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Michael A. Esfeld
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On the Possibility of a Realist Ontological Commitment in Quantum Mechanics
Text.Serial.Journal
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2018-06-01
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Michael A. Esfeld
2018/06/01
10.13135/2036-542X/8030
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2036-542X
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On the Possibility of a Realist Ontological Commitment in Quantum Mechanics
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                                                                                     – ISBN 978-88-255-1945-7
                                                          DOI 10.4399/97888255194572 – pag. 11-33 (giugno 2018)




                  On the Possibility of a Realist Ontological
                     Commitment in Quantum Mechanics
                                         A O, M A. E*




A: This paper reviews the structure of standard quantum mechanics,
introducing the basics of the von Neumann–Dirac axiomatic formulation
as well as the well–known Copenhagen interpretation. We review also the
major conceptual difficulties arising from this theory, first and foremost,
the well–known measurement problem. The main aim of this essay is to
show the possibility to solve the conundrums affecting quantum mechanics
via the methodology provided by the primitive ontology approach. Using
Bohmian mechanics as an example, the paper argues for a realist attitude
towards quantum theory. In the second place, it discusses the Quinean crite-
rion for ontology and its limits when it comes to quantum physics, arguing
that the primitive ontology programme should be considered as an improve-
ment on Quine’s method in determining the ontological commitments of
a theory.

K: Quantum Mechanics; Copenhagen Interpretation; Ontological
Commitment; Primitive Ontology; Bohmian Mechanics.


. Introduction

In the context of naturalized metaphysics we are typically guided by the
best scientific theories available to inform our ontology; for instance, one
may be ontologically committed to the existence of genes, molecules and
elementary particles, accepting the content of biological, chemical and
physical theories.
   Notoriously, this received view on ontological commitment in analytic
philosophy was proposed by (Quine ). In this seminal essay the author
claimed that ontological questions must be answered by looking at the
   ∗
       Université de Lausanne; Michael-Andreas.Esfeld@unil.ch; Andrea.Oldofredi@unil.ch.


                                                                                                            
      Andrea Oldofredi, Michael A. Esfeld


content of the most advanced scientific theories at our disposal, proposing
the following method:

     ) Select a set of statements considered true from the best scientific
        theories available;
     ) Express these statements in the language of first–order predicate logic
        (regimentation procedure);
     ) Be ontologically committed to all and only the bound variables
        appearing in existentially quantified sentences which make them
        true.

   With Quine’s words: «A theory is committed to those and only those
entities to which the bound variables of the theory must be capable
of referring in order that the affirmations made in the theory be true»
(Quine : ).
   To take an elementary example, endorsing what our most advanced
physical theories claim, we accept the existence of electrons and hence,
we consider true the quantified sentence “Electrons exist”. Its translation
into the language of first–order logic ∃xE(x) is useful to make clear what
entities we are ontologically committed to: in order for this sentence to
be true, there must be in the world a physical entity to which the bound
variable x refers which satisfies the predicate E (x), thus accepting this
sentence we are committed to the existence of (at least) one electron.
   Interestingly, the first step requires that, relatively to a specific domain,
there is a decisional element to establish which is the best theory among
possible competitors in order to choose the sentences to regiment. Alter-
natively stated, before the regimentation procedure we need to select a
theory which we endorse as (approximately) true among a set of possible
rival frameworks. Although the application of this method to classical
physical theories is quite unproblematic, since these frameworks are
ontologically well–defined theories, in the quantum context, as we will
explain, one needs to do some work not only in order to choose what is
the best framework to apply it, but also to spell out its limits.
   Analyzing the structures of classical theories, though remarkably
different from one another, it is possible to state that (i) these theories
provide a clear ontology, specifying the fundamental entities describing
matter in space — typical examples are the particle ontology instantiated
by Newtonian mechanics or the field ontology proposed in Maxwell’s
     . The expression “classical physical theories” refers to a set of theories covering an exten-
sive spectrum of scales, from descriptions of microscopic gases provided by classical statistical
mechanics, to cosmological effects treated by Einstein’s theory of gravitation. Broadly speaking,
they apply up to the length/energy scales where quantum interferences cannot be neglected.
     On the Possibility of a Realist Ontological Commitment in Quantum Mechanics   


electromagnetic theory — and that (ii) physical objects can be described
individually by the maximal set of properties they instantiate, whose
values are independent of the performance of observations and always
well–defined.
   Against this background, the physics concerned with molecules, atoms
and sub–atomic particles described by Quantum Mechanics (QM) calls
into question a classical world–view. Contrary to the classical case, in
quantum theory physical objects cannot be specified in terms of a maxi-
mal set of properties, since their values are not determinate until a mea-
surement of a particular magnitude is performed: isolated systems in QM
are objects with indeterminate properties. For instance, reconsidering the
previous example, the sentence ∃xE (x) commits us to the existence of
quantum objects without well defined position and momentum in space,
since Heisenberg’s uncertainty relations forbid the possibility to simulta-
neously measure with arbitrary precision the values of these magnitudes,
concluding that electrons (and all quantum particles) do not have both a
well defined position and a definite momentum. Thus, given the truth
conditions imposed by QM, the application of Quine’s method would im-
ply a commitment to a quite peculiar ontology. However, one may object
that it is not necessary to expect the retention of a classical world–view
in quantum contexts. This objection, although fair, cannot resist that the
peculiarities of QM are derived from conceptual and technical difficulties
affecting its mathematical structure, like the measurement problem or
the presence of mathematically ambiguous notions appearing within its
axioms. Hence, these conundrums prevent us from considering it as a
coherent description of the physical phenomena existing at the quantum
length–scales.
   These facts suggest that one should look for a quantum theory that is
immune to the conceptual difficulties affecting its standard formulation.
To overcome the problematic aspects of QM a significant number of
interpretations and alternative formulations have been presented.
   In this paper we will consider the Primitive Ontology (PO) approach,
a philosophical perspective which tries to solve the quantum puzzles
by providing a precise methodology to construct physical theories with
a well–defined ontology — that is, theories which specify the theoret-
ical entities representing matter in space and how they behave. These
frameworks recover by construction the explanatory scheme of classical
physical theories providing descriptions and explanations of quantum
phenomena and processes in terms of well–defined entities moving and
interacting in space and time, therefore avoiding obscure ontological state-

