Modal Interpretation of Quantum Mechanics
MODAL INTERPRETATION OF QUANTUM MECHANICS
The term modal interpretation is ambiguous. It is a proper name that refers to a number of particular interpretations of quantum mechanics. And it is a term that singles out a class of conceptually similar interpretations, which includes proposals that are not generally referred to as modal ones.
This ambiguity was already present when Bas C. van Fraassen coined the term in the 1970s by transposing the semantic analysis of modal logics to quantum logic. The resulting modal interpretation of quantum logic defined a class of interpretations of quantum mechanics, of which van Fraassen developed one instance in detail, called the Copenhagen modal interpretation. In the 1980s Simon Kochen and Dennis Dieks developed independently an interpretation of quantum mechanics that became known as the modal interpretation, turning the term into a proper name. In the 1990s further research produced new proposals, broadening attention to the class of modal interpretations.
The development of modal interpretations can be positioned as attempts to understand quantum mechanics as a theory according to which some but not all observables of physical systems have definite values. Quantum mechanics predicts the outcomes of measurements of observables pertaining to systems and is typically silent about whether these observables have values themselves. Attempts to add to quantum mechanics descriptions of systems in which all quantummechanical observables have values became deadlocked in the 1960s: Kochen and Ernst Specker's nogo theorem proved that such descriptions are inconsistent if these values have to comply to the same mathematical relations as the observables themselves; John S. Bell's inequalities showed that the descriptions easily lead to nonlocal phenomena at odds with relatively theory (Redhead 1987). Modal interpretations add descriptions to quantum mechanics according to which only a few preferred observables have values, and avoid in this way specifically the KochenSpecker theorem.
A second common element is that modal interpretations do not ascribe one state to a system, as quantum mechanics does, but two: a dynamical state and a value state. By doing so another peculiarity of quantum mechanics is overcome, namely that states of systems evolve alternately by two mutually incompatible laws: the Schrödinger equation that yields smooth state evolution in between measurements, and the projection postulate that yields discontinuous evolution at measurements. In modal interpretations dynamical states of systems evolve with the Schrödinger equation only, and value states evolve typically discontinuously. A particular modal interpretation is now characterized by the value states it assigns to systems; value states fix the preferred definitevalued observables and their values.
Finally there is the claim that modal interpretations stay close to quantum mechanics. The dynamical states that modal interpretations assign can be taken as the states that quantum mechanics assigns, the only difference being that the former do not evolve by the projection postulate. Modal interpretations may thus be said to incorporate quantum mechanics instead of replacing it, as some hiddenvariables theories do.
QuantumMechanical HilbertSpace Mathematics
In quantum mechanics the state and observables of a physical system are represented by mathematical entities defined on a Hilbert space associated with the system. A Hilbert space H contains vectors ψ 〉, and if it is an n dimensional space, there exist sets {e _{1}〉,e _{2}〉, … e_{n} 〉} of n vectors that are pairwise orthogonal. Such a set is called a basis of the space, which means that any vector ψ 〉 in H can be decomposed as a weighted sum of the elements of the basis: ψ 〉=∑_{i}c_{i} e_{i} 〉. The Hilbert space associated with two disjoint physical systems consists of the tensor product H _{1}⊗H _{2} of the Hilbert spaces associated with the separate systems. If {e _{1}〉, … e_{n} 〉} is a basis of H _{1} and {f _{1}〉, … f_{m} 〉} a basis of H _{2}, then any vector Ψ 〉part of H _{1}⊗H _{2} can be decomposed as a sum Ψ 〉=∑_{i,j}C_{ij} e_{i} 〉⊗f_{j} 〉 (a double summation).
Linear operators A on a Hilbert space are linear mappings within that space. The operator that projects any vector on the vector ψ 〉 is called a projector and is written as ψ 〉〈ψ . In quantum mechanics the state of a system is represented by such a projector, or by a density projector W which is a complex sum ∑_{i}λ_{i} ψ_{i} 〉〈ψ_{i}  of projectors. An observable pertaining to a system (e.g., its momentum or spin) is represented by a selfadjoint operator A. Selfadjoint operators and density operators can be decomposed in terms of their eigenvalues a_{i} and projectors on their pairwise orthogonal eigenvectors a_{i} 〉, that is, A =∑_{i}a_{i} a_{i} 〉〈a_{i} . (Complications due to degeneracies, phase factors, and infinities are ignored.)
