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6 February
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1995
-33
ELSEVIER
PHYSICS
LETTERS
A
PhysicsLettersA 197 (1995) 367-371
Physical motivation of the modal interpretation
of quantum mechanics
Dennis Dieks zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJ
FounaWions of Science Unit, Vniversiteit
Received 18 July 1994; revised manuscript
Vtrecht, PO. Box 80.000, 3.508 TA Vtrecht, The Netherlands
received 25 November 1994; accepted for publication 12 December 1994
Communicated by P.R. Holland zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON
Abstract
We show that the modal interpretation of quantum mechanics follows from essentially two demands: ( 1) Definite properties
are to be ascribed to physical systems in such a way that the magnitudes which are definite are definable solely from the
quantum state and the structure of Hilbert space; (2) There is a one-to-one relation between properties possessed by a system
and properties of its environment.
1. Introduction
vector of a composite system. Consider such a system &p represented by It,@) E W @ Yip; I-P and
7@ are the Hilbert spaces associated with o and p,
respectively. Consider the bi-orthonormal
decomposition of I@@),
The modal interpretation of quantum mechanics [ 11
is a realist interpretation of the quantum mechanical
formalism. It is realist in the sense that it assigns definite values to a set of magnitudes pertaining to a
physical system. In other words, the interpretation is
(1.1)
about zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
properties of physical systems, not about meaj
surement results. Measurement outcomes are just specific examples of properties; as in the case of pointer
where{h>Ij and{IPj>Ij are subsets of orthonormal
positions, exhibited by a measuring device. A second
bases of 7-P and 7@, respectively. This decomposiimportant feature of the interpretation is that it is eni
tion generates two sets of projectors operating on 7-P
tirely based on the usual formalism of quantum meandonWs,respectively:
{~~~)(~~J}j~d{~~j)(~j~}~~
chanics - nothing is added to the mathematical strucIf there is no degeneracy among the numbers { Icj 12}
ture. The new aspect of the interpretation is exactly
these sets of one-dimensional
projectors are uniquely
that this well-known structure is not regarded merely
determined by the decomposition.
If degeneracy ocas a codification of possible measurement results and
curs this is no longer so; but the projectors belongtheir probabilities,
but is also considered to contain
ing to one value of { ]cj12} can be added to form a
information about physical properties and the probamulti-dimensional
projector. In this way a new set of
bilities of their presence.
projectors, including multi-dimensional
ones, can be
The property
ascription
is based on the bidefined. This set is again uniquely determined by the
orthonormal
(Schmidt)
decomposition
of the state
bi-orthonormal decomposition.
IS”‘>=~cjlaj)lP.i)~
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368
D. Dieks /Physics
Letters A 197 (I 995) 367-371
The modal interpretation assigns definite values to
the same set of properties cannot be definite in all cirthe physical magnitudes represented by the projectors
cumstances (as will turn out to be the case): it will
in the just-described
sets. Also all functions of them
depend on the entangled state of system plus environare taken to represent magnitudes with definite valment which magnitudes become definite.
ues (“well-defined”
or “applicable” physical magniThe charge has been leveled that the modal intertudes). zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Which value of all possible values of a given
pretation’s way of determining the set of definite magapplicable magnitude is actually realized is not fixed
nitudes is ad hoc. Why for example, so the objection
by the interpretation: the interpretation is probabilistic.
goes, concentrate on the bi-orthonormal
decomposiFor each possible value of an applicable magnitude a
tion and not on one of the countless other possible
probability is specified. The probability that the magrepresentations of the total state vector? Our result can
nitude represented by 1cq) (q 1 (and also Ipi) (pi I) has
be seen as a justification of the modal prescription. Inthe value 1 is given by Ici12. In the case of degenerdeed, the first of the above requirements leading to the
acy it is stipulated that the magnitude represented by
modal imerpretation seems natural in giving any interCiEh IczJ(czil has value 1 with probability CiEll Ici12
pretation to the mathematical formalism of quantum
(Zl is an index-set containing indices j, k such that zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG
mechanics without adding anything to the mathematical structure. The second requirement can be seen as
lC j12
= IC k12).