   . (J ) and (L ) discuss several proposals in detail.
      Andrea Oldofredi, Michael A. Esfeld


ments about the indefiniteness of quantum objects. Notable examples,
among others, are the Ghirardi–Rimini–Weber spontaneous collapse
theories, in particular the variants GRWm and GRWf implementing a
matter density field or flash ontology respectively, Nelsonian mechanics
or Bohmian Mechanics (BM). In this essay we will concentrate on the
latter proposal to show a possible way to recover a clear ontological com-
mitment in the quantum realm. We will argue that since BM provides
better explanations and descriptions of physical objects and processes
at the quantum length scales, there are arguments to prefer BM over
QM, and thus to consider the former a better candidate to apply Quine’s
method in the quantum context. Nevertheless, we will also discuss the
general validity of this method and its limitations in the context of quan-
tum physics, arguing that the PO perspective improves on this method to
establish one’s ontological commitments.



. Ontological Commitments in Standard Quantum Mechanics

Although the appearance of the quantum hypothesis dates back to the
first decade of the XX century, with Planck’s heuristic solution to the
black–body radiation and Einstein’s subsequent application to explain
observational data concerning the photoelectric effect, the first coherent
formulations of QM were proposed only in the mid–Twenties. On the
one hand, Heisenberg, Born and Jordan developed matrix mechanics in
their Drei–Männer–Arbeit, an algebraic formulation of quantum theory
where physical magnitudes are expressed by means of matrices evolving
in time, on the other, starting from de Broglie’s relation, Schrödinger
developed his wave mechanics, where physical states are represented
by wave functions evolving in space and time according to a diffusion
equation.
    Although the mathematical structures and the starting assumptions
of these representations are profoundly different, their physical equiva-
lence has been proved soon after. However, for a rigorous and systematic
presentation of QM one should have waited until the textbooks of Dirac
(The Principles of Quantum Mechanics, ) and von Neumann (Math-
ematische Grundlagen der Quantenmechanik, ). Remarkably, the von
Neumann–Dirac formulation of quantum theory is, with minimal modifi-
cations, currently referred to as the “standard” or “textbook” formulation

    . For a detailed account of historical developments of QM the reader should refer to (J
) and (S ).
        On the Possibility of a Realist Ontological Commitment in Quantum Mechanics        


of QM. In this paper we stick to this interpretation to introduce the basic
principles of the theory.

.. The von Neumann–Dirac Principles of Quantum Mechanics

According to this formulation of QM, also known as Princeton interpreta-
tion (see Jaeger ):), a quantum system is described by a state vector
| ψ〉 , element of a complex vector space called Hilbert space H, providing
a complete specification of its properties. In QM only operationally ac-
cessible properties are considered magnitudes of quantum systems; they
are represented by positive, linear Hermitian operators A acting on H.
Given a measurable quantity, its possible values are the eigenvalues (real
numbers) of the associated operator A, whereas possible states in which a
system may be found after a measurement of A are represented by the
eigenvectors of A. Implicitly we have defined the eigenvalue–eigenstate
link, a core tenet of the von Neumann–Dirac formulation. With Dirac’s
words:

   The expression that an observable ‘has a particular value’ for a particular state
   is permissible in quantum mechanics in the special case when a measurement
   of the observable is certain to lead to the particular value, so that the state is in
   an eigenstate of the observable [. . . ]. In the general case we cannot speak of an
   observable having a value for a particular state, but we can speak of its having an
   average value for the state. We can go further and speak of the probability of its
   having any specified value for the state, meaning the probability of this specified
   value being obtained when one makes a measurement of the observable (Dirac
   :).

    As correctly pointed out by Jaeger, this quote eloquently characterizes
the idea that quantum systems in isolation do not instantiate properties
with definite values; consequently, quantum mechanical objects possess
indefinite properties prior experimental observation, a significant ontologi-
cal difference w.r.t. classical objects. Unlike the classical case, (i) properties
of quantum systems are not independent of observations, which become
central in the microphysical context, and (ii) the theory provides only prob-
abilities for the possible measurement outcomes, as clearly expressed by
the next postulate, the Born’s rule. This axiom is crucial to connect the
abstract Hilbert space formulation of QM with experimental results, and
states that if the vectors | ψ〉 , | ϕ〉 represent two different physical states
of a given system, there exists a probability p | ψ〉 , | ϕ〉 to find | ψ〉 in

