# Modal Interpretations of Quantum Mechanics

*First published Tue 12 Nov, 2002*

The original ‘modal interpretation’ of quantum theory was born in the early 1970s, and at that time the phrase referred to a single interpretation, due to van Fraassen. The phrase now encompasses a wide class of interpretations, and is better taken to refer to a general approach to the analysis of the structure -- both conceptual and mathematical -- of quantum theory. I shall describe the history of modal interpretations, how the phrase has come to be (best) used in this way, and the general program of (at least some of) those who advocate this approach.

- 1. The Copenhagen Variant
- 2. Kochen-Dieks-Healey Interpretations
- 3. Motivating Early Modal Interpretations
- 4. Reality Sets in: The Problem of Imperfect Measurement
- 5. The Algebraic Approach
- 6. Dynamics
- 7. Open Projects
- Bibliography
- Other Internet Resources
- Related Entries

## 1. The Copenhagen Variant

By the early 1970s, researchers in philosophy of physics had become painfully aware of the nonlocality inherent in standard quantum theory. It arises most dramatically, perhaps, in the context of the projection postulate, which asserts that upon measurement of a physical system, its state will ‘collapse’ (or be ‘projected’) to one of the possible values of the measured quantity. This postulate is difficult to accept in any case (what effects this discontinuous change in the physical state of a system? what exactly is a ‘measurement’?), but it is especially worrying when applied to entangled compound systems whose components are well-separated in space. The classic example is the Einstein-Podolsky-Rosen-type experiment, in which two particles initially interacting are separated. Their quantum-mechanical state is ‘entangled’, which means, for our purposes, that the collapse resulting from a measurement on one of them simultaneously (and instantaneously) affects the other.

A possible way clear of this problem was noticed by van Fraassen (1972, 1974), who proposed to eliminate the projection postulate from the theory. Of course, others had made this proposal before. Bohm's (1952) theory (itself preceded by de Broglie's proposals from the 1920s) eliminates the projection postulate, as do the various many-worlds (and relative-state) interpretations. Van Fraassen's proposal was, however, apparently quite different from these other approaches. It relied, in particular, on a distinction between the ‘value state’ of a system, and the ‘dynamical state’ of a system. The value state describes the system's properties, while the dynamical state determines which properties the system might have at a later time. (More specifically, the dynamical state determines how the system ‘reacts to perturbations’, as van Fraassen sometimes says, such information being what we need to make predictions about its future value states.)

The dynamical state is just the quantum state, and it never collapses. The value state is (typically) something other than the quantum state. Indeed, it is something other than the dynamical state exactly when the dynamical state is not a pure state. In these cases (i.e., when the dynamical, or quantum, state of a system is mixed), the dynamical state constrains the possible value states (and empirical adequacy requires that it also generate the correct frequencies for those value states that are observable).

Van Fraassen's proposal thus violates the so-called ‘eigenstate-eigenvalue link’, which asserts that a system has the value state corresponding to a given eigenvalue (of a given observable) if and only if its quantum state is an eigenstate of the observable corresponding to that eigenvalue. Van Fraassen accepts the ‘if’ part, but denies the ‘only if’ part.

What are the possible ‘value states’ for a given system at
a given time? For van Fraassen, the form of the value states is
restricted in an important, and somewhat conservative, way:
propositions about a physical system cannot be jointly true unless
they can be jointly certain. In other words, the uncertainty
principle imposes a limit not only on our knowledge of the properties
of a system, but also on the properties themselves. It must be
*possible* for a dynamical state to assign a value state
probability 1. This restriction motivates van Fraassen to term his
interpretation the ‘Copenhagen Variant’ of the modal
interpretation. Other variants (for example, van Fraassen identifies
an ‘Anti-Copenhagen’ variant, which he attributes to Arthur
Fine) would impose less restrictive conditions on the form of the
value states.

Finally, ‘value states’ are maximal with respect to the
restriction just noted. It follows immediately that value states are
representable as pure states. But *which* pure states are the
possible value states at a given moment? Van Fraassen formulates a
very permissive criterion, which other authors have found too
permissive, the reason stemming from his ‘constructive
empiricist’ philosophy of science. He is more concerned with the
possibility of giving an interpretation of the theory in which the
theory is empirically adequate, i.e. compatible with all observable
phenomena (in the sense used by van Fraassen), in particular the
observable phenomenon that (what we normally call) measurements do
have results, rather than the specification of a theory that will
predict that such measurements have results. Indeed, if we apply the
eigenstate-eigenvalue link to the theory (which, recall, is quantum
mechanics without the projection postulate), it turns out that
measurements do not have results. But an ‘interpretation' that
predicts that measurements will have results, especially if it
predicts also what the probabilities will be for these results (Born
rule), and how the future evolution of the system will be affected by
these results (projection postulate), is presumably not a mere
interpretation of the given theory, but a new theory altogether (a
hidden variables theory). The account given by van Fraassen is much
more modest, prescribing as an interpretational rule only that the
potential value states are the states that appear in the various
decompositions of the (generally mixed) dynamical state.

Of course, empirical adequacy does require that, in cases of measurement, the actual value state of the apparatus be one describing a definite measurement result; and thus the observed value states, in these cases, are only a very restricted subset of the possible value states according to van Fraassen. Observation tells us also that, in these cases, the dynamical state generates a probability measure over this more restricted set, thus enabling us to make predictions about the result. In the end, van Fraassen is forced to give an account of measurements, and the account is considered by many to be problematic.

