Possible Worlds in the Modal Interpretation
Author(s): Meir Hemmo
Source: Philosophy of Science, Vol. 63, Supplement. Proceedings of the 1996 Biennial Meetings
of the Philosophy of Science Association. Part I: Contributed Papers (Sep., 1996), pp. S330-S337
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POSSIBLE WORLDS IN THE MODAL INTERPRETATION
MEIR HEMMOt:
CambridgeUniversity
An outlinefor a modalinterpretation
in termsof possibleworldsis presented.
The so-calledSchmidthistoriesaretakento correspondto the physicallypossible
worlds.Thedecoherencefunctiondefinedin the historiesformulationof quantum
theoryis takento prescribea non-classicalprobabilitymeasureoverthe set of the
possibleworlds.This is shownto yield dynamicsin the form of transitionprobabilitiesfor occurrenteventsin eachworld.The role of the consistencycondition
is discussed.
1. Introduction.In the modal interpretation' of quantum theory, the Schrodinger
equation is taken to describe in full the time evolution of the quantum state of any
closed system. No collapse of the state or state vector reduction is required. It is
assumed that at any time t there is a preferred set of states defining which states
are physically accessible at t to every subsystem of the universe. This set is determined by the projections P,(t)p(t) of the universal state p(t) appearing in the diagonal form of the reduced density operator of each subsystem (call these projections Schmidt events). It is further assumed that the probability at each time t that
one of the Schmidt events is occurrent at t is equal to
Tr(p(t)Pi(t)).
(1)
This probability measure is instantaneous since it is defined for single moments
of time. However, a description of a system at more than one time requires a notion
of a history, that is a time-ordered sequence of the Schmidt events, and a definition
of a joint probability distribution for such histories. In the absence of general multitime probabilities, (1) seems to correspond to an instantaneous chance process in
which one event amongst the physically possible events is selected at each moment
of time independently of the selections in the past.2 There are some proposals for
multi-time probabilities (Vermaas 1995, Bacciagaluppi and Dickson 1995) which
make the modal interpretation a prototype of a stochastic hidden variables theory
(Bub 1994, Bacciagaluppi and Hemmo 1995).
In this paper I want to propose a different approach. I shall outline a generalization of the modal interpretation that includes histories and multi-time probabilities as defined in the consistent histories formulation of quantum theory.3 This
generalization will be given in terms of possible worlds in the sense used for intl would like to thank Jeremy Butterfield,Itamar Pitowsky, Pieter Vermaas, and especially,
Guido Bacciagaluppifor many discussionsand valuablecomments.
IDepartmentof Philosophy,CambridgeUniversity,Wolfson College,Cambridge,EnglandCB3
9BB.
'The presentationof the modal interpretationis not meant to be exhaustive.See Vermaasand
Dieks 1995 for a generalizedversion and references.
2Forwhy this is wanting, see Bell 1981, Hartle 1989, Maudlin 1994.
3SeeGell-Mannand Hartle 1993 for one version of the consistenthistoriesformulationand for
references.
Philosophyof Science,63 (Proceedings)
pp. S330-S337.0031-8248/96/63supp-0039$2.00
Copyright1996by the Philosophyof ScienceAssociation.All rightsreserved.
S330
POSSIBLE WORLDS IN THE MODAL INTERPRETATION
S331
stance by Lewis (1986). The framework advanced here is close in some respects to
ideas suggested by Zeh (1973), Healey (1984), Kochen (1985) and Deutsch (1985).
The paper is structured as follows. In section 2, I summarize basic ideas in the
consistent histories formulation. In section 3, I define the notion of a possible world
and present the possible worlds version of the modal interpretation. Some problems and consequences are discussed in Sections 4 and 5.
2. Quantum Histories. In the consistent histories formulation a quantum history
is defined to be a finite time-ordered and discrete sequence of arbitraryHeisenberg
projections:
, = (p : P,(t ),..., Pi,(t)),
(2)
where p is the initial state. The string of times t, < ...
< t, is called the temporal
support of the history. If a quantum history has a probability, then its probability
is defined to be
Prob,(o)
= Tr(P,(t)
. . . P,(t)pP,(t)
. . . Pi(tn)).
