## 1. The origin of the modal approach



In traditional approaches to quantum measurement theory a central role
is played by the projection postulate, which asserts that upon
measurement the state of a physical system undergoes a discontinuous
change: it is projected ("collapses") onto an eigenstate
corresponding to the measurement result. This postulate leads to many
difficulties: When exactly does a collapse takes place? What defines a
"measurement" as opposed to an ordinary physical
interaction? The postulate is especially worrying when applied to
entangled compound systems whose components are well-separated in
space. For example, in the Einstein-Podolsky-Rosen (EPR) experiment
there are strict correlations between two systems that have interacted
in the past, in spite of the fact that the correlated quantities are
not sharply defined in the individual systems. The projection
postulate in this case implies that the collapse resulting from a
measurement on one of the systems instantaneously defines a sharp
property in the distant other system. (See the discussion of the
collapse or projection postulate in the entry on
 philosophical issues in quantum theory.)



A possible way out of these problems was noticed by van Fraassen
(1972, 1974, 1991), who proposed to eliminate the projection postulate
from the theory and to give a probabilistic interpretation to the
remaining formalism of unitary quantum mechanics. Van Fraassen's
interpretation relied on the distinction between the "dynamical
state" and the "value state" of a system at any
instant:


* The *dynamical state* determines what *may be* the
case: which physical properties the system may possess, and what
probabilities are attached to these possibilities.
* The *value state* represents what *actually is* the
case, that is, all the system's physical properties that are
sharply defined at the instant in question.



The dynamical state is the ordinary quantum state that occurs in the
Schrodinger equation (in non-relativistic quantum mechanics), i.
e. a vector or density matrix in Hilbert space, with the proviso that
it always evolves unitarily, so never collapses.



The motivation for distinguishing between dynamical and value state is
the idea that a system may have a sharp value of an observable even if
its unitarily evolving state is not an eigenstate of that observable.
This thought contradicts the so-called "eigenstate-eigenvalue
link", which says that a system possesses a sharp value of an
observable (namely, one of its eigenvalues) if and only if its quantum
state is the corresponding eigenstate. The modal approach accepts the
"if" part, but denies the "only if" part of
this statement. Van Fraassen accordingly introduced the notion of
"value state": a system's physical properties (the
sharply defined observables) are not one-to-one represented by the
dynamical state, which occurs in the evolution equation, but by the
value state, which typically differs from the dynamical state.



Van Fraassen called his proposal "modal" because it
naturally connects to a modal logic of quantum propositions. Indeed,
the dynamical state in general only tells us what is possible and
specifies probabilities for the various possibilities. According to
van Fraassen, this modal and probabilistic aspect is fundamental to
quantum mechanics and should not be viewed as arising from an
incompleteness of the description (see Bueno 2014 for the relation
between this and van Fraassen's constructive empiricism).



What then are the definite quantities and value states for a given
system at a given time? Empirical adequacy requires that, in cases of
measurement, the after-measurement value state of the measuring device
should correspond to a definite measurement result. Therefore, the
scheme minimally has to generate the Born probability measure over
value states that represent the possible measurement results. Van
Fraassen's interpretation did not go further than this; it did
not specify precise value states outside of measurement situations and
defended agnosticism about the properties of physical systems in
general (see Ruetsche 1996 for critical discussion concerning this
point).



It is easy to see how these ideas could motivate a program for
providing a more elaborate modal interpretation of quantum theory that
assigns definite properties to physical systems even when no
measurements are taking place. To this more general modal program we
now turn.


## 2. General features of modal interpretations



In the 1980s several authors presented realist interpretations which
can be regarded as variations on the just-mentioned modal themes (for
an overview and references, see Dieks and Vermaas 1998). There are
differences between these modal interpretations, but all agree on the
following points:


* The interpretation should be based on the standard formalism of
quantum mechanics without the projection postulate: a modal
interpretation is a "non-collapse interpretation", that
is, an interpretation of unitary quantum mechanics.
* The interpretation should be "realist" in the sense
that it assumes that quantum systems always possess a number of
definite properties, which may change with time.
* Quantum mechanics is taken to be universal: it applies both to
microscopic and to macroscopic systems.
* The unitarily evolving state of the system contains information
about the possible physical properties of the system and the
corresponding probabilities. So, there is a probabilistic relation
between the dynamical state and physical possibilities (represented by
possible value states).
* A quantum measurement is an ordinary physical interaction, during
which the dynamical state always evolves unitarily.



The Kochen-Specker theorem (1967) is an obstacle to any traditional
realist interpretation of quantum mechanics, since it proves the
impossibility of ascribing precise values to *all* physical
quantities (observables) of a quantum system simultaneously, while
preserving the functional relations between commuting observables.
(See the entry on the
 the Kochen-Specker theorem.)
 Therefore, realist non-collapse interpretations are committed to
selecting a (Boolean) subset of definite-valued observables out of all
observables. Each modal interpretation thus supplies a "rule of
definite-value ascription" or "actualization rule",
which picks out, from the set of all the observables of a quantum
system, the subset of definite-valued properties, which may change in
time.



What should this actualization rule look like? In the mid-1990s a
series of authors confronted this question (Clifton 1995a, 1995b,
Dickson 1995a, 1995b). These authors proposed conditions
to be satisfied by an actualization rule that defines a set of
possibilities purely from the dynamical state, and concluded that the
possible value states of the components of a two-part composite system
should be given by the states occurring in the Schmidt (bi-orthogonal)
decomposition of the dynamical state, or, equivalently, by the
projectors occurring in the spectral decomposition of the density
matrices representing partial systems (obtained by partial tracing)
--see Section 4 for details.



The problem of definite-valued properties was approached in a more
general way by Bub and Clifton (1996); for an improved version, see
Bub, Clifton and Goldstein 2000). These authors started from the
assumption that the definite-valued subset is determined by the
dynamical state \(\ket{\phi}\) *plus* a "privileged
observable" \(\boldsymbol{R}\), representing a property that is
always definite-valued *a priori*.
From this perspective, Bub (1992, 1994, 1997) drew the conclusion that
with hindsight a number of traditional interpretations of quantum
theory might be viewed as modal interpretations. Among them are (what
Bub took to be) the Dirac-von Neumann interpretation and Bohr's
interpretation; and, as a more clear-cut case, Bohm's theory.
Indeed, this theory (Bohm 1952) is a modal non-collapse interpretation
with *position* as the privileged observable
\(\boldsymbol{R}\). A more recent proposal for a modal scheme using a
privileged observable is the Modal-Hamiltonian interpretation (see
Section 8).


## 3. Relation to other non-collapse interpretations



The proposal to do without the projection postulate was also made by
others, before van Fraassen published his work. In particular,
Bohm's hidden-variables theory (itself preceded by de
Broglie's proposals from the 1920s) and Everett's (1957)
relative-state scheme (elaborated by De Witt (1970) into the
many-worlds interpretation) followed this path. According to
Bohm's theory, physical systems are built up from particles that
always possess definite positions, while the wave function specifies
the probabilities associated with the various possible values these
positions may have. There are no collapses in Bohm's theory: the
wave function evolves unitarily at all times. Therefore, the theory
qualifies as a modal interpretation of quantum mechanics: the state
determines a range of possible particle configurations and specifies
the probabilities for these possibilities to be actually realized.



