## 1. Introduction



Holism has often been taken as the thesis that the whole is more than
the sum of its parts. Several different interpretations of this
epigram prove relevant to physics, as we shall see. Here is a
correspondingly vague initial statement of nonseparability: The state
of the whole is not constituted by states of its parts. It is already
apparent both that holism and nonseparability are related notions and
that their exact relation needs to be clarified.



In one interpretation, holism is a methodological thesis
 (Section 2),
 to the effect that the best way to study the behavior of a complex
system is to treat it as a whole, and not merely to analyze the
structure and behavior of its component parts. Alternatively, holism
may be taken as a metaphysical thesis
 (Section 3):
 There are some wholes whose natures are simply not determined by the
nature of their parts. Methodological holism stands opposed to
methodological reductionism, in physics as well as in other sciences.
But it is a certain variety of metaphysical holism that is more
closely related to nonseparability. What is at issue here is the
extent to which the properties of the whole are determined by the
properties of its parts: property holism
 (Section 4)
 denies such determination, and thereby comes very close to a thesis
of nonseparability. In turn, nonseparability can be analyzed either as
state nonseparability
 (Section 5),
 or as spatiotemporal nonseparability
 (Section 6).
 By and large, a system in classical physics can be analyzed into
parts, whose states and properties determine those of the whole they
compose
 (Section 7).
 But all hands agree that the state of a system in quantum theory
resists such analysis. The quantum state of a system specifies its
chances of exhibiting various properties on measurement. In ordinary
quantum mechanics, the most complete such specification is given by
what is called a pure state. Even when a compound system has a pure
state, some of its subsystems may not have their own pure states.
Emphasizing this characteristic of quantum mechanics, Schrodinger
described such component states as "entangled"
 (Section 8).
 Superficially, such entanglement of states already demonstrates
nonseparability. At a deeper level, it has been maintained that the
puzzling statistics that arise from measurements on entangled quantum
systems either demonstrate, or are explicable in terms of, holism or
nonseparability rather than any problematic action at a distance
 (Sections 8,
 9). The Aharonov-Bohm effect
 (Section 10)
 also appears to exhibit action at a distance, as the behavior of
electrons is modified by a magnetic field they never experience. But
this effect may be understood instead as a result of the local action
of nonseparable electromagnetism. And indeed, this type of holism is
not rooted in quantum mechanics or entanglement: though the
measurement of the Aharonov-Bohm effect uses quantum ingredients, it
builds on the existence of non-separable quantities in classical gauge
theory (e.g. electromagnetism) in a vacuum. But even focussing on
theories without gauge symmetry, we can obtain puzzling correlations
between distant simultaneous measurements in the vacuum, according to
quantum field theory
 (Section 12).
 A form of quantum theory used to study them represents systems by
algebras of operators with new kinds of states defined on them,
thereby making room for failures of state and system separability with
no analogs in ordinary quantum mechanics. String theory
 (Section 13)
 is an ambitious research program in the framework of quantum field
theory. According to string theory, all fundamental particles can be
considered to be excitations of underlying non-pointlike entities in a
multi-dimensional space. The particles' intrinsic charge, mass
and spin may then arise as nonseparable features of the world at the
deepest level.


## 2. Methodological Holism



Methodologically, holism stands opposed to reductionism, somewhat as
follows.



> 
> *Methodological Holism*: An understanding of a certain kind of
> complex system is best sought at the level of principles governing the
> behavior of the whole system, and not at the level of the structure
> and behavior of its component parts.
> 



> 
> *Methodological Reductionism*: An understanding of a complex
> system is best sought at the level of the structure and behavior of
> its component parts.
> 



This seems to capture much of what is at stake in debates about holism
in social and biological science. In social science, societies are the
complex systems, composed of individuals; while in biology, the
complex systems are organisms, composed of cells, and ultimately of
proteins, DNA and other molecules. A methodological individualist
maintains that the right way to approach the study of a society is to
investigate the behavior of the individual people that compose it. A
methodological holist, on the other hand, believes that such an
investigation will fail to shed much light on the nature and
development of society as a whole. There is a corresponding debate
within physics. Methodological reductionists favor an approach to
(say) condensed matter physics which seeks to understand the behavior
of a solid or liquid by applying quantum mechanics (say) to its
component molecules, atoms, ions or electrons. Methodological holists
think this approach is misguided: As one condensed matter physicist
put it "the most important advances in this area come about by
the emergence of qualitatively new concepts at the intermediate or
macroscopic levels--concepts which, one hopes, will be compatible
with one's information about the microscopic constituents, but
which are in no sense logically dependent on it." (Leggett 1987,
p.113)



It is surprisingly difficult to find methodological reductionists
among physicists. The elementary particle physicist Steven Weinberg,
for example, is an avowed reductionist. He believes that by asking any
sequence of deeper and deeper why-questions one will arrive ultimately
at the same fundamental laws of physics. But this explanatory
reductionism is metaphysical in so far as he takes explanation to be
an ontic rather than a pragmatic category. On this view, it is not
physicists but the fundamental laws themselves that explain why
"higher level" scientific principles are the way they are.
Weinberg (1992) explicitly distinguishes his view from methodological
reductionism by saying that there is no reason to suppose that the
convergence of scientific explanations must lead to a convergence of
scientific methods.


## 3. Metaphysical Holism



The metaphysical holist believes that the nature of some wholes is not
determined by that of their parts. One may distinguish three varieties
of metaphysical holism: ontological, property and nomological
holism.



> 
> *Ontological Holism*:
>  Some objects are not wholly composed of basic physical parts.
> 



> 
> *Property Holism*: Some objects have properties that are not
> determined by physical properties of their basic physical parts.
> 



> 
> *Nomological Holism*: Some objects obey laws that are not
> determined by fundamental physical laws governing the structure and
> behavior of their basic physical parts.
> 



All three theses require an adequate clarification of the notion of a
 basic physical part.
 One way to do this would be to consider objects as basic, relative to
a given class of objects subjected only to a certain kind of process,
just in case every object in that class continues to be wholly
composed of a fixed set of these (basic) objects. Thus, atoms would
count as basic parts of hydrogen if it is burnt to form water, but not
if it is converted into helium by a thermonuclear reaction. But this
way excludes consideration of time-slices and point events (for
example) as basic (spatio)temporal parts of an object. What counts as
a part, and what parts are basic, are matters best settled in a
particular context of enquiry.