(3)
This probability generalizes the standard Born rule which is taken to specify the
probability of histories consisting of a single event.
The fundamental axiom of the consistent histories formulation defines the dynamical conditions under which it is meaningful to assign a probability to a quantum history. A history in a set of alternative histories is assigned a probability
provided the histories in the set satisfy a consistency (or decoherence) condition. I
shall use what Gell-Mann and Hartle (1993) call medium decoherence (call it the
consistency condition). A set of histories is said to be consistent if
D,pH(J, a,) = Tr(P,,(t)
. . . P,(t,)pPi(t)
. . . P,i(t,)) = 0,
(4)
for any ik -t ik (where H is the global Hamiltonian). If p is pure, this condition is
equivalent to the lack of interference between the alternative histories in the set.
Note that the consistency condition encodes the information about both the
initial state p and the Hamiltonian H: in this sense, consistency is a dynamical
condition on a set of histories. In the consistent histories formulation (4) functions
as a condition for selection of preferred sets of histories to which probabilities can
be assigned.
The crucial consequence of (4) is that it implies that the probabilities assigned
to the histories satisfy all probability sum rules of standard probability theory
(Hartle 1989, Gell-Mann and Hartle 1993). For example, if a set of histories is
consistent, then the sum rule eliminating all projections at one time
,i Prob,(P,(t),
...,
P,(t), ...
P,i(t))
= Probp(P(t1) . . . Pij_,(tj_), Pi+,(tj+1),.... Pi,(t))
(5)
is satisfied. In fact, (4) is taken as a constraint on the assignment of probabilities
to histories in a given set, precisely because satisfaction of the probability sum
rules is supposed to be a necessary condition for interpreting (3) as a probability
function.
I see two problems emerging from the consistency condition (4) in the context
of a many (or possible) worlds interpretation. First, (4) is too weak, if it is supposed
to be a sufficient condition for selecting preferred sets of histories. There are infinitely many sets of histories for which (4) is satisfied, but which are inequivalent
in the sense that the projections in one set will in general correspond to arbitrary
superpositions of the projections in another set. Also, many of these sets need not
S332
MEIRHEMMO
correspond to the variables that typically decohere in standard models of environment-induced decoherence (see Zurek 1993).
Second, (4) is too strong, if it is taken to be a necessary condition for probability
assignment. Physical processes of environment-induced decoherence that are supposed to induce consistency of histories typically yield only approximate consistency.4 In these processes the probability sum rules will not be exactly satisfied.
But then, the numbers attached by (3) to the histories cannot be meaningfully
interpreted as probabilities. At best, (3) may be taken to represent approximate
probabilities.
3. The modal interpretationas a theory about possible worlds. I shall now present
a generalization of the modal interpretation to quantum histories. I shall use specific sets of histories called Schmidt histories. A Schmidt history is a history that
contains at its temporal support projections of the universal state p(t) onto the
states that diagonalize the reduced density operators of the systems defining the
history.5 I assume that the universal state p(t) is pure IP(t))(P(t)l. The axioms are
the following.
(A) Preferredhistories. All and only Schmidt histories correspond to the physically possible worlds. A Schmidt history
, = (p: P,,(t1), P(t2),
...
Pk(tk)),
(6)
specifies the possible world in which at all t < t, the true events are given (in
the Schr6dinger picture) by the projections onto the time-dependent universal
state IT(t)). At the times ti - t, the true events are determined by the projections of IT(t)) onto the corresponding Schmidt subspaces P,(t). At the intermediate times between ti and ti 1 the true events are given by the Schrodinger
evolutes of the projected state P,(t)lT(ti)).
Thus every Schmidt history describes infull a possible world.6 By construction,
there is a one-to-one correspondence between any Schmidt history and a possible
world, so that the set of all the physically possible worlds is given by the set of all
Schmidt histories. This defines an infinite class of physically possible worlds.7
(B) Multi-time probabilities. The probability of a possible world described by
a Schmidt history ai is given by the Gell-Mann and Hartle probability function:
. . . Pi(tk)).