Van Fraassen's account is ‘modal’ because it leads, in
a relatively straightforward way, to a modal logic of quantum
propositions. For van Fraassen, perhaps the most important point is
that one should not presume this modal logic to arise from ignorance
about the actual state of affairs, which is the aim of science to
uncover. In other words, we do not say that a system with dynamical
state *W* possibly has some value state
*V*_{1}, *V*_{2},…, and we need to
find out which one, or which one with which probability. What is
important is that there are possible value states for all physical
systems (i.e. possible stories about the world) that are compatible
with all the observable data. On the other hand, it is quite easy to
see how van Fraassen's account gave rise to a program that is
largely concerned with providing a ‘realistic’
interpretation of quantum theory, a program to which we now turn.

## 2. Kochen-Dieks-Healey Interpretations

The basic outlines of that program are already apparent in van Fraassen's work (or in what may be considered its limitations). The main idea is to define a set of possible properties for a physical system, then to assert that the ‘dynamical’ (i.e., quantum-mechanical) state generates an ignorance-interpretable probability measure over this set. More precisely, one defines an ignorance-interpretable probability measure over value states, which themselves assign ‘possessed’ or ‘not possessed’ to each property in the set. The fact that one uses the quantum-mechanical probability measure guarantees a kind of minimal empirical adequacy with respect to the properties in the set of possible properties. If, in addition, one finds a world described in terms of these properties (and only them) to be plausibly our world, then one might rest satisfied with this interpretation. (The exception might be a desire for an explicit dynamical picture of the possessed properties. See below.)

In the late 1980s, various other philosophers -- typically, as noted above, with a more realistic bent than van Fraassen -- realized that the central features of this approach can be used in the service of interpretations of quantum theory with other philosophical interests. Here we shall consider three cases, albeit briefly and largely without reference to their ‘background’ philosophical motivations (which are all realist, though in very different ways): Kochen, Dieks, and Healey.

Kochen's (1985) modal interpretation sparked a series of proposals that lasted through the 1980s and 1990s. His proposal is based on the polar decomposition theorem (see Reed and Simon (1979, pp. 197-198) for a statement and proof), but is somewhat easier to understand in terms of the so-called ‘biorthogonal decomposition theorem’:

Biorthogonal Decomposition Theorem:

Given a vector, |v>, in a tensor-product Hilbert space,H_{1}H_{2}, there exist bases {|e_{i}>} and {|f_{j}>} forH_{1}andH_{2}respectively such that |v> can be written as a linear combination of terms of the form |e_{i}> |f_{i}>. If the absolute value (modulus) of the coefficients in this linear combination are all unequal then the bases are unique.

In other words, the state of a two-particle system picks out (in most cases, uniquely) a basis (and therefore an observable) for each of the component systems. (See, for example, Schrödinger (1935) for a proof of this theorem.)

Recall from the previous section that van Fraassen was not (for philosophical reasons) compelled to provide a very restrictive prescription for deciding what the possible value states for a given system might be. We see in the biorthogonal decomposition theorem a way to choose the possible value states from one single decomposition of the (mixed) dynamical state of a system: let them be (for each component system) the elements of the basis picked out by the theorem. It is manifest that the dynamical state generates a probability measure over this set of possible value states, namely the standard quantum mechanical measure.

Taking this view, our interpretation is essentially connected with
the existence of two-component compound systems. In one sense, this
feature is not really a noticeable departure from van Fraassen's
view, because recall that for van Fraassen, only systems that are in
mixed (indeed, improperly mixed) dynamical states have value states
that differ from their dynamical states. This situation will
typically occur for systems that are components of a compound
system. (There is this difference: the biorthogonal decomposition
theorem holds only for *two-component* compound systems, while
subsystems of arbitrary compound systems will typically have for
their dynamical state an improper mixture.) This formal similarity
between Kochen and van Fraassen masks a rather significant
philosophical difference, however. Kochen's account is meant to
be perspectival or relational, meaning that a system has a property
only in relation to other systems (see below).

Indeed, in a typical measurement situation, Kochen's
prescription is the same as van Fraassen's. For consider a
typical measurement in which a ‘pointer’ becomes correlated
with the value that some ‘measured’ system has for a given
observable. Letting the |*e*_{i}> represent the
possible ‘indicator states’ of the pointer, and
|*f*_{j}> represent the possible values that the
system might have for the measured observable, the final state of the
compound system should indeed take the form of a linear combination
of terms of the form |*e*_{i}>
|*f*_{i}>, so that by
Kochen's (*and* van Fraassen's) prescription, the
pointer indeed does have its indicator states as possible value
states. (More precisely, for van Fraassen the pointer's indicator
states are among the potential value states; as discussed above, van
Fraassen is concerned only to establish this fact, and not the fact
that they *are* the value states even when they are
unobservable.)

For Kochen, the fact that the application of the interpretation is
restricted to subsystems of a two-component compound system is not a
problem. Indeed, he appears to adopt a metaphysics of properties in
which systems do not have intrinsic properties: all properties are
relational. Kochen calls the relation ‘witnessing’.
Consider again the measurement described above. In this case, the
pointer (at the end of the measurement) may be said to
‘indicate’ (or, as Kochen prefers, ‘witness’) the
result, i.e., the value that the measured system has for the measured
observable. Now, because Kochen intends his interpretation to apply
in *all* circumstances (not only in measurements), we must
abstract the idea of ‘indication’ or ‘witnessing’
away from the context of measurements, and whatever notion we end up
with is supposed to apply to all cases of possession of
properties. Kochen's interpretation is therefore highly
‘perspectival’: systems do not possess properties
intrinsically, but relative to the ‘perspective’ of another
system that ‘witnesses’ it to posses the property in
question.

Other authors, notably Dieks, prefer (or at least did originally prefer) a metaphysics of intrinsically possessed properties. They are therefore faced with two questions.

- What can one say (if anything) about the properties of subsystems that are not components of a two-component system in a pure state?

To raise the second question, note that a three-component compound system may be divided into pairs of subsystems in several ways. Consider, for example, the compound system A&B&C. We could arrive at properties for A by applying the biorthogonal decomposition theorem to the two-component system A&(B&C). We could also apply the theorem to (for example) B&(C&A) or C&(A&B).