Weinberg's (1992) reductionism is opposed to nomological holism
in science. He claims, in particular, that thermodynamics has been
explained in terms of particles and forces, which could hardly be the
case if thermodynamic laws were autonomous. In fact thermodynamics
presents a fascinating but complex test case for the theses both of
property holism and of nomological holism. One source of complexity is
the variety of distinct concepts of both temperature and entropy that
figure in both classical thermodynamics and statistical mechanics.
Another is the large number of quite differently constituted systems
to which thermodynamics can be applied, including not just gases and
electromagnetic radiation but also magnets, chemical reactions, star
clusters and black holes. Both sources of complexity require a careful
examination of the extent to which thermodynamic properties are
determined by the physical properties of the basic parts of
thermodynamic systems. A third difficulty stems from the problematic
status of the probability assumptions that are required in addition to
the basic mechanical laws in order to recover thermodynamic principles
within statistical mechanics. (An important example is the assumption
that the micro-canonical ensemble is to be assigned the standard,
invariant, probability distribution.) Since the basic laws of
mechanics do not determine the principles of thermodynamics without
some such assumptions (however weak), there may well be at least one
interesting sense in which thermodynamics establishes nomological
holism. The related entry
 philosophy of statistical mechanics
 contains further discussion of these difficulties, especially in
Section 6.


## 4. Property/Relational Holism



While some form of
 ontological holism
 has occasionally been considered, the variety of metaphysical holism
most clearly at issue in quantum mechanics and in gauge theory is
property holism. But to see just what the issue is we need a more
careful formulation of that thesis.



First the thesis should be contextualized to *physical*
properties of composite *physical* objects. We are interested
here in how far a physical object's properties are fixed by
those of its parts, not in some more general determinationist
physicalism. Next, to arrive at an interesting formulation of property
holism we must accept that this thesis is not only concerned with
properties, and not concerned with all properties. The properties of a
whole will typically depend upon *relations* among its proper
parts as well as on properties of the individual parts. But if we are
permitted to consider *all* properties and relations among the
parts, then these trivially determine the properties of the whole they
compose. For one relation among the parts is what we might call the
complete composition relation--that relation among the parts
which holds just in case they compose this very whole with all its
properties.



Let us call a canonical set of properties and relations of the parts
which may or may not determine the properties and relations of the
whole the supervenience basis. To avoid trivializing the theses we are
trying to formulate, only certain properties and relations can be
allowed in the supervenience basis. The intuition as to which these
are is simple--the supervenience basis is to include just the
qualitative intrinsic properties and relations of the parts, i.e., the
properties and relations which these bear in and of themselves,
without regard to any other objects, and irrespective of any further
consequences of their bearing these properties for the properties of
any wholes they might compose. Unfortunately, this simple intuition
resists precise formulation. It is notoriously difficult to say
precisely what is meant either by an
 intrinsic property
 or relation, or by a purely qualitative property or relation. And the
other notions appealed to in expressing the simple intuition are
hardly less problematic. But, imprecise as it is, this statement
serves already to exclude certain unwanted properties and relations,
including the complete composition relation, from the supervenience
basis.



Finally, we arrive at the following opposing theses:



> 
> *Physical Property Determination*:
>  Every qualitative intrinsic physical property and relation of a set
> of physical objects from any domain *D* subject only
> to type *P* processes supervenes on qualitative
> intrinsic physical properties and relations in the supervenience basis
> of their
>  basic physical parts
>  relative to *D* and *P*.
> 



> 
> *Physical Property Holism*:
>  There is some set of physical objects from a domain *D*
> subject only to type *P* processes,
> not all of whose qualitative intrinsic physical properties and
> relations supervene on qualitative intrinsic physical properties and
> relations in the supervenience basis of their
>  basic physical parts
>  (relative to *D* and \(P)\).
> 



If we take the real state of a set of physical objects to be given by
their qualitative intrinsic physical properties and relations, then
physical property determination says (while physical property holism
denies) that the real state of wholes is determined by the real state
of their parts.



There is some residual unclarity in the notion of supervenience that
figures in these theses. The idea is familiar enough--that there
can be no relevant difference in objects in *D*
without a relevant difference in their basic physical parts. I take it
that the modality involved here is not logical but broadly physical.
One might try to explicate the notion of supervenience here in terms
of models of a true, descriptively complete, physical theory. At issue
is whether such a physical theory has two models which agree on the
qualitative intrinsic physical properties and relations of the basic
parts of one or more objects in *D* but disagree on
some qualitative intrinsic property or relation of these objects.



Teller (1989) has introduced the related idea of what he calls
relational holism.



> 
> *Relational Holism*:
>  There are non-supervening relations--that is, relations that do
> not supervene on the nonrelational properties of the relata. (p. 214)
> 



Within physics, this specializes to a close relative of physical
property holism, namely:



> 
> *Physical Relational Holism*: There are physical relations
> between some physical objects that do not supervene on their
> qualitative intrinsic physical properties.
> 



Physical property holism entails physical relational holism, but not
vice versa. For suppose that *F* is some qualitative
intrinsic physical property or relation of one or more elements of *D*
that fails to supervene on qualitative intrinsic
physical properties and relations in the supervenience basis of their
basic physical parts. We may define a (non-intrinsic) physical
relation \(R\_{F}\) to hold of the basic physical parts of elements of
*D* if and only if *F* holds of
these elements. Clearly \(R\_{F}\) does not supervene on the
qualitative intrinsic physical properties of these parts. So physical
property holism entails physical relational holism. But the converse
entailment fails. For let \(R\_{G}\) be a physical relation that holds
between the basic parts of some elements in *D* when
and only when those elements are in the relation \(S\_{G}\). \(R\_{G}\)
may fail to supervene on the qualitative intrinsic physical properties
of these basic parts, even though all qualitative intrinsic physical
properties and relations of elements of *D*
(including \(S\_{G}\)) supervene on the qualitative intrinsic physical
properties *and relations* of their basic parts.



Physical relational holism seems at first sight too weak to capture
any distinctive feature of quantum phenomena: even in classical
physics the spatiotemporal relations between physical objects seem not
to supervene on their qualitative intrinsic physical properties. But
when he introduced relational holism Teller (1987) maintained a view
of spacetime as a quantity: On this view spatiotemporal relations do
in fact supervene on qualitative intrinsic physical properties of
ordinary physical objects, since these include their spatiotemporal
properties.


## 5. State Nonseparability



Physics treats systems by assigning them states. The thermodynamic
state of a gas specifies its pressure, volume and temperature. The
state of a system of classical particles is represented as a point in
a phase space coordinatized by their positions and momenta. One
expects that if a physical system is composed of physical subsystems,
then both the composite system and its subsystems will be assigned
states by the relevant physical theory. One further expects that the
state of the whole will not be independent of those of its parts, and
specifically that if a system is composed of two subsystems, *A*
and *B*, then it will satisfy a
principle formulated by Einstein (1935). Howard (1985, p.180) gives
the following translation of this principle, which I'll call
the



> 
> *Real State Separability Principle*:
>  The real state of the pair \(AB\) consists precisely of the real
> state of *A* and the real state of *B*,
> which states have nothing to do with one another.
> 



But the assignment of states to systems in quantum mechanics seems not
to conform to these expectations (see related entry
 quantum mechanics).
 Recall that the quantum state of a system specifies its chances of
exhibiting various properties on measurement. At least in ordinary
quantum mechanics, the mathematical representative of this state is an
object defined in a Hilbert space--a kind of vector space. This
is analogous in some ways to the representation of the state of a
system of particles in classical mechanics in a phase space. Let us
formulate a principle of



> 
> *State Separability*:
>  The state assigned to a compound physical system at any time is
> supervenient on the states then assigned to its component subsystems.
> 



This principle could fail in one of two ways: the subsystems may
simply not be assigned any states of their own, or else the states
they are assigned may fail to determine the state of the system they
compose. Interestingly, state assignments in quantum mechanics have
been taken to violate state separability in both ways.



