ready>*AR*) denotes the state of the
L-apparatus (R-apparatus) being ready to measure *z*-spin. In
this state, the L- and the R-apparatus are in the definite state of
being ready to measure *z*-spin, and (similarly to the state
|ps9>) the L- and the R-particle have definite
*z*-spin properties: Either the L-particle has *z*-spin
'up' and the R-particle has *z*-spin
'down,' or the L-particle has *z*-spin
'down' and the R-particle has *z*-spin
 'up,'[24] 
 where the probability of the realization of each of these
possibilities is approximately 1/2. In the (earlier) *z*-spin
measurement on the L-particle, the state of the particle pair and the
apparatuses evolves to the state:




> 
>  |ps11> = ((1/[?]2+e) 
> |*z*-up>*L*|up>*AL*|*z*-down>*R* [?] (1/[?]2-e') 
> |*z*-down>*L*|down>*AL*|*z*-up>*R*) |ready>*AR*
> 






where (as before) |up>*AL* and
|down>*AL* denote the states of the L-apparatus
pointing to the outcomes *z*-spin 'up' and
*z*-spin 'down', respectively. In this state,
either the L-particle has a *z*-spin 'up' and the
L-apparatus points to 'up,' or the L-particle has
*z*-spin 'down' and the L-apparatus points to
'down.' And, again, the probability of each of these
possibilities is approximately 1/2. The evolution of the properties
from the state |ps10> to the state
|ps11> depends on the dynamical laws. In almost all
modal interpretations, if the particles have definite *z*-spin
properties before the measurements, the outcomes of *z*-spin
measurements will reflect these properties. That is, the evolution of
the properties of the particles and the measurement apparatuses will
be deterministic, so that the spin properties of the particles do not
change in the L-measurement and the pointer of the L-apparatus comes
to display the outcome that corresponds to the *z*-spin
property that the L-particle had before the measurement. If, for
example, before the measurements the L- and the R-particle have
respectively the properties *z*-spin 'up' and
*z*-spin 'down', the (earlier) *z*-spin
measurement on the L-particle will yield the outcome 'up'
and the spin properties of the particles will remain
unchanged. Accordingly, a *z*-spin measurement on the
R-particle will yield the outcome 'down'. Thus, in this
case the modal interpretation involves neither action at a distance
nor action\* at a distance.




However, if the measurement apparatuses are set up to measure
*x*-spin rather than *z*-spin, the evolution of the
properties of the L-particle and the L-apparatus will be
indeterministic. As before, the L-measurement will not cause any change
in the actual spin properties of the R-particle. But the L-measurement
outcome will cause an instant change in the spin dispositions of the
R-particle and the R-measurement apparatus. If, for example, the
L-measurement outcome is *x*-spin 'up' and the
L-particle comes to posses *x*-spin 'up,' then the
R-particle and the R-apparatus will have respectively
the dispositions to possess *x*-spin 'down' and to
display the outcome *x*-spin 'down' on a
*x*-spin measurement. Thus, like the minimal Bohm theory, the
modal interpretation may involve action\* at a distance in the EPR/B
experiment. But, unlike the minimal Bohm theory, here spins are
intrinsic properties of particles.




In the above modal interpretation, property composition fails: The
properties of composite systems are not decomposable into the
properties of their subsystems. Consider, again, the state
|ps10>. As 'separated' systems (i.e., in
the decompositions of the composite system of the particle
pair+apparatuses into the L-particle and the R-particle+apparatuses
and into the R-particle and the L-particle+apparatuses) the L- and the
R-particle have definite *z*-spin properties. But, as
subsystems of the composite system of the particle pair (e.g. in the
decomposition of the composite system of the particle pair+apparatuses
into the particle pair and the apparatuses), they have no definite
*z*-spin properties.




A failure of property composition occurs also in the state 
|ps11>, where the L- and the R-particle have
definite *z*-spin properties both as 'separated'
systems and as subsystems of the particle pair (though in contrast with 
|ps10>, in |ps11> the range
of the possible properties of the particles as separated systems and
as subsystems of the pair is the same). For nothing in the above property
assignment implies that in |ps11> the spin
properties that the L-particle has as a 'separated' system
and the spin properties that it has as a subsystem of the particle pair
be the same: The L-particle may have *z*-spin 'up'
as a separated system and *z*-spin 'down' as a
subsystem of the particle pair.




Furthermore, the dynamics of the properties that the L-particle
(R-particle) has as a separated system and the dynamics of its
properties as a subsystem of the particle pair are generally
 different.[25]
 Consider, again, the state |ps10>. In the (earlier)
*z*-spin measurement on the L-particle, the spin properties
that the L-particle has as a separated system follow a deterministic
evolution -- the L-particle has either *z*-spin
'up' or *z*-spin 'down' before and
after the L-measurement; whereas as a subsystem of the particle pair,
the spin properties of the L-particle follow an indeterministic
evolution -- the L-particle has no definite spin properties
before the L-measurement and either *z*-spin 'up'
(with approximately chance 1/2) or *z*-spin
'down' (with approximately chance 1/2) after the
L-measurement.




The failure of property composition implies that the quantum realm as
depicted by the above version of the modal interpretation involves
state non-separability and property and relational holism. State
separability fails because the state of the particle pair is not
generally determined by the separate states of the particles. Indeed,
as is easily shown, the actual properties that the L- and the
R-particle each has in the state |ps9> are also
compatible with product states in which the L- and the R-particle are
not entangled. Property and relational holism fail because in the
state |ps9> the properties of the pair do not
supervene upon the properties of its subsystems and the spatiotemporal
relations between them. Furthermore, process separability fails for
similar reasons. 




The failure of property composition in the modal interpretation
calls for explanation. It may be tempting to postulate that the
properties that a system (e.g. the L-particle) has, as a separated
system, are the same as the properties that it has as a subsystem of
composite systems. But, as Bacciagaluppi (1995) and Clifton (1996a)
have shown, such property assignment will be inconsistent: It will be
subject to a Kochen and Specker-type contradiction. Furthermore, as
Vermaas (1997) demonstrates, the properties of composite systems and
the properties of their subsystems cannot be correlated (in ways
compatible with the Born rule).




For what follows in the rest of this subsection, the views of
different authors differ widely. Several variants of modal
interpretations were developed in order to fix the problem of the
failure of property composition. The most natural explanation of the
failure of property composition is that quantum states assign
relational rather than intrinsic properties to systems (see Kochen
1985, Bene and Dieks 2002, and Berkovitz and Hemmo 2006a,b). For
example, in the relational modal interpretation proposed by Berkovitz
and Hemmo (2006a,b), the main idea is that quantum states assign
properties to systems only relative to other systems, and properties
of a system that are related to different systems are generally
different. In particular, in the state |ps10> the
L-particle has a definite *z*-spin property relative to the
R-particle, the measurement apparatuses and the rest of the universe,
but (as a subsystem of the particle pair) it has no definite
*z*-spin relative to the measurement apparatuses and the rest
of
 universe.[26]
 On this interpretation, the properties of systems are highly
non-local by their very nature. Properties like pointing to
'up' and pointing to 'down' are not intrinsic
to the measurement apparatuses. Rather, they are relations between
the apparatuses and other systems. For example, the property of the
L-apparatus pointing to 'up' relative to the particle
pair, the R-apparatus and the rest of the universe is not intrinsic to
the L-apparatus; it is a relation between the L-apparatus and the
particle pair, the R-apparatus and the rest of the universe. As such,
this property is highly non-local: It is located in neither the L-wing
nor any other subregion of the universe. Yet, due to the dynamical
laws, properties like the position of pointers of measurement
apparatuses, which appear to us to be local, behave like local
properties in any experimental circumstances, and accordingly this
radical type of non-locality is unobservable (for more details, see
Berkovitz and Hemmo 2006b, sections 8.1 and 9).




Another way to try to explain the failure of property composition is
to interpret the properties of composite systems as holistic,
non-decomposable properties. On this interpretation, the
*z*-spin 'up' property that the L-particle has as a
subsystem of the particle pair in the state |ps9> is
completely different from the *z*-spin 'up'
property that the L-particle has as a separated system, and the use of
the term '*z*-spin up' in both cases is misleading
(for more details, see Berkovitz and Hemmo 
 2006a).[27]




The relational and holistic interpretations of properties mark a
radical shift from the standard interpretation of properties in
orthodox quantum mechanics. Other advocates of the modal interpretation
have chosen not to follow this interpretation, and opted for a modal
interpretation that does not violate property composition. While the
property assignment above does not assume any preferred partition of
the universe (the partition of the universe into a particle pair and
the rest of the universe is as good as the partition of the universe
into the L-particle and the rest of the universe), proponents of
property composition postulated that there is a preferred partition of
the universe into 'atomic' systems and accordingly a
preferred factorization of the Hilbert space of the universe. This
preferred factorization is supposed to be the basis for the
'core' property assignment: Properties are prescribed to
atomic systems according to a property assignment that is a
generalization of the bi-orthogonal decomposition property
 assignment.[28]
 And the properties of complex systems are postulated to be
compositions of the properties of their atomic systems (see the entry
on modal interpretations of quantum theory, section 2, and
Bacciagaluppi and Dickson 1999). The challenge for this atomic modal
interpretation is to justify the assumption that there is a preferred
partition of the universe, and to provide some idea about how such
factorization should look like.