    . (B ) labeled this interpretation the “basic approach” to QM. (L :),
provides a concise discussion of the differences between Dirac’s and von Neumann’s approach.
      Andrea Oldofredi, Michael A. Esfeld


the state | ϕ〉 , given by the squared modulus of their scalar product on
H p | ψ〉 , | ϕ〉 = | 〈ψ |ϕ 〉|  .
   The last two axioms of QM concern the dynamical evolution of physical
systems. In the first place, the evolution in space and time of quantum
systems is governed by a deterministic partial differential equation, the
Schrödinger Equation (SE):
                                               ∂
                                         ih        |ψ〉 = H
                                                          b |ψ〉.                                      ()
                                               ∂t
    SE has several properties among which linearity. It implies that if two
state vectors | ψ 〉 , | ψ 〉 are both possible solutions of the same SE, then
their linear combination (superposition) ψ〉s = α | ψ 〉 + β | ψ 〉 is still a
solution of the same SE. Thus, the new superposed state ψ〉s is also a
consistent representation of the system.
    This fact is the peculiarity of quantum mechanics, since in classical con-
text ψ〉s would represent a different state w.r.t. its superposed components.
To better understand the effect of this shift, suppose to measure the z–spin
of a quantum particle: the eigenstates in which it is possible to find it are
only “z–spin–up” and “z–spin–down”, but from linearity follows that also
the superposition “z–spin–up and z–spin–down” is a consistent description
of the state in which the particle may be. This entails that prior to a spin
measurement the particle has indefinite spin, being neither in the z–spin–up
state nor in the z–spin–down state .
    However, since in experimental observations such superpositions are
never revealed, von Neumann and Dirac introduced the notion of pro-
jection of the state vector, which is distinctive of this formulation of QM.
Suppose we measure a certain quantity represented by the operator A with
eigenvalues ai and eigenvectors | j〉 : if the system is described by |ψ〉 , the
Born’s rule gives the probability to obtain a specific aj as measurement out-
come, p(aj ) = | 〈j |ψ 〉|  . After the measurement’s performance the system
is projected into one of the possible eigenstates. Considering our previous
example, the observation of the particle’s z–spin (the interaction between
the quantum particle and the experimental apparatus) causes a suppression
of the SE and of the superposition with the consequent stochastic jump of
the system in one of the possible spin eigenstates. These stochastic “jumps”

    . In SE i is the imaginary unit,  h = h/π is the reduced Planck constant and H           b is the
Hamiltonian operator representing the sum of kinetic and potential energy of the system.
    . Here |α| , |β| ∈C represent the probabilities to find the system in | ψ 〉 , | ψ 〉 respectively.
The normalization |α| + |β| =  means that with certainty we will find the system in one of the
possible eigenstates.
    . (H ) contains famous no–go theorems which prove the contextual nature of
quantum theory.
         On the Possibility of a Realist Ontological Commitment in Quantum Mechanics            


make QM inherently probabilistic; Dirac viewed them as “unavoidable
disturbance” of quantum systems in measurement situations:
                                                                                         0
    When we measure a real dynamical variable ξ , belonging to the eigenvalue ξ , the
    disturbance involved in the act of measurement causes a jump in the state of the
    dynamical system. From physical continuity, if we make a second measurement of
    the same dynamical variable immediately after the first, the result of the second
    measurement must be the same as the first. Thus after the first measurement has
    been made, there is no indeterminacy in the result of the second. Hence after the
    first measurement is made, the system is in an eigenstate of the dynamical variable
    ξ , [. . . ]. In this way, we see that a measurement always causes the system to jump
    into an eigenstate of the dynamical variable that is being measured, the eigenvalue
    this eigenstate belongs to being equal to the result of the first measurement (Dirac
    :, reported in ( Jaeger :).


    Again, before the first measurement, the system’s state is inherently inde-
terminate, since its properties depend strictly upon the act of observation. It
is worth saying that the projection postulate has been introduced to recon-
cile the postulates of quantum theory with experimental evidence: assuming
(i) that state vectors provide a complete description of quantum systems
and (ii) that their motion would have been entirely described by the SE
alone, the superposed states would have been amplified to the macroscopic
scale. Thus, also macroscopic objects could have been in a superposition,
contradicting the uniqueness and definiteness of measurement outcomes.
This is the essence of the famous Measurement Problem (MP) of quantum
theory, which can be stated more clearly as follows:

    ) State vectors provide a complete description of quantum systems;
    ) State vectors evolve according to a linear dynamical equation (SE);
    ) Measurements have a unique determinate outcome.

    Any pair of these statements is consistent and entails the falsity of the
third one, but their conjunction generates inconsistencies with experimental
evidence. Reconsidering the z–spin example, the device we use must have a
ready state pointing in a neutral direction before the measurement, and two
different pointers indicating different directions, let’s say left ( L) and right (
R), representing the possible eigenstates in which we may find the system
after the observation, z–spin up and z–spin down respectively. Feeding a
particle into the classical device one obtains the following superposition in
virtue of the SE:

     . For a detailed discussion of the MP one may refer to (M ) and (L ). For a
historical presentation of the quantum theory of measurement (J ) is still unsurpassed.
      Andrea Oldofredi, Michael A. Esfeld


                                      
                                    p (|zup , L〉 + |zdown , R〉);
                                       
    Thus, a microscopic superposition is amplified to macroscopic scale until a
measurement is performed. However, since we never observe macroscopic
superpositions, SE cannot provide a complete dynamical story for quantum
objects. Experimental practice suggests that quantum systems are also sub-
jected to stochastic jumps when they interact with measurement devices
causing the instantaneously suppression of the unitary evolution. These inter-
actions clearly have a macroscopic effect, namely the state of the experimental
apparatus will be correlated with the eigenstate (and relative eigenvalue) in
which the quantum system will be found, so that we can actually observe a
definite measurement result. This is the content of the projection postulate.
    Nonetheless, although the projection postulate makes the quantum
formalism consistent with experimental evidence, it does not provide a good
solution to the MP. In the first place it implies an indispensable, arbitrary and
not precisely defined division between the observed system, which could
also include the experimental apparatus, and the observer, who concludes
every act of observation: not only nothing in the formalism refers to the
notion of observer, though it plays a fundamental role in the theory, but
also the axioms of QM do not provide any description or explanation of
the processes responsible for the suppression of the deterministic evolution
provided by the SE. Thus, the notion of measurement, albeit pivotal within
these axioms and taken as an unexplained primitive concept, is neither
mathematically, nor physically well–defined.