The quantum state of a system may be either pure or mixed (see related
entry
 quantum mechanics).
 In ordinary quantum mechanics a pure state is represented by a vector
in the system's Hilbert space. On one common understanding, any
 entangled quantum systems
 violate state separability in so far as a vector representing the
state of the system they compose does not factorize into a product of
vectors, one in the Hilbert space of each individual subsystem, that
could be taken to represent their pure states. On the other hand, in
such a case each subsystem may be uniquely assigned what is called a
mixed state represented in its Hilbert space not by a vector but by a
more general object--a so-called von Neumann density operator.
But then state separability fails for a different reason: the
subsystem mixed states do not uniquely determine the compound
system's state. A failure of state separability may not occasion
much surprise if states are thought of merely in their role of
specifying a system's chances of exhibiting various possible
properties on measurement. But it becomes more puzzling if one
believes that a system's quantum state also has a role in
specifying some or all of its categorical properties. For that role
may connect a failure of state separability to metaphysical holism and
nonseparability.


## 6. Spatial and Spatiotemporal Nonseparability



The idea is familiar (particularly to Lego enthusiasts!) that if one
constructs a physical object by assembling its physical parts, then
the physical properties of that object are wholly determined by the
properties of the parts and the way it is put together from them. A
principle of spatial separability tries to capture that idea.



> 
> *Spatial Separability*:
>  The qualitative intrinsic physical properties of a compound system
> are supervenient on those of its spatially separated component systems
> together with the spatial relations among these component systems.
> 



If we identify the real state of a system with its qualitative
intrinsic physical properties, then spatial separability is related to
a separability principle stated by Howard (1985, p. 173) to the effect
that any two spatially separated systems possess their own separate
real states. It is even more closely related to Einstein's
(1935)
 real state separability principle.
 Indeed, Einstein formulated this principle in the context of a pair
\(A, B\) of spatially separated systems.



Spatial nonseparability
 --the denial of spatial separability--is also closely
related to
 physical property holism.
 At least classically, spatial relations are the only clear examples
of qualitative intrinsic physical relations required in the
supervenience basis for physical property determination/holism: other
intrinsic physical relations seem to supervene on them, while any
instance of physical property holism due to the spatial separation of
basic physical parts would entail spatial nonseparability. But if one
thought a spatially localized object has a determinate value for a
magnitude like mass only by virtue of its mass relations to other such
objects elsewhere then one might decide to include those relations in
the supervenience basis also (see Dasgupta (2013)).



If we take a spacetime perspective, then spatial separability
naturally generalizes to



> 
> *Spatiotemporal Separability*:
>  Any physical process occupying spacetime region *R*
> supervenes upon an assignment of qualitative intrinsic physical
> properties at spacetime points in *R*.
> 



Spatiotemporal separability is a natural restriction to physics of
David Lewis's (1986, p. x) principle of Humean supervenience. It
is also closely related to another principle formulated by Einstein
(1948, pp. 233-234 of Howard's (1989) translation) in the
following words: "An essential aspect of [the] arrangement of
things in physics is that they lay claim, at a certain time, to an
existence independent of one another, provided these objects
'are situated in different parts of space'" (the
context of the quote suggests that Einstein intended his principle to
apply to objects provided they occupy spacelike separated regions
of spacetime).



As Healey (1991, p. 411) shows, spatiotemporal separability entails
spatial separability, and so spatial nonseparability entails
spatiotemporal nonseparability. Because it is both more general and
more consonant with a geometric spacetime viewpoint, it seems
reasonable to consider spatiotemporal separability to be the primary
notion. Accordingly, separability without further qualification will
mean spatiotemporal separability in what follows, and nonseparability
will be understood as its denial.



> 
> *Nonseparability*:
>  Some physical process occupying a region *R* of
> spacetime is not supervenient upon an assignment of qualitative
> intrinsic physical properties at spacetime points in *R*.
> 



It is important to note that nonseparability entails neither
 physical property holism
 nor
 spatial nonseparability:
 a process may be nonseparable even though it involves objects without
proper parts. But this section has explained that either of the latter
principles entails nonseparability under quite weak assumptions.


## 7. Holism and Nonseparability in Classical Physics



Classical physics presents no definitive examples of either
 physical property holism
 or
 nonseparability.
 As section 6 explained, almost any instance of physical property
holism would demonstrate nonseparability. This justifies restricting
attention to the latter notion. Now the assumption that all physical
processes are completely described by a local assignment of magnitudes
forms part of the metaphysical background to classical physics. In
Newtonian spacetime, the kinematical behavior of a system of point
particles under the action of finite forces is supervenient upon
ascriptions of particular values of position and momentum to the
particles along their trajectories. This supervenience on local
magnitudes extends also to dynamics if the forces on the particles
arise from fields defined at each spacetime point.



The boiling of a kettle of water is an example of a more complex
physical process. It consists in the increased kinetic energy of its
constituent molecules permitting each to overcome the short range
attractive forces which otherwise hold it in the liquid. It thus
supervenes on the assignment, at each spacetime point on the
trajectory of each molecule, of physical magnitudes to that molecule
(such as its kinetic energy), as well as to the fields that give rise
to the attractive force acting on the molecule at that point.



As an example of a process in Minkowski spacetime (the spacetime
framework for Einstein's special theory of relativity), consider
the propagation of an electromagnetic wave through empty space. This
is supervenient upon an ascription of the electromagnetic field tensor
at each point in the spacetime.



But it does not follow that classical processes like these are
separable. For one may question whether an assignment of basic
magnitudes at spacetime points amounts to or results from an
assignment of qualitative intrinsic properties at those points. Take
instantaneous velocity, for example: this is usually defined as the
limit of average velocities over successively smaller temporal
neighborhoods of that point. This provides a reason to deny that the
instantaneous velocity of a particle at a point supervenes on
qualitative intrinsic properties assigned at that point. Similar
skeptical doubts can be raised about the intrinsic character of other
"local" magnitudes such as the density of a fluid, the
value of an electromagnetic field, or the metric and curvature of
spacetime (see Butterfield (2006)).