(7)
Probp(ro)= Tr(Pk(t) . . . P,(t1)pP,(t)
This defines a probability measure over the set of the possible worlds that is
non-classical, since every Schmidt history is assigned a probability irrespective of
whether the history is contained in a consistent set of histories (see Section 4). In
this I rely on a strategy suggested by Hemmo and Bacciagaluppi (1995) in the
context of the many worlds interpretation. As in the standard modal interpreta-
4See Hartle 1989, Zurek 1993 for derivationsof consistencyfrom environment-induceddecoherencein measurementsituations.
5Propertiesof Schmidthistoriesare describedin Albrecht 1993, Kent 1995, Bacciagaluppiand
Hemmo 1995b,AppendixA).
6Ifp(t) is a mixed state, the interpretationof a history might be different,but this depends on
the meaningof assigninga mixed state to the whole universe.
7Notice that every sequence of Schmidt projections(in any factorization)and every string of
times are allowed.
POSSIBLE WORLDS IN THE MODAL INTERPRETATION
S333
tion, one may take the probability function (7) to correspond to a genuine chance
process in which one history in a given set of histories becomes occurrent. But,
there might be different interpretations of what this selection means.
One can derive now the conditional (or transition) probability of occurrent
events within each world. Take a world described by the history ai =(
p. Pi,(t1) ...
Pi(tk)).
The probability at time tk of an event P(t) occurring at time t
> t, conditional upon the past history ai is
Ch tk()
))
Tr(P(t)P(tk)
* . Pi,(t)pPi,(t1)
.
. . .
Pik(t)P(t))
Tr(Pik(tk)... Pi,(tI)pPi,(t) ... Pik(tk)
This function may be taken to define the chance of an event in a possible world
given the world's history, and it can be trivially generalized to define the chance
of past and future histories. Notice that (8) is the usual conditional probability
prescribed in the histories formulation for prediction. Similarly, one can define
retrodiction.
I shall take the probability function (8) to be the fundamental law describing
the time evolution of the occurrent events within every possible world. This law is
formally a mixture of the Schrodinger equation and of the collapse law of standard
quantum theory. The projections Pi(t~)p(ti) correspond to the events that are in
general chancy in the world's history. At the intermediate times between ti and ti+I,
the true events are the Schr6dinger evolutes of the P(t)p (t), so the chance specified
by (8) reduces to deterministic (probability zero and one) transitions. Thus, we
have a full description of the behavior in time of the events occurring in each
possible world.
Notice that in this framework the universal state p(ti) determines both which
are the occurrent events in each world (via the Schmidt projections) and what are
their probabilities (via (7)). But, as in the standard modal interpretation, there is
no collapse of the state. This is manifested in the fact that sets of Schmidt histories
may exhibit interference.
J(tk
4. Discussion. I will discuss now two consequences which might be seen as problematic. The first concerns the inconsistency of general sets of Schmidt histories.
The second involves a certain arbitrarinessin the way Schmidt histories have been
taken to fix the possible worlds (Axiom (A)). I start with the inconsistency.
Axiom (B) assigns a probability via (7) to every possible world. This is independent of the consistency of the set in which the history is embedded. General
sets of Schmidt histories significantly violate the consistency condition (4) (Albrecht 1993, Kent 1995). This means that the multi-time probability (7) (and the
induced chance (8)) will in general violate the probability sum rules of standard
probability theory. Notice that the multi-time probabilities (7) will have as marginals the single-time probabilities (1) only for sets of Schmidt histories that are
consistent. In spite of these non-classical features, I will now argue that (7) can be
taken to define a probability measure over the worlds.
The function (7) defines a measure that is positive and generally non-normalizable. However, (7) is additive on disjoint sets of worlds, whereas additivity of the
measure in the sense of the sum rules is not needed. To see why, notice that Axiom
(A) takes a Schmidt history to be a primitive corresponding uniquely to a possible
world. In particular, a history consisting of sums of projections does not specify
the union of the corresponding worlds: the quantumlogical disjunctioncorresponds
to a single world and not to the set theoretic union of the corresponding sets of
worlds. This means that by allowing violations of the sum rules one merely denies
S334
MEIR HEMMO
that the logical structure across the worlds is described by quantum logic (though
quantum logic is valid within a world): which is a point one can live with, to say
the least. One can interpretthe non-classicality of the measure (7) as entirely emerging from its non-normalizability. Consistency of histories need not be relevant.