Finally, while the modal interpretation was designed to solve the
measurement problem and reconcile quantum mechanics with special
relativity, it faces challenges on both accounts. First, in certain
imprefect measurements (where there are imprefections in the coupling
between the measured system and the pointer of the measurement
apparatus and/or the pointer and the environment), modal
interpretations that are based on the Schmidt
biorthogonal-decomposition theorem (and more generally the spectral
decomposition theorem) fail to account for definite measurement
outcomes, in contradiction to our experience (see Bacciagaluppi and
Hemmo 1996 and Bacciagaluppi 2000). For versions of the modal
interpretations that seem to escape this problem, see Van Fraassen
(1973, 1991), Bub (1992, 1997), Bene and Dieks (2002) and Berkovitz
and Hemmo (2006a,b). Second, as we shall see in section 10.2, a number
of no-go theorems challenge the view that modal interpretations could
be genuinely relativistic.


#### 5.3.3 Everett-like interpretations




In 1957, Everett proposed a new no-collapse interpretation of orthodox
quantum mechanics (see Everett 1957a,b, 1973, Barrett 1999, the entry
on Everett's relative-state formulation of quantum mechanics, the
entry on the many-worlds interpretation of quantum mechanics, and
references therein). The Everett interpretation is a no-collapse
interpretation of quantum mechanics, where the evolution of quantum
states is always according to unitary and linear dynamical equations
(the Schrodinger equation in the non-relativistic case). In this
interpretation, quantum states are fundamentally relative. Systems have
relative states, which are derivable from the various branches of the
entangled states. For example, consider again |ps11>. 




> 
>  |ps11> = (1/[?]2+e) 
> |*z*-up>*L*|up>*AL*|*z*-down>*R* |ready>*AR*
> [?] (1/[?]2-e') 
> |*z*-down>*L*|down>*AL*|*z*-up>*R* |ready>*AR*.
>  






In this quantum-mechanical state, the L-apparatus is in the state of
pointing to the outcome *z*-spin 'up'
*relative* to the L-particle being in the state *z*-spin
'up,' the R-particle being in the state *z*-spin
'down' and the R-apparatus being ready to measure
*z*-spin; and in the state of pointing to the outcome
*z*-spin 'down' *relative* to the L-particle
being in the state *z*-spin 'down,' the R-particle
being in the state *z*-spin 'up' and the
R-apparatus being ready to measure *z*-spin. Likewise, the
L-particle is in the state *z*-spin 'up' relative
to the L-apparatus being in the state of pointing to the outcome
*z*-spin 'up,' the R-particle being in the state
*z*-spin 'down' and the R-apparatus being ready to
measure *z*-spin; and in the state *z*-spin
'down' relative to the L-apparatus being in the state of
pointing to the outcome *z*-spin 'down,' the
R-particle being in the state *z*-spin 'up' and the
R-apparatus being ready to measure *z*-spin. And similarly,
*mutatis mutandis*, for the relative state of the R-particle
and the R-apparatus.




Everett's original formulation left the exact meaning of these
relative states and their relations to observers' experience and
beliefs open, and there have been different Everett-like
interpretations of these states. Probably the most popular reading of
Everett is the splitting-worlds interpretation (see DeWitt 1971,
Everett's relative-state formulation of quantum mechanics, Barrett
1999, and references therein). In the splitting-worlds interpretation,
each of the branches of the state |ps11> refers to a
different class of worlds (all of which are real) where the states of
the L-apparatus, R-apparatus and the particles are all separable:
Class-1 worlds in which the L-particle is in the state *z*-spin
'up,' the R-particle is in the state *z*-spin
'down,' the L-apparatus is in the state of pointing to the
outcome *z*-spin 'up' and the R-apparatus in the
state of being ready to measure *z*-spin; and class-2 worlds in
which the L-particle is in the state *z*-spin
'down,' the R-particle is in the state *z*-spin
'up,' the L-apparatus is in the state of pointing to the
outcome *z*-spin 'down' and the R-apparatus is in
the state of being ready to measure *z*-spin. More generally,
each term in state of the universe, as represented in a certain
preferred basis, reflects the states of its systems in some class of
worlds; where the range of the different classes of worlds increases
whenever the number of the terms in the quantum state (in the
preferred basis) increases (this process is called
'splitting').




The splitting-worlds reading of Everett faces a number of challenges.
First, supporters of the Everett interpretation frequently motivate
their interpretation by arguing that it postulates the existence of
neither a controversial wave collapse nor hidden variables, and it
leaves the simple and elegant mathematical structure of quantum
mechanics intact. But, the splitting-worlds interpretation adds extra
structure to no-collapse orthodox quantum mechanics. Further, this
interpretation marks a radical shift from orthodox quantum
mechanics. A scientific theory is not constituted only by its
mathematical formalism, but also by the ontology it postulates, the
way it depicts the physical realm and the way it accounts for our
experience. The many parallel worlds ontology of the splitting-worlds
interpretation and its account of our experience are radically
different from the ontology of the intended interpretation of orthodox
quantum mechanics and its account for our experience. Second, relative
states are well defined in any basis, and the question arises as to
which basis should be preferred and the motivation for selecting one
particular basis over others. Third, in the splitting-worlds
interpretation each of the worlds in the universe may split into two
or more worlds, and the problem is that (similarly to the collapse in
orthodox collapse quantum mechanics) there are no clear criteria for
when a splitting occurs and how long it takes. Fourth, there is the
question of how the splitting-worlds interpretation accounts for the
statistical predictions of the orthodox theory. In the Everett-like
interpretations in general, and in the splitting-worlds interpretation
in particular, all the possible measurement outcomes in the EPR/B
experiment are realized and may be observed. Thus, the question
arises as to the meaning of probabilities in this interpretation. For
example, what is the meaning of the statement that in the state
|ps10> (see section 5.3.2) the probability of the
L-measurement apparatus pointing to the outcome 'up' in an
earlier *z*-spin measurement on the L-particle is
(approximately) 1/2? In the splitting-worlds interpretation the
probability of that outcome appears to be 1! Furthermore, setting
aside the problem of interpretation, there is also the question of
whether the splitting-worlds interpretation, and more generally
Everett-like interpretations, can account for the particular values of
the quantum probabilities of measurement outcomes. Everett claimed to
derive the Born probabilities in the context of his
interpretation. But this derivation has been controversial. (For
discussions of the meaning of probabilities, or more precisely the
meaning of the coefficients of the various terms in quantum states, in
Everett-like interpretations, see Butterfield 1996, Lockwood 1996a,b,
Saunders 1998, Vaidman 1998, Barnum et al. 2000, Bacciagaluppi 2002,
Gill 2003, Hemmo and Pitowsky 2003, 2005, Wallace 2002, 2003, 2005a,b,
Greaves 2004 and Saunders 2004, 2005.) 




Other readings of Everett include the many-minds interpretation
(Albert and Loewer 1988, Barrett 1999, chapter 7), the
consistent-histories approach (Gell-Mann and Hartle 1990), the
Everett-like relational interpretation (Saunders 1995, Mermin 1998)
and (what may be called) the many-structures interpretation (Wallace
2005c). While these readings address more or less successfully the
problems of the preferred basis and splitting, except for the
many-minds interpretation of Albert and Loewer the question of whether
there could be a satisfactory interpretation of probabilities in the
context of these theories and the adequacy of the derivation of the
Born probabilities are still a controversial issue (see Deutsch 1999,
Wallace 2002, 2003, Lewis 2003, Graves 2004, Saunders 2004, Hemmo and
Pitowsky 2005, and Price 2006).




What kind of non-locality do Everett-like interpretations involve?
Unfortunately, the answer to this question is not straightforward, as
it depends on one's particular reading of the Everett
interpretation. Indeed, all the above readings of Everett seem to treat
the no-collapse wave function of the universe as a real physical entity
that reflects the non-separable state of the universe, and accordingly
they involve state non-separability. But, one may reasonably expect
that different readings depict different pictures of physical reality
and accordingly might postulate different kinds of non-locality. Thus,
any further analysis of the type of non-locality postulated by each of
these readings requires a detailed study of their ontology (which we
plan to conduct in future updates of this entry).