Everett (1957) also eliminated collapses, so that after a measurement
object system and measuring device (or observer) end up in an
entangled superposition. To reconcile this description with the
uniqueness of our experience, Everett introduced the notion of
*relative state*: individual terms of the superposition
represent different definite measurement results, and *relative
to* the corresponding device states the object systems possess
definite properties as well. This interpretative move was worked out
by DeWitt and others into the idea that *all* measurement
results represented in the post-measurement superposition are
realized: in a measurement worlds are multiplied, and in each world a
different measurement outcome becomes determinate.



As a matter of historical fact, it is unclear whether this idea of
many co-existing worlds accords with Everett's own view: Everett
stated that the wave function serves the purpose of providing us with
a set of factual and counterfactual descriptions of the world we live
in, and this suggests that he rather saw his relative-state
interpretation as a theory about one single world. This would make
Everett's interpretation a modal interpretation (see for
extensive discussion Dieks 2021).



However this may be, the feature that distinguishes modal
interpretations from other non-collapse interpretations is that modal
interpretations pertain to one single world, about which they make
"modal statements", i.e. statements concerning
possibilities and their probabilities.


## 4. Biorthogonal-decomposition and spectral-decomposition modal interpretations



We now turn to those modal interpretations in which the set of
physical possibilities is determined by the (dynamical) quantum state
of a system (see the penultimate paragraph of Section 2).



In the biorthogonal-decomposition interpretation (BDMI, sometimes
known as "Kochen-Dieks modal interpretation", Kochen 1985,
Dieks 1988, 1989), the definite-valued observables are
picked out by the biorthogonal (Schmidt) decomposition of the
state.



**Biorthogonal Decomposition Theorem**:
  

Given a vector \(\ket{\psi}\) in a tensor-product Hilbert space
\(\mathcal{H}^1 \otimes \mathcal{H}^2\), there exist bases
\(\{\ket{a\_i}\}\) and \(\{\ket{p\_i}\}\) for \(\mathcal{H}^1\) and
\(\mathcal{H}^2\) respectively, such that \(\ket{\psi}\) can be
written as a linear combination of terms of the form \(\ket{a\_i}
\otimes \ket{p\_i}\). If the absolute values (modulus) of the
coefficients in this linear combination are all unequal, then the
bases are unique (see, for example, Schrodinger 1935 for a
proof). 



Applied to quantum mechanics the theorem says that, given a composite
system consisting of two subsystems, its state picks out (in many
cases, uniquely) a basis for each of the subsystems. In BDMI, those
bases are taken to generate the definite-valued properties of the
corresponding subsystems: the definite-valued observables are those
that commute with all the projections on these basis vectors.



BDMI can be applied directly to the quantum measurement problem. Let
us consider an ideal measurement according to the standard von Neumann
model, in which a quantum measurement is an interaction between a
system \(S\) and a measuring apparatus \(M\). Before the interaction,
\(M\) is prepared in a ready-to-measure state \(\ket{p\_0}\),
eigenvector of the pointer observable \(P\) of \(M\), and the state of
\(S\) is a superposition of the eigenstates \(\ket{a\_i}\) of an
observable \(A\) of \(S\). The interaction introduces a correlation
between the eigenstates \(\ket{a\_i}\) of \(A\) and the eigenstates
\(\ket{p\_i}\) of \(P\):

 

\[
\ket{\psi\_0} = \sum\_{i} c\_i \ket{a\_i} \otimes \ket{p\_0} \rightarrow 
\ket{\psi} = \sum\_i c\_i \ket{a\_i} \otimes \ket{p\_i}
\]


In this case, according to the BDMI actualization rule, the preferred
context of the measured object system \(S\) is defined by the set
\(\{\ket{a\_i}\}\) and the preferred context of the measuring apparatus
\(M\) is defined by the set \(\{\ket{p\_i}\}\). Therefore, the pointer
position is a definite-valued property of the apparatus: as a result
of the interaction with the object system the pointer acquires one of
its possible values (eigenvalues) \(p\_i\). Likewise, after the
interaction the measured observable is a definite-valued property of
the object system, and acquires one of its possible values
(eigenvalues) \(a\_i\).



In spite of the fact that this modal interpretation is characterized
by the central role played by biorthogonal decomposition, two
different versions can be distinguished. One of them adopts a
metaphysics in which all properties are relational and, as a
consequence, the fact that the application of the interpretation is
restricted to subsystems of a two-component compound system is not a
problem (Kochen 1985). This relation has been called
"witnessing": properties are not possessed by the system
absolutely, but only when it is "witnessed" by another
system. Consider the measurement described above: the pointer
"witnesses" the value acquired by the measured observable
of the measured system.



By contrast, according to the other version the
properties ascribed to the system do not have a relational character.
This proposal therefore faces consistency questions about the
assignments of definite values to observables according to different
ways of splitting up the total system into components. Consider, for
example, the three-component composite system \(\alpha \beta \chi\).
We could apply the biorthogonal decomposition theorem to the
two-component system (i) \(\alpha(\beta \chi)\), or (ii) \(\beta(\chi
\alpha)\) or (iii) \(\chi(\alpha \beta)\). Suppose that, as a result
of this, in case (i) the system \(\alpha\) has the definite-valued
property \(P\), in case (ii) the system \(\beta\) has the
definite-valued property \(Q\), and in case (iii) the system \(\alpha
\beta\) has the definite-valued property \(R\). How do the
definite-valued properties of \(\alpha\) and \(\beta\) relate to those
of \(\alpha \beta\)? Are the definite-values properties of system
\(\alpha \beta\) \(P \amp Q\), or \(R\), or both?



A generalization is the spectral-decomposition modal interpretation
(SDMI, Vermaas and Dieks 1995). SDMI is based on the spectral
decomposition of the reduced density operator: the definite-valued
properties \(\Pi\_i\) of a system and their corresponding probabilities
\(\mathrm{Pr}\_i\) are given by the non-zero diagonal elements of the
spectral decomposition of the system's state,

 

\[
 \varrho = \sum\_{i} \alpha\_i\Pi\_i
\qquad 
 \mathrm{Pr}\_i = \mathrm{Tr}(\varrho \Pi\_i)
\]


This new proposal matches the old one in cases where the old one
applies, and generalizes it by fixing the definite-valued properties
in terms of multi-dimensional projectors when the biorthogonal
decomposition is degenerate: definite-valued properties need not
always be represented by one-dimensional
vectors--higher-dimensional subspaces of the Hilbert space can
also occur.