One response to such doubts is to admit to a minor consequent
violation of separability while introducing a weaker notion,
namely



> 
> *Weak Separability:* Any physical process occupying spacetime
> region *R* supervenes upon an assignment of
> qualitative intrinsic physical properties at points of *R*
> and/or in arbitrarily small neighborhoods of those
> points.
> 



Along with a correspondingly strengthened notion of



> 
> *Strong Nonseparability:* Some physical process occupying a
> region *R* of spacetime is not supervenient upon an
> assignment of qualitative intrinsic physical properties at points of
> *R* and/or in arbitrarily small neighborhoods of
> those points.
> 



No holism need be involved in a process that is nonseparable, but not
strongly so, as long as the basic parts of the objects involved in it
are themselves taken to be associated with arbitrarily small
neighborhoods rather than points. Versions of *Weak
Separability* are offered by both Belot (1998, p. 540), whose term
is Synchronic Locality, and by Myrvold (2011, p. 425), who calls it
Patchy Separability. The idea is that the state of a region supervenes
on assignments of intrinsic properties to patches of the region, where
the patches may be taken to be arbitrarily small.



Any physical process fully described by a local spacetime theory will
be at least weakly separable. For such a theory proceeds by assigning
geometric objects (such as vectors or tensors) at each point in
spacetime to represent physical fields, and then requiring that these
satisfy certain field equations. But processes fully described by
theories of other forms will also be separable. These include many
theories which assign magnitudes to particles at each point on their
trajectories. Of familiar classical theories, it is only theories
involving direct action between spatially separated particles which
involve nonseparability in their description of the dynamical
histories of individual particles. But such processes are weakly
separable within spacetime regions that are large enough to include
all sources of forces acting on these particles, so that the
appearance of strong nonseparability may be attributed to a mistakenly
narrow understanding of the spacetime region these processes actually
occupy.



The propagation of gravitational energy according to general
relativity apparently involves strongly nonseparable processes, since
gravitational energy cannot be localized (it does not contribute to
the stress-energy tensor defined at each point of spacetime as do
other forms of energy). But even a non-locally-defined gravitational
energy will still be supervenient upon the metric tensor defined at
each point of the spacetime, and so the process of its propagation
will be weakly separable.



The definition of nonseparability becomes problematic in general
relativity, since its application requires that one identify the same
region *R* in possible spacetimes with different
geometries. But while there is no generally applicable algorithm for
making a uniquely appropriate identification, some identification may
appear salient in a particular case. For example, one can meaningfully
discuss whether or not the field is the same everywhere in the region
outside the solenoid in the
 Aharonov-Bohm effect
 with an increased current flowing, even though the size of the
current will have a (tiny) influence on the geometry of that region.
Note that the definition of nonseparability does not require that one
identify the same point in spacetimes of distinct geometries.



While strictly outside the domain of classical physics, quantum
phenomena such as the
 Aharonov-Bohm effect
 may be considered manifestations of nonseparability and holism even
in *classical* electromagnetism. Nonseparability would be a
trivial notion if no qualitative intrinsic physical properties were
ever assigned at spacetime points or in their neighborhoods. But this
would require a thorough-going relationism that took not just
geometric but all local features to be irreducibly relational (cf.
Esfeld (2004)). We will return to this point in our discussion of the
 Aharonov-Bohm effect.


## 8. The Quantum Physics of Entangled Systems



Quantum entanglement is in the first instance a relation between not
physical but mathematical objects representing the states of quantum
systems. Different forms of quantum theory represent quantum states of
various systems by different kinds of mathematical object. So the
concept of quantum entanglement has been expressed by a family of
definitions, each appropriate to a specific form and application of
quantum theory (see Earman (2015)). The first definition
(Schrodinger (1935)) was developed in the context of applications
of ordinary non-relativistic quantum mechanics to pairs of
distinguishable particles that have interacted, such as an electron
and proton.



A hydrogen atom may be represented in ordinary non-relativistic
quantum mechanics as a quantum system composed of two subsystems: an
electron *e* and a nuclear proton *p*.
When isolated, its quantum state may be represented
by a vector \(\Psi\) in a space *H* constructed as a
tensor product of spaces \(H\_{p}\) and \(H\_{e}\) used to represent
states of \(e, p\) respectively. The states of \(e, p\) are then
defined as entangled if and only if
 


 

\[ \Psi \ne \Psi\_{p} \otimes \Psi\_{e}\]


for *every* pair of vectors \(\Psi\_{p}, \Psi\_{e}\) in
\(H\_{p}\), \(H\_{e}\) respectively. This definition naturally
generalizes to systems composed of *n*
distinguishable particles. But alternative definitions seem preferable
for a collection of indistinguishable particles--of electrons or
of photons for example (see Ghirardi *et al.* (2002), Ladyman
*et al.* (2013)).



It follows that the states of electron and proton in an isolated
hydrogen atom are entangled. But one may also represent the hydrogen
atom as composed of a center-of-mass subsystem *C*
and a relative subsystem *R* represented by vector
states \(\Psi\_{C}\), \(\Psi\_{R}\) in \(H\_{C}, H\_{R}\) respectively
such that
 


 

\[ \Psi = \Psi\_{C} \otimes \Psi\_{R}\]


If the state of the hydrogen atom is represented by \(\Psi\) then the
states of quantum subsystems \(C, R\) are not entangled but the states
of quantum subsystems \(p, e\) are entangled. This illustrates the
important point that one cannot draw metaphysical conclusions from a
mathematical condition of quantum entanglement without first deciding
which quantum systems are physical parts composing some physical
whole. It may seem natural to take the physical parts of a hydrogen
atom to be an electron and a proton. But note that the state of an
isolated hydrogen atom is usually represented by \(\Psi\_{R}\), and not
by \(\Psi\) or \(\Psi\_{e}\).



Viewed as basic physical parts of a hydrogen atom represented by state
\(\Psi\), its electron and proton may be considered entangled physical
parts since \(\Psi\) cannot be expressed as a product of vectors
representing the state of each. The electron and the proton may each
be assigned a mixed state, but these do not uniquely determine the
state \(\Psi\): state separability is violated. This may occasion no
surprise if a system's state merely specifies its chances of
exhibiting various possible properties on measurement. But it may have
metaphysical significance if a system's quantum state plays a
role in specifying its categorical properties--its real state, so
that the
 real state separability principle
 is threatened. His commitment to this principle is one reason why
Einstein denied that a physical system's real state is given by
its quantum state (though it's not clear what he thought its
real state consisted in). But according to (one variant of) the rival
Copenhagen interpretation, the quantum state gives a physical
system's real dynamical state by specifying that it contains
just those qualitative intrinsic quantum dynamical properties to which
it assigns probability 1. On this last interpretation, violation of
state separability in quantum mechanics leads to
 physical property holism:
 it implies, for example, that a pair of fundamental particles may
have the intrinsic property of being spinless even though this is not
determined by the intrinsic properties and relations of its component
particles.