For example, the question of action at a distance in the EPR/B
experiment may arise in the context of the splitting-worlds interpretation,
but not in the context of Albert and Loewer's many-minds
interpretation. Albert and Loewer's interpretation takes the bare
no-collapse orthodox quantum mechanics to be the complete theory of the
physical realm. Accordingly, the L-apparatus in the state 
|ps11> does not display any definite outcome. Yet,
in order to account for our experience of a classical-like world, where
at the end of measurements observers are typically in mental states of
perceiving definite outcomes, the many-minds interpretation appeals to
a dualism of mind-body. Each observer is associated with a continuous
infinity of non-physical minds. And while the physical state of the
world evolves in a completely deterministic manner according to the
Schrodinger evolution, and the pointers of the measurement
apparatuses in the EPR/B experiment display no definite outcomes,
states of minds evolve in a genuinely indeterministic fashion so as to
yield an experience of perceiving definite measurement outcomes. For
example, consider again, the state |ps10>. While
in a first *z*-spin L-measurement, this state evolves
deterministically into the state |ps11>, minds of
observers evolve indeterministically into either the state of
perceiving the outcome *z*-spin 'up' or the state of
perceiving the outcome *z*-spin 'down' with the
usual Born-rule probabilities (approximately 50% chance for each of
these outcomes). Since in this state the L-particle has no definite
spin properties and the L-apparatus points to no definite measurement
outcome, and since in the later *z*-spin measurement on the
R-particle the R-particle does not come to possess any definite spin
properties and the R-apparatus points to no definite spin outcome, the
question of whether there is action at a distance between the L-particle and the
L-apparatus on the one hand and the R-particle and the R-apparatus on
the other does not arise.


## 6. Superluminal causation




In all the above interpretations of quantum mechanics, the failure of
factorizability (i.e., the failure of the joint probability of the
measurement outcomes in the EPR/B experiment to factorize into their
single probabilities) involves non-separability, holism and/or some
type of action at a distance. As we shall see below,
non-factorizability also implies superluminal causal dependence
according to certain accounts of causation.




First, as is not difficult to show, the failure of factorizability
implies superluminal causation according to various probabilistic
accounts of causation that satisfy Reichenbach's (1956, section 19)
principle of the common cause (for a review of this principle, see the
entry on Reichenbach's principle of the common cause).




Here is why. Reichenbach's principle may be formulated as follows:



> 
> 
> **PCC (Principle of the Common Cause).** For any
> correlation between two (distinct) events which do not cause each
> other, there is a common cause that screens them off from each other.
> Or formally: If distinct events *x* and *y* are
> correlated, i.e.,
> 
> 
> 
> > 
> >  (Correlation)   
> > 
> > *P*(*x* & *y*) 
> > [?] *P*(*x*) * *P*(*y*),
> >  
> 
> 
> 
> 
> and they do not cause each other, then their common cause,
> *CC*(*x*,*y*), screens them off from each other, i.e.,
> 
> 
> 
> > 
> > (Screening Off)   
> > 
> > 
> > 
> > |  |  |
> > | --- | --- |
> > | *P**CC*(*x*,*y*)(*x*/*y*) =
> > *P**CC*(*x*,*y*)(*x*) | *P**CC*(*x*,*y*)(*y*)
> >  [?] 0 |
> > | *P**CC*(*x*,*y*)(*y*/*x*) =
> > *P**CC*(*x*,*y*)(*y*) | *P**CC*(*x*,*y*)(*x*) [?] 
> >  0.[29] |
> > 
> > 
> > 
> 
> 
> 
> 
> Accordingly, *CC*(*x*,*y*) renders *x* and
> *y* probabilistically independent, and the joint probability of
> *x* and *y* factorizes upon
> *CC*(*x*,*y*):
> 
> 
> 
> > 
> > *P**CC*(*x*,*y*)(*x* & *y*) =
> > *P**CC*(*x*,*y*)(*x*) *
> > *P**CC*(*x*,*y*)(*y*).
> >  
> 
> 
> 




The above formulation of PCC is mainly intended to cover cases in
which *x* and *y* have no partial, non-common causes. But
PCC can be generalized as follows:



> 
> 
> **PCC\*.** The joint probability of any distinct,
> correlated events, *x* and *y*, which are not causally
> connected to each other, factorizes upon the union of their partial
> (separate) causes and their common cause. That is, let
> *CC*(*x*,*y*) denote the common causes of
> *x* and *y*, and *PC*(*x*) and
> *PC*(*y*) denote respectively their partial causes. Then,
> the joint probability of *x* and *y* *factorizes*
> *upon the Union of their Causal Pasts* (henceforth, FactorUCP),
> i.e., on the union of *PC*(*x*), *PC*(*y*)
> and *CC*(*x*,*y*):
> 
> 
> 
> 
> **FactorUCP**   
> 
> *P**PC*(*x*) *PC*(*y*)
>  *CC*(*x*,*y*)
>  (*x* & *y*) =
> *P**PC*(*x*)
>  *CC*(*x*,*y*) (*x*) *
> *P**PC*(*y*) 
> *CC*(*x*,*y*) (*y*).
> 
> 
> 
> 
> 




Like PCC, the basic idea of FactorUCP is that the objective
probabilities of events that do not cause each other are determined by
their causal pasts, and given these causal pasts they are
probabilistically independent of each other. As is not difficult to
see, factorizability is a special case of FactorUCP. That is, to obtain
factorizability from FactorUCP, substitute l for
*CC*(*x*,*y*), *l* for
*PC*(*x*) and *r* for
*PC*(*y*). FactorUCP and the assumption that the
probabilities of the measurement-outcomes in the EPR/B experiment are
determined by the pair's state and the settings of the
measurement apparatuses jointly imply factorizability. Thus, given this
later assumption, the failure of factorizability implies superluminal
causation between the distant outcomes in the EPR/B experiment
according to any account of causation that satisfies FactorUCP (for
some examples of such accounts, see Butterfield 1989 and Berkovitz
1995a, 1995b, section 6.7, 
 1998b).[30]




Superluminal causation between the distant outcomes also exists
according to various counterfactual accounts of causation, including
accounts that do not satisfy FactorUCP. In particular, in Lewis's
(1986) influential account, counterfactual dependence between distinct
events implies causal dependence between them. And as Butterfield
(1992b) and Berkovitz (1998b) demonstrate, the violation of
Factorizability involves a counterfactual dependence between the
distant measurement outcomes in the EPR/B experiment.




But the violation of factorizability does not imply superluminal
causation according to some other accounts of causation. In
particular, in process accounts of causation there is no superluminal
causation in the EPR/B experiment. In such accounts, causal dependence
between events is explicated in terms of continuous processes in space
and time that transmit 'marks' or conserved quantities
from the cause to the effect (see Salmon 1998, chapters 1, 12, 16 and
18, Dowe 2000, the entry on causal processes, and references
therein). Thus, recalling (sections 1, 2, 4 and 5) that none of the
interpretations of quantum mechanics and alternative quantum theories
postulates any (direct) continuous process between the distant
measurement events in the EPR/B experiment, there is no superluminal
causation between them according to process accounts of causation.


## 7. Superluminal signaling




Whether or not the non-locality predicted by quantum theories may be
classified as action at a distance or superluminal causation, the
question arises as to whether this non-locality could be exploited to
allow superluminal (i.e., faster-than-light) signaling of information.
This question is of particular importance for those who interpret
relativity as prohibiting any such superluminal signaling. (We shall
return to discuss this interpretation in section 10.)




Superluminal signaling would require that the state of nearby
controllable physical objects (say, a keyboard in my computer)
superluminally influence distant observable physical phenomena (e.g. a
pattern on a computer screen light years away). The influence may be
deterministic or indeterministic, but in any case it should cause a
detectable change in the statistics of some distant physical
quantities.




It is commonly agreed that in quantum phenomena, superluminal
signaling is impossible in practice. Moreover, many believe that such
signaling is excluded in principle by the so-called 'no-signaling
theorem' (for proofs of this theorem, see Eberhard 1978,
Ghirardi, Rimini and Weber 1980, Jordan 1983, Shimony 1984, Redhead
1987, pp. 113-116 and 118). It is thus frequently claimed with respect
to EPR/B experiments that there is no such thing as a Bell telephone,
namely a telephone that could exploit the violation of the Bell
inequalities for superluminal signaling of
 information.[31]




The no-signaling theorem demonstrates that orthodox quantum
mechanics excludes any possibility of superluminal signaling in the
EPR/B experiment. According to this theory, no controllable physical
factor in the L-wing, such as the setting of the L-measurement
apparatus, can take advantage of the entanglement between the systems
in the L- and the R-wing to influence the statistics of the measurement
outcomes (or any other observable) in the R-wing. As we have seen in
section 5.1.1, the orthodox theory is at best incomplete. Thus, the
fact that it excludes superluminal signaling does not imply that other
quantum theories or interpretations of the orthodox theory also exclude
such signaling. Yet, if the orthodox theory is empirically adequate, as
the consensus has it, its statistical predictions obtain, and
accordingly superluminal signaling will be excluded as a matter of
fact; for if this theory is empirically adequate, any quantum theory
will have to reproduce its statistics, including the exclusion of any
actual superluminal signaling.