The SDMI also has a direct application to the measurement situation.
Consider quantum measurement as described above, where the reduced
states of the measured system \(S\) and the measuring apparatus \(M\)
are

 

\[\begin{align\*}
\varrho\_r^S &= \mathrm{Tr}\_{(M)} \ket{\psi} \bra{\psi} 
 = \sum\_i \norm{c\_i}^2 \ket{a\_i} \bra{a\_i} 
 = \sum\_i \norm{c\_i}^2 \Pi\_i^a \\
\varrho\_r^M &= \mathrm{Tr}\_{(S)}\ket{\psi}\bra{\psi} 
 = \sum\_i \norm{c\_i}^2 \ket{p\_i} \bra{p\_i} 
 = \sum\_i \norm{c\_i}^2 \Pi\_i^p
\end{align\*}\]


According to SDMI, the preferred context of \(S\) is defined by the
projectors \(\Pi\_i^a\) and the preferred context of \(M\) is defined
by the projectors \(\Pi\_i^p\). Therefore, also in SDMI, the
observables \(A\) of \(S\) and \(P\) of \(M\) acquire actual definite
values, whose probabilities are given by the diagonal elements of the
diagonalized reduced states.



Healey (1989), who was among the first to make use of the biorthogonal
decomposition theorem for interpretational purposes, has proposed a
somewhat different modal scheme. After using the biorthogonal
decomposition theorem, Healey modifies the set of possible properties
in order to fulfill a variety of additional desiderata, e.g.
pertaining to the relationship between composite systems and their
subsystems. The structure of definite-valued properties that emerges
from these conditions is rather complicated. Some progress has been
made since Healey's book was published (see Reeder and Clifton
1995) but, in general, it remains difficult to see what the set of
definite-valued properties is according to this approach.


## 5. Property composition and atomic modal interpretation



The modal property attributions as reviewed above raise consistency
questions when a system can be split up into components in different
ways. Consider, for example, the three-component composite system
\(\alpha \beta \chi\). We could apply the biorthogonal decomposition
theorem to the two-component system (i) \(\alpha(\beta \chi)\), or
(ii) \(\beta(\chi \alpha)\), or (iii) \(\chi(\alpha \beta)\). Suppose
that, as a result of this, in case (i) the system \(\alpha\) has the
definite-valued property \(P\), in case (ii) the system \(\beta\) has
the definite-valued property \(Q\), and in case (iii) the system
\(\alpha \beta\) has the definite-valued property \(R\). How do the
definite-valued properties of \(\alpha\) and \(\beta\) relate to those
of \(\alpha \beta\)? Are the definite-values properties of system
\(\alpha \beta\) \(P \amp Q\), or \(R\), or both? This problem was
addressed by different authors during the 1990s (see Vermaas 1999,
Bacciagaluppi 1996).



Elaborating on this, let us take a composite system \(\alpha \beta\),
whose component subsystems \(\alpha\) and \(\beta\) are represented by
the Hilbert spaces \(\mathcal{H}^{\alpha}\) and
\(\mathcal{H}^{\beta}\), respectively, and consider a property
represented by the projector \(\Pi^{\alpha}\) defined on
\(\mathcal{H}^{\alpha}\). It is usual to assume that \(\Pi^{\alpha}\)
represents the same property as that represented by \(\Pi^{\alpha}
\otimes I^{\beta}\) defined on \(\mathcal{H}^{\alpha} \otimes
\mathcal{H}^{\beta}\), where \(I^{\beta}\) is the identity operator on
\(\mathcal{H}^{\beta}\). This assumption is based on the observational
indistinguishability of the magnitudes represented by \(\Pi^{\alpha}\)
and \(\Pi^{\alpha} \otimes I^{\beta}\): if the
\(\Pi^{\alpha}\)-measurement has a certain outcome, then the
\(\Pi^{\alpha} \otimes I^{\beta}\)-measurement has exactly the same
outcome.



The question now arises whether the rules of BDMI and SDMI, applied to
\(\alpha\), assign a value to \(\Pi^{\alpha}\) that is the same as the
value assigned to \(\Pi^{\alpha} \otimes I^{\beta}\) when the rules
are applied to the composite system \(\alpha \beta\) (a condition
known as Property Composition), and vice versa (Property
Decomposition). The answer to this question is negative: BDMI and SDMI
violate Property Composition and Property Decomposition (for a proof,
see Vermaas 1998).



If one maintains that the projectors \(\Pi^{\alpha}\) and
\(\Pi^{\alpha} \otimes I^{\beta}\) must represent the same property,
the violation of Property Composition and Property Decomposition is a
serious problem. Thus, Arntzenius (1990) finds the violation of
Property Composition bizarre, since different truth values are
assigned to propositions like 'the left-hand side of a table is
green' and 'the table has a green left-hand side'; a
similar argument is put forward by Clifton (1996, see also Clifton
1995c).



However, Vermaas (1998) argues that the observational
indistinguishability of the magnitudes represented by \(\Pi^{\alpha}\)
and \(\Pi^{\alpha} \otimes I^{\beta}\) does not force one to consider
these two projectors as representing the same property: in fact, they
are different from a theoretical viewpoint, since they are defined on
different Hilbert spaces. Moreover, he argues that the examples
developed by Arntzenius and Clifton sound bizarre precisely in the
light of Property Composition and Property Decomposition. But in the
quantum realm we must accept that the questions of which properties
are possessed by a system and which by its subsystems are different
questions: the properties of a composite system \(\alpha\beta\)
don't always fix the properties of subsystem \(\alpha\), and
vice versa. Vermaas concludes that the tenet that \(\Pi^{\alpha}\) and
\(\Pi^{\alpha} \otimes I^{\beta}\) do represent the same property can
be viewed as an addition to quantum mechanics, which can be denied as,
for instance, van Fraassen (1991) did.



A similar problem is due to the fact that the factorization of Hilbert
spaces is not unique: any given factorization \(\mathcal{H} =
\mathcal{H}^1 \otimes \mathcal{H}^2\) can be "rotated" to
produce different factorizations \(\mathcal{H}' =
\mathcal{H}^1{}^\prime \otimes \mathcal{H}^2{}^\prime\). Are we to
apply the same rules to each such factorization? How are the results
related, if at all? A theorem due to Bacciagaluppi (1995, see also
Vermaas 1997) shows that if one applies SDMI to every factorization
and insists that the values of observables that occur in the different
factorizations are factorization-independent, then one will end up
with a mathematical contradiction of the Kochen-Specker type.



One possible reaction to these difficulties is to assume that there is
a preferred factorization. (One may also regard the above problems as
an indication that systems possess their properties in a relational
way, see Section 5). The Atomic Modal Interpretation (AMI,
Bacciagaluppi and Dickson 1999) thus assumes that nature fixes a set
of mutually disjoint atomic quantum systems \(S^j\) that constitute
the building blocks of all other quantum systems. From the
mathematical point of view, this means that the Hilbert space
\(\mathcal{H}^{\univ}\) of the entire universe can only be
meaningfully factorized in a single way, which defines a preferred
factorization. If each atomic quantum system \(S^j\) corresponds to a
Hilbert space \(\mathcal{H}^j\), then the Hilbert space
\(\mathcal{H}^{\univ}\) of the universe must be written as



\[
\mathcal{H}^{\univ} = \mathcal{H}^1 \otimes \mathcal{H}^2 \otimes \cdots \otimes \mathcal{H}^j \otimes \cdots
\]


The appeal of this idea is that it is in consonance with the standard
model of particle physics, where the fundamental blocks of nature are
the elemental particles, e.g., quarks, electrons, photons, etc., and
their interactions. Moreover, the problems mentioned above can be
evaded in AMI: once the atomic building blocks have been assigned
definite properties, one may stipulate that composite systems have
properties that follow from property composition. Some of these ideas
are implemented in the Modal-Hamiltonian Interpretation, to be
discussed in Section 8. The property ascription to the atomic quantum
systems in the AMI further follows the general idea of modal
interpretations, that is, the ascription depends via a fixed rule on
the dynamical state of the system.