     . It should also be clear why quantum probabilities naturally emerge from the theory’s formalism:
if observations suppress SE leading to stochastic jumps, and if the state vectors provide complete
descriptions of the quantum system, then the measurements’ results must be inherently probabilistic,
since they do not reveal pre–existing values of some property instantiated by quantum systems. Then,
indeterminacy must be interpreted as an intrinsic feature of the quantum realm and should not be
interpreted as “lack of knowledge” about initial conditions of quantum systems.
    . However, von Neumann explicitly denied the active role of the observer’s consciousness in
measurement situations: «no matter how far we calculate [. . . ] at some time we must say: and this is
perceived by the observer. That is, we must always divide the world into two parts, the one being
the observed system, the other the observer. In the former, we can follow up all physical processes
(in principle at least) arbitrarily precisely. In the latter it is meaningless» ( N :). It
is sufficient, then, to say that the observer’s experience must be consistent with registered events
in experimental situations. Indeed, even though observers play a crucial role in von Neumann’s
view, (W :) notes that this interpretation is fully consistent with metaphysical realism:
«the classical tradition of simply located objects characterized independently of experiment, was
presupposed by Born and von Neumann and imposed on the data with the help of an informal
language of ‘particles’ and ‘states’». Thus, although conceptually problematic, von Neumann’s theory
does not attribute to observers any creative role. Be that as it may, despite the heterogeneous personal
beliefs on ontological matters there was no consensus among the fathers of the theory about its
ontological content; see (G and I ) for historical details.
         On the Possibility of a Realist Ontological Commitment in Quantum Mechanics              


   In sum, the projection postulates introduces an inconsistency between
the dynamical laws governing the temporal evolution of quantum states,
since it is not clear why and how measurement interactions cause the inter-
ruption of the continuous motion provided by the SE; in other words, the
reasons which make “being observed” and “not–being observed” a relevant
physical distinction for quantum systems remain entirely obscure. In addi-
tion to this, there is no explanation of what distinguishes a measurement
interaction from a non–measurement interaction:
    [w]hat the traditional theory did not do is state, in clear physical terms, the con-
    ditions under which the non–linear evolution takes place. There were, of course,
    theorems that if one puts in collapses somewhere between the microscopic and
    the macroscopic, then, for all practical purposes, it doesn’t much matter where
    they are put in. But if the linear evolution which governs the development of the
    fundamental object in one’s physical theory occasionally breaks down or suspends
    itself in favor of a radically different evolution, then it is a physical question of
    the first order exactly under what circumstances, and in what way, the breakdown
    occurs. The traditional theory papered over this defect by describing the collapses
    in terms of imprecise notions such as “observation” or “measurement” (Maudlin
    :).

   Hence, having underlined these problematic aspects of QM, to apply
Quine’s method will lead to be ontologically committed to a peculiar world,
where the nature of molecular, atomic and subatomic objects is not only
inherently indeterminate, but this indeterminateness is moreover due to
the problems related with the notion of measurement and observer. One
should therefore look for a better theory to regiment.

.. Princeton and Copenhagen

Under many respects the von–Neumann–Dirac formulation of QM differs
remarkably from the Copenhagen interpretation, which was mainly devel-
oped by Bohr and Heisenberg. In the first place, Bohr never introduced
explicitly the projection postulate, nor did he apply the quantum formalism
to experimental devices being strongly convinced that one should describe
them classically, in virtue of the limited knowledge available of quantum
systems expressed by the uncertainty relations; thus, quantum descriptions
must be supplemented by classical concepts. Furthermore, not only the
results of quantum measurements are necessarily expressed in terms of
arrangements of macroscopic objects, but also the experimental procedure
    . Interestingly, it is also controversial whether one may properly individuate a unitary set of
accepted theses, given the significant divergences between the supporters of this ‘interpretation”.
In this regard the reader may refer to (H ) and (B ). For lack of space, here we
concentrate only on a few aspects concerning the theory of measurement.
       Andrea Oldofredi, Michael A. Esfeld


must be controllable and communicable in order to provide an objective de-
scription of quantum phenomena: «it is decisive to recognize that, however
far the phenomena transcend the scope of classical physical explanation, the
account of all evidence must be expressed in classical terms» (Bohr :).
    Hence, measurement is a central notion in QM also according to Bohr’s
view, since properties of quantum objects strictly depend on the devices
used in measurement situations: the changing experimental set–up will
necessarily affect the nature of quantum systems. With Bohr’s words:

     The unambiguous account of proper quantum phenomena must, in principle, in-
     clude a description of all relevant features of the experimental arrangement [. . . ]. In
     the case of quantum phenomena, the unlimited divisibility of events implied in such
     an account is, in principle, excluded by the requirement to specify the experimental
     conditions. Indeed, the feature of wholeness typical of proper quantum phenomena
     finds its logical expression in the circumstance that any attempt at a well–defined
     subdivision would demand a change in the experimental arrangement incompatible
     with the definition of the phenomena under investigation (Bohr :).