But the no-signaling theorem does not demonstrate that superluminal
signaling would be impossible if orthodox quantum mechanics were not
empirically adequate. Furthermore, this theorem does not show that
superluminal signaling is in principle impossible in the quantum realm
as depicted by other theories, which *actually* reproduce the
statistics of orthodox quantum mechanics but do not prohibit in theory
the violation of this statistics. In sections 7.2-7.3, we shall
consider the in-principle possibility of superluminal signaling in
certain collapse and no-collapse interpretations of quantum mechanics.
But, first, we need to consider the necessary and sufficient conditions
for superluminal signaling.


### 7.1 Necessary and sufficient conditions for superluminal signaling




To simplify things, in our discussion we shall focus on
non-factorizable models of the EPR/B experiment that satisfy
l-independence (i.e., the assumption that the distribution of
the states l is independent of the settings of the measurement
apparatuses). Superluminal signaling in the EPR/B experiment would be
possible in theory just in case the value of some controllable
physical quantity in the nearby wing could influence the statistics of
measurement outcomes in the distant wing. And in non-factorizable
models that satisfy l-independence this could happen just in
case the following conditions obtained:



> 
> **Controllable probabilistic dependence.** The
> probabilities of distant measurement outcomes depend on some nearby
> *controllable* physical quantity. 
>  



> 
> **l****-distribution.** There can be in theory an ensemble of particle pairs the states of which deviate from the quantum-equilibrium distribution; where the quantum-equilibrium distribution of pairs' states is the distribution that reproduces the predictions of orthodox quantum mechanics.




Four comments: (i) In controllable probabilistic dependence, the term
'probabilities of measurement outcomes' refers to the model
probabilities, i.e., the probabilities that the states l
prescribe for measurement outcomes. 

  
 

(ii) Our discussion in this entry focuses on models of the EPR/B
experiment in which probabilities of measurement outcomes depend only
on the pair's state l and the settings of the measurement
apparatuses to measure certain properties. In such models, parameter
dependence (i.e., the dependence of the probability of the distant
measurement outcome on the setting of the nearby measurement
apparatus) is a necessary and sufficient condition for controllable
probabilistic dependence. But, recall (footnote 3) that in some models
of the EPR/B experiment, in addition to the pair's state and the
setting of the L- (R-) measurement apparatus there are other local
physical quantities that may be relevant for the probability of the L-
(R-) measurement outcome. In such models, parameter dependence is not
a necessary condition for controllable probabilistic dependence. Some
other physical quantities in the nearby wing may be relevant for the
probability of the distant measurement outcome. (That is, let a
and b denote all the relevant local physical quantities, other
than the settings of the measurement apparatuses, that may be relevant
for the probability of the L- and the R-outcome, respectively. Then,
controllable probabilistic dependence would obtain if for some pairs'
states l, L-setting *l*, R-setting *r* and local
physical quantities a and b, *P*l
*l r a b*(*yr*) [?]
*P*l *l r
b*(*yr*) obtained.) For the relevance
of such models to the question of the in-principle possibility of
superluminal signalling in some current interpretations of quantum
mechanics, see sections 7.3 and 7.4. 

  
 

(iii) The quantum-equilibrium distribution will not be the same in all
models of the EPR/B experiment; for in general the states l
will not be the same in different models. 

  
 

(iv) In models that actually violate both controllable probabilistic
dependence and l-distribution, the occurrence of controllable
probabilistic dependence would render the actual distribution of
l states as non-equilbrium distribution. Thus, if controllable
probabilistic dependence occurred in such models, the actual
distribution of l states would satisfy
l-distribution. 




The argument for the necessity of controllable probabilistic
dependence and l-distribution is straightforward. Granted
l-independence, if the probabilistic dependence of the distant
outcome on a nearby physical quantity is not controllable, there can
be no way to manipulate the statistics of the distant outcome so as to
deviate from the statistical predictions of quantum
mechanics. Accordingly, superluminal transmission of information will
be impossible even in theory. And if l-distribution does not
hold, i.e., if the quantum-equilibrium distribution holds, controllable
probabilistic dependence will be of no use for superluminal
transmission of information. For, averaging over the model
probabilities according to the quantum-equilbrium distribution, the
model will reproduce the statistics of orthodox quantum
mechanics. That is, the distribution of the l-states will be
such that the probabilistic dependence of the distant outcome on the
nearby controllable factor will be washed out: In some states the
nearby controllable factor will raise the probability of the distant
outcome and in others it will decrease this probability, so that on
average the overall statistics of the distant outcome will be
independent of the nearby controllable factor (i.e., the same as the
statistics of orthodox quantum mechanics). Accordingly, superluminal
signaling will be impossible.




The argument for the sufficiency of these conditions is also
straightforward. If l-distribution held, it would be possible
in theory to arrange ensembles of particle pairs in which controllable
probabilistic dependence would not be washed out, and accordingly the
statistics of distant outcomes would depend on the nearby controllable
factor. (For a proof that these conditions are sufficient for
superluminal signaling in certain deterministic hidden variables
theories, see Valentini 2002.)




Note that the necessary and sufficient conditions for superluminal
signaling are different in models that do not exclude in theory the
violation of l-independence. In such models controllable
probabilistic dependence is not a necessary condition for superluminal
signaling. The reasoning is as follows. Consider any empirically
adequate model of the EPR/B experiment in which the pair's state and
the settings of the measurement apparatuses are the only relevant
factors for the probabilities of measurement outcomes, and the
quantum-equilibrium distribution is l-independent. In such a
model, parameter independence implies the failure of controllable
probabilistic dependence, yet the violation of l-independence
would imply the possibility of superluminal signaling: If
l-independence failed, a change in the setting of the nearby
measurement apparatus would cause a change in the distribution of the
states l, and a change in this distribution would induce a
change in the statistics of the distant (space-like separated)
measurement outcome. 




Leaving aside models that violate l-independence, we now turn
to consider the prospects of controllable probabilistic dependence and
l-distribution, starting with no-collapse interpretations.


### 7.2 No-collapse theories


#### 7.2.1 Bohm's Theory




Bohm's theory involves parameter dependence and thus controllable
probabilistic dependence: The probabilities of distant outcomes depend
on the setting of the nearby apparatus. In some pairs' states
l, i.e., in some configurations of the positions of the particle
pair, a change in the apparatus setting of the (earlier) say
L-measurement will induce an immediate change in the probability of
the *R*-outcome: e.g. the probability of *R*-outcome
*z*-spin 'up' will be 1 if the L-apparatus is set
to measure *z*-spin and 0 if the L-apparatus is switched off
(see section 5.3.1). Thus, the question of superluminal signaling
turns on whether l-distribution obtains.




Now, recall (section 5.3.1) that Bohm's theory reproduces the quantum
statistics by postulating the quantum-equilibrium distribution over
the positions of particles. If this distribution is not an accidental
fact about our universe, but rather obtains as a matter of law,
superluminal signaling will be impossible in principle. Durr,
Goldstein and Zanghi (1992a,b, 1996, fn. 15) argue that, while
the quantum-equilibrium distribution is not a matter a law, other
distributions will be possible but atypical. Thus, they conclude that
although superluminal signaling is not impossible in theory, it may
occur only in atypical worlds. On the other hand, Valentini (1991a,b,
1992, 1996, 2002) and Valentini and Westman 2004) argue that there are
good reasons to think that our universe may well have started off in a
state of quantum non-equilibrium and is now approaching gradually a
state of equilibrium, so that even today some residual non-equilibrium
must be
 present.[32] 
 Yet, even if such residual non-equilbrium existed, the question is
whether it would be possible to access any ensemble of systems in a
non-equilbrium distribution. 


#### 7.2.2 Modal interpretations




The presence or absence of parameter independence (and accordingly the
presence or absence of controllable probabilistic dependence) in the
modal interprtation is a matter of controversy, perhaps due in part to
the multiplicity of versions of this interpretation. Whether or not
modal interpretations involve parameter dependence would probably
depend on the dynamics of the possessed properties. At least some of
the current modal interpretations seem to involve no parameter
dependence. But, as the subject editor pointed out to the author, some
think that the no-go theorem for relativistic modal interpretation due
to Dickson and Clifton (1998) implies the existence of parameter
dependence in all the interpretations to which this theorem is
applicable. Do modal interpretations satisfy l-distribution?
The prospects of this condition depend on whether the possessed
properties that the modal interpretation assigns in addition to the
properties prescribed by the orthodox interpretation, are
controllable. If these properties were controllable at least in
theory, l-distribution would be possible. For example, if the
possessed spin properties that the particles have at the emission from
the source in the EPR/B experiment were controllable, then
l-distribution would be possible. The common view seems to be
that these properties are uncontrollable. 