The main challenge for AMI is to justify the assumption that there is
a preferred partition of the universe and to provide some idea about
what this factorization should look like. AMI also faces another
problem. A non-atomic quantum system \(S^{\sigma}\), defined as
composite of atomic quantum systems, does not necessarily have
properties that correspond to the outcomes of measurements. The reason
is that the system \(S^{\sigma}\) might be in the quantum state
\(\varrho^{\sigma}\) with an eigenprojector \(\Pi^{\sigma}\) such that
\(\mathrm{Tr}^{\sigma}(\varrho^{\sigma}\Pi^{\sigma}) = 1\). This
implies that if one measured the property represented by
\(\Pi^{\sigma}\), one would obtain a positive outcome with probability
1. But it may be the case that the projector \(\Pi^{\sigma}\) is not a
composite of atomic properties and, therefore, according to AMI, it is
not a property possessed by the composite quantum system
\(S^{\sigma}\). 



Two answers to this difficulty have been proposed. The first allows
the existence of dispositional properties in addition to ordinary
properties (Clifton 1996). According to the second answer, the outcome
1 of the projector \(\Pi^{\sigma}\) is a collective dynamical effect
onto the measurement device (Dieks 1998): the composite quantum system
may behave as a collective entity, screening off the atomic quantum
systems. This implies that a non-atomic quantum system \(S^{\sigma}\)
may sometimes be taken as if it were atomic within a coarse-grained
description.


## 6. Dynamics of properties



As we have seen, modal interpretations intend to provide, for every
instant, a set of definite-valued properties and their probabilities.
Some advocates of modal interpretations are willing to leave the
matter at that. Others take it to be crucial that also questions of
the following form are answered: given that the property \(P\) of a
system has the actual value \(\alpha\) at time \(t\_0\), what is the
probability that its property \(P'\) has the actual value \(\beta\) at
time \(t\_1 \gt t\_0\)? In other words, they want a *dynamics* of
actual properties.



There are arguments on both sides (van Fraassen 1997). Those who argue for the necessity
of such a dynamics maintain that we have to assure that the
*trajectories* of actual properties *really are*, at
least for macroscopic objects, like we see them to be, i.e., like the
records contained in memories. For example, we should require not only
that the book at rest on the desk possesses a definite location, but
also that, if undisturbed, its location relative to the desk does not
change in time. Accordingly, one cannot get away with simply
specifying possible definite properties at each instant of time. We
need also to show that this specification is at least compatible with
a reasonable dynamics; better still, to specify this dynamics
explicitly.



Those who consider a dynamics of actual properties to be superfluous
reply that such a dynamics is more than what an interpretation of
quantum mechanics needs to provide. Memory contents for each instant
are enough to make empirical adequacy possible.



Ruetsche (2003) argues that it is important in this connection whether
the modal interpretation is viewed as leading to a
*hidden-variables theory*, in which value states are added as
hidden variables to the original formalism in order to obtain a full
description of the physical situation, or rather as an interpretive
move that equips the original formalism with a new semantics (see also
Section 11). In the first case one would expect a full dynamics of
actual properties, in the second this is not so clear.



Of course, modal interpretations admit a trivial dynamics, namely, one
in which there is no correlation from one time to the next. In this
case, the probability of the transition from the property \(P\) having
the actual value \(\alpha\) at \(t\_0\) to the property \(P'\) having
the actual value \(\beta\) at \(t\_1 \gt t\_0\) is just the single-time
probability for \(P'\) having \(\beta\) at \(t\_1\). However, this
dynamics is unlikely to interest those who feel the need for a
dynamics at all. Several researchers have contributed to the project
of constructing a more interesting form of dynamics for modal
interpretations (see Vermaas 1996, 1998). An important account is due
to Bacciagaluppi and Dickson (1999, see also Bacciagaluppi 1998). That
work shows the most significant challenges that the construction of a
dynamics of actual properties must face.



The first challenge is posed by the fact that the set of
definite-valued properties--let us call it
'\(S\)'--may change over time. One therefore has to
define a family of maps, each one being a 1-1 map from \(S\_0\)
at time \(t\_0\) to a different \(S\_t\) at time \(t\), for any time.
With such a family of maps, one can effectively define conditional
probabilities within a single state space, and then translate them
into "transition" probabilities. For this technique to
work, \(S\_t\) must have the same cardinality at any time. However, in
general this is not the case: for instance, in SDMI, the number of
different projectors appearing in the spectral decomposition of the
density matrix may vary with time.



A way out of this is to augment \(S\) at each time so that its
cardinality matches the highest cardinality that \(S\) ever achieves.
Of course, one hopes to do so in a way that is not completely ad hoc.
For example, in the context of SDMI, Bacciagaluppi, Donald and Vermaas
(1995) show that the "trajectory" through Hilbert space of
the spectral components of the reduced state of a physical system
will, under reasonable conditions, be continuous, or have only
isolated discontinuities, so that the trajectory can be naturally
extended to a continuous trajectory (see also Donald 1998). This
result suggests a natural family of maps as discussed above, which map
each spectral component at one time to its unique continuous evolved
component at later times.



The second challenge to the construction of a dynamics arises from the
fact that one wants to define transition probabilities over
infinitesimal units of time, and then derive the finite-time
transition probabilities from them. Bacciagaluppi and Dickson (1999)
argue that, adapting results from the theory of stochastic processes,
one can show that the procedure may be carried out for modal
interpretations of at least some varieties.



Finally, one must define infinitesimal transition probabilities that
will give rise to the proper quantum-mechanical probabilities at each
time. Following earlier papers by Bell (1984), Vink (1993), and
others, Bacciagaluppi and Dickson (1999) define an infinite class of
such infinitesimal transition probabilities, such that all of them
generate the correct single-time probabilities, which arguably are all
we can really test. However, Sudbery (2002) has contended that the
form of the transition probabilities *would* be relevant to the
precise form of spontaneous decay or the "Dehmelt quantum
jumps"; he independently develops the dynamics of Bacciagaluppi
and Dickson and applies it in such a way that it leads to the correct
predictions for these experiments. Gambetta and Wiseman (2003, 2004)
have developed a dynamical modal account in the form of a
non-Markovian process with noise, also extending their approach to
positive operator-valued measures (POVMs). More recently, Hollowood
(2013a, 2013b, 2014) has elaborated the idea that the dynamics of
value states can be modeled by a discrete-time Markov chain.