(See Hemmo and Bacciagaluppi 1995, Hemmo 1996 for more detailed discussions
of what violations of the sum rules might mean.)
Consider for example the violation of the sum rule (5) in the specific context of
the modal interpretation. Take a generic Schmidt history
ai = (p: (Pi,(tl)
* ., Pis-_(t- 1), Pi,+l(tj+
* . . , Pin(t))
(9)
describing a world in which the true events at time tj are the time-evolved projections of P _,(t-_i)p(tj-_). The history ai labels a single world, not the quantum
logical disjunction of the alternative projections Pi(tj) at the omitted time tj. In
general, at time tj each of the projections onto the Pi(t) is completely different
from the time evolute of P_ ,(tj_i)p(tj_l). The projection onto Pi(tj) is a Schmidt
event, whereas the time evolute of P _,(tj_ )p(tj_1)need not be so, and will in general
correspond to an arbitrary superposition of P(t). So from the point of view of
the modal interpretation the probability sum rules are irrelevant.8A similar argument applies with respect to the violations of the marginal probability rule.
Thus I argue that the consistency condition (4), taken as an a priori constraint
on the assignment of probabilities to histories, is unjustified in a possible worlds
framework. The arguments above show that it is possible to take (4) to define a
multi-time probability measure over the possible worlds, so that any world labeled
by a Schmidt history is assigned a probability regardless of (the degree of) the
inconsistency of the set in which the history is embedded. This has the advantage
that approximate consistency and approximate probabilities play no fundamental
role.
Furthermore, although in principle violations of the sum rules are allowed, they
will not be observed in the statistics of the results of ordinary quantum mechanical
measurements. The standard modal interpretation can be shown to recover the
Born probabilities as two-time transition probabilities between measurement outcomes (Bacciagaluppi and Hemmo 1995; 1996, Appendix A). Since the Gell-Mann
and Hurtle probabilities correspond to a sequential application of the Born rule,
this result can be adapted to show that sets of Schmidt histories representing sequences of measurement outcomes form approximately consistent sets of histories.
This has two consequences. First, one can take the approximate consistency as
an empirical explanation of why we have the impression that the sum rules of
standard probability theory are never violated, even though, in general, they are.
Second, this result has a straightforward application to the question of relative
frequencies within a world. Namely, the relative frequencies of sequences of measurement outcomes in the corresponding Schmidt worlds will approximately coincide with the predictions of standard quantum theory (with the collapse postulate), except for a set of worlds with measure zero.
Finally, I turn to the problem of arbitrariness. Consider the correspondence
between a Schmidt history and a possible world (Axiom (A)). A Schmidt history
ai is taken to describe a single world in which at the times t1 < . .. < tk the Pi,(t),
P,(t2) ..., Pik(tk)are, by construction, Schmidt events. These are, in general,
8A special case in which the sum rule (5) is satisfiedis when the probabilityof the Pij(t) will
reduce to zero or one. In this case, however, the projectionsonto the Pi(t,) will coincide with the
deterministic evolutes of Pj_ (tP_-)p(t- 1).
POSSIBLE WORLDS IN THE MODAL INTERPRETATION
S335
chance events. At the intermediate times, between tj and t+ 1,the true events in the
world are represented by the Schrodinger evolutes of the Pi(t). These are the deterministic events in the world. However, in general, for any intermediate time t
t tj these time-evolved events will not coincide with the Schmidt events at t. Furthermore, for any time t # tj, they may correspond to arbitrary superpositions of
the Schmidt events at this time.9 This seems to undermine the idea that Schmidt
histories are physically preferred.
To minimize the arbitrariness,I shall rely on Kochen's (1985) idea of witnessing.