### 7.3 Collapse theories


#### 7.3.1 Dynamical models for state-vector reduction




In the GRW/Pearle collapse models, wave functions represent the most
exhaustive, complete specification of states of individual
systems. Thus, pairs prepared with the same wave function have always
the same l state -- a state that represents their
quantum-equilbrium distribution for the EPR/B experiment. Accordingly,
l-distribution fails. Do these models involve controllable
probabilistic dependence?




Recall (section 5.1.2) that there are several models of state
reduction in the literature. One of these models is the so-called
non-linear Continuous Stochastic Localization (CSL) models (see Pearle
1989, Ghirardi, Pearle and Rimini 1990, Butterfield et al. 1993, and
Ghirardi et al. 1993). Butterfield et al. (1993) argue that in these
models there is a probabilistic dependence of the outcome of the
R-measurement on the process that leads to the (earlier) outcome of
the L-measurement. In these models, the process leading to the
L-outcome (either *z*-spin 'up' or *z*-spin
'down') depends on the interaction between the L-particle
and the L-apparatus (which results in an entangled state), and the
specific realization of the stochastic process that strives to
collapse this macroscopic superposition into a product state in which
the L-apparatus displays a definite outcome. And the probability of
the R-outcome depends on this process. For example, if this process is
one that gives rise to a *z*-spin 'up' (or renders
that outcome more likely), the probability of R-outcome
*z*-spin 'up' is 0 (more likely to be 0); and if
this process is one that gives rise to a *z*-spin
'down' (or renders that outcome more likely), the
probability of R-outcome *z*-spin 'down' is 0 (more
likely to be 0). The question is whether there are controllable
factors that influence the probability of realizations of stochastic
processes that lead to a specific L-outcome, so that it would be
possible to increase or decrease the probability of the R-outcome. If
such factors existed, controllable probabilistic dependence would be
possible at least in theory. And if this kind of controllable
probabilistic dependence existed, l-distribution would also
obtain; for if such dependence existed, the actual distribution of
pairs' states (in which the pair always have the same state, the
quantum-mechanical state) would cease to be the quantum-equilbrium
distribution.


### 7.4 The prospects of *controllable probabilistic dependence*




In section 7.3.1, we discussed the question of the in-principle
controllability of local measurement processes and in particular the
probability of their outcome, and the implications of such
controllability for the in-principle possibility of superluminal
signaling in the context of the CSL models. But this question is not
specific to the CSL model and (more generally) the dynamical models
for state-vector reduction. It seems likely to arise also in other
quantum theories that model measurements realistically. Here is
why. Real measurements take time. And during that time, some physical
variable, other than the state of the measured system and the setting
of the measurement apparatus, might influence the chance (i.e., the
single-case objective probability) of the measurement outcome. In
particular, during the L-measurement in the EPR/B experiment, the
chance of the L-outcome *z*-spin 'up'
('down') might depend on the value of some physical
variable in the L-wing, other than the state of the particle pair and
the setting of the L-measurement apparatus. If so, it will follow from
the familiar perfect anti-correlation of the singlet state that the
chance of R-outcome *z*-spin 'up'
('down') will depend on the value of such variable (for
details, see Kronz 1990a,b, Jones and Clifton 1993, pp. 304-305, and
Berkovitz 1998a, section 4.3.4). Thus, if the value of such a variable
were controllable, controllable probabilistic dependence would
obtain.


### 7.5 Superluminal signaling and action-at-a-distance




If superluminal signaling were possible in the EPR/B experiment in any
of the above theories, it would not require any continuous process in
spacetime to mediate the influences between the two distant
wings. Indeed, in all the current quantum theories in which the
probability of the R-outcome depends on some controllable physical
variable in the L-wing, this dependence is not due to a continuous
process. Rather, it is due to some type of 'action' or (to
use Shimony's (1984) terminology) 'passion' at a distance,
which is the 'result' of the holistic nature of the
quantum realm, the non-separability of the state of entangled systems,
or the non-separable nature of the evolution of the properties of
systems. 


## 8. The analysis of factorizability: implications for quantum non-locality




In sections 5-7, we considered the nature of quantum non-locality as
depicted by theories that violate factorizability, i.e., the assumption
that the probability of joint measurement outcomes factorizes into the
single probabilities of these outcomes. Recalling section 3,
factorizability can be analyzed into a conjunction of two conditions:
OI (outcome independence)--the probability of a distant
measurement outcome in the EPR/B experiment is independent of the
nearby measurement outcome; and PI (parameter independence)--the
probability of a distant measurement outcome in the EPR/B experiment
is independent of the setting of the nearby measurement
apparatus. Bohm's theory violates PI, whereas other mainstream quantum
theories satisfy this condition but violate OI. The question arises as
to whether violations of PI involve a different kind of non-locality
than violations of OI. So far, our methodology was to study the
nature of quantum non-locality by analyzing the way various quantum
theories account for the curious correlations in the EPR/B
experiment. In this section, we shall focus on the question of whether
quantum non-locality can be studied in a more general way, namely by
analyzing the types of non-locality involved in violations of PI and
in violations of OI, independently of how these violations are
realized.


### 8.1 Non-separability, holism and action at a distance




It is frequently argued or maintained that violations of OI involve
state non-separability and/or some type of holism, whereas violations
of PI involve action at a distance. For notable examples, Howard
(1989) argues that spatiotemporal separability (see section 4.3)
implies OI, and accordingly a violation of it implies spatiotemporal
non-separability; Teller (1989) argues that particularism (see section
4.3) implies OI, and thus a violation of it implies relational holism;
and Jarrett (1984, 1989) argues that a violation of PI involves some
type of action at a distance. These views are controversial,
however.




First, as we have seen in section 5, in quantum theories the
violation of either of these conditions involves some type of
non-separability and/or holism.




Second, the explicit attempts to derive OI from separability or
particularism seem to rely (implicitly) on some locality conditions.
Maudlin (1998, p. 98) and Berkovitz (1998a, section 6.1) argue that
Howard's precise formulation of spatiotemporal separability embodies
both separability and locality conditions, and Berkovitz (1998a,
section 6.2) argues that Teller's derivation of OI from particularism
implicitly relies on locality conditions. Thus, the violation of OI
*per se* does not imply non-separability or holism.




Third, a factorizable model, i.e., model that satisfies OI, may be
non-separable (Berkovitz 1995b, section 6.5). Thus, OI cannot be simply
identified with separability.




Fourth, Howard's spatiotemporal separability condition (see section
4.3) requires that states of composite systems be determined by the
states of their subsystems. In particular, spatiotemporal separability
requires that joint probabilities of outcomes be determined as some
function of the single probabilities of these outcomes. Winsberg and
Fine (2003) object that as a separability condition, OI arbitrarily
restricts this function to be a product function. And they argue that
on a weakened formalization of separability, a violation of OI is
compatible with separability. Fogel (2004) agrees that Winsberg and
Fine's weakened formalization of separability is correct, but argues
that, when supplemented by a certain 'isotropy' condition,
OI implies this weakened separability condition. Fogel believes that
his suggested 'isotropy' condition is very plausible, but,
as he acknowledges, this condition involves a nontrivial measurement
 context-independence.[33]




Fifth, as the analysis in section 5 demonstrates, violations of OI
might involve action at a distance. Also, while the minimal Bohm
theory violates PI and arguably some modal interpretations do not, the
type of action at a distance they postulate, namely action\* at a
distance (see section 5.2), is similar: In both cases, an earlier
spin-measurement in (say) the L-wing does not induce any immediate
change in the intrinsic properties of the R-particle. The
L-measurement only causes an immediate change in the dispositions of
the R-particle--a change that may influence the behavior of the
R-particle in future spin-measurements in the R-wing. But, this change
of dispositions does not involve any change of local properties in the
R-wing, as these dispositions are relational (rather than intrinsic)
properties of the R-particle. Furthermore, the action at a distance
predicated by the minimal Bohm theory is weaker than the one
predicated by orthodox collapse quantum mechanics and the GRW/Pearle
collapse models; for in contrast to the minimal Bohm theory, in these
theories the measurement on the L-particle induces a change in the
intrinsic properties of the R-particle, independently of whether or
not the R-particle undergoes a measurement. Thus, if the R-particle
comes to possess (momentarily) a definite position, the EPR/B
experiment as described by these theories involves action at a
distance -- a stronger kind of action than the action\* at a
distance predicated by the minimal Bohm theory.