   Following this interpretation of QM, in agreement with the Princeton
school, quantum objects have indeterminate properties in isolation. How-
ever, a central tenet of Bohr’s view absent in the previous formulation
of quantum theory is the complementarity principle. In QM information
concerning quantum systems is obtainable only through measurements,
however, there are pieces of information that cannot be obtained simulta-
neously given the incompatibility of experimental protocols, so that they
cannot be represented by a unique quantum state of the examined system.
Thus, the information obtainable by incompatible experiments is comple-
mentary. For instance, quantum objects have been claimed to have both
corpuscular and wave descriptions, although it is not possible to show both
these traits in a single observation, given the incompatibility of experimental
procedures. To this regard, (S :) claims that «any preparation
protocol that is maximally complete, in the sense that all the procedures are
mutually compatible and are such that no further procedure can add any
more information, can be represented by a quantum state, and that state
represents in a mathematical form all the conceivable knowledge about the
object that experiments can reveal to us». Since quantum states represent
the complete description of physical systems, it is clear that the nature of
quantum objects essentially depends on experimental protocols and mea-
suring devices and that observations do not reveal any pre–existing values of
properties attributed to quantum systems. Interestingly, pairs of properties

    . (L and L ) in their monumental textbook on QM ironically recognized
that classical mechanics should emerge as a limit of QM, but the latter needs the former for its
foundations.
         On the Possibility of a Realist Ontological Commitment in Quantum Mechanics             


measurable with incompatible experimental protocols, such as momentum
and position, cannot be revealed simultaneously, in agreement with Heisen-
berg’s uncertainty relations. Instead the Princeton interpretation provides a
different account of experimental situations: on the one hand, experimental
protocols do not play a role in determining the properties of a quantum
object, on the other, non–commuting observables cannot be measured si-
multaneously not having a set of common eigenstates. Thus, clearly, the von
Neumann–Dirac interpretation provides a remarkably different explanation
for the indeterminateness of quantum objects. Be that as it may, the theory
remains silent about the ontological status of quantum objects in isolation:

    The quantum postulate implies that any observation of atomic phenomena will in-
    volve an interaction with an agency of observation not to be neglected. Accordingly,
    an independent reality in the ordinary physical sense can neither be ascribed to the
    phenomena nor to the agencies of observation. After all, the concept of observation
    is in so far arbitrary as it depends on which objects are included in the system to be
    observed (Bohr : ).

    As clearly expressed, also in this interpretation the division between
classical and quantum regimes is as vague as the definition of the physical
processes taking place during the interaction between quantum systems and
classical devices. Furthermore, also in this case, the stochastic interruption
of the SE in measurement situations does not find any precise description,
nor explanation, so that it does not provide any satisfactory solution to
the MP. Thus, also this interpretation of QM being based on the notion of
measurement, inherits every conceptual problem presented in the previous
paragraph.
    Taking into account a different Copenhagenist understanding of the
quantum state, it is interesting to consider Heisenberg’s subjective interpre-
tation, which was mainly advanced in the Fifties. According to this account,
the quantum state is a representation of the experimenter’s knowledge of a
particular quantum system. The later Heisenberg said that:

    When we are observing objects of our daily experience, the physical process
    transmitting the observation of course plays only a secondary role. However,
    for the smallest building blocks of matter every process of observation causes a
    major disturbance; it turns out that we can no longer talk of the behavior of the

   . (J ) and references therein provide a careful reconstruction of Bohr’s philosophy.
   . The young Heisenberg was committed to a much more objective view of QM, being closer to
Bohr ideas. In the Twenties he relegated the subjective element of quantum theory to the ignorance
expressed by the probability function, which provides only a description of an ensemble of possible
events, or potentialities, borrowing the Aristotelian terminology. These potentialities disappear in
measurement situations. However, the subjective element in Heisenberg’s view does not refer to the
consciousness of the observer which is inactive in observational processes.
       Andrea Oldofredi, Michael A. Esfeld


     particle apart from the process of observation. In consequence, we are finally led
     to believe that the laws of nature which we formulate mathematically in quantum
     theory deal no longer with the particles themselves but with our knowledge of the
     elementary particles. The question whether these particles exist in space and time
     “in themselves” can thus no longer be posed in this form. We can only talk about
     the processes that occur when, through the interaction of the particle with some
     other physical system such as a measuring instrument, the behavior of the particle
     is to be disclosed (Heisenberg :–).

    This reading of the quantum state was also shared by other notable
supporters of the Copenhagen interpretation, like Pauli or Peierls, who
explicitly claimed that the statements of QM express fundamentally our
knowledge of quantum systems.
    Accepting this interpretation of QM implies not only that a description of
isolated quantum objects is unobtainable, but also that quantum mechanics
is no longer a mechanical theory, i.e. a framework providing descriptions
and explanations of quantum phenomena in terms of objects moving and
interacting in space and time. QM becomes an epistemological theory
concerned the notion of knowledge of human observers implying a shift
from an object–oriented ontology to a framework which describes the
evolution (the updating) of agents’ knowledge of some physical system.
Therefore, the ontological commitment provided by a subjective reading
of the wave function concerns agents’ beliefs and remains silent on the
ontological status of the physical systems these beliefs are about.
    In sum, if one applies Quine’s method to these interpretations of QM ei-
ther one is committed to the existence of objects with indefinite properties,
or one is led to a subjective interpretation of QM that does not specify what
kind of objects populate the quantum realm.


. A Realist View in QM: the Theory of Local Beables

In order to solve the conundrums afflicting the standard formulation of
QM, (Bell (),Chapert ) proposed to focus the attention of physicists
and philosophers working on the foundations of QM on the ontological
problems deriving from its axioms. To formulate a set of principles not
containing ill–defined notions, Bell’s proposal was to spell out clearly,
also in the quantum context, the ontology of a given physical theory T
via the specification of the set of theoretical entities with a reference
to physical objects localized in space–time. Thus, he introduced the

   . This view contributed also to the development of Quantum Bayesianism, the most prominent
contemporary subjective interpretation of QM.
         On the Possibility of a Realist Ontological Commitment in Quantum Mechanics                  


neologism beable, from the English verb to be, indicating what elements
of a given physical theory T refer to or represent real objects in the world
and their properties, in opposition to what is only observable, which
plays a dominant role in the standard formulation of QM. To state it
concisely, Bell’s theory provides a specification of the primitive ontology
of a given theoretical framework.
    Considering a physical theory T, its primitive ontology is a metaphys-
ical assumption defining the entities which cannot be further analyzed
and in terms of more elementary notions. These entities are the variables
appearing in T’s equations with a direct physical meaning, i.e. referring to
(what according to T are considered) real objects precisely localized and
moving in –dimensional physical space (or in –dimensional space–time).
Furthermore, every physical phenomenon included in T’s domain is on-
tologically reduced and explained via the dynamical evolution in space of
these fundamental objects according to the particular laws of motion gov-
erning the behaviour of the PO, recovering in the quantum context the
satisfactory and efficient explanatory scheme of classical physical theories.
In this respect, we may also claim with Bell’s words that

    [t]he beables must include the settings of switches and knobs on experimental equip-
    ment, the currents in coils, and the reading of instruments. “Observables” must be
    made, somehow, out of beables. The theory of local beables should contain, and give
    precise physical meaning to, the algebra of local observables (Bell : ).