Take a universe c4factorized as l ?0 // in a pure state IT(t))('(t)l
JI(t)) =
A,(t)lVi(t))? JDi(t)),
(10)
and suppose that the expansion (10) is the Schmidt form. This expansion defines
a symmetry between the possible properties lqi(t)) of,/, and the possible properties
1i,(t)) of /2which Kochen denotes by witnessing. Kochen's witnessing can be
interpreted as defining via the one-to-one correlation in (10) the conditions under
which each of the alternatives l,i(t)) becomes a memory (or a record) of the corresponding |(1i(t)),and vice versa.
Witnessing plays a central role in Kochen's interpretation in defining the sets of
states that are physically possible. Those states (and their memories) are in fact
identified with the branches |Vi(t)) (2) I(Ii(t))defined by (10), which are thus given
an independent existence. However, since these branches are defined only at single
moments of time, Kochen's notion of witnessing is instantaneous. Thus it is not
suitable to define the conditions under which memories emerge and persist in time
which, I take it, require certain dynamical constraints of stability over time.
In this framework, it is natural to formulate stability conditions in terms of noreinterferenceof the corresponding branches. That is, I shall require that the oI)(t))
will be memories of the IVi(t))only if the branches IVi(t))(? JI)i(t))do not reinterfere
as (10) evolves in time. This condition can be generalized to include sequences of
states that would be simultaneous memories of full history, as long as the histories
do not reinterfere. In fact, this condition (call it the witnessing condition) can be
relaxed by requiring only approximate no-reinterference between the histories
(Hemmo and Bacciagaluppi 1995, Hemmo 1996).
The witnessing condition has two important consequences. First, as long as
interference between the branches remains negligible, the Schrodinger evolutes at
the intermediate times t will approximately coincide with the projections that
would be obtained upon evolving the total state according to the Schrodinger
equation until time t and then taking the Schmidt projections at t. In this case the
intermediate events will be approximately given by the Schmidt projections at the
intermediate times.
Second, if the witnessing condition is satisfied, then there exist in the present
memories of past events, and then it should be possible to make reliable retrodictions of the events in the past on the basis of measurements in the present. GellMann and Hartle (1993) (and Halliwell 1993) prove that the existence of (alternative) memories of the (alternative) events in a set of histories is equivalentto the
consistency of the set. In fact, for a pure universal state l1(t)) the consistency of a
set of histories is equivalent to the vanishing of interference terms between the
different branches of 1@(t))corresponding to the alternative histories in the set.
Similarly, if the witnessing condition does not rule out negligible interference be9However,by construction,the intermediateeventsare one-to-onecorrelatedwith some Schmidt
projections.
S336
MEIRHEMMO
tween the branches, then the existence of memories will be equivalent to approximate consistency of the corresponding set of histories.
5. Conclusion. There are two ways in which one could interpret the witnessing
condition. One way is to take witnessing literally as a condition for the existence
of memories in the possible worlds. On this view, every Schmidt history will be
assigned a probability regardless of witnessing or consistency. Approximate consistency will be a dynamically emerging property of sets of worlds that contain
memories. Further, worlds in which the Schrodinger evolution yields events that
significantly differ from the Schmidt events will be necessarily restricted to sets of
histories which have no memories. In this sense, these worlds will contain events
that are unobservable.
Alternatively, one can cut off these unobservable events from the ontology by
taking the witnessing condition as a fundamental constraint on the set of the physically possible worlds. This view is more similar to the Everett interpretation
(Deutsch 1985, Saunders 1993). Then, the set of the physically possible worlds will
be reduced to only include worlds labeled by Schmidt histories which in addition
contain memories of these histories. Approximate consistency will be given again
a fundamental role, but now it might not be problematic.
To sum up: obviously, I have left many open questions. But I hope to have
convinced you that histories and multi-time probabilities as defined in the histories
formulation may be put to use in the modal interpretation. Insofar as the physics
is concerned, no extra structure in addition to the Schrodinger equation must be
used, except for the restriction to a preferred set of histories. This might have an
advantage in the as yet unsuccessful attempts to generalize the modal interpretation to a relativistic theory.
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