Particular Modal Interpretations
In all interpretations named modal, the dynamical state of a system is represented by a density operator W on the system's Hilbert space. This dynamical state evolves with the Schrödinger equation and has the usual quantummechanical meaning in terms of measurement outcomes: If observable A is measured, its eigenvalue a_{i} is found with probability p (a_{i} )=〈a_{i} W a_{i} 〉.
The value state of a system is represented by a vector v 〉 and determines the values of observables by the rule: A has value a_{i} iff v 〉 is equal to the eigenvector a_{i} 〉 of A. This rule leaves many observables without values; a specific value state is an eigenvector of only a few operators, which then represent the preferred observables. Particular modal interpretations fix the value states of systems differently.
In van Fraassen's (1973, 1991) Copenhagen modal interpretation v 〉 is a vector in the support of the dynamical state (which implies that W can be written as a convex sum of v 〉〈v  and other projectors). Van Fraassen is more specific about value states after measurements. If an observable A of a system is measured, the dynamical state of the composite of system and measurement device may become Ψ 〉〈Ψ , with Ψ 〉=∑_{i}c_{i} a_{i} 〉⊗R_{i} 〉. The vectors a_{i} 〉 are eigenvectors of the measured observable, and the R_{i} 〉's are eigenvectors of a device observable that represents the outcomes (the pointer readings). The value states after this measurement are, according to van Fraassen, with probability c_{i} ^{2} simultaneously given by a_{i} 〉 for the system and by R_{i} 〉 for the measurement device, respectively.
The decomposition Ψ 〉=∑_{i}c_{i} a_{i} 〉⊗R_{i} 〉 is mathematically special because it contains one summation (as said, a decomposition of a vector Ψ 〉 in a product space H _{1}⊗H _{2} relative to bases of the separate Hilbert spaces has usually a double summation). This special singlesum decomposition is called the biorthogonal decomposition of Ψ 〉, and a theorem (Schrödinger 1935) states that every vector Ψ 〉 in H _{1}⊗H _{2} determines exactly one basis {e _{1}〉, … e_{n} 〉} for H _{1} and one basis {f_{1} 〉, … f_{m} 〉} for H_{2} for which its decomposition becomes such a biorthogonal decomposition.
Kochen (1985) and Dieks (1989) use this decomposition to define value states in their modal interpretation: If two disjoint systems have a composite dynamical state Ψ 〉〈Ψ  and the biorthogonal decomposition of the vector Ψ 〉 is Ψ 〉=∑_{i}c_{i} e_{i} 〉⊗f_{i} 〉, then the value states are with probability c_{i} ^{2} simultaneously e_{i} 〉 for the first system and f_{i} 〉 for the second. Kochen adds a perspectival twist to this proposal, absent in Dieks's earlier writing: For Kochen the first system witnesses the second to have value state f_{i} 〉 iff it has itself value state e_{i} 〉 (which is the case with probability c_{i} ^{2}) and the second system then witnesses, conversely, the first to have value state e_{i} 〉.
The KochenDieks proposal applies to two systems with a composite dynamical state represented by a projector Ψ 〉〈Ψ  only. The spectral modal interpretation by Pieter Vermaas and Dieks (1995) generalizes this proposal to n disjoint systems with an arbitrary composite dynamical state W. This composite state fixes the dynamical states of all subsystems. Let W (x ) be the dynamical state of the x th system part of the composite and let it have an eigenvalueeigenvector decomposition W (x )=∑_{i}w_{i} (x )w_{i} (x )〉〈w_{i} (x ). The value state of this x th system is then w_{i} (x )〉 with probability w_{i} (x ). Vermaas and Dieks gave, moreover, joint probabilities that the disjoint systems have simultaneously their value states w_{i} (1)〉, w_{j} (2)〉, etcetera.
In the spectral modal interpretation a composite system, say, system 1+2 composed of the disjoint systems 1 and 2, has an eigenvector w_{k} (1+2)〉 of its dynamical state W (1+2) as its value state. The atomic modal interpretation by Guido Bacciagaluppi and Michael Dickson (1999) fixes the value states of such composite systems differently. Bacciagaluppi and Dickson assume that there exists a set of disjoint atomic systems, for which the value states are determined similarly as in the spectral modal interpretation, and propose that the value states of composites of those atoms are tensor products of the value states of the atoms: the value state of the composite of atoms 1 and 2 is w_{i} (1)〉⊗w_{j} (2)〉 iff the value states of the atoms are w_{i} (1)〉 and w_{j} (2)〉, respectively.