If an entangled pure vector state of a pair of quantum systems
violates state separability then there are measurements of dynamical
variables (one on each subsystem) whose joint quantum probability
distribution cannot be expressed as a product of probability
distributions for separate measurements of each variable. Quantum
theory predicts such a probability distribution for each of many types
of spatially separated measurements of variables including spin and
polarization components on a pair of entangled physical entities
assigned such a state, and many of these distributions have been
experimentally verified. If one thought that quantum theory treats
each dynamical variable by replacing a precise real value assignment
by a probability distribution for the results of measurements of that
dynamical variable, one might take this already to violate the real
state separability principle. But if one entertains a theory that
supplements the quantum state by values of additional
"hidden" variables, then the quantum probabilities would
be taken to arise from averaging over many distinct hidden states. In
that case, it would rather be the probability distribution conditional
on a complete specification of the values of the hidden variables that
should be taken to specify the underlying chances of system and
subsystems exhibiting various possible properties on measurement. The
real state might then include all these conditional probability
distributions. The best known example of such a theory is the Bohm
theory (see the entry on
 Bohmian mechanics),
 where the "hidden" variables are spatial positions. In
each specific experimental context all conditional probabilities are 0
or 1, so the joint conditional probability distributions indeed
factorize. But the outcome of a measurement of a chosen dynamical
variable on one subsystem depends on what dynamical variable is chosen
and measured on the other, no matter just when or how far apart these
measurements are selected and made.



Bell (1964, [2004]) reasoned that any *local* hidden variable
theory must yield conditional probabilities of 0 or 1 for each local
outcome in order to reproduce all quantum predictions but could not
permit these to depend on the choice of distant measurement. He then
proved that the probabilistic predictions of any local hidden variable
theory must satisfy particular inequalities violated by predictions of
quantum theory for certain entangled state assignments (see entry
 Bell's Theorem).
 In later work Bell (1990, [2004]) generalized this argument to apply
to any theory of a certain type meeting a condition he called Local
Causality which, he claimed, quantum mechanics does not meet. Howard
(1989, 1992) took outcome independence -- the probabilistic
independence of the outcomes of a given pair of measurements, one on
each of a pair of entangled systems, conditional on definite values
for any assumed hidden variables on the joint system--as a
separability condition. Outcome independence may be contrasted with
parameter independence--the condition that, given a definite
hidden variable assignment, the outcome of a measurement on one of a
pair of entangled systems is probabilistically independent of what
measurement, if any, is made on the other system. Together with
parameter independence, outcome independence implies the factorization
of conditional probabilities that leads to what are now called Bell
inequalities. These inequalities constrain the patterns of statistical
correlations to be expected between the results of measurements of
variables such as spin and polarization on a pair of entangled systems
in any quantum state. Quantum mechanics predicts, and experiment
confirms, that such Bell inequalities do not always hold. The Bohm
theory accommodates this fact by its violation of parameter
dependence, and hence Local Causality. But Howard (1989), as well as
Teller (1989), suggested that we appeal instead to a failure of
outcome independence to understand why Bell inequalities do not always
hold, and that this failure is associated rather with holism or
nonseparability. Howard (1989) blamed the violation of Bell
inequalities on the violation of his separability condition: Teller
(1989) took it to be a manifestation of
 relational holism.
 They both acquit parameter independence of blame because they believe
that (at least when the measurement events on the entangled systems
are spacelike separated) parameter independence (unlike outcome
independence) is a consequence of relativity theory: (note that the
Bohm theory requires a preferred frame not provided by relativity
theory).



Henson (2013) and others have questioned this line of reasoning,
including the conclusion that its appeal to holism or nonseparability
helps one to understand how these correlations involving entangled
systems come about without any action at a distance that violates
relativity theory, Local Causality or Einstein's (1948)



> 
> *Principle of Local Action*:
>  If *A* and *B* are spatially
> distant things, then an external influence on *A* has
> no *immediate* effect on *B*.
> 



Howard's (1989,1992) identification of outcome independence with
a separability condition has proved controversial, as has
Teller's (1989) claim that violations of Bell inequalities are
no longer puzzling if one embraces (physical) relational holism
(Laudisa 1995; Berkowitz 1998; Henson 2013). Winsberg and Fine (2003)
object that separability requires only that the conditional joint
probabilities be determined as some function of the marginal
probabilities, whereas outcome independence arbitrarily restricts this
to be the product function. By allowing other kinds of functional
dependence they are able to construct models of experiments whose
results would exhibit violations of Bell inequalities. They claim that
these models are both local and separable even though they violate
outcome independence. But Fogel (2007) presents alternative
formalizations of the separability condition, of which several indeed
imply outcome independence. The view that violations of outcome
independence are consistent with relativity theory, while violations
of parameter independence are not, has also been criticized (Jones
& Clifton 1993; Maudlin 2011). But Myrvold (2016) has responded by
arguing that locally-initiated state-vector collapse in violation of
outcome dependence may be perfectly compatible with relativity.



While diverging from the Copenhagen prescription mentioned above, some
 modal interpretations
 take real states of systems to be closely enough related to quantum
states that entangled systems' violation of quantum state
separability implies some kind of holism or nonseparability. Van
Fraassen (1991, p. 294), for example, sees his modal interpretation as
committed to "a strange holism" because it entails that a
compound system may fail to have a property corresponding to a tensor
product projection operator \(P\otimes I\) even though its first
component has a property corresponding to *P*. In
fact, a clearer case of holism would arise in a modal interpretation
that implied that the component lacked *P* while the
compound had \(P\otimes I\): *ceteris paribus*, that would
provide an instance of physical property holism. Healey (1989,1994)
offered a modal interpretation and used it to present a model account
of the puzzling correlations which portrays them as resulting from the
operation of a process that violates both
 spatial
 and
 spatiotemporal separability.
 He argued that, on this interpretation, the
 nonseparability
 of the process is a consequence of
 physical property holism;
 and that the resulting account yields genuine understanding of how
the correlations come about without any violation of relativity theory
or
 Local Action.
 But subsequent work by Clifton and Dickson (1998) and Myrvold (2001)
cast doubt on whether the account can be squared with relativity
theory's requirement of Lorentz invariance. More recently Healey
(2016) has given a different account of how quantum theory may be used
to explain violations of Bell inequalities consistent with Lorentz
invariance and Local Action. This account involves no metaphysical
holism or nonseparability.



Esfeld (2001) takes holism, in the quantum domain and elsewhere, to
involve more than just a failure of supervenience. He maintains that a
compound system is holistic in that its subsystems themselves count as
quantum systems only by virtue of their relations to other subsystems
together with which they compose the whole.


## 9. Ontological Holism in Quantum Mechanics?



As applied to physics,
 ontological holism
 is the thesis that there are physical objects that are not wholly
composed of basic physical parts. Views of Bohr, Bohm and others may
be interpreted as endorsing some version of this thesis. In no case is
it claimed that any physical object has *non*physical parts.
The idea is rather that some physical entities that we take to be
wholly composed of a particular set of basic physical parts are in
fact not so composed.