- How are the properties of A and B related to those of A&B?

Suppose, for example, that A has the property *P* and B has
the property *Q*. Should one ascribe the property
*P*&*Q* to A&B, or should A&B have some
property that it gets from applying the biorthogonal decomposition to
C&(A&B), or both?

Although in his early proposals Dieks (1988, 1989) did not answer
these questions, his later work, together with (Vermaas and Dieks,
1995) addresses them. (The fullest account is in Vermaas (1999). See
also Bacciagaluppi (forthcoming).) Dieks' answer to the first
question relies on the fact that the density operator (reduced state)
of a single component of a two-component system has for its spectral
resolution exactly the projections spanned by the basis elements
picked out by the biorthogonal decomposition theorem, in the case
when the decomposition is unique. One may then generalize the
original proposal by supposing that the possible value states for
*every* system are just the elements in its (typically,
improper and reduced) density operator's spectral decomposition,
whose existence and uniqueness is guaranteed by the spectral theorem
(stated and proved in just about any textbook of functional
analysis). This new proposal matches the old one in cases where the
old one applies, i.e., in cases where the biorthogonal decomposition
applies, and guarantees a unique biorthogonal decomposition.

The answer to the second question is somewhat more involved. In fact,
we can make the issue even more complicated by noting that a given
tensor-product Hilbert space can be factored in many ways. In
essence, the factorization of a given Hilbert space, *H*, into
two factors, *H*_{1} and *H*_{2}, can
be ‘rotated’ to produce additional factorizations into
*H*_{1} and
*H*_{2}.
There is a continuous infinity of such possibilities. Are we to
apply the proposal to each such factorization? How are the results
related, if at all?

A theorem due to Bacciagaluppi (1995) shows, in essence, that if one applies Dieks' proposal to the ‘subsystems’ obtained in every factorization and insists that the results be comparable (i.e., that the subsystems thus obtained do not have their properties ‘relative to a factorization’ but instead have them absolutely), then one will be led to a mathematical contradiction of the Kochen-Specker variety. While one could adopt the view that subsystems have their properties ‘relative to a factorization’, most advocates of modal interpretations have instead adopted the view that there is a ‘preferred factorization’ of the universal Hilbert space into subsystems. This assumption amounts to the adoption of the existence of fixed ‘atomic’ degrees of freedom of the universe.

One is still faced, however, with the question how degrees of freedom
for a compound system are related to those of its components. The
answer to this question depends, finally, on another issue. Is
Dieks' proposal to be applied to the ‘atoms’ only, or
to any subsystem whatever? For example, do we apply the proposal
(from our schematic example above) to A&B&C as well as to
A&B? Vermaas (1997) has shown that doing so makes question 2
unanswerable in general: one cannot define generally valid
correlations between a composite system and its components in this
case. (If one is willing to adopt perspectivalism -- as Kochen is,
for example -- then one can perhaps justify the lack of such
correlations.) Unless one is willing to adopt some form of
perspectivalism, then, one is apparently led to the atomic modal
interpretation (see, for example, Bacciagaluppi and Dickson, 1999),
according to which the basic proposal is applied *only* to the
‘atomic’ subsystems of the universe. The properties of all
other (compound) systems are inherited from their subsystems. There
are connections here with discussions in metaphysics about the
possibility of the existence of ‘non-supervenient’
properties. (Clifton (1995c) also offers an important theorem
concerning this issue.)

Richard Healey (1989) was also among the first to make use of the biorthogonal decomposition theorem, taking Kochen's ideas in a somewhat different direction. Healey's main concern was indeed the apparent nonlocality of quantum theory. Healey's intuition about the way a 'modal' interpretation based on the biorthogonal decomposition theorem would be applied to, say, an EPR experiment, was as follows. The theorem is applied first to the composite of apparatus on one side and EPR pair (while the EPR pair has not yet interacted with the apparatus on the other side). Thus the EPR pair acquires a 'holistic' property, which can then explain why the apparatus on the other side acquires a property that is correlated to the result on the other side.

Irrespective of whether this picture is general enough for its
intended purpose, it shows that Healey does not subscribe to an
'atomic' modal interpretation, since it is crucial for him that the
EPR pair as a whole be assigned a (non-product)
property. Healey's proposal *begins* with the atomic
interpretation, making use of the biorthogonal decomposition theorem,
but the set of possible properties is then expanded (and subsequently
restricted) by a number of conditions. Healey's aim is
apparently to walk a thin line amongst a variety of desiderata. The
first is consistency. As shown by (for example) the theorems of
Bacciagaluppi and Vermaas, mentioned above -- not to mention the
Kochen-Specker theorem itself -- given certain conditions on the set
of possibly-possessed properties, one cannot add properties to this
set willy-nilly. A second is to maintain a plausible theory of the
relationship between composite systems and their subsystems. A third
is to maintain a plausible account of the relations among possessed
properties at a given time. A fourth is to maintain a plausible
account of the relations among possessed properties at different
times.

The structure of possibly-possessed properties that emerges from Healey's conditions is (for this author at least) extremely difficult to grasp. Some progress has been made since Healey's book was published (see for example Reeder and Clifton, 1995), but in general, it remains difficult to see what the set of possibly-possessed properties is, for Healey.

## 3. Motivating Early Modal Interpretations

On the other hand, the clear advantage that Healey's approach had over others of roughly the same period was motivation. The necessity (or indeed plausibility) of Healey's conditions might be debatable, but it is clear what they assert, and why one might want to make such an assertion -- Healey himself gives reasons. Still, there remains the fundamental question: why begin with the biorthogonal decomposition (or more generally, the spectral decomposition) in the first place? For those interpretations that say little beyond the application of these decompositions to determine the set of possibly-possessed properties, the question is all the more pressing, and the interpretations are all the more lacking in direct physical motivation.