### 8.2 Superluminal signaling




It was also argued, notably by Jarrett 1984 and 1989 and Shimony 1984,
that in contrast to violations of OI, violations of PI may give rise
(at least in principle) to superluminal signaling. Indeed, as is not
difficult to see from section 7.1, in theories that satisfy
l-independence there is an asymmetry between failures of PI and
failures of OI with respect to superluminal signaling: whereas
l-distribution and the failure of PI are sufficient conditions
for the in-principle possibility of superluminal signaling,
l-distribution and the failure of OI are not. Thus, the
prospects of superluminal signaling look better in parameter-dependent
theories, i.e., theories that violate PI. Yet, as we have seen in
section 7.2.1, if the Bohmian quantum-equilbrium distribution obtains,
then Bohm's theory, the paradigm of parameter dependent theories,
prohibits superluminal signaling. And if this distribution is obtained
as a matter of law, then Bohm's theory prohibits superluminal
signaling even in theory. Furthermore, as we remarked in section 7.1,
if the in-principle possibility of violating l-independence is
not excluded, superluminal signaling may exist in theories that
satisfy PI and violate OI. In fact, as section 7.4 seems to suggest,
the possibility of superluminal signaling in theories that satisfy PI
but violate OI cannot be discounted even when l-independence is
impossible.


### 8.3 Relativity




Jarrett (1984, 1989), Ballentine and Jarrett (1997) and Shimony (1984)
hold that superluminal signaling is incompatible with relativity
theory. Accordingly, they conclude that violations of PI are
incompatible with relativity theory, whereas violations of OI may be
compatible with this theory. Furthermore, Sutherland (1985, 1989)
argues that deterministic, relativistic parameter-dependent theories
(i.e., relativistic, deterministic theories that violate PI) would
plausibly require retro-causal influences, and in certain experimental
circumstances this type of influences would give rise to causal
paradoxes, i.e., inconsistent closed causal loops (where effects
undermine their very causes). And Arntzenius (1994) argues that all
relativistic parameter-dependent theories are impossible on pain of
causal paradoxes. That is, he argues that in certain experimental
circumstances any relativistic, parameter-dependent theory would give
rise to closed causal loops in which violations of PI could not
obtain. 




It is noteworthy that the view that relativity *per se* is
incompatible with superluminal signaling is disputable (for more
details, see section 10). Anyway, recalling (section 8.2), if
l-distribution is excluded as a matter of law, it will be
impossible even in theory to exploit the violation of PI to give rise
to superluminal signaling, in which case the possibility of
relativistic parameter-dependent theories could not be discounted on
the basis of superluminal signaling.




Furthermore, as mentioned in section 7.1 and 7.4, the in-principle
possibility of superluminal signaling in theories that satisfy PI and
violate OI cannot be excluded *a priori*. Thus, if relativity
theory excludes superluminal signaling, the argument from superluminal
signaling may also be applied to exclude the possibility of some
relativistic outcome-dependent theories. 




Finally, Berkovitz (2002) argues that Arntzenius's argument for the
impossibility of relativistic theories that violate PI is based on
assumptions about probabilities that are common in linear causal
situations but are unwarranted in causal loops, and that the real
challenge for these theories is that in such loops their predictive
power is undermined (for more details, see section 10.3). 


### 8.4 Superluminal causation




In various counterfactual and probabilistic accounts of causation
violations of PI entail superluminal causation between the setting of
the nearby measurement apparatus and the distant measurement outcome,
whereas violations of OI entail superluminal causation between the
distant measurement outcomes (see Butterfield 1992b, 1994, Berkovitz
1998b, section 2). Thus, it seems that theories that violate PI
postulate a different type of superluminal causation than theories
that violate OI. Yet, as Berkovitz (1998b, section 2.4) argues, the
violation of PI in Bohm's theory *does* involve some type of
outcome dependence, which may be interpreted as a generalization of
the violation of OI. In this theory, the specific R-measurement
outcome in the EPR/B experiment depends on the specific L-measurement
outcome: For any three different directions *x*, *y*,
*z*, if the probabilities of *x*-spin 'up'
and *y*-spin 'up' are non-zero, the probability of
R-outcome *z*-spin 'up' will generally depend on
whether the L-outcome is *x*-spin 'up' or
*y*-spin 'up'. Yet, due to the determinism that
Bohm's theory postulates, OI trivially obtains. Put it another way, OI
does not reflect all the types of outcome independence that may exist
between distant outcomes. Accordingly, the fact that a theory
satisfies OI does not entail that it does not involve some other type
of outcome dependence. Indeed, in all the current quantum theories
that violate factorizability there are correlations between distant
specific measurement outcomes -- correlations that may well be
interpreted as an indication of counterfactual superluminal causation
between these outcomes.


### 8.5 On the origin and nature of parameter dependence




Parameter dependence (PI) postulates that in the EPR/B experiment the
probability of the later, distant measurement outcome depends on the
setting of the apparatus of the nearby, earlier measurement. It may be
tempting to assume that this dependence is due to a direct influence
of the nearby setting on the (probability of the) distant outcome. But
a little reflection on the failure of PI in Bohm's theory, which is
the paradigm for parameter dependence, demonstrates that the setting
of the nearby apparatus *per se* has no influence on the
distant measurement outcome. Rather, it is because the setting of the
nearby measurmenent apparatus influences the nearby measurement
outcome and the nearby outcome influences the distant outcome that the
setting of the nearby apparatus can have an influence on the distant
outcome. For, as is not difficult to see from the analysis of the
nature of non-locality in the minimal Bohm theory (see section 5.3.1),
the setting of the apparatus of the nearby (earlier) measurement in
the EPR/B experiment influences the outcome the nearby measurement,
and this outcome influences the guiding field of the distant particle
and accordingly the outcome of a measurement on that particle.




While the influence of the nearby setting on the nearby outcome is
necessary for parameter dependence, it is not sufficient for it. In
all the current quantum theories, the probabilities of joint outcomes
in the EPR/B experiment depend on the settings of both measurement
apparatuses: The probability that the L-outcome is *l*-spin
'up' and the R-outcome is *r*-spin 'up'
and the probability that the L-outcome is *l*-spin
'up' and the R-outcome is *r*-spin
'down' both depend on (*l* [?] *r*),
i.e., the distance between the angles *l* and *r*. In
theories in which the sum of these joint probabilities is invariant
with respect to the value of (*l* [?] *r*),
parameter independence obtains: for all pairs' states l,
L-setting *l*, and R-settings *r* and *r*',
L-outcome *xl*, and R-outcomes
*y**r* and *y**r*' :




> 
> (PI)   
> 
> *P*l *l* *r*(*xl* & 
> *yr*) + 
> *P*l *l* *r*(*xl* &
> !*yr*) =
> *P*l *l* *r*' (*xl* 
> & *y**r*' ) +
> *P*l *l* *r*' (*xl* 
> & !*y**r*' ).
>  






Parameter dependence is a violation of this invariance condition.


## 9. Can there be 'local' quantum theories?




The focus of this entry has been on exploring the nature of the
non-local influences in the quantum realm as depicted by quantum
theories that violate factorizability, i.e., theories in which the
joint probability of the distant outcomes in the EPR/B experiment do
not factorize into the product of the single probabilities of these
outcomes. The motivation for this focus was that, granted plausible
assumptions, factorizability must fail (see section 2), and its
failure implies some type of non-locality (see sections 2-8). But if
any of these plausible assumptions failed, it may be possible to
account for the EPR/B experiment (and more generally for all other
quantum phenomena) without postulating any non-local influences. Let
us then consider the main arguments for the view that quantum
phenomena need not involve non-locality.



 In arguments for the failure of factorizability, it is presupposed
that the distant measurement outcomes in the EPR/B experiment are real
physical events. Recall (section 5.3.3) that in Albert and Loewer's
(1988) many-minds interpretation this is not the case. In this
interpretation, definite measurement outcomes are (typically) not
physical events. In particular, the pointers of the measurement
apparatuses in the EPR/B experiment do not display any definite
outcomes. Measurement outcomes in the EPR/B experiment exist only as
(non-physical) mental states in observers' minds (which are
postulated to be non-physical entities). So sacrificing some of our
most fundamental presuppositions about the physical reality and
assuming a controversial mind-body dualism, the many-minds
interpretation of quantum mechanics does not postulate any action at a
distance or superluminal causation between the distant wings of the
EPR/B experiment. Yet, as quantum-mechanical states of systems are
assumed to reflect their physical states, the many-minds theory does
postulate some type of non-locality, namely state non-separability and
property and relational holism.