It was Bohr's (1934) view that one can meaningfully ascribe
properties such as position or momentum to a quantum system only in
the context of some well-defined experimental arrangement suitable for
measuring the corresponding property. He used the expression
'quantum phenomenon' to describe what happens in such an
arrangement. In his view, then, although a quantum phenomenon is
purely physical, it is not composed of distinct happenings involving
independently characterizable physical objects--the quantum
system on the one hand, and the classical apparatus on the other. And
even if the quantum system may be taken to exist outside the context
of a quantum phenomenon, little or nothing can then be meaningfully
said about its properties. It would therefore be a mistake to consider
a quantum object to be an independently existing component part of the
apparatus-object whole.



Bohm's (1980, 1993) reflections on quantum mechanics led him to
adopt a more general holism. He believed that not just quantum object
and apparatus, but any collection of quantum objects by themselves,
constitute an indivisible whole. This may be made precise in the
context of Bohm's (1952) interpretation of quantum mechanics by
noting that a complete specification of the state of the
"undivided universe" requires not only a listing of all
its constituent particles and their positions, but also of a field
associated with the wave-function that guides their trajectories. If
one assumes that the basic physical parts of the universe are just the
particles it contains, then this establishes ontological holism in the
context of Bohm's interpretation. But there are alternative
views of the ontology of the Bohm theory (see the entry
 Bohmian mechanics).



Some (Howard 1989; Dickson 1998) have connected the failure of a
principle of separability to ontological holism in the context of
violations of Bell inequalities. Howard (1989) formulates the
following separability principle (pp. 225-6)



> 
> *Howard Separability*:
>  The contents of any two regions of space-time separated by a
> nonvanishing spatiotemporal interval constitute separable physical
> systems, in the sense that (1) each possesses its own, distinct
> physical state, and (2) the joint state of the two systems is wholly
> determined by these separate states.
> 



He takes Einstein to defend this as a principle of individuation of
physical systems, without which physical thought "in the sense
familiar to us" would not be possible. Howard himself
contemplates the possible failure of this principle for
 entangled quantum systems,
 with the consequence that these could no longer be taken to be wholly
composed of what are typically regarded as their subsystems. Dickson
(1998), on the other hand, argues that such holism is not "a
tenable scientific doctrine, much less an explanatory one" (p.
156).



One may try to avoid the conclusion that experimental violations of
Bell inequalities manifest a failure of
 Local Action
 by invoking ontological holism for events. The idea would be to deny
that these experiments involve distinct, spatiotemporally separate,
measurement events, and to maintain instead that what we usually
describe as separate measurements involving an entangled system in
fact constitute one indivisible, spatiotemporally disconnected, event
with no spatiotemporal parts. But such ontological holism conflicts
with the criteria of individuation of events inherent in both quantum
theory and experimental practice.


## 10. The Aharonov-Bohm Effect and Field Holonomies



Aharonov and Bohm (1959) drew attention to the quantum mechanical
prediction that an interference pattern due to a beam of charged
particles could be produced or altered by the presence of a constant
magnetic field in a region from which the particles were excluded.
This effect has since been experimentally demonstrated. At first
sight, the Aharonov-Bohm effect seems to involve action at a distance.
It seems clear that the (electro-)magnetic field acts on the particles
since it affects the interference pattern they produce; and this must
be action at a distance since the particles pass through a region from
which that field is absent. But alternative accounts of the phenomenon
are possible which portray it rather as a manifestation of (strong)
 nonseparability
 (Healey 1997). There need be no action at a distance if the behavior
both of the charged particles and of electromagnetism are nonseparable
processes. While such a treatment of electromagnetism (and other gauge
theories) is increasingly common in physics, to treat the motion of
the charged particles as a nonseparable process is to endorse a
particular position on how quantum mechanics is to be interpreted.



An interpretation of quantum mechanics that ascribes a nonlocalized
position to a charged particle on its way through the apparatus is
committed to a violation of
 spatiotemporal separability
 in the Aharonov-Bohm effect, since the particle's passage
constitutes a nonseparable process. To see why the electromagnetism
that acts on the particles during their passage may also be taken to
be nonseparable it is necessary to consider contemporary
representations of electromagnetism in terms of neither fields nor
vector potentials.



Following Wu and Yang's (1975) analysis of the Aharonov-Bohm
effect, it has become common to consider vacuum electromagnetism to be
completely and nonredundantly described neither by the electromagnetic
field, nor by its vector potential, but rather by the so-called Dirac
phase factor:

 

\[ \exp[(ie/\hbar) \oint\_C A\_{\mu}(x^{\mu}.dx^{\mu}] \]


where \(A\_{\mu}\) is the electromagnetic potential at spacetime point
\(x^{\mu}\), *e* is the particles' charge, and
the integral is taken over each closed loop *C* in
spacetime. This can be seen as an instance of the more general notion
of the holonomy of a closed curve, a notion that has come to the fore
in contemporary formulations of gauge theories including
electromagnetism in terms of fiber bundles (Healey (2007)). Applied to
the Aharonov-Bohm case, this means that the constant magnetic field is
accompanied by an association of a phase factor \(S(C)\) with all
closed curves *C* in space, where \(S(C)\) is defined
by

 

\[ S(C) = \exp[-(ie/\hbar) \oint\_C \mathbf{A(r).}d\mathbf{r}] \]


(where \(\mathbf{A}(\mathbf{r})\) is the magnetic vector potential at
point \(\mathbf{r}\) of space). This approach has the advantage that
since \(S(C)\) is gauge-invariant, it may readily be considered a
physically real quantity. Moreover, the effects of electromagnetism in
the field-free region may be attributed to the fact that \(S(C)\) is
nonvanishing for certain closed curves *C* within
that region. But it is significant that, unlike the magnetic field and
its potential, \(S(C)\) is not defined at each point of space at each
moment of time.



Can \(S(C)\) at some time be taken to represent an intrinsic property
of a region of space corresponding to the curve *C*?
There are two difficulties with this suggestion. The first is that the
presence of the quantity *e* in the definition of
\(S(C)\) appears to indicate that \(S(C)\) rather codes the effect of
electromagnetism on objects with that specific charge. If in fact
*all* charges are multiples of some minimal value *e*,
then this would no longer be a problem: the value of
\(S(C)\) for this minimal charge could then be viewed as representing
an intrinsic property of a region of space corresponding to the curve
*C*. If not, one could rather take

 

\[ I(C) = \oint\_C \mathbf{A(r).}d\mathbf{r} \]


to represent an intrinsic property of *C*. The second
difficulty is that closed curves do not correspond uniquely to regions
of space: e.g., circling the region in which there is a magnetic field
twice on the same circle will produce a different curve from circling
it once. But this does not prevent one from taking \(S(C)\) at some
time to represent an intrinsic property of a loop--the oriented
region of space traced out by a closed curve *C* that
intersects itself only at its end-point.