A series of theorems from the mid-1990s proposes to answer (or to begin to answer) this question. The first of these theorems was due to Clifton (1995a), the title of the paper indicating the project: “Independently motivating the Kochen-Dieks Modal Interpretation of Quantum Mechanics”. A series of related results followed, including those by Clifton (1995b), Dickson (1995a, 1995b) and Bub and Clifton (1996). Here I shall discuss Clifton's original paper, and Bub and Clifton's theorem, the former to indicate the general thrust of these earlier arguments, and the latter as a way to introduce Bub's own modal interpretation.

The theorem discussed here is not quite Clifton's, which is slightly
stronger (because its assumptions are slightly weaker), but the
reader should be able to grasp the general idea of these theorems
from the following discussion. They take the general form, for some
mathematically-stated (but hopefully physically motivated) conditions
A, B, C, etc.: If one wants a set of possibly-possessed properties to
obey conditions A, B, C, etc., then the set *must* take the
form asserted by the spectral-decomposition form of the modal
interpretation. That form, more precisely, is the following. Consider
a system in the (in general reduced and improper) mixed state,
*W*. Let {*P*_{i}} be the set of *W*'s
spectral projections, and let
*B*<*P*_{i}> be the Boolean algebra
generated by the *P*_{i}, which is in this case just
the set of all sums of elements of {*P*_{i}}. Finally,
let *Q* be the null space of *W*, that is, orthogonal
to each *P*_{i}. Then the set of all possibly-valued
projections, *P*, for our system is the set

{P|P=P_{j}+Q, whereP_{i}is inB<P_{i}> andQis contained inQ}.

So theorems of the sort proven by Clifton and others take the form: sets of the form given above are the only sets that fulfill conditions A, B, C, etc.

In one such theorem, roughly the one proven originally by Clifton, the conditions are:

**Closure**: the set of all possibly-possessed properties is closed under conjunction, disjunction, and negation (suitably understood in quantum-logical terms).**Classicality**: the quantum-mechanical probability measure (generated by the reduced state*W*) over the set of all possibly-possessed properties obeys all of the laws of classical probability, and -- crucially -- it is ‘ignorance-interpretable’.**Certainty**: for any property R, if the reduced state W assigns probability 1 or probability 0 to R, then R is in the set of all possibly-possessed properties.**Ignorance**: each member of the spectral resolution of W is in the set of all possibly-possessed properties.

The final condition is probably the most difficult to justify, though one should note that Clifton's theorem actually relies on a considerably weaker condition. (Of course in conjunction with the other conditions, it implies the condition given here).

The theorem of Bub and Clifton (1996) (in a slightly improved version
from Bub, Clifton, and Goldstein (2000)) concerns a set of
possibly-possessed properties that is characterized somewhat
differently. Specifically, it is characterized in terms of (in the
simplest case) a pure state, |*v*> and an observable,
**R**. The pure state is the quantum-mechanical state of
the system, while the observable is supposed to be
‘definitely-valued’; i.e., whatever else gets a value, the
spectral projections of **R** must.

The conditions of the Bub-Clifton theorem are the following:

**Closure**: as above.**Truth and Probability**: essentially the same condition as ‘classicality’, above.**R-preferred**: the eigenspaces of**R**are among the set of possibly-possessed properties.**|**: the set of possibly-possessed properties are definable solely in terms of the pure state |*v*>,R-definability*v*> and the observable**R**.**Maximality**: the set of possibly-possessed properties is maximal with respect to the conditions above.

The idea, then, is to find a (maximal) set of possibly-possessed
properties that admits an empirically adequate, but
ignorance-interpretable, probability measure, makes
**R** definite-valued, and is fixed by the state of the
system, |*v*>, and **R**. Again, these
conditions are supposed to be intuitively clear, if not
compelling. The most controversial is surely
‘**R**-preferred’, for it is unclear why there
should be some ‘preferred’ observable in this sense, and
especially how it might be picked out. One would not like the
observable to be picked out by fiat, for example. If we were willing
to choose an observable and stipulate in a more or less ad hoc manner
that it must have a value, then it is unclear why we would be
concerned about the interpretation of quantum theory in the first
place.

Bub and Clifton prove the rather remarkable result that the
conditions above give rise to a unique set of possibly-possessed
properties, defined as follows. Let {*P*_{i}} be the
set of projections onto the vectors, |*v _{i}*>,
which are the projections of |

*v*> onto the eigenspaces of

**R**. Then the set is as defined above for Clifton's theorem concerning spectral-decomposition modal interpretations.

Bub is not silent on the issue raised above, of why there should be a
preferred observable, and how it might get chosen. First, he notes
that a number of traditional interpretations of quantum theory can be
characterized in this way. Notable among them are the Dirac-von
Neumann interpretation, (what Bub takes to be) Bohr's
interpretation, and, perhaps, Bohm's theory. In the last case,
Bub (following Vink, 1993) argues that Bohm's theory can in a
sense be recovered as a kind of limit of modal interpretations in
which the **R**s are chosen to be discretized position
observables. Second, Bub argues (especially in his 1997) that
**R** could be picked out by the physical process of
decoherence. We shall have to leave this suggestion as a tantalizing
possibility.

## 4. Reality Sets in: The Problem of Imperfect Measurement

Earlier I suggested that the spectral-decomposition (and the biorthogonal-decomposition) modal interpretations solve the measurement problem in a particularly direct way: at the end of a typical measurement, the compound system (apparatus plus measured system) is in a state such that the possible properties as picked out by these modal interpretations include exactly the ‘pointer’ states of the apparatus. Hence these interpretations assign ‘the right’ state to apparatuses.