The Class of Modal Interpretations
The class of modal interpretations comprises those proposals according to which only a few observables have values, and that can be formulated in terms of dynamical and value states. The interpretations by Richard Healey (1989) and by Jeffrey Bub (1997) have this structure quite explicitly and are therefore often called modal ones (Healey's proposal has a number of similarities with the KochenDieks proposal; in Bub's the value state of a system is an eigenvector of an observable fixed independently of the system's dynamical state). One may argue that David Bohm's mechanics (1952) is also a modal interpretation.
Results
The development and application of modal interpretations have led to mixed results. The maximum set of observables that can have values by modal interpretations without falling prey to the KochenSpecker theorem has been determined (Vermaas 1999). Bub and Rob Clifton showed that this set is the only one that satisfies a series of natural assumptions on descriptions of single systems (Bub, Clifton, and Goldstein 2000). The evolution of value states, which determines the description of systems over time, can be given (Bacciagaluppi and Dickson 1999). This evolution was, however, shown not to be Lorentzcovariant for the spectral and atomic modal interpretations and, to a lesser extent, for Bub's interpretation, revealing that the assumption that only a few quantummechanical observables have values, still may lead to problems with relatively theory (Dickson and Clifton 1998, Myrvold 2002).
Moreover, even though this assumption yields consistent descriptions of single systems, joint descriptions of systems were still proved to be problematic. First, it is commonly assumed in quantum mechanics that the observable of a system 1 represented by the operator A defined on H _{1}, and the observable of a composite system 1+2 represented by the operator A _{1}⊗I _{2} on H _{1}⊗H _{2} (I _{2} is the identity operator on H _{2}) are one and the same observable. The Copenhagen, KochenDieks, and spectral modal interpretations have the debatable consequence that these observables should be distinguished (Clifton 1996). Second, the spectral modal interpretation cannot give joint probabilities that systems 1, 2, … , and their composites, 1+2, … , have simultaneously their value states w_{i} (1)〉, w_{j} (2)〉, w_{k} (1+2)〉, etcetera (Vermaas 1999, ch. 6).
These negative results motivated in part the formulation of the atomic modal interpretation but can also be avoided by adopting Kochen's perspectivalism, which implies that one accepts constraints on describing different systems simultaneously. Finally, the KochenDieks, spectral, and atomic modal interpretations have problems with properly describing measurements, doubting their empirical adequacy. David Albert and Barry Loewer (1990) argued that after a measurement, the dynamical state of the systemdevice composite need not be Ψ 〉〈Ψ  with Ψ 〉=∑_{i}c_{i} a_{i} 〉⊗R_{i} 〉, and that the mentioned interpretations then need not yield descriptions in which the device displays an outcome (Bacciagaluppi and Hemmo 1996).
Assessment
These results allow critical conclusions about particular modal interpretations and raise doubts about the viability of the class of modal interpretations. Three remarks can be made about this assessment.
First, an evaluation of the results may depend on what one expects from interpretations. If interpretations are to provide descriptions that allow realist positions about quantum mechanics, the inability of, say, the spectral modal interpretation to give joint probabilities that systems have simultaneously value states, proves this interpretation problematic. But if interpretations, in line with van Fraassen's view, are to yield understanding of what quantum mechanics means, this inability of the spectral modal interpretation is an interesting conclusion about how quantummechanical descriptions of systems differ from those of other physical theories. The result that some modal interpretations may be empirical inadequate, is, however, fatal independently of one's expectations for interpretations.
Second, the set of particular modal interpretations that is analyzed so far does not exhaust the class of modal interpretations. Research therefore continues (e.g., Bene and Dieks 2002).
Third, these results are relevant to the project of interpreting quantum mechanics in general. Existing and new interpretations, modal or not, according to which only some observables have definite values, are constrained by the negative results and can now be assessed as such; and existing and new interpretations may benefit from the positive results about modal interpretations.
See also Bell, John, and Bell's Theorem; Bohm, David; Quantum Mechanics; Van Fraassen, Bas.
Bibliography
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Pieter E. Vermaas (2005)
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