Once these difficulties have been handled, it is indeed possible to
consider electromagnetism in the Aharonov-Bohm effect as faithfully
represented at a time by a set of intrinsic properties of loops in
space (or more generally space-time). But if one does so, then
electromagnetism itself manifests (strong)
 nonseparability.
 For these intrinsic properties do not supervene on any assignment of
qualitative intrinsic physical properties at spacetime points in the
region concerned, nor even in arbitrarily small neighborhoods of those
points. Whether the magnetic field remains constant or changes, the
associated electromagnetism constitutes a nonseparable process, and so
the Aharonov-Bohm effect violates
 spatiotemporal separability.
 If the motion of the particles through the apparatus is a
nonseparable process, then it is possible to account for the \(AB\)
effect in terms of a purely local interaction between electromagnetism
and this process. For the particles effectively traverse loops traced
out by closed curves *C* on their nonlocalized
"trajectories", and so they interact with electromagnetism
precisely where this is defined.



Assuming it exhibits nonseparability, does the Aharonov-Bohm effect
involve some kind of holism? The states of the particles need not be
entangled with each other. But the state of the field may be thought
to be holistic, in so far as the electromagnetic properties of loops
do not supervene on properties (such as electric and magnetic field
strengths) at the points that make up those loops. Since these are
classical fields, the Aharonov-Bohm effect may be taken to demonstrate
holism as well as nonseparability even in *classical* physics.
However, a loop may also be thought of as traced out by a curve
composed by "stringing together" a set of curves that
trace out smaller loops, and removing segments traversed by two such
curves in opposite directions. In that case, in a simply-connected
region, the holonomy properties of any loop will be determined by
those of an arbitrary set of smaller loops that compose it in this
way. But when spacetime is not simply connected, there are loops that
cannot be decomposed in this way: e.g. the loop that encompasses
"the hole" in space cannot be further decomposed.



Myrvold (2011) interprets this result as saying that Weak Separability
(what he calls Global Patchy Separability) fails for non-simply
connected spacetimes, even though locally--over simply-connected
patches--separability holds. Moreover, he here sees a marked
distinction between the holonomy and the field formalism for gauge
theories: in which supposedly separability always holds.



But Gomes (2021) effaces this distinction: he argues that a field
formalism maintains separability, but only at the cost of dealing with
gauge *variant* field variables. Once we restrict attention to
the gauge-*invariant* quantities, the nonseparability
illustrated above (using the holonomy variables) is restored.
Moreover, he argues that only in a vacuum is nonseparability
conditional on the topology of spacetime. In the presence of charged
matter, there may be gauge-invariant quantities that cannot be
decomposed into smaller patches, even for simply-connected regions.
And this is easy to illustrate: suppose that, instead of the phase
factor \(S(C)\), for a closed loop C, we write \(S(I)\) for an open
segment *I*, that begins at point *x* and ends
at point *y*. For a (Klein-Gordon) charged scalar
field \(\varphi\) that vanishes everywhere except at small
neighborhoods \(\nu\_x, \nu\_y\), of *x* and *y*,
the function:



\[
 \bar\varphi(x)S(I)\varphi(y)
\]


is gauge-invariant, and it cannot be further decomposed into
quantities with support on patches that do not include the entirety of
\(I. \)For instance, if there are two patches, each containing either
\(\nu\_x\) or \(\nu\_y\), then that patch has insufficient charges to
complete \(S(I)\) into a gauge-invariant function. This example is in
line with the physics literature, which largely takes gauge-invariance
(and in particular the existence of a generalized Gauss constraint),
to signal some general type of nonlocality (cf. Strocchi (2013, Ch.
7)).


## 11. Alternative Approaches



This entry has focused mainly on
 metaphysical holism
 and its relation to nonseparability. That there is a variety of
alternative ways of understanding holism in physics is illustrated by
a special issue of the journal *Studies in the History and
Philosophy of Physics* (2004) devoted to this topic.



Seevinck (2004) proposes an epistemological criterion of holism and
illustrates its application to physical theories. A physical theory
counts as holistic by this criterion if and only if it is impossible
in principle to infer the global properties, as assigned in the
theory, by local resources available to an agent, where these include
(at least) all local operations and classical communication. To apply
this criterion it is necessary to specify how a theory assigns
properties, a matter on which different interpretations of the theory
may disagree. Seevinck (2004) argues that neither classical physics
nor Bohmian mechanics is holistic in this sense. Applying the
eigenvalue-eigenstate link to a particular state of a bipartite
quantum system he then shows that this manifests epistemological
holism, even when the state of the system is not
 entangled.



Placek (2004) understands quantum state holism as involving a thesis
about probabilities: that the probability of a joint result of a
combined measurement on a pair of entangled quantum systems is not
determined by the probabilities of the two results. But he takes this
to be only one ingredient of a more complete conception, whose
formulation and analysis requires a modal framework combining
indeterminism, (rudiments of) relativistic space-time, and
probability--Belnap's (1992) theory of branching
space-times.



Esfeld (2004) argues for a metaphysics of relations based on a
characterization of quantum entanglement in terms of non-separability,
thereby regarding entanglement as a sort of holism. He there
characterizes non-separability as follows:



> 
> *Non-separability:* The states of two or more systems are
> non-separable if and only if it is only the joint state of the whole
> that completely determines the state-dependent properties of each
> system and the correlations among these systems (to the extent that
> these are determined at all).
> 



He takes this to imply that any case of quantum entanglement is a case
of non-separability, and non-separability is the reason why quantum
entanglement is a sort of holism. (He discusses the relation between
non-separability and holism in chapter 8 of Esfeld (2001).)



Lyre (2004) and Healey (2004) see electromagnetism and other gauge
theories as manifesting nonseparability for reasons different from
those arising from quantum entanglement (cf.
 The Aharonov-Bohm Effect).
 Lyre takes this to be a variant of spatiotemporal holism, and
connects it to structural realism. Healey (2007) argues that general
relativity does not manifest this kind of nonseparability even though
it may be formulated as a gauge theory. Weatherall (2016) argues that
a distinction exists, but it is not enough to rule out spatiotemporal
holism in general relativity. If there are more than one spatial
dimensions, an analogue of the Aharonov-Bohm effect in general
relativity (using the directions of parallel-propagated vectors)
theoretically exhibits a similar type of nonseparability as that
observed in gauge theory.


## 12. Quantum Field Theory



Certain phenomena that arise within quantum field theory have been
taken to challenge principles of separability or to involve holism.
These have been analyzed most intensively by mathematical physicists
and philosophers taking an algebraic approach to quantum theory, even
though many empirical successes of quantum field theory have been
achieved by following other approaches.