The above prescription for finding the definite quana way to generalize (and make rigorous) a significant
tities pertaining to a system can also be given in a form
part of Bohr’s interpretation of quantum mechanics.
in which the Schmidt decomposition and the state vecAccording to Bohr the applicability of concepts detor of the composite system do not occur explicitly
pends on the type of macroscopic measuring device
[4] : the definite properties of a system can be dethat is present; given a “phenomenon” there is a onetermined from the spectral resolution of the reduced
to-one correspondence between the properties of the
density operator of the system alone.
measuring device and those of the object system. Our
Physical magnitudes are represented in the usual
second requirement implements this idea also in sitway by operators in Hilbert space. There is conseuations in which there is no macroscopic measuring
quently no contextualism
in the sense that the repdevice, but only a correlation with the (possibly miresentation of physical magnitudes by operators decroscopic) environment.
pends on the context. However, there is contextualThe idea that there is a correspondence
between
ism in another sense: the set of definite properties
properties of a system and those of its environment
is also physically motivated by other approaches to
picked out depends on how the system under considthe interpretation of quantum mechanics, especially
eration is “entangled” with its surroundings. In fact,
we shall demonstrate that the modal interpretation is
the decoherence approach (the “monitoring” of a systhe only possible realist interpretation satisfying estem by its environment, see Ref. [5] and references
therein).
sentially two demands, one of which embodies this
“entanglement”.
However, it should be noted that the modal interThe first demand is that the Hilbert space formalpretation can also be justified in another way, indeism, with the usual representation of physical magnipendent of the second of the above requirements. As
shown by Clifton [3] (see also Ref. [2] ) the modal
tudes by observables, should be completely respected;
no additional structure should be used in defining the
interpretation is uniquely determined by a number of
applicable physical magnitudes (it seems appropriate
natural assumptions about the structure of the set of
definite properties which is definable from the statisto call this a “no-hidden-variables”
demand). We will
tical operator W pertaining to a system. Our alternatake this requirement to mean that each individual deftive approach is meant to highlight the specific way
inite magnitude should be definable from the quantum
in which the modal interpretation is contextual, and to
state and the structure of Hilbert space alone.
show that it is the natural realist interpretation to be
The second requirement says that there should be
combined with decoherence or “Copenhagen” ideas.
a one-to-one correlation between the definite properties of a system and the definite properties of its environment. This will lead to a type of contextuality if
D. Dieks / Physics Letters A 197 (1995) 367-371
2. The problem
369 zyxwvut
that I$) represents a composite system, we can take it
that the Hilbert space has a dimension greater than 2.
In that case Gleason’s theorem says that every probability measure on the closed subspaces of Hilbert space
has the form p(P) = Tr WP, with P a projection operator (p(P)
is the probability assigned to the subspace on which P projects) and W a semi-definitely
positive Hermitian operator with trace 1. Now, the requirement of definability in terms of I+) has the consequence that all probabilities should remain the same
under any unitary transformation of zyxwvutsrqponmlkjihgf
W that leaves I$)
invariant. That means that W itself must be invariant
under any such transformation. The subspaces of the
Hilbert space on which the projection operators in the
spectral resolution of W project should consequently
also be invariant. However, the group of unitary transformations which leave I+) invariant has only two invariant subspaces: the one-dimensional
subspace ‘FI+
spanned by ]I+%)and the subspace 7-$ orthogonal to
I@). It follows that W must have the following form:
W = qP+ + c2Pi/rn, with P$ and Pk the projection
Every realist interpretation of quantum mechanics
has to specify which physical magnitudes possess definite values. Our aim is to formulate a rule for such a
specification which is based on the standard formalism of quantum mechanics alone. In line with this we
assume that all physical magnitudes are represented
in the standard way by Hermitian operators on Hilbert
space. Not all physical magnitudes thus represented
can simultaneously possess definite values: that would
lead to inconsistencies,
for example via Gleason’s theorem. Therefore, a selection has to be made from the
set of all magnitudes. Our objective is to define the
definite physical magnitudes from the state vector and
the structure of Hilbert space.
In our argument we will make use of probabilities
defined on Hilbert space. In accordance with an unproblematical and generally accepted part of the standard interpretation of quantum mechanics, we assume
that a ket I$) induces the usual probability measure
on the (closed) subspaces of Hilbert space. This is
operators on 7-L*and 7-L;, respectively, m the number
equivalent to assuming that all values of all physical
of dimensions of Xi, and with ct + c:! = 1, cl, c2 >
magnitudes are assigned a probability in the situation
0. This general form of W is sufficient for our further
described by I$). As already stated, we do not wish
argument.