There are two problems with this claim, which by itself is true. First, not everything to which one might want to assign a definite property is (or is obviously) an apparatus at the end of a measurement. Second, measurements in the real world do not satisfy the ideal model that I described earlier. In particular, they do not effect a perfect correlation between the apparatus and the measured system -- measuring apparatuses are imperfect. But then it is far from clear that the biorthogonal (or spectral) decomposition picks out the right properties for the apparatus. (A related problem faces Bub: does decoherence always pick out appropriate observables as definite-valued?)

This problem was first raised by Albert and Loewer (1990, 1991),
later developed by Elby (1993), and it sparked considerable
discussion. Before we turn to the reply, we note that in fact the
problem is unavoidable in the context of quantum theory. It is not
due merely to the fact that measuring apparatuses are
inaccurate. Rather, the quantum-mechanical formalism itself does not
permit perfect correlations. Consider, for example, a standard
Stern-Gerlach measurement of the spin of a particle. Immediately
after the interaction between the particle and the magnets -- and
*even if* that interaction were to produce, initially, a
perfect separation between the spin-up particles and the spin-down
particles -- the wavefunction for the particle emerging from the
magnets would spread instantaneously to cover all of space. The
particle will necessarily have a non-zero probability of turning up
in the ‘wrong’ region. (See Dickson (1994) for a longer
discussion of this point.) So the problem we are facing here is not a
problem of engineering alone; it is intrinsic to quantum theory. (For
this reason, we might expect to learn something by examining it,
whether modal interpretations survive the problem or not.)

The response of modal interpretations to this problem of intrinsic
‘inaccuracy’ in measurements comes in three stages. First,
we may notice that the ‘error terms’ in the state of the
compound (apparatus plus measured) system would typically be very
small, so that the true final state would be extremely close to the
ideal state (in the sense that their inner product would be very
close to one). In that case, one might expect that the spectral
decompositions (of the reduced states for the apparatus and measured
system) would pick out states for the two systems that are extremely
close to the ‘ideal’ states. Specifically, the
‘real’ possibly possessed properties of the apparatus would
be very close (in Hilbert space) to the ‘ideal’ possibly
possessed properties. One interesting issue that arises here is
whether ‘close’ is good enough. Whatever one's answer,
it is crucial to realize that modal interpretations are not here
proposing a FAPP (‘for all practical purposes’) solution to
the measurement problem. No, they assert that the *real* state
of the apparatus is ‘close’ to the ideally expected state,
and that there is no empirical problem with making this
assertion.

However, before one can settle into debating that issue, we must face
up to two more stages in the modal interpretation's response to
the problem of imperfect measurements. The first arises from the
realization that when the final state of the compound system is very
nearly degenerate (when written in the basis given by the measured
observable and the apparatus' ‘pointer’ observable --
i.e., when the probabilities for the various results are nearly
equal), the spectral decomposition does *not*, in fact, choose
a basis that is even close to the ideally expected result. This
point was discussed in greatest detail by Bacciagaluppi and Hemmo
(1996). Relying on the (near) ubiquity of decoherence in the
macroscopic realm, they argue that when the apparatus is considered
as a finite-dimensional system (more precisely, when the apparatus is
modelled by a finite-dimensional Hilbert space), decoherence more or
less guarantees that the spectral decomposition of the (reduced)
state of any macroscopic object will be very close to the ideally
expected result. For example, pointers will be well-localized in
position.

The final stage of the modal interpretation's response to the
problem involves consideration of the (probably more realistic) case
of infinitely many distinct states for the apparatus. 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 does *not* pick out states that are highly
localized. This result applies more generally to other cases where a
macroscopic system (not idealized as finite-dimensional) experiences
decoherence due to interaction with its environment (see also Donald
(1998)).

Two suspicions arise immediately from these results. The first is that the modal interpretation, as stated thus far, was never in a position to deal with quantum mechanics in infinite-dimensional Hilbert spaces. The second (related to the first) is that the spectral decomposition is in any case not the right tool to use to pick out the possibly possessed properties. These problems are serious. Even standard non-relativistic quantum theory occurs in the arena of infinite-dimensional Hilbert spaces, not to mention quantum field theory. Indeed, in the latter case, most of what we have said to this point would have to be significantly revised, or simply thrown away.

## 5 The Algebraic Approach

The Algebraic approach to modal interpretations addresses these concerns by, first, aiming for a formalism that is significantly more general than that developed thus far -- one that can apply to quantum theory in infinite-dimensional Hilbert space, and to quantum field theory -- and second, abstracting away from a particular choice for the possibly possessed properties. (Note that two alternative variants of modal interpretations, which also aim to address some of the above concerns, have been recently proposed by Spekkens and Sipe (2001a,b) and by Bene and Dieks (2002).)

The rudiments of an algebraic approach are already present in the
work of those who, in the mid 1990s, aimed to provide a motivation for
modal interpretations. We saw there that modal interpretations were
described in more or less algebraic terms, namely, as a certain set
closed under algebraic operations (the operations of meet, join, and
orthocomplement on the lattice of projections on a Hilbert space, for
example). Indeed, Bub defines his interpretation in these terms: his
set of possible possessed properties is defined algebraically, in terms
of an arbitrarily chosen observable (above, **R**).

While it was recognized by early workers (Bub, Clifton, Dickson, and others) that the set of possibly possessed properties can be characterized in interesting algebraic ways, the first serious algebraic work on modal interpretations was done by Bell and Clifton (1995), who defined the notion of a ‘quasiBoolean algebra’. These algebras are ‘almost’ distributive, in a well-defined sense. It is their ‘near’ distributivity that permits the definition of classical probability measures over them, which in many interpreters' eyes is the precondition for adopting an ignorance interpretation of probabilities.

Following on this work, Zimba and Clifton (1998) changed tack a bit, and considered not algebras (or lattices) of projection operators, but algebras of observables. The advantages of this approach are many. First, there is a well-developed theory of operator algebras upon which one can draw. Second, it allows one, in principle, to deal with observables generally, including those that do not have (proper) eigenspaces. Third, it provides a possibly more compelling justification for the kinds of ‘closure’ condition that have been mentioned above.