    This is to say that the primitive variables must connect T to our macro-
scopic ontology. It is worth noting that the theory of local beables follows
the Bohrian idea for which “the account of all evidence must be expressed
in classical terms”: via the specification of the PO the vague expression
“classical terms” acquires a precise meaning, since also the classical devices
used to measure quantum properties are constituted by the beables of the
theory, and are therefore treated in a mathematically and physically rigorous
way as clearly stated in the above quotation.
    The central tenet of the PO approach to quantum physics is that every
well–defined physical theory must satisfy the following requirements:

    . Here we consider the expressions “primitive ontology” and “local beables” synonyms
since a precise assessment of their differences goes beyond the scope of the present paper.
    . The reader may refer to (A ) and (E a) for a more detailed explanation of
this notion.
    . The PO contextually depends on the theoretical framework in which it is assumed. To this
regard, (A : ) explicitly claims that «there is no rule to determine the primitive ontology of
a theory. Rather, it is just a matter of understanding how the theory was introduced, it has developed,
and how its explanatory scheme works».
      Andrea Oldofredi, Michael A. Esfeld


     ) A physical theory T aims to provide a careful description of a specific
        domain of our world and to explain a specific set of phenomena;
     ) To provide this description, T must implement a specific primitive
        ontology of objects moving in physical space (–dimensional or –
        dimensional). These objects are fundamental in two senses: (i) they
        are not reducible to more elementary notions, (ii) the macroscopic
        objects of our ordinary experience must be reduced to the motion in
        space of the primitive variables;
     ) T must provide a set of dynamical laws governing the motion of the
        PO;
     ) The mathematical structure of a given theory is naturally divided
        in two sub–structures: on the one hand, there are objets with a
        direct physical meaning, i.e. those entities referring to real objects in
        physical space, and on the other there are mathematical structures
        without a direct reference to physical objects.

    These features constitute what (Allori ) defines as the common struc-
ture shared by the PO theories. Bohmian mechanics is an explicit example of
such a theory. We will employ it to illustrate how the theory of local beables
allows to solve the problems affecting the standard formulation of QM.

.. An explicit Example: Bohmian Mechanics

Bohmian mechanics is an alternative formulation of QM fully developed
by the physicist David Bohm in two fundamental papers Bohm (a)
and Bohm (b) and nowadays used by several theoretical physicists and
quantum chemists.
    BM is a deterministic quantum theory of particles which move in three–
dimensional physical space and follow continuous trajectories. Albeit this
theory is statistically equivalent to the standard QM, their physical content
is remarkably different since the former makes a precise metaphysical
hypothesis concerning the intrinsic corpuscular nature of matter. In BM
every physical fact is reduced to the motion of the Bohmian particles, which
always have definite positions independently of any observation.
    According to this theory, physical systems are described by a couple
(ψ, Q), where the first element is the usual wave function and the second
represents a specific N–particle configuration with positions (Q , . . ., QN );
these positions constitute the additional variables introduced by BM. The
    . The reader may refer to (B and V ) for a historical reconstruction
of the theory and to (F ) for a sociological analysis of the elements involved in the abandon-
ment of the Bohmian ideas. (D and T ) is an excellent mathematical exposition of BM,
whereas (O and M ) is concerned with some specific applications.
         On the Possibility of a Realist Ontological Commitment in Quantum Mechanics              


dynamics is composed by two laws of motion: on the one hand, the wave
function evolves according to the usual SE (), on the other, the motion of
Bohmian particles is governed by the so–called guiding equation:
                                          dQ
                                       = υψt (Q).                             ()
                                    dt
    The vector velocity field on the right–hand–side in () explicitly depends
on the wave function, whose role is to guide the motion of the particles.
The solutions of the guiding equation are integral curves corresponding to
particles’ trajectories.
    From () it is easy to note that BM is a non–local theory, in perfect
agreement with Bell’s theorem. However, it is important to stress that there
is no inconsistency with special relativity since Bohmian particles do not
travel faster than light, thus, these non–local effects cannot be used to send
signals at superluminal speed. Therefore, no operational contradiction with
relativity arises.
    The last step we have to make in order to complete our brief presenta-
tion of BM is to guarantee the empirical, or statistical, equivalence w.r.t. the
predictions of QM. The empirical equivalence is achieved via equivariance:
if at any arbitrary initial time t the particle configuration is distributed
according to | ψ |  , then it will be so distributed for any later time t, pre-
serving the Born’s distribution (see Dürr et al. (), Chapter  for the
mathematical justification of this statement).
    The motivations to consider BM as a serious alternative to the standard
quantum theory are very well known: not only the notorious measurement
problem vanishes, but also BM does not rely on physically ill–defined no-
tions such as measurement and observer, present instead in the axioms of
standard QM. It is important to state that BM restores on the one hand an
ontology of particles, and on the other an epistemic interpretation of the
quantum probabilities. The stochastic nature of the theory is a manifestation
of our ignorance concerning the exact positions of the particles: according
to BM, the maximal information at our disposal is always given by | ψ|  ,
and randomness must be interpreted as a lack of knowledge absolutely
detached from whatever sort of ontological indeterminacy of the quantum
particles.
    Having qualitatively introduced the bare bones of the theory, let us
now discuss how it solves the MP. Consider a wave function which is in
a superposition of two possible eigenstates of a two–valued operator (for
instance the spin of a particle in one of the three possible directions):