to be committed to the position that all physical magnitudes actually possess a value in that situation - it
3. Determination of the physical magnitudes that
should be stressed that our assumption about the expossess a definite value
istence of a probability measure does not imply that
position. We can just follow the standard interpretaOur second requirement directs our attention to systion, namely that the probabilities pertain to the outtems plus environment. We therefore consider a quancomes of possible ideal measurements on the system
tum state I$@) of a composite system, an “object”
described by the ket I+). When, at a later stage, we will
have determined the restricted set of magnitudes that
system (Yand its environment p. (This will also cover
the generalized formulation of the modal interpretado possess definite values in the. situation described
tion discussed in Ref. 141, as long as a pure state is
by I+), the probabilities of the values of this special
assumed to apply at some level of “compositeness”.)
set zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
will be interpreted as probabilities of a value actul
To determine the definite properties of our system a,
ally being there (for those properties that were already
we look for projection operators Pa that are candidefinite before the measurement, ideal measurements
dates for representing such properties. Because our
will just reveal the pre-existing values).
aim is to determine a description of our physical sysIn fact, a weaker assumption would suffice for our
tem
which is as “fine-grained” as possible, we are insubsequent determination
of the set of definite magterested
in projection operators with a minimum numnitudes. We could just demand that a probability disber of dimensions - preferably one-dimensional
protribution on Hilbert space is induced by I$), without
jectors. Because a and p occur in the formalism in
specifying its form. The first of our above main requirements then says that the probabilities should be
a completely symmetrical way, there should also be
projection operators Pp representing properties of the
definable in terms of I$) and the structure of Hilbert
environment /3.
space only. Because we will be considering the case
370
D. Dieks /Phy sics Letters A 197 (1995) 367- 371
Making the assumption of the previous section, we
pend on whether all coefficients in the bi-orthonormal
find that the probability of joint occurrence (Pa and zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
decomposition of Iqafi) are unique or whether degenPp both taking the value 1) can be calculated as
eracies occur. To show this, we make use of the following. If an applicable physical magnitude can be de(V?P”P%@V
( or, with the more general form
of W: Tr(WPaPp)
= zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
c~(s,b” filPaP~lqV@) + cz( 1 fined solely from I@@) and the structure of the Hilbert
space
3-1”@ 7-@, all transformations which leave these
(V?PcuP%@))).
We now make use of the idea that there is a oneelements invariant also leave the magnitude in questo-one correlation between the properties of the systion invariant. All applicable magnitudes should theretem and its environment. We express this demand as
fore be invariant under all unitary transformations
of
follows: P(P;)
= P(Pf)
= P(PfPf)
for two corthe form Un @ Up that leave I+@) invariant (Ua is
a unitary operator in 7-i*, Up a unitary operator in
related properties represented by Pf and Pf, respec7#). To determine the class of such operators, apply
tively. It follows, if we use the above expression for
u= @ up to IQ. (l.l),
joint probabilities
(either the usual one or the one
in terms of W), that (+I”plPfPkpI$@)
= (+I”fljPt @
lly@) = (+@I1123
P,Pl$,nP). Write P; = ~ao)(~l
U” @ UpI+ap) = CCj
j
U” lfXj) @ U’ lpj).
(3.1)
and P,f = Ibo)(bol, and I@)
= zijcijlai)Ibj),
with
{IQ)} and { lbj)} orthogonal bases in 7-f” and HP, reBy virtue of the unitarity of U” and U* the right-hand
spectively. The correlation condition yields [coal2 =
side of Eq. (3.1) is again a bi-orthonormal decompoCjJCjO12, SO CjfJ= COj = 0 ifj
# 0. We
zjlCO j12 = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
sition. If U” @3Up]@@) = I@@) it follows from the
find therefore that either P (PF) = P( Pkp) = 0, or
uniqueness properties of the bi-orthonormal
decom11/I@) = clac)Ibc) + c’l&@) with lc12 > 0 and with
position that Vlaj)
= Iaj) for each j belonging to a
Ic/@) a ket whose decomposition in the product basis
value of Icj12 which occurs only once in the decomonly contains components in which
{l”i>l~j)}
O f zap
position. However, if lck12 = (cl12 for k,Z E Zm, with
j a~) or Ibo) do not occur. The ket IqF@) of course has
Z, an index-set, the restriction of U” to the subspace
a bi-orthogonal decomposition; from the above it folspanned by IaJ, i E Zmr can be an arbitrary unitary
lows that all kets from 3-1” occurring in this decomoperator.
position are orthogonal to I%), and all kets from tip
It follows that the one-dimensional
projectors assoorthogonal to Ibo). The bi-orthogonal decomposition
ciated with non-degenerate values of Icj12 are invariof I@p), completed with the term c IUO)Ibo), therefore
ant under the unitary transformations
leaving I@@)
is a bi-orthogonal
decomposiEion of I$@). In other
invariant. However, in the case of degeneracy only the
words, the term lua)lbu) occurs in the bi-orthogonal
multi-dimensional
projector on the subspace spanned
decomposition of I@@). Repeating this argument for
by IQJ, corresponding to one single value of IcJ2, is
other possible definite projectors, we find that all oneinvariant. All smaller projectors will change under the
dimensional projection operators that may represent
transformations Ua .