## 7. Perspectival modal interpretation



In Section 5 it was discussed how the attribution of physical
properties to systems in an absolute (monadic) sense leads to
consistency questions. Such questions may be avoided if one adopts a
metaphysics in which properties are *relational*. In fact, it
can be argued that the mathematical structure of non-collapse quantum
mechanics strongly suggests a relational picture anyhow, as evidenced,
for example, by Everett's (1957) reasons for introducing his
*relative* states (these reasons relate to the non-collapse
treatment of Wigner's friend paradox, see also Dieks 2019,
2021).



An early relational modal interpretation was proposed by Kochen
(1985). According to Kochen, in a bi-partite system one system possess
properties "as witnessed" by the other system.



Bene and Dieks (2002) have developed a more general perspectival modal
interpretation (PMI), in which the properties of any physical system
are defined with respect to arbitrary other physical systems that
serve as "reference systems" (Bene 1997). PMI is closely
related to SDMI since similar rules are used to assign properties to
quantum systems.



In more detail, the state of \(S\) with respect to the reference
system \(R\) is denoted by \(\varrho\_{R}^{S}\). In the special case in
which \(R\) coincides with \(S\), the state \(\varrho\_{S}^{S}\) is
called "*the state of S with respect to itself*".
If the system \(S\) is contained in a system \(A\), the state
\(\varrho\_{A}^{S}\) is defined as the density operator that can be
derived from \(\varrho\_{A}^{A}\) by taking the partial trace over the
degrees of freedom in \(A\) that do not pertain to \(S\):



\[
\varrho\_{A}^{S} = \mathrm{Tr}\_{(A\setminus S)} \varrho\_{A}^{A}
\]


PMI starts with the dynamical state of the whole universe with respect
to itself, which it is assumed to be a pure state \(\varrho\_{U}^{U} =
\ket{\psi} \bra{\psi}\), evolving unitarily according to the
Schrodinger equation. For any system \(S\) contained in the
universe, its value state with respect to itself \(\varrho\_{S}^{S}\)
is postulated to be one of the projectors of the spectral resolution
of \(\varrho\_{U}^{S} = \mathrm{Tr}\_{(U\setminus S)} \varrho\_{U}^{U} =
\mathrm{Tr}\_{(U\setminus S)} \ket{\psi} \bra{\psi}\). In particular,
if there is no degeneracy among the eigenvalues of
\(\varrho\_{U}^{S}\), \(\varrho\_{S}^{S}\) is the one-dimensional
projector \(\ket{\psi\_{S}} \bra{\psi\_{S}}\).



Within PMI it may happen that a system is localized from the
perspective of one observer and, nevertheless, is delocalized from a
different perspective. However, when different observers look at the
same macroscopic object, this effect will be suppressed due to
decoherence effects.



The "metaphysical" idea behind this perspectival
interpretation is that different relational descriptions, given from
different perspectives, are to be considered as equally objective;
physical reality has a relational character and perspectival states
cannot be reduced to more fundamental non-relational states.



In this relational/perspectival interpretation the course of events in
experiments of the EPR-type may be described in a local manner.
Indeed, the change in the relational state of particle 2 with respect
to particle 1 can be understood as a consequence of the change in the
reference system brought about by the local measurement interaction
between particle 1 and the measuring device. This local measurement is
responsible for the creation of a new perspective, and from this new
perspective there is a new relational state of particle 2 (Dieks 2009, 2019).



PMI agrees with Bohr's statement that the definition of physical
reality in the quantum realm should include the experimental context
(see Dieks 2017 for a detailed discussion of Bohr's views and a
comparison with the modal approach). However, PMI is more general and
technically precise: PMI defines the state of a system with respect to
arbitrary contexts, not necessarily related to measurements. This
removes the threat of subjectivism, as the relational states follow
unambiguously from the quantum formalism and the physics of the
situation.



Berkovitz and Hemmo (2006) suggest that a relational approach is
needed to generalize modal interpretations to the relativistic case.
Dieks (2021) provides a detailed scheme with the help of which PMI can
be made relativistically covariant (borrowing ideas from the
many-worlds interpretation).



Rovelli and coworkers have also proposed a relational interpretation
of quantum mechanics in which there is the possibility of different
perspective-dependent descriptions of a physical system (Rovelli 1996,
Rovelli and Smerlak 2007, Laudisa and Rovelli 2021, see also van
Fraassen 2010 and the entry
 relational quantum mechanics.)
 There are points of contact between PMI and Rovelli's
relational interpretation, but also differences. In Rovelli's
proposal, discrete interaction events (some of which correspond to
measurements) are fundamental and the state has to be updated every
time such an interaction event occurs. By contrast, in PMI the state
evolution is always unitary.


## 8. Modal-Hamiltonian interpretation



As Bub (1997) points out (see Section 2), in most modal
interpretations the preferred context of definite-valued observables
is determined solely by the dynamical state of the system; but one
could also introduce a preferred definite-valued observable. An
example is Bohmian mechanics, in which the preferred context is a
priori defined by the position observable; in this case, property
composition and property decomposition hold. But this is not the only
reasonable possibility for a modal interpretation with a fixed
preferred observable. In fact, the modal-Hamiltonian interpretation
(MHI, Lombardi and Castagnino 2008) endows the Hamiltonian of a system
with a determining role, both in the definition of systems and
subsystems and in the selection of the preferred context.



The MHI is based on the following postulates:



**Systems postulate (SP):**
  

A quantum system \(S\) is represented by a pair \((\mathcal{O}, H)\)
such that (i) \(\mathcal{O}\) is a space of self-adjoint operators on
a Hilbert space, representing the observables of the system, (ii) \(H
\in \mathcal{O}\) is the time-independent Hamiltonian of the system
\(S\), and (iii) if \(\varrho\_0 \in \mathcal{O}'\) (where
\(\mathcal{O}'\) is the dual space of \(\mathcal{O})\) is the initial
state of \(S\), it evolves according to the Schrodinger
equation.



Although any quantum system can be decomposed in parts in many ways,
according to MHI a decomposition leads to parts which are also quantum
systems only when the components' behaviors are dynamically
independent of each other, that is, when there is no interaction among
the subsystems:



**Composite systems postulate (CSP):**
  

A quantum system represented by \(S: (\mathcal{O}, H)\), with initial
state \(\varrho\_0 \in \mathcal{O}'\), is *composite* when it
can be partitioned into two quantum systems \(S^1 : (\mathcal{O}^1,
H^1)\) and \(S^2 : (\mathcal{O}^2, H^2)\) such that (i) \(\mathcal{O}
= \mathcal{O}^1 \otimes \mathcal{O}^2\), and (ii) \(H = H^1 \otimes
I^2 + I^1 \otimes H^2\) (where \(I^1\) and \(I^2\) are the identity
operators in the corresponding tensor product spaces). In this case,
we say that \(S^1\) and \(S^2\) are *subsystems* of the
composite system \(S = S^1 \cup S^2\). If the system is not composite,
it is *elemental*. 



With respect to the preferred context, the basic idea of MHI is that
the Hamiltonian of the system defines actualization. Any observable
that does not have the symmetries of the Hamiltonian cannot acquire a
definite actual value, since this actualization would break the
symmetry of the system in an arbitrary way.