   . For details (D et al. (), Chapter , sec. –. For a recent review of the literature on
quantum probabilities in BM see (O et al. ).
     Andrea Oldofredi, Michael A. Esfeld



                                    ψ = α ψ + α ψ
with | α |  + | α |  = . Before the measurement we assume that the pointer
of the experimental device is in the ready state Φ , where the possible
pointer positions will be Φ , Φ , indicating the possible eigenstates in which
we may find the system after the observation. We label them position  and
 for simplicity.
    According to quantum theory we know that, given the deterministic
evolution provided by the SE, from the initial state in which the system
under consideration and the apparatus are independent (and described by a
product wave function), we obtain a macroscopic superposition:

                               αi ψi Φ Schroedinger           αi ψi Φi .
                       X                               X

                       i=,               −→          i=,

     The great merit of Bohmian mechanics comes from a simple idea: to
describe quantum mechanically even the experimental apparatus, so that
also macroscopic objects as the measurement devices are composed of
Bohmian particles. This is in perfect agreement with Bell’s theory of local
beables introduced a few lines above.
     In BM this experimental situation is described by a couple (X , Y ) where
the former variable refers to the initial configuration of the particles of the
system and the latter to the configuration of the particles that compose the
apparatus at the initial time t . Then, given () and (), the total configuration
of particles (X , Y ) evolves into another configuration (Xt , Yt ) at time t > t
which is one of the possible eigenstates of the measured operator: the
pointer will point to position  with probability | α |  and to position  with
| α |  . (For a detailed exposition see (Dürr and Teufel ), Chapter )
     Considering the theory of measurement in BM, it would be immediately
clear how measurement results of whatever observable depend strictly upon
(i) the initial particle configuration of the system, which comprehends the
observed system and the experimental apparatus (which receives a quantum
mechanical treatment in this context), and (ii) the dynamical equations of the
theory, so that the final outcome is completely determined by the evolution
in space and time of the total configuration of the particles involved (for
details see Chap.  of Dürr and Teufel () and Bohm (b) sections
, ). Furthermore, contrary to the case of standard QM, in BM there is
no collapse of the wave function: the dynamics of the system depends on
the laws of motion governing the beables of the theory which prescribe a
continuous spatiotemporal evolution. Once one looks at the final configuration
of particles, then one finds the pointer in one of the possible (macroscopic)
positions corresponding to one of the possible outcomes. This means that
        On the Possibility of a Realist Ontological Commitment in Quantum Mechanics          


within this theoretical framework nothing induces or causes stochastically
the result of a given measurement. Thus, Bohmian mechanics avoids by
construction any reference to ill–defined notions: measurement outcomes
receive an explanation in terms of the motion of the beables in space, becoming
functions of the primitive ontology. An explicit example in the context of BM is
contained in Dürr et al. (), where the authors offer a carefully analysis of the
reduction of spin measurements to position measurements, and consequently
to the dynamical evolution of particles (for details see especially sec. ).
    The case of BM is generalizable to every theory implementing a clear
PO, since these frameworks provide a detailed theory of measurement
to supply a rigorous description of the physical processes taking place in
measurements situations, and therefore a detailed explanation of the ob-
tained outcomes. This particular feature of the PO theories is crucial, since
measurement constitutes the only connection between a given theoreti-
cal framework and experience: «its analysis is therefore one of the most
sensitive parts of any interpretation [of QM]» ( Jammer : ).
    Ultimately, the macroscopic ontology of ordinary experience, to which
experimental outcomes belong, is grounded in the primitive variables of a
given theory and their motion in space and time, so that every physical fact
within the theory’s domain is reduced to them. In sum, we can agree with
Bell in claiming that notions such as measurement, apparatus, observable
and so on should be derived from the primitive ontology. Observation
is always theory–laden and a specific theory of measurement should be
inferred from the fundamental structure of a theory, after all «[d]oes not
any analysis of measurement require concepts more fundamental than the
measurement? And should not the fundamental theory be about these
more fundamental concepts?» (Bell : ).

.. The status of the Wave Function in BM

In contrast to the Copenhagen and the Princeton interpretations, BM offers
an unambiguous formulation of quantum mechanics in which neither vague
notions appear, nor the MP arises. Nonetheless, there is an open debate
about its ontology (see e.g. (Esfeld et al.)): if we apply the Quinean
programme, the result is that we are committed to the existence of the
particles and the wave function, since the theory, if put in first order logic,
quantifies over both of them. Indeed, according to Bohm himself and some
contemporary supporters of his pilot–wave theory, like A. Valentini and
P. Holland among others, the wave function is conceived as a field in
physical space. Hence, the sentence “there exists an electron”, ∃xE(x),