Zimba and Clifton focus largely on this last issue, considering a
number of closure conditions on the set of definite-valued
observables. For example, should the set be closed under taking real
linear combinations? (In this case, one assumes that a real linear
combination of observables that are definite-valued is itself
definite-valued.) Arbitrary algebraic combinations? Arbitrary
(‘self-adjoint’) functions? Zimba and Clifton prove a
number of interesting results concerning the algebra of
*observables* picked out by modal interpretations. (Their
results are not all applicable to the infinite-dimensional case,
however). Somewhat more precisely, one begins with a quasiBoolean
algebra of projections -- not necessarily one picked out by any of
the prescriptions we have discussed, but just any quasiBoolean
algebra -- and then considers the *observables* that are
definite-valued in virtue of this quasiBoolean algebra's
constituting an algebra of possibly-possessed properties. Following
Zimba and Clifton, let us call such an algebra of observables
**D**. Zimba and Clifton then consider whether there
exist valuations on **D** (i.e., assignments of values
to all observables in **D**) that respect arbitrary
(self-adjoint) functional relationships among the observables in
**D**. That is, letting *v*[**A**]
represent the value of **A** (for **A** in
**D**), and letting *f* be any (self-adjoint)
function, we require that *f*(*v*[**A**])
= *v*[*f*(**A**)]. The answer is
‘yes’. More importantly, they show that there are
sufficiently many such valuations that the quantum-mechanical
probabilities over **D** can be recovered from a
classical probability measure over all such valuations. In other
words, one can understand quantum-mechanical probabilities as
ignorance about which values the observables in **D**
actually have.

The latest installment of this line of reasoning is due to Halvorson and Clifton (1999). They extend results from Zimba and Clifton to the case of unbounded observables. However, there remain open questions about this case.

## 6. Dynamics

As we have seen, modal interpretations propose to provide, for every
moment in time, a set of possibly-possessed properties (or
definite-valued observables) and probabilities for possession of
these properties (or for values of these observables). Some advocates
of modal interpretations may be willing to leave the matter, more or
less, at that. Others take it to be crucial for any modal
interpretation that it also answer questions of the form: Given that
a system possesses property *P* at time *s*, what is
the probability that it will possess property
*P*
at time *t* (*t* > *s*)? In other words,
they want a *dynamics* of possessed properties. (It is clear
for instance that Healey's account requires some such dynamics.)

There are arguments on both sides. Those who consider a dynamics of
possessed properties to be superfluous might ask whether quantum
mechanics could not get away with just single-time probabilities. Why
can we not settle for an interpretation that supplements standard
quantum mechanics *only* by providing in a systematic way a
set (the set of possibly possessed properties) over which its
single-time probabilities are defined? If we require of this set that
it include the everyday properties of macroscopic objects, then what
more do we need? Arguably, Van Fraassen has a similar position,
considering a dynamics of value states to be more than what an
interpretation of quantum mechanics should provide.

Those who argue for the necessity of dynamics reply that we need an
assurance that the *trajectories* of possessed properties are,
at least for macroscopic objects, more or less as we see them to be.
For example, we should require not only that the book at rest on the
desk have a definite location, but also that, if undisturbed, its
location relative to the desk does not change in time. Hence one
cannot get away with simply specifying the definite properties at
each time. We need also to be shown that this specification is at
least *compatible* with a reasonable dynamics. Even better, we
would like to see the dynamics explicitly.

The issue comes down to what one considers to be ‘the phenomena that need saving’ by an interpretation. Those who believe that the phenomena in question include dynamical phenomena will be inclined to search for a dynamics of possessed properties (or definite values). Others might not.

Of course, modal interpretations do admit -- trivially -- an
unreasonable dynamics, namely, one in which there is no correlation
from one time to the next. (In this case, probability of a transition
from the property *P* at s to
*P*
at t is just the single-time probability for
*P*
at *t*.) In such a case, the book on the table might
*not* remain at rest relative to the table, even if
undisturbed. Such dynamics are unlikely to interest those who feel
the need for a dynamics. As we just saw, their motivation is
(probably) to provide an assurance that modal interpretations can
describe the world more or less as we think it is.

Several researchers have contributed to the project of constructing a dynamics for modal interpretations. The most complete account is from Bacciagaluppi and Dickson (1999). That work answers most of the significant challenges facing the construction of a dynamics, though it does leave some important questions open.

The first challenge is posed by the fact that the set of possibly
possessed properties -- let us call it ‘*S*’ -- can
change over time. In other words, the ‘state space’
(*S*) over which we wish to define transition probabilities is
itself time-dependent. The solution is to define a family of maps,
each one being a 1-1 map from *S* at one time to (a
different!) *S* at another time. With such a family of maps,
one can effectively define conditional probabilities within a single
state space, then translate them into ‘transition’
probabilities by means of this family of maps. Of course, for this
technique to work, *S* must have the same cardinality at each
time. In general (for example, in those interpretations that rely on
the spectral-decomposition), it does not. One must, then, 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 the spectral decomposition version of the modal interpretation, 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). This result suggests a natural family of maps as discussed above: map each spectral component at one time to its unique (continuous) evolute 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, then derive the finite-time transition probabilities from them. This problem is central to the theory of stochastic processes. Adapting results from the theory of stochastic processes, one can show that the procedure can, more or less, be carried out for modal interpretations of at least some varieties.