**Actualization rule (AR):**
  

Given an elemental quantum system represented by \(S: (\mathcal{O},
H)\), the actual-valued observables of \(S\) are \(H\) and all the
observables commuting with \(H\) and having, at least, the same
symmetries as \(H\).



The selection of the preferred context exclusively on the basis of a
preferred observable has been criticized by arguing that in the
Hilbert space formalism all observables are on an equal footing.
However, quantum mechanics is not just Hilbert space mathematics: it
is a physical theory that includes a dynamical law in which the
Hamiltonian is singled out to play a central role.


 The justification for selecting the Hamiltonian as the preferred
observable ultimately lies in the success of MHI and its ability to
solve interpretive difficulties (see Lombardi and Castagnino 2008,
Sections 5 and 6). With respect to the first point: the scheme has
been applied to several well-known physical situations (free particle
with spin, harmonic oscillator, free hydrogen atom, Zeeman effect,
fine structure, the Born-Oppenheimer approximation), leading to
results consistent with empirical evidence. With respect to
interpretation, MHI confronts quantum contextuality by selecting a
preferred context, and has proved to be able to supply an account of
the measurement problem, both in its ideal and its non-ideal versions;
moreover, in the non-ideal case it gives a criterion to distinguish
between reliable and non-reliable measurements, a criterion that can
be generalized when expressed in informational terms (Lombardi,
Fortin, and Lopez 2015).



In MHI property composition and property decomposition hold because
the actualization rule only applies to *elemental* systems: the
definite-valued properties of composite systems are selected on the
basis of those of the elemental components and following the usual
quantum assumption according to which the observable \(A^1\) of a
subsystem \(S^1\) and the observable \(A = A^1 \otimes I^2\) of the
composite system \(S = S^1 S^2\) represent the same property (Ardenghi
and Lombardi 2011).



The preferred context of MHI does not change with time: the
definite-valued observables always commute with the Hamiltonian and,
therefore, they are constants of motion of the system. This means that
they are the same during the whole "life" of the quantum
system as a closed system, since its initial "birth", when
it arises as a result of an interaction, up to its final
"death", when it disappears by interacting with another
system. As a consequence, there is no need of accounting for the
dynamics of the actual properties as in BDMI and SDMI.



In more recent years, MHI has extended its applications to further
situations, such as the non-collapse account of consecutive
measurements in physics (Ardenghi, Lombardi, and Narvaja 2013) and the
problem of optical isomerism in chemistry (Fortin, Lombardi, and
Martinez Gonzalez 2018).


## 9. Non-ideal measurements



Above we suggested that BDMI and SDMI immediately solve the
measurement problem. This is right in the case of ideal von Neumann
measurements, in which the eigenstates \(\ket{a\_i}\) of the measured
observable \(A\) of the object system \(S\) become perfectly
correlated with the eigenstates \(\ket{p\_i}\) of the pointer \(P\) of
the measuring apparatus \(M\). However, ideal measurements can never
be achieved in practice: the interaction between \(S\) and \(M\) never
produces a completely perfect correlation between \(A\) and \(P\). Two
kinds of non-ideal measurements are usually distinguished in the
literature:


* *Imperfect measurement* (*first kind*)
  

\(\sum\_i c\_i \ket{a\_i} \otimes \ket{p\_0} \rightarrow \sum\_{ij} d\_{ij}
\ket{a\_i} \otimes \ket{p\_j}\) (in general, \(d\_{ij} \ne 0\) with \(i
\ne j)\)
* *Disturbing measurement* (*second kind*)
  

\(\sum\_i c\_i \ket{a\_i} \otimes \ket{p\_0} \rightarrow \sum\_i c\_i
\ket{a\_i^d} \otimes \ket{p\_i}\) (in general, \(\braket{a\_i^d}{a\_j^d}
\ne \delta\_{ij}\))



Note, however, that disturbing measurement can be rewritten as
imperfect measurements (and vice versa).



Imperfect measurements pose a challenge to BDMI and SDMI, since their
rules for selecting the definite-valued properties in this case do not
pick out the right properties for the apparatus (see Albert and Loewer
1990, 1991, 1993; also Ruetsche 1995). An example that clearly brings
out the difficulties was formulated in the context of Stern-Gerlach
experiments (Elby 1993). This argument uses the fact that the
wavefunctions in the \(z\)-variable typically have infinite
"tails" that introduce non-zero cross-terms; therefore,
the "tail" of the wavefunction of the "down"
beam may produce detection in the upper detector, and vice versa (see
Dickson 1994 for a detailed discussion).



If the biorthogonal decomposition is applied to the non-perfectly
correlated state \(\sum\_{ij} d\_{ij} \ket{a\_i} \otimes \ket{p\_j} =
\sum\_i c\_i' \ket{a\_i'} \otimes \ket{p\_i'}\), according to BDMI the
result does not select the pointer \(P\) as a definite-valued
property, but a different observable \(P'\) with eigenstates
\(\ket{p\_i'}\). In this case, in which the selected definite-valued
properties are different from those expected, the question arises how
different they are. In the case of an imperfect measurement, it may be
assumed that the \(d\_{ij} \ne 0\), with \(i \ne j\), be small; then,
the difference might be also small. But in the case of a disturbing
measurement, the \(d\_{ij} \ne 0\), with \(i \ne j\), need not be small
and, as a consequence, the disagreement between the modal
interpretation assignment and the experimental result might be
unacceptable (see a full discussion in Bacciagaluppi and Hemmo 1996).
This fact has been considered as a "silver bullet" for
killing the modal interpretations of the BDMI and SDMI types (Harvey
Brown, cited in Bacciagaluppi and Hemmo 1996). This is premature: a
fuller analysis involving environmental decoherence is able to remove
many of these problems--see below.



Another problem for the BDMI and SDMI interpretations relates to
degeneracy. When the final state of the composite system (measured
system plus measuring device) is very nearly degenerate when written
in the basis given by the measured observable and the apparatus'
pointer (that is, when the probabilities for the various results are
almost equal), the spectral decomposition need not, in general, select
definite-valued properties that are close to those ideally expected.
In fact, the observables so selected may be incompatible
(non-commuting) with the observables that we expect on the basis of
observation (Bacciagaluppi and Hemmo 1994, 1996).



In order to solve these problems, several authors have appealed to the
phenomenon of decoherence; this will now be discussed.


## 10. The role of decoherence



According to the environment-induced approach to decoherence (Zurek
1981, 2003; see also Schlosshauer 2007), any macroscopic measuring
apparatus is an open system in continuous interaction with its
environment; as a consequence of this interaction, the reduced state
of the apparatus becomes, almost instantaneously, indistinguishable
from a state that would represent an ignorance mixture ("proper
mixture") over unknown but definite positions of the
apparatus' macroscopic pointer. The idea that decoherence might
play an important role in modal interpretations, by making macroscopic
quantities definite, was suggested by several authors early on (Dieks
1989, Healey 1995). Consideration of decoherence has acquired central
relevance in the discussion of non-ideal measurements.