 Another way to get around Bell's argument for non-locality in the
EPR/B experiment is to construct a model of this experiment that
satisfies factorizability but violates l-independence (i.e.,
the assumption that the distribution of all the possible pairs' states
in the EPR/B experiment is independent of the measured quantities). In
section 2, we mentioned two possible causal explanations for the
failure of l-independence. The first is to postulate that
pairs' states and apparatus settings share a common cause, which
correlates certain types of pairs' states with certain types of
settings (e.g. states of type l1 are correlated with
settings of type *l* and *r*, whereas states of type
l2 are correlated with settings of type
*l'* and *r'*, etc.). As we noted, thinking
about all the various ways one can measure properties, this
explanation seems conspiratorial. Furthermore, it runs counter to one
of the most fundamental presuppositions of empirical science, namely
that in experiments preparations of sources and settings of
measurement apparatuses are typically independent of each other. The
second possible explanation is to postulate causation from the
measurement events backward to the source at the emission time. (For
advocates of this way out of non-locality, see Costa de Beauregard
1977, 1979, 1985, Sutherland 1983, 1998, 2006 and Price 1984, 1994,
1996, chapters 3, 8 and 9.) Maudlin (1994, p. 197-201) argues that
theories that postulate such causal mechanism are
inconsistent. Berkovitz (2002, section 5) argues that Maudlin's line
of reasoning is based on unwarranted premises. Yet, as we shall see in
section 10.3, this way out of non-locality faces some
challenges. Furthermore, while a violation of l-independence
provides a way out of Bell's theorem, it does not necessarily imply
locality; for the violation of l-independence is compatible
with the failure of factorizability. 




A third way around non-locality is to 'exploit' the
inefficiency of measurement devices or (more generally) measurement
set-ups. In any actual EPR/B experiment, many of the particle pairs
emitted from the source fail to be detected, so that only a sample of
the particle pairs is observed. Assuming that the observed samples are
not biased, it is now generally agreed that the statistical
predictions of orthodox quantum mechanics have been vindicated (for a
review of these experiments, see Redhead 1987, section 4.5). But if
this assumption is abandoned, there are perfectly local causal
explanations for the actual experimental results (Clauser and Horne
1974, Fine 1982b, 1989a). Many believe that this way out of
non-locality is *ad hoc*, at least in light of our current
knowledge. Moreover, this strategy would fail if the efficiency of
measurement devices exceeded a certain threshold (for more details,
see Fine 1989a, Maudlin 1994, chapter 6, Larsson and Semitecolos 2000
and Larsson 2002). 




Finally, there are those who question the assumption that
factorizability is a locality condition (Fine 1981, 1986, pp. 59-60,
1989b, Cartwright 1989, chaps. 3 and 6, Chang and Cartwright 1993).
Accordingly, they deny that non-factorizability implies non-locality.
The main thrust of this line of reasoning is that the principle of the
common cause is not generally valid. Some, notably Cartwright (1989)
and Chang and Cartwright (1993), challenge the assumption that common
causes always screen off the correlation between their effects, and
accordingly they question the idea that non-factorizability implies
non-locality. Others, notably Fine, deny that correlations must have
causal explanation. 




While these arguments challenge the view that the quantum realm as
depicted by non-factorizable models for the EPR/B experiment
*must* involve non-locality, they do not show that viable
local, non-factorizable models of the EPR/B experiment (i.e., viable
models which do not postulate any non-locality) are possible. Indeed,
so far none of the attempts to construct local, non-factorisable
models for EPR/B experiments has been successful. 


## 10. Can quantum non-locality be reconciled with relativity?




The question of the compatibility of quantum mechanics with the
special theory of relativity is very difficult to resolve. (The
question of the compatibility of quantum mechanics with the general
theory of relativity is even more involved.) The answer to this
question depends on the interpretation of special relativity and the
nature of the exact constraints it imposes on influences between
events. 




A popular view has it that special relativity prohibits any
superluminal influences, whereas theories that violate factorizability
seem to involve such influences. Accordingly, it is held that quantum
mechanics is incompatible with relativity. Another common view has it
that special relativity prohibits only certain types of superluminal
influence. Many believe that relativity prohibits superluminal
signaling of information. Some also believe that this theory prohibits
superluminal transport of matter-energy and/or
action-at-a-distance. On the other hand, there is the view that
relativity *per se* prohibits only superluminal influences that
are incompatible with the special-relativistic space-time, the
so-called 'Minkowski space-time,' and that this
prohibition is compatible with certain types of superluminal
influences and superluminal signaling (for a comprehensive discussion
of this issue, see Maudlin, 1994, 1996, section
 2).[34]





It is commonly agreed that relativity requires that the descriptions
of physical reality (i.e., the states of systems, their properties,
dynamical laws, etc.) in different coordinate systems should be
compatible with each other. In particular, descriptions of the state of
systems in different foliations of spacetime into parallel spacelike
hyperplanes, which correspond to different inertial reference frames,
are to be related to each other by the Lorentz transformations. If this
requirement is to reflect the structure of the Minkowski spacetime,
these transformations must hold at the level of individual processes,
and not only at the level of ensembles of processes (i.e., at the
statistical level) or observed phenomena. Indeed, Bohm's theory,
which is manifestly non-relativistic, satisfies the requirement that
the Lorentz transformations obtain at the level of the observed
phenomena. 




However, satisfying the Lorentz transformations at the level of
individual processes is not sufficient for compatibility with
Minkowski spacetime; for the Lorentz transformations may also be
satisfied at the level of individual processes in theories that
postulate a preferred inertial reference frame (Bell 1976). Maudlin
(1996, section 2) suggests that a theory is genuinely relativistic
(both in spirit and letter) if it can be formulated without ascribing
to spacetime any more, or different intrinsic structure than the
relativistic
 metrics.[35]
The question of the compatibility of
relativity with quantum mechanics may be presented as follows: Could a
quantum theory that does not encounter the measurement problem be
relativistic in that sense? 


### 10.1 Collapse theories




The main problem in reconciling collapse theories with special
relativity is that it seems very difficult to make state collapse
(modeled as a real physical process) compatible with the structure of
the Minkowski spacetime. In non-relativistic quantum mechanics, the
earlier L-measurement in the EPR/B experiment induces a collapse of the
entangled state of the particle pair and the L-measurement apparatus.
Assuming (for the sake of simplicity) that measurement events occur
instantaneously, state collapse occurs along a single spacelike
hyperplane that intersects the spacetime region of the L-measurement
event--the hyperplane that represents the (absolute) time of the
collapse. But this type of collapse dynamics would involve a preferred
foliation of spacetime, in violation of the spirit, if not the letter
of the Minkowski spacetime. 




The current dynamical collapse models are not genuinely
relativistic, and attempts to generalize them to the special
relativistic domain have encountered difficulties (see, for
example, the entry on collapse theories, Ghirardi 1996, Pearle 1996,
and references therein). A more recent attempt to address these
difficulties due to Tumulka (2004) seems more promising.




In an attempt to reconcile state collapse with special relativity,
Fleming (1989, 1992, 1996) and Fleming and Bennett (1989) suggested
radical hyperplane dependence. In their theory, state collapse occurs
along an infinite number of spacelike hyperplanes that intersect the
spacetime region of the measurements. That is, in the EPR/B experiment
a collapse occurs along all the hyperplanes of simultaneity that
intersect the spacetime region of the L-measurement. Similarly, a
collapse occurs along all the hyperplanes of simultaneity that
intersect the distant (space-like separated) spacetime region of the
R-measurement. Accordingly, the hyperplane-dependent theory does not
pick out any reference frame as preferred, and the dynamics of the
quantum states of systems and their properties can be reconciled with
the Minkowski spacetime. Further, since all the multiple collapses are
supposed to be real (Fleming 1992, p. 109), the predictions of
orthodox quantum mechanics are reproduced in each reference
frame. 




The hyperplane-dependent theory is genuinely relativistic. But the
theory does not offer any mechanism for state collapses, and it does
not explain how the multiple collapses are related to each other and
how our experience is accounted for in light of this multiplicity.




Myrvold (2002b) argues that state collapses can be reconciled with
Minkowski spacetime even without postulating multiple different
collapses corresponding to different reference frames. That is, he
argues with respect to the EPR/B experiment that the collapses induced
by the L- and the R-measurement are local events in the L- and the
R-wing respectively, and that the supposedly different collapses
(corresponding to different reference frames) postulated by the
hyperplane-dependent theory are only different descriptions of the
same local collapse events. Focusing on the state of the particle
pair, the main idea is that the collapse event in the L-wing is
modeled by a (one parameter) family of operators (the identity
operator before the L-measurement and a projection to the collapsed
state after the L-measurement), and it is local in the sense that it
is a projection on the Hilbert space of the L-particle; and similarly,
*mutatis mutandis*, for the R-particle. Yet, if the quantum
state of the particle pair represents their complete state (as the
case is in the orthodox theory and the GRW/Pearle collapse models),
these collapse events seem non-local. While the collapse in the L-wing
may be said to be local in the above technical sense, it is by
definition a change of local as well as distant (spacelike)
properties. The operator that models the collapse in the L-wing
transforms the entangled state of the particle pair--a state in
which the particles have no definite spins--into a product of
non-entangled states in which both particles have definite spins, and
accordingly it causes a change of intrinsic properties in both the L-
and the R-wing.