Finally, one must actually define infinitesimal transition probabilities that will give rise to the proper quantum-mechanical probabilities at a time. Following earlier work by Bell (1984) and Vink (1993) and others, Bacciagaluppi and Dickson define in fact an infinite class of such infinitesimal transition probabilities. Some of them might be considered more ‘quantum-mechanical’ than others, but all of them generate the correct single-time probabilities, which are, one might argue, strictly speaking all we can really test. On the other hand, Sudbery (2000) has argued that the form of the transition probabilities would be relevant to the precise form of spontaneous decay or the 'Dehmelt quantum jumps' (otherwise known as 'quantum telegraph' or 'intermittent fluorescence'). Indeed, he independently develops the dynamics of Bacciagaluppi and Dickson and applies it in such a way that with the 'standard' choice of transition probabilities it leads to the correct predictions for these experiments. Such somewhat more unusual experiments may well turn out to be crucial testing grounds for alternative 'interpretations' of quantum mechanics, as emphasised by Shimony (1990).

## 7. Open Projects

A number of open projects and problems face modal interpretations.
Above we saw that the spectral decomposition version is unlikely to
be adequate. The more recent algebraic work abstracts away from
specific choices, but in the end one feels compelled to return to
this issue. Indeed, at the very least one would like to know that
some choice or other can at least capture what we believe to be true
about the world. We have noted a number of theorems of the form
‘the largest set of observables that can be made simultaneously
definite (subject to some conditions) is *S*’ for some
*S*. Must we suppose that nature has been so kind as to make
all statements simultaneously true of her describable in terms of one
of these sets? Without a demonstration that the answer is
(plausibly, at least) ‘yes’, one might wonder.

Other fundamental questions arise, which we have not discussed at all
here. For example, is the very idea behind modal interpretations
reasonable? Some have argued that quantum theory should not be viewed
in terms of ‘operators’ and ‘quantum states’.
They even question the fundamentality of the Hilbert space formalism,
which modal interpretations take quite seriously. For example, Daumer
*et al.* (1996) argue that one should not naïvely take
operators to represent physical quantities (though it is
controversial whether modal interpretations in fact do so, or in any
case do so in the naïve sense that they dislike). On the other
hand, Brown, Suárez, and Bacciagaluppi (1998) argue that there
is *more* to quantum reality than what is usually described by
operators and quantum states: gauges and coordinate systems are
crucial to our description of physical reality, while modal
interpretations have standardly not taken such things into
consideration.

The recent algebraic work is itself a source of several open
questions. Halvorson and Clifton (1999) themselves mention
several. One may also ask more fundamental questions about the
algebraic approach itself. For example, what is the motivation for
the algebraic closure conditions? Do the functional operations
correspond to well-defined empirical operations? If the physical
meaning of the observable **A** is well-understood, do
we thereby understand what *f*(**A**) means? (If
no, then why insist on functional closure? If yes, what does it
mean?)

In the realm of dynamics, Bacciagaluppi and Dickson (1999) raise a number of outstanding questions. In addition to these, the issue whether a dynamics is really needed is still a topic of discussion among researchers. Related to these questions is the question of Lorentz-invariance. Dickson and Clifton (1998) have shown that a large class of modal interpretations cannot be made Lorentz-invariant. Are all modal interpretations subject to this or similar results? If so, what do we make of that situation?

Many more open questions and problems face modal interpretations. Whatever their merits in the end, one can at least say that they have given rise to a serious and fruitful series of investigations into the nature of quantum theory.