As we have seen, in BDMI and SDMI, the biorthogonal and spectral
decompositions do generally not pick out the right properties for the
apparatus in non-ideal measurements. However, Bacciagaluppi and Hemmo
(1996) show that, when the apparatus is a finite-dimensional system in
interaction with an environment containing a large number of degrees
of freedom, decoherence guarantees that the spectral decomposition of
the reduced apparatus' state will be very close to the ideally
expected result and, as a consequence, an observable very close to and
indistinguishable from the apparatus pointer will be selected as
definite-valued. Alternatively, Bub (1997) suggests that instead of
decoherence--with the "tracing out" of the
environment and the diagonalization of the reduced state of the
apparatus--the triorthogonal or \(n\)-orthogonal decomposition
theorem involving the environment may be used to single out a unique
pointer basis for the apparatus.



In either case, interaction with the environment solves the problem of
non-ideal measurements in BDMI and SDMI for the case of
finite-dimensional macroscopic apparatuses. However, the case of
*infinitely* many distinct states for the apparatus is more
troublesome. Bacciagaluppi (2000) has analyzed this situation, using a
continuous model of the apparatus' interaction with the
environment. He concludes that in this case the spectral decomposition
of the reduced state of the apparatus generally does not pick out
properties that are close enough to empirically required ones. This
result seems to apply generally to cases where a macroscopic system
with infinitely many degrees of freedom experiences decoherence due to
interaction with its environment (see Donald 1998). However, model
calculations by Hollowood (2013, 2014) indicate that the problem may
be less severe in realistic circumstances than originally supposed,
and the issue needs further investigation.



Modal interpretations with an a priori fixed preferred observable
(MHI, and Bohm's interpretation) are immune to the above
objections: decoherence by the environment need not be invoked in
order to account for the definiteness of the apparatus' pointer
position (neither in ideal nor in non-ideal measurements). However, in
MHI there still is a relation with the decoherence program. In fact,
the measuring apparatus is always a macroscopic system with a huge
number of degrees of freedom, and the pointer must therefore be a
"collective" and empirically accessible observable; as a
consequence, the many degrees of freedom --and the corresponding
degeneracies-- of the pointer play the role of a decohering
"internal environment" (for details, see Lombardi et
al. 2011). The role of decoherence in the MHI becomes clearer when the
phenomenon of decoherence is understood from a closed-system
perspective (Castagnino, Fortin, and Lombardi 2010). (See the entry on
the role of decoherence in quantum mechanics.)


## 11. The measurement problem revisited: completeness and hidden variables



The measurement problem is usually given the form of the
incompatibility of three assertions of the following form (cf. Maudlin
1995, Myrvold 2018):


1. The quantum state is "complete".
2. The quantum state always evolves in accordance with a linear
dynamical equation (e.g., the Schrodinger equation).
3. Successful measurements have unique and determinate outcomes.



To avoid inconsistency three interpretive strategies are standardly
considered: denying completeness (a), which is often taken to lead to
hidden-variable theories; denying (b), which results in theories
according to which collapses take place; or denying (c), which leads
to many worlds.


 
Modal interpretations accept both (b) and (c), so that it follows
that they must violate (a). Thus, it seems clear that modal
interpretations have to view quantum mechanics as
"incomplete". Indeed, it is true that according to modal
interpretations the quantum state does not fix all actual physical
properties (i.e., the actual values of all definite-valued physical
quantities) of a system, which is the criterion of incompleteness
presupposed in (a).


 
However, many proponents of modal schemes argue that the latter
feature is not appropriately expressed by saying that quantum
mechanics, modally conceived, is incomplete. They object that this
incompleteness terminology is apt to suggest a defect of the theory,
and the need for completion (by hidden variables). But modal theorists
maintain that their interpretations of quantum mechanics are not in
need of additions, because these interpretations already contain
everything that can reasonably be expected from theories describing a
fundamentally probabilistic (indeterministic) world. First, modal
schemes completely specify which observables are
definite-valued. Second, they fix the possible values that may be
taken by these observables and they provide the probabilities with
which these values may be realized. Since it is a core modal principle
that the actualization of properties in the quantum world is a
fundamentally probabilistic (indeterministic) process, nothing more
than this probabilistic description is possible. From this viewpoint,
insisting that modally interpreted quantum mechanics is necessarily
incomplete is tantamount to stipulating that any probabilistic theory
must be incomplete by definition--even if it applies to a
fundamentally probabilistic world. But that appears an inappropriate
use of terminology.


## 12. The interpretation of probability



The leading ideas of modal interpretations are thus
*indeterminism* and *probabilism*: quantum mechanics
does not describe actuality but provides us with a list of
possibilities and their probabilities. This central position of the
concept of probability raises two issues: the formal treatment of
probabilities and the interpretation of probability.



Since the set of all projector operators on a given Hilbert space does
not have a Boolean structure, the Born probability (which is defined
over these projectors) does not satisfy the definition of probability
of Kolmogorov (which applies to a Boolean algebra of events). For this
reason, some authors define a generalized non-Kolmogorovian
probability function over the ortho-algebra of quantum events (Hughes
1989, Cohen 1989). Modal interpretations do not follow this path: they
conceive probabilities as represented by a Kolmogorovian measure on
the Boolean algebra representing the definite-valued quantities,
generated by mutually commuting projectors. As we have seen, the
various modal interpretations differ from each other in their
definitions of the preferred context on which the Kolmogorovian
probability is defined.



In general, a context can be described as a complete set of orthogonal
projectors \(\{\Pi\_{\alpha}\}\), such that \(\sum\_{i} \Pi\_{i} = I\)
and \(\Pi\_{i}\Pi\_j = \delta\_{ij}\Pi\_{i}\), where \(I\) is the identity
operator in \(\mathcal{H} \otimes \mathcal{H}\). Each such context
generates a Boolean structure, and the state of the system defines a
Kolmogorovian probability function on each individual
context. However, only the probabilities defined on the preferred
context selected by the modal interpretation at hand correspond to
physical possibilities, one of which is actual.



So we are dealing with possible events, among which only one is
actual. The fact that the actual event is not singled out by modal
interpretations, as a point of principle, is what makes them
fundamentally probabilistic. This aspect distinguishes modal
interpretations from many-worlds interpretations, where all events are
actual and in which probability must consequently be a problematic
notion. Nevertheless, this modal unanimity about the fundamental role
of probability does not mean that all modal interpretations agree
about the interpretation of probability.



In the context of BDMI, SDMI and PMI it is usually claimed that, given
the space of possible events, the state generates an
ignorance-interpretable probability measure over this set: quantum
probabilities quantify the ignorance of the observer about the actual
values acquired by the system's observables (see, e.g. Dieks
1988, Clifton 1995a, Vermaas 1999, Bene and Dieks 2002).



By contrast to this "actualism"--the conception that
reduces possibility to actuality (see Bueno
2014)--some modal interpretations, in particular MHI, adopt a
possibilist conception, according to which possible
events--*possibilia*--constitute a basic ontological
category (see Menzel 2018). The probability measure is in this case
seen as a representation of an ontological propensity of a possible
quantum event to become actual (Suarez 2004; Terekhovich 2019 
[Other Internet Resources]).