   . See (H and R ) for a recent defense of such a view about the wave function.
    Andrea Oldofredi, Michael A. Esfeld


in this formulation of BM indicates that there exists a particle with well–
defined values for position, mass and charge in physical space guided by a
relative wave field, the wave function of the electron in question.
    However, for most interpreters this result is puzzling: the wave function
is defined not on three–dimensional space, but on configuration space — the
mathematical space each point of which represents a possible configuration of
particles in physical space (for N particles, configuration space accordingly has
N dimensions). The puzzle then is how an object that exists in configuration
space could influence the behaviour of objects in physical space.
    Since the debate about the status of the wave function is far from be
solved, one may conclude that, once again, answering this ontological ques-
tion requires genuine philosophical work instead of simply bringing these
theories in the form of first order logic: before the regimentation of the sen-
tences of BM, therefore, we need a precise answer to the metaphysical issue
concerning the nature of the wave function. In this respect, the primitive
ontology programme can be conceived as an improvement on the Quinean
criterion for ontology.
    The PO, defined by the four criteria listed at the beginning of this section,
is primitive also in the sense that the entities that it poses enter a physical
theory T as ultimate referents of T, which are not defined by their function
for something else. In brief, the primitive ontology of a given theoretical
framework is constituted by the set of those entities that, according to T,
exist simply in the world. According to this definition of primitiveness, the
PO stands in contrast to the mathematical structures of a theory without
a direct reference to physical objects; these latter ones are labeled non–
primitive variables in Allori et al. (), where the authors claim that the
PO does not exhaust the entire ontology of a physical theory, since it
comprehends also the structures in the T’s formalism whose function is to
govern the behavior of the beables. Here, we follow a more recent literature
so that we call these variables the dynamical structure of T, which is made
up by all those parameters that are introduced through their functional role
for the evolution of those entities that constitute the primitive ontology.
The wave function is the central element of the dynamical structure of
BM, because it is introduced in terms of its functional or causal role for the
evolution of the particle configuration.
    The dynamical structure is nomological in the sense that it represents
the behaviour of the elements of the primitive ontology, containing the
parameters that are needed to formulate laws for the evolution of the beables.
The distinction between primitive ontology and dynamical structure makes
evident why the ontological commitment points towards the PO in the first
place and why the ontological commitment to the dynamical structure is
an open issue, as stressed a few lines above.
       On the Possibility of a Realist Ontological Commitment in Quantum Mechanics   


    In order to answer the question regarding the ontological status of the
wave function and the other non–primitive variables, it is certainly pos-
sible to apply any one of the philosophical stances w.r.t. the natural laws
to the dynamical structure as a whole: primitivism, according to which
the dynamical structure is a further primitive over and above the primitive
ontology (see e.g. (Maudlin ) for that attitude with respect to the wave
function); dispositionalism, a view in which the dynamical structure refers
to dispositional properties of the physical objects in which the laws are
anchored, think e.g. of gravitational mass in classical mechanics (disposi-
tionalism with respect to the wave function in BM is set out in (Belot )
and in (Esfeld et al. )); Humeanism, where the dynamical structure is
part of the best system — that is, the system that achieves the best balance
between simplicity and informational content in representing the evolution
of the elements of the primitive ontology — and hence does not call for
an additional ontological commitment (that attitude is developed with re-
spect to the wave function in (Miller ), (Esfeld b), (Callender ),
(Bhogal and Perry )).
    In a further step, one may then move on from the primitive ontology of
a given theory to primitive ontology tout court, that is, seek to work out a
proposal about what the entities are that simply exist in the world, given
our best physical theories. The rationale for doing so is that naturalistic
metaphysics strives for an ontology of the natural world that is not relative
to particular physical theories. One may even go as far as claiming that it is
inappropriate to speak of the ontology of this or that theory. Ontology is
about what there is. It goes without saying that our access to what there is
comes through the representations that we conceive in terms of physical
theories. But this does not necessarily imply that ontology is relative to
particular theories. In other words, the idea for a fundamental ontology
is to search for an answer to the following question: What is a minimal
set of entities that form an ontology of the natural world, given our well–
established physical theories? An accredited candidate for an answer to that
question, both in classical and in quantum physics, is an ontology composed
by point particles that are characterized only by their relative positions (that
is, by their distances to each other) and the change in position (see (Esfeld
and Deckert ) for making the case for that answer).


. Conclusion

In this paper we have argued that the PO approach provides a sound method-
ology to construct theories avoiding the difficulties affecting the standard
formulation of QM: according to this framework, the structure of a physical
    Andrea Oldofredi, Michael A. Esfeld


theory is well–defined if and only if its basic entities are specified and their
equations of motion are given. In this manner, one is able to obtain an ex-
planation of physical phenomena in terms of the evolution of the primitive
ontology in space.
   From our discussion it emerged that Quine’s method is not rich enough
to distinguish between the mathematical structures provided with a direct
physical meaning and those which are not; therefore, it cannot be alone a
secure guide to one’s ontological commitment in microphysical contexts,
not only given the pitfalls inherent to the quantum formalism, but also since
the ontological status of the wave function is not definitively established, so
that it still depends contextually on one’s interpretation of ψ. It is clear, then,
that one’s ontological commitment depends on the answer given to this
question. To this regard, however, the major methodological contribution
conveyed by the PO is that the mathematical, physical and philosophical
aspects of a particular theoretical framework should be clearly separated,
so that it is always possible to interpret its elements (primitive and non–
primitive) unambiguously. The main message of this paper, therefore, is to
consider the PO programme as a useful guide in search for an ontology in
the quantum realm to be added to Quine’s method.
   Furthermore, this perspective has been recently extended also to the
realm of Quantum Field Theory (see (Struyve ) and references therein
for an overview as well as, most recently, (Deckert et al. )) where, once
again, Bohmian theories show the concrete possibility to consistently apply
the PO methodology to recover QFT’s phenomenology via the definition
of ontologically clear alternative formulations. Although these theories
may be still speculative or involve a partial reformulation of quantum field
theory, they have the merit to show possible solutions to the conceptual
issues present in the standard formulation, indicating a way to find a clear
ontology also in the quantum theory of fields.

A: A. Oldofredi is grateful to the Swiss National Science
Foundation for financial support (Grant No. –).
        On the Possibility of a Realist Ontological Commitment in Quantum Mechanics   


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