## Bibliography

- Albert, D. and Loewer, B. 1990. “Wanted dead or alive: two
attempts to solve Schrödinger's paradox,”
*Proceedings of the PSA 1990*, Fine, A., Forbes, M., and Wessels, L. (eds), Vol. 1, pp. 277-285. East Lansing, Michigan: Philosophy of Science Association. - -----. 1991. “Some alleged solutions to
the measurement problem,”
*Synthese*, 88, 87-98. - Bacciagaluppi, G. 1995. “A Kochen-Specker theorem in the
modal interpretation of quantum mechanics,”
*International Journal of Theoretical Physics*, 34, 1205-1216. - Bacciagaluppi, G. 2000. “Delocalized properties in the modal
interpretation of a continuous model of decoherence”
*Foundations of Physics*, 30, 1431-1444. - -----. forthcoming.
*Modal Interpretations of Quantum Mechanics*. Cambridge: Cambridge University Press. - -----. and Dickson, M. 1999. “Dynamics for modal
interpretations,”
*Foundations of Physics*, 29, 1165-1201. - -----., Donald, M., and Vermaas, P. 1995.
“Continuity and discontinuity of definite properties in the modal
interpretation,”
*Helvetica Physica Acta*, 68, 679-704. - -----. and Hemmo, M. 1996. “Modal interpretations,
decoherence and measurements,”
*Studies in History and Philosophy of Modern Physics*, 27, 239-277. - Bell, J. L. and Clifton, R. 1995. “QuasiBoolean algebras and
simultaneously definite properties in quantum mechanics,”
*International Journal of Theoretical Physics*, 34, 2409-2421. - Bell, J. S. 1984. “Beables for quantum field theory,”
*Speakable and Unspeakable in Quantum Mechanics*(1987), pp. 173-180. Cambridge: Cambridge University Press. - Bene, G. and Dieks, D. 2002. “A perspectival version of the modal interpretation of quantum mechanics and the origin of macroscopic behavior,” Foundations of Physics, 32, 645-671.
- Bohm, D. 1952. “A suggested interpretation of the quantum
theory in terms of ‘hidden’ variables, I and II,”
*Physical Review*, 85, 166-193. - Brown, H., Suárez, M., and Bacciagaluppi, G. 1998.
“Are ‘sharp values’ of observables always objective
elements of reality?”
*The Modal Interpretation of Quantum Mechanics*, Dieks, D. and Vermaas, P. (eds), pp. 69-101. Dordrecht: Kluwer Academic Publishers. - Bub, J. 1997.
*Interpreting the Quantum World*. Cambridge: Cambridge University Press. - Bub, J. and Clifton, R. 1996. “A uniqueness theorem for
interpretations of quantum mechanics,”
*Studies in History and Philosophy of Modern Physics*, 27B, 181-219. - Bub, J., Clifton, R., and Goldstein, S. 2000. “Revised proof
of the uniqueness theorem for ‘no collapse’ interpretations
of quantum mechanics,”
*Studies in History and Philosophy of Modern Physics*, 31B, 95-98. - Clifton, R. 1995a. “Independently motivating the Kochen-Dieks
modal interpretation of quantum mechanics,”
*British Journal for the Philosophy of Science*, 46, 33-57. - -----. 1995b. “Making sense of the Kochen-Dieks
‘no-collapse’ interpretation of quantum mechanics
independent of the measurement problem,”
*Annals of the New York Academy of Science*, 755, 570-578. - -----. 1995c. “Why modal interpretations of quantum
mechanics must abandon classical reasoning about the physical
properties,”
*International Journal of Theoretical Physics*, 34, 1302-1312. - Daumer, M., Dürr, D., Goldstein, S., and Zanghì, N.
1996. “Naïve Realism about Operators,”
*Erkenntnis*, 45, 379-397. - Dickson, M. 1994. “Wavefunction tails in the modal
interpretation,”
*Proceedings of the PSA 1994*, Hull, D., Forbes, M., and Burian, R. (eds), Vol. 1, pp. 366-376. East Lansing, Michigan: Philosophy of Science Association. - -----. 1995a. “Faux-Boolean algebras, classical
probability, and determinism,”
*Foundations of Physics Letters*, 8, 231-242. - Dickson, M. 1995b. “Faux-Boolean algebras and classical
models,”
*Foundations of Physics Letters*, 8, 401-415. - Dickson, M. and Clifton, R. 1998. “Lorentz-invariance in
modal interpretations”
*The Modal Interpretation of Quantum Mechanics*, Dieks, D. and Vermaas, P. (eds), pp. 9-48. Dordrecht: Kluwer Academic Publishers. - Dieks, D. 1988. “The formalism of quantum theory: an
objective description of reality?,”
*Annalen der Physik*, 7, 174-190. - -----. 1989. “Quantum mechanics without the projection
postulate and its realistic interpretation,”
*Foundations of Physics*, 19, 1397-1423. - Donald, M. 1998. “Discontinuity and continuity of definite
properties in the modal interpretation” pp. 213-222 in
*The Modal Interpretation of Quantum Mechanics*, Dieks, D., and Vermaas, P. (eds.). Dordrecht: Kluwer Academic Publishers. - Elby, A. 1993. “Why ‘modal’ interpretations of
quantum mechanics don't solve the measurement problem,”
*Foundations of Physics Letters*, 6, 5-19. - Halvorson, H. and Clifton, R. 1999. “Maximal beable
subalgebras of quantum mechanical observables,”
*International Journal of Theoretical Physics*, 38, 2441-2484. - Healey, R. 1989.
*The Philosophy of Quantum Mechanics: An Interactive Interpretation*. Cambridge: Cambridge University Press. - Kochen, S. 1985. “A new interpretation of quantum
mechanics,” in
*Symposium on the Foundations of Modern Physics 1985*, Mittelstaedt, P. and Lahti, P. (eds), pp. 151-169. Singapore: World Scientific. - Reed, M. and Simon, B. 1979.
*Methods of Modern Mathematical Physics*, Volume 1. San Diego: Academic Press. - Reeder, N. and Clifton, R. 1995. “Uniqueness of prime
factorizations of linear operators in quantum mechanics,”
*Physics Letters A*, 204, 198-204. - Schrödinger, E. 1935. “Discussion of probability
relations between separated systems,”
*Proceedings of the Cambridge Philosophical Society*, 31, 555-563. - Shimony, A. 1990. “Desiderata for a modified quantum dynamics,” PSA 1990, Vol. 2, 49-59. Repr. in Search for a Naturalistic Worldview, Vol. 2, Shimony, A., pp. 55-67. Cambridge: Cambridge University Press.
- Spekkens, R. W. and Sipe, J. E. 2001a. “Non-orthogonal core projectors for modal interpretations of quantum mechanics,” Foundations of Physics, 31, 1403-1430.
- -----. 2001b. “A modal interpretation of quantum mechanics based on a principle of entropy minimization,” Foundations of Physics, 31, 1431-1464.
- Sudbery, A. 2000. “Diese verdammte Quantenspringerei,” quant-ph/0011082.
- van Fraassen, B. 1972. “A formal approach to the philosophy
of science,” in
*Paradigms and Paradoxes: The Philosophical Challenge of the Quantum Domain*, Colodny, R. (ed.), pp. 303-366. Pittsburgh: University of Pittsburgh Press. - -----. 1974. “The Einstein-Podolsky-Rosen paradox,”
*Synthese*, 29, 291-309. - Vermaas, P. and Dieks, D. 1995. “The modal interpretation of
quantum mechanics and its generalization to density operators,”
*Foundations of Physics*, 25, 145-158. - Vermaas, P. 1997. “A no-go theorem for joint property
ascriptions in modal interpretations of quantum mechanics,”
*Physical Review Letters*, 78, 2033-2037. - -----. 1999.
*A Philosopher's Look at Quantum Mechanics: Possibilities and Impossibilities of a Modal Interpretation*. Cambridge: Cambridge University Press. - Vink, J. 1993. “Quantum mechanics in terms of discrete
beables,”
*Physical Review A*, 48, 1808-1818. - Zimba, J. and Clifton, R. 1998. “Valuations on functionally
closed sets of quantum mechanical observables and von Neumann's
‘no-hidden-variables’ theorem,”
*The Modal Interpretation of Quantum Mechanics*, Dieks, D. and Vermaas, P. (eds), pp. 69-101. Dordrecht: Kluwer Academic Publishers.