Algebraic quantum field theory (AQFT) represents the state in a region
of spacetime by means of a function from an algebra of associated
"field" or "observable" operators: the value
of this function for a self-adjoint operator represents the expected
result of a measurement of the corresponding observable on that
region. A state is said to be decomposable (some say separable) across
algebras \(R\_{A}, R\_{B}\) associated with regions \(A, B\) if its
restriction \(\omega\) to the algebra \(R\_{AB}\) generated by \(R\_{A},
R\_{B}\) is a product state--i.e. satisfies
\(\omega(XY)=\omega(X)\omega(Y)\), for all \(X\in R\_{A}, Y\in R\_{B}\);
or if \(\omega\) is a limit of convex combinations of product states:
otherwise, it is said to be entangled across \(R\_{AB}\) (see e.g.
Valente 2010, pp. 1031-2). This is a natural reformulation of a
generalization to mixed states of the first condition for entanglement
given in
 section 8.



Entanglement is endemic in AQFT. Summers and Werner (1985) proved that
the vacuum state of a quantum field is not only entangled across
algebras associated with certain spacelike separated regions of
Minkowski spacetime, but that it also maximally violates Bell
inequalities for algebras associated with these regions. They also
proved (1988) that *every* state on a pair of space-like
separated open regions whose closures share a single point is
maximally entangled across their algebras. For each state, the degree
of entanglement decreases rapidly with spatial separation. But if and
only if \(R\_{A}, R\_{B}\) possess what is called the *split
property* is any state decomposable across these algebras.



The split property (Valente 2010, p. 1035) is a strengthening of the
condition of microcausality (observables on space-like separated
regions commute). Summers (2009) argues that it is meaningful to speak
of independent subsystems in relativistic quantum theory if they can
be localized in spacetime regions \(A, B\) whose algebras \(R\_{A},
R\_{B}\) posses the split property; and that most, if not all,
physically relevant models of quantum field theories have this
property (for sufficiently space-like separated regions \(A, B)\).



The split property is a kind of independence condition. Redei
(2010) argues that, by conforming to this and other independence
conditions, an AQFT can meet all the requirements Einstein (1948)
considered necessary for a quantum theory satisfactorily to realize
the field-theoretic ideal. These were the requirement that physical
things be arranged in a space-time continuum (Spatiotemporality); that
things located in space-like separated regions have their own distinct
states (Independence); and that if \(a, b\) are located in space-like
separated regions \(A, B\) respectively, then an external influence on
*a* has no *immediate* effect on *b*
(Local Action).
 (The first two names are Redei's: the last is
Einstein's.) Redei takes AQFT to satisfy
Spatiotemporality because of its basic assumption that observables are
localized in space-time regions; that an AQFT's satisfaction of
the split property and other members of a hierarchy of independence
conditions establishes Independence; and that an AQFT obeys Local
Action insofar as it meets a condition he calls operational
separability.



In evaluating Redei's argument it is important to ask
what counts as a physical thing. Einstein mentioned two possible
candidates: bodies and fields.
 Howard's separability principle
 permits a natural transposition of Einstein's
 real state separability principle
 to field theory. Entangled states in AQFT violate section 5's
 state separability
 principle just as they do in nonrelativistic quantum mechanics even
though the split property and related independence conditions on their
algebras hold. So if the content of a space-time region were specified
by its algebra in AQFT, considered as a physical system with real
physical state given by a state on that algebra, then Howard's
separability principle would fail. (Although the failure of the split
property or other algebraic independence conditions for certain
regions in a quantum field theory would represent a more radical
threat to the separate existence of such physical systems in those
regions than mere entanglement). But it is doubtful that Einstein
would have counted observables or their algebras as physical things.
If instead these are taken to represent magnitudes of physical fields
or the spatio-temporal region on which they are defined then meeting
Redei's Independence requirement is still consistent with
failure of Howard's (stronger) separability principle. Finally
meeting Redei's operational separability condition only
for *non-selective* operations would not suffice to ensure
conformity to Local Action. Einstein's reasons for rejecting the
completeness of quantum mechanical description are naturally extended
to AQFT: if a state on its local algebra completely specifies the real
state of a space-time region then either the natural extension of his
 real state separability principle
 or his principle of
 Local Action
 fails.



Metaphysical holism
 presupposes a division of a whole into parts. To apply the part/whole
distinction here one must address the ontology of quantum field
theory. Taking spacetime regions to be the relevant physical objects,
one could understand the system/subsystem and part/whole relations in
terms of spatiotemporal inclusion. To assess
 physical property holism
 or
 nonseparability
 we need to determine the qualitative intrinsic properties and
relations pertaining to spacetime regions in quantum field theory.



Arageorgis (2013) gives an example of quantum field states entangled
across two regions which nevertheless, he argues, fail to exhibit the
same kind of
 state nonseparability
 as singlet and triplet spin states of a pair of quantum particles
(see Maudlin 1998). But he suggests that his example does exhibit a
kind of epistemological state nonseparability, in so far as an agent
confined to a single region cannot determine its state by operations
confined to that region. By applying the eigenvalue-eigenstate link to
his example, Arageorgis (2013) argues that the energy of a particular
compound quantum field system is not determined by the energies (or
any other qualitative intrinsic properties and relations) of its
component subsystems. He concludes that this example manifests
physical property holism.



Wayne (2002) has suggested that quantum field theory is best
interpreted as postulating extensive holism or nonseparability. On
this interpretation, the fundamental quantities in quantum field
theory are vacuum expectation values of products of field operators
defined at various spacetime points. The field can be reconstructed
out of all of these. Nonseparability supposedly arises because the
vacuum expectation value of a product of field operators defined at an
*n*-tuple of distinct spacetime points does not
supervene on qualitative intrinsic physical properties defined at
those *n* points, together with the spatiotemporal
relations among the points. But it is not clear that vacuum
expectation values of products of field operators defined at *n*-tuples
of distinct spacetime points represent either
qualitative intrinsic physical properties of these *n*-tuples
or physical relations between them. Improved
assessment of the extent to which quantum field theory illustrates
 holism
 or
 nonseparability
 must await further progress in the interpretation of quantum field
theory. (Kuhlman, Lyre and Wayne (2002) represents a relevant first
step: but see also Fraser (2008), Baker (2009).)


## 13. String Theory



String theory (or its descendant, *M*-theory) has
emerged as a speculative candidate for unifying much of fundamental
physics, including quantum mechanics and general relativity. Existing
string theories proceed by quantizing classical theories of basic
entities that are extended in one or more dimensions of a space that
has 6 or 7 tiny compact dimensions in addition to the three spatial
dimensions of ordinary geometry. If these additional dimensions are
appropriately considered spatial, then it is natural to extend the
concepts of
 spatial
 and
 spatiotemporal
 separability to encompass them. In that case, processes involving
classical strings (or *p*-branes with \(p \gt 0\))
would count as (spatiotemporally) nonseparable, even though all
particles and their properties conform to spatial separability.



The status of
 nonseparability
 within a quantized string field theory is not so easy to assess,
because of the general problems associated with deciding what the
ontology of any relativistic quantum field theory should be taken to
be.