It is to be noted that the indeterminism of modal quantum mechanics
does not imply that the present quantum state of a system fails to
determine its future quantum states. The Schrodinger equation is
deterministic, and the probabilities encoded in the quantum state
therefore evolve in a deterministic way. Indeterminism comes in when
we consider the actual values of observables, which are not
predetermined even if the quantum state is completely fixed: modal
interpretations take processes of actualization to be essentially
probabilistic.



In this context, the two mentioned views about probability (actualism
versus possibilism) do not exclude each other. That probabilities
quantify ignorance about actual values of observables (as in
actualism) does not entail that this ignorance is due to a lack of
knowledge about underlying deterministic processes. Possibilism adds
that quantum probabilities represent propensities and that our
ignorance is a necessary consequence of the indeterministic ontology
of the physical world, in which propensities constitute a fundamental
category.


## 13. Open problems and perspectives



Many quantum physicists implicitly subscribe to modal ideas in their
work, in the sense that they employ the formalism of unitary quantum
mechanics in order to make probabilistic statements about one single
world (our own). This use of unitary quantum mechanics is especially
popular in quantum information theory. There is also a growing
interest in questions concerning the conceptual consequences of a
consistent application of unitary (noncollapse) quantum mechanics to
the physical world (see, e.g., Frauchiger and Renner 2018 and ensuing
discussions). These questions find their natural place within the
modal interpretation program.



During the last decade, modal interpretations have also more
explicitly drawn the attention of practicing physicists and
mathematicians interested in foundational matters. For instance,
Hollowood (2014) offers an interpretation of quantum mechanics
inspired by PMI: in this account the state of an open system
represents its properties as seen from the perspective of the closed
system of which it is a subsystem. Barandes and Kagan (2014a, 2014b,
2020) propose a "minimal modal interpretation", inspired
by SDMI, according to which the preferred context is given by the
evolving reduced state of the open system. Nakayama (2014a, 2014b) has
explored connections between the modal interpretation and the
framework of topos theory.



There are also several specifically philosophical issues connected to
the modal program: about the nature of the "things" to
which quantum mechanics refers, that is, about the basic categories of
the quantum ontology. As we have seen, in general the properties of
quantum systems are considered to be monadic in modal interpretations,
with the exception of the perspectival version PMI where these
properties are relational. In both cases, it may be asked whether a
quantum system has to be conceived as an individual substratum
supporting properties or as a mere "bundle" of properties
which do not need to constitute an individual in the ordinary sense.
Following an original idea of the MHI, da
Costa, Lombardi and Lastiri (2013) and da Costa and Lombardi (2014)
have suggested that, in the modal context, the bundle view is the most
appropriate one to supply an answer to the problem of
indistinguishability of identical quantum particles (French and Krause
2006). As noticed by Steven French (2020 [Other Internet Resources]),
this view of non-individual bundles finds a natural resonance with
ontic structural realism, which was mainly motivated by the
ontological challenges of quantum mechanics (Ladyman 1998). Moreover,
this ontology of properties has been taken as a basis for a
metaphysical interpretation of relational quantum mechanics (Oldofredi
2021). It should be noted that this bundle ontology does not
contradict the emergence of classical distinguishable particles in
certain situations, e.g. the classical limiting case (Lombardi and
Dieks 2016).



Also other ontological topics relating to modal interpretations have
recently become subjects of discussion. Thus, Calosi (2018) argues
that BDMI and AMI pose significant challenges to quantum monism, that
is, the view according to which there is only one fundamental entity
--the universe as a whole. In the context of the discussion about
the classification between \({\psi}\)-ontic and \({\psi}\)-epistemic
models, Oldofredi and Lopez (2020) show that such a classification is
too narrow to include relational interpretations such as PMI.



There are also more technical issues that await further discussion.
Modal interpretations as discussed here are based on the standard
Hilbert space formalism of quantum mechanics. However, Brown,
Suarez, and Bacciagaluppi (1998) argue that there is more to
quantum reality than what is described by operators and quantum
states: they claim that gauges and coordinate systems are important to
our description of physical reality as well, while modal
interpretations (AM, BDMI and SDMI) have standardly not taken such
things into consideration. In a similar vein, in the context of the
MHI it has been argued that Galilean space-time symmetries endow the
formal skeleton of quantum mechanics with physical flesh and blood and
identify the fundamental physical magnitudes that allow the theory to
be applied to concrete physical situations. If the set of
definite-valued observables of a system is postulated to be invariant
under Galilean transformations (the intuition being that a mere change
in inertial reference frame does not change this set), the MHI rule of
actualization can be reformulated in an explicitly invariant form, in
terms of the Casimir operators of the Galilean group (Ardenghi,
Castagnino, and Lombardi 2009, Lombardi, Castagnino, and Ardenghi
2010).



Another fundamental question is the relativistic extension of the
modal approach. Dickson and Clifton (1998) have shown that a large
class of modal interpretations of ordinary quantum mechanics cannot be
made Lorentz-invariant in a straightforward way (see also Myrvold
2002). With respect to the extension to algebraic quantum field theory
(see Dieks 2002, Kitajima 2004), Clifton (2000) proposed a natural
generalization of the non-relativistic modal scheme, but Earman and
Ruetsche (2005) showed that it is not yet clear whether it will be
able to deal with measurement situations and whether it is empirically
adequate. The problems revealed by these investigations relate to the
fact that modal interpretations employ the non-relativistic concept of
a global state, of an extended system, at one instant. In a local
field-theoretic context this becomes different, and this may help
avoiding conflicts with relativity (Earman and Ruetsche 2005).
Berkovitz and Hemmo (2005) and Hemmo and Berkovitz (2005) propose a
different way out: they argue that perspectivalism can come to the
rescue here (see also Berkovitz and Hemmo 2006). The idea that
perspectivalism is essential for ensuring compatibility with special
relativity is elaborated by Dieks (2021), who puts forward a
relativistically covariant treatment of experiments of the EPR-type,
based on PMI.



In the context of MHI an alternative program for achieving
relativistic covariance has been developed, based on the idea that the
actualization rule, expressed in terms of the Casimir operators of the
Galilean group in non-relativistic quantum mechanics, can be
transferred to the relativistic domain by changing the symmetry group
accordingly. In this case the definite-valued observables of a system
become those represented by the Casimir operators of the
Poincare group. Since the mass operator and the squared spin
operator are the only Casimir operators of the Poincare group,
these then are always definite-valued observables. This conclusion is
in agreement with a usual assumption in quantum field theory:
elemental particles always have definite values of mass and spin, and
those values are precisely what define the different kinds of
elemental particles of the theory (Lombardi, Castagnino, and Ardenghi
2010).



These various proposals and developments have arisen in the context of
detailed technical investigations. This illustrates two features of
the modal approach: on the one hand it makes use of a precise
mathematical framework; on the other hand it is flexible and
open-ended, and can accommodate different views concerning the rules
determining the set of definite-valued observables. It is natural to
expect new offshoots; for example, it seems plausible that strategies
followed in many-worlds interpretations to define different
"branches" can also be incorporated in the modal approach
in order to define single-world possibilities. Whatever these
prospects, one can at least say that the modal approach has given rise
to a fruitful series of investigations into the structure of quantum
theory.