 

In any case, Myrvold's proposal demonstrates that even if state
collapses are not hyperplane dependent, they need not be incompatible
with relativity theory.


### 10.2 No-collapse theories




Recall (section 5.3) that in no-collapse theories, quantum-mechanical
states always evolve according to a unitary and linear equation of
motion (the Schrodinger equation in the non-relativistic case),
and accordingly they never collapse. Since the wave function has a
covariant dynamics, the question of the compatibility with relativity
turns on the dynamics of the additional properties --the
so-called 'hidden variables'-- that no-collapse
theories typically postulate. In Albert and Loewer's many-minds theory
(see section 5.3.3), the wave function has covariant dynamics, and no
additional physical properties are postulated. Accordingly, the theory
is genuinely relativistic. Yet, as the compatibility with relativity
is achieved at the cost of postulating that outcomes of measurements
(and, typically, any other perceived properties) are mental rather
than physical properties, many find this way of reconciling quantum
mechanics with relativity unsatisfactory. 




Other Everett-like interpretations attempt to reconcile quantum
mechanics with the special theory of relativity without postulating
such a controversial mind-body dualism. Similarly to the many-minds
interpretation of Albert and Loewer, and contrary to Bohm's theory and
modal interpretations, on the face of it these interpretations do not
postulate the existence of 'hidden variables.' But
(recalling section 5.3.3) these Everett-like interpretations face the
challenge of making sense of our experience and the probabilities of
outcomes, and critics of these interpretations argue that this
challenge cannot be met without adding some extra structure to the
Everett interpretation (see Albert and Loewer 1988, Albert 1992,
pp. 114-5, Albert and Loewer 1996, Price 1996, pp. 226-227, and
Barrett 1999, pp. 163-173); a structure that may render these
interpretations incompatible with relativity. Supporters of the
Everett interpretation disagree. Recently, Deutsch (1999), Wallace
(2002, 2003, 2005a,b) and Greaves (2004) have suggested that
Everettians can make sense of the quantum-mechanical probabilities by
appealing to decision-theoretical considerations. But this line of
reasoning has been disputed (see Barnum *et al.* 2000, Lewis
2003b, Hemmo and Pitowsky 2005 and Price 2006). 




Modal interpretations constitute another class of no-collapse
interpretations of quantum mechanics that were developed to reconcile
quantum mechanics with relativity (and to solve the measurement
problem). Yet, as the no-go theorems by Dickson and Clifton (1998),
Arntzenius (1998) and Myrvold (2002) demonstrate, the earlier versions
of the modal interpretation are not genuinely compatible with
relativity theory. Further, Earman and Ruetsche (2005) argue that a
quantum-field version of the modal interpretation (which is set in the
context of relativistic quantum-field theory), like the one proposed
by Clifton (2000), would be subject to serious challenges. Berkovitz
and Hemmo (2006a,b) develop a relational modal interpretation that
escapes all the above no-go theorems and to that extent seems to
provide better prospects for reconciling quantum mechanics with
special relativity. 


### 10.3 Quantum causal loops and relativity




Recall (section 8) that many believe that parameter-dependent theories
(i.e., theories that violate parameter independence) are more difficult
or even impossible to reconcile with relativity. Recall also that one
of the lines of argument for the impossibility of relativistic
parameter-dependent theories is that such theories would give rise to
causal paradoxes. In our discussion, we focused on EPR/B experiments
in which the measurements are distant (spacelike separated). In a
relativistic parameter-dependent theory, the setting of the nearby
measurement apparatus in the EPR/B experiment would influence the
probability of the distant (spacelike separated) measurement
outcome. Sutherland (1985, 1989) argues that it is plausible to
suppose that the realization of parameter dependence would be the same
in EPR/B experiments in which the measurements are not distant from
each other (i.e., when the measurements are timelike separated). If so,
relativistic parameter-dependent theories would involve backward
causal influences. But, he argues, in deterministic, relativistic
parameter-dependent theories these influences would give rise to
causal paradoxes, i.e., inconsistent closed causal loops.


 

Furthermore, Arntzenius (1994) argues that all relativistic
parameter-dependent theories are impossible on pain of causal
paradoxes. In his argument, he considers the probabilities of
measurement outcomes in a setup in which two EPR/B experiments are
causally connected to each other, so that the L-measurement outcome of
the first EPR/B experiment determines the setting of the L-apparatus
of the second EPR/B experiment and the R-measurement outcome of the
second EPR/B experiment determines the setting of the R-apparatus of
the first EPR/B experiment. And he argues that in this experiment,
relativistic parameter-dependent theories (deterministic or
indeterministic) would give rise to closed causal loops in which
parameter dependence would be impossible. Thus, he concludes that
relativistic, parameter-dependent theories are impossible. (Stairs
(1989) anticipates the argument that the above experimental setup may
give rise to causal paradoxes in relativistic, parameter-dependent
theories, but he stops short of arguing that such theories are
impossible.)




Berkovitz (1998b, section 3.2, 2002, section 4) argues that
Arntzenius's line of reasoning fails because it is based on untenable
assumptions about the nature of probabilities in closed causal
loops--assumptions that are very natural in linear causal
situations (where effects do not cause their causes), but untenable in
causal loops. (For an analysis of the nature of probabilities in
causal loops, see Berkovitz 2001 and 2002, section 2.) Thus, he
concludes that the consistency of relativistic parameter-dependent
theories cannot be excluded on the grounds of causal paradoxes. He
also argues that the real challenge for relativistic
parameter-dependent theories is concerned with their predictive
power. In the causal loops predicted by relativistic
parameter-dependent theories in Arntzenius's suggested experiment,
there is no known way to compute the frequency of events from the
probabilities that the theories prescribe. Accordingly, such theories
would fail to predict any definite statistics of measurement outcomes
for that experiment. This lack of predictability may also present some
new opportunities. Due to this unpredictability, there may be an
empirical way for arbitrating between these theories and quantum
theories that do not predicate the existence such causal loops in
Arntzenius's experiment. 




Another attempt to demonstrate the impossibility of certain
relativistic quantum theories on the grounds of causal paradoxes is
advanced by Maudlin (1994, pp. 195-201). (Maudlin does not present his
argument in these terms, but the argument is in effect based on such
grounds.) Recall (sections 2 and 9) that a way to try to reconcile
quantum mechanics with relativity is to account for the curious
correlations between distant systems by local backward influences
rather than non-local influences. In particular, one may postulate
that the correlations between the distant measurement outcomes in the
EPR/B experiment are due to local influences from the measurement
events backward to the state of the particle pair at the source. In
such models of the EPR/B experiment, influences on events are always
confined to events that occur in their past or future light cones, and
no non-locality is postulated. Maudlin argues that theories that
postulate such backward causation will be inconsistent. More
particularly, he argues that a plausible reading of Cramer's (1980,
1986) transactional interpretation, and any other theory that
similarly attempts to account for the EPR/B correlations by
postulating causation from the measurement events backward to the
source, will be inconsistent. 




Berkovitz (2002, section 5) argues that Maudlin's argument is, in
effect, that if such retro-causal theories were true, they would
involve closed causal loops in which the probabilities of outcomes
that these theories assign will certainly deviate from the statistics
of these outcomes. And, similarly to Arntzenius's argument, Maudlin's
argument also rests on untenable assumptions about the nature of
probabilities in causal loops (for a further discussion of Maudlin's
and Berkovitz's arguments and, more generally, the prospects of
Cramer's theory, see Kastner 2004). Furthermore, Berkovitz (2002,
sections 2 and 5.4) argues that, similarly to relativistic
parameter-dependent theories, the main challenge for theories that
postulate retro-causality is not causal paradoxes, but rather the fact
that their predictive power may be undermined. That is, the
probabilities assigned by such theories may fail to predict the
frequency of events in the loops they predicate. In particular, the
local retro-causal theories that Maudlin considers fail to assign any
definite predictions for the frequency of measurement outcomes in
certain experiments. Yet, some other theories that predicate the
existence of causal loops, such as Sutherland's (2006) local
time-symmetric Bohmian interpretation of quantum mechanics, seem not
to suffer from this problem.