definite properties, with a non-zero probability of takSo we have now found the one-dimensional
projecing the value 1, project on mutually orthogonal kets
tion operators corresponding to the terms in the biIaj), and lpi), that occur in the decomposition
of
orthonormal decomposition with unique coefficients,
I@“P) in the way shown in Eq. ( 1.1) .
and the multi-dimensional
projection operators proAs a result, of all one-dimensional
projectors only
jecting on the subspaces associated with the terms with
the ones which project on kets which occur in the
equal coefficients, as the “finest-grained”
candidates
bi-orthogonal decomposition of IJ,@), with non-zero
for applicable physical magnitudes. Conversely, it is
coefficients, are candidates for representing definite
clear that these operators indeed satisfy our correlamagnitudes with a non-zero probability of being 1;
tion requirement and can be individually defined in
conversely, it is clear that all such projectors satisfy
terms of I$“fi) and the structure of the Hilbert space
our correlation requirement.
7-Z”@ tip alone - they are the projectors occurring in
But are all these one-dimensional
projection operathe spectral resolution of the reduced state W” (obtors in fact definable from \lfi@)? This turns out to detained from Ifi@) by partial tracing). These projec-
D. Dieks /Phy sics Letters A 197 (1995) 367- 371
371
tom can be ranked (and thus individually identified)
this property ascription the bi-orthonormal
decompoby the associated values of Icj12.
sition of the state vector of a composite system plays
Finally, straightforward repetition of the correlation
a crucial role. It follows from the above that this role
argument for higher-dimensional
projectors shows that
of the bi-orthonormal decomposition (or, equivalently,
the only projectors pertaining to a and p, respectively,
the spectral resolution of the reduced density operaand possessing a l-l correlation are sums of the protor) need not be postulated in an ad hoc way, but can
jectors we already have found; this set is therefore
be derived from physically motivated requirements.
complete.
We still have to discuss the case in which the probAcknowledgement
ability of a projection operator taking the value 1 is
zero. Here the situation is the same as in the case of
I am much indebted to Guido Bacciagaluppi, Tim
degeneracy we have just discussed. The set of all uniBudden,
Pieter Vermaas and especially Rob Clifton
tary transformations
leaving leap) invariant, and refor
their
suggestions
and criticism of earlier versions
specting the factorisation of 1-l”s into ‘FI” and ‘7-f@,
of
this
paper.
includes arbitrary transformations
U” and Up in the
null-spaces of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
W” and Wp, respectively. Only the operators projecting on the whole of these null-spaces
References
can therefore represent definite properties ’ .
[ 11 D. Dieks, Phys. Rev. A 49 (1994) 2290, and references
therein,
[2] J. Bub, On the structure of quanta1 proposition systems, to
be published in Found. Phys. (1994).
[3] R. Clifton, Independently motivating the Kochen-Dieks
modal interpretation of quantum mechanics, to be published
4. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Conclusion
in Brit. J. Philos. Sci. ( 1994).
[4] P.R. Vermaas, and D. Dieks, The modal interpretation
of quantum mechanics and its generalization to density
We have shown that the property ascription furoperators, to be published in Found. Phys. ( 1994).
nished by the modal interpretation is the only one that
[5] W.H. Zumk, and J.P. Paz, Decoherence, chaos, the quantum
solely relies on the Hilbert space structure and the
and the classical, in: Symposium on the foundations of
state vector in assigning properties that are in one-tomodem physics, eds. l? Busch, P. Lahti and I? Mittelstaedt
one relation to the properties of the environment. In
(World Scientific, Singapore, 1993).
1This is in agreement with Ref. [ 11, and deviates slightly from
the proposal by Clifton [ 31. Clifton takes all one-dimensional
projectors in the null-spaces of Wn and Wp as definite, on the
grounds that their definiteness best explains the fact that the value 0
will be found with certainty in a measurement
of these observables.
It seems, however, that the fact that the “bigger” projector on the
total null-space has the value 0 with certainty can serve the same
explanatory purpose. For example, if a system with certainty has
the property that its position is outside a given region, this by itself
seems sufficient to explain that a measurement which is sensitive
to individual positions inside that region will with certainty give
a null-result.