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The different versions

Pieter E. Vermaas

Published Feb 5, 2010• 0 citations

Abstract

In this chapter I introduce the different versions of the modal interpretation, including the three on which I focus in this book. The property ascription of these versions is characterised by the map W α ↦ {〈p j , C α j 〉} j which I call the core property ascription. In the next chapter I discuss how this core property ascription determines the full property ascription W α ↦ The best modal interpretation The best imaginable modal interpretation is, I guess, an interpretation which (A) ascribes at all times all the properties to a system which pertain to that system, and (B) ascribes these properties such that the classical logical relations between the negation, conjunction and disjunction of properties are satisfied. The content of the first requirement (A) is clear: assuming that every projection onto a subspace of a Hilbert space ℋ α represents one and only one property of α, it follows that all sets of definite-valued projections should contain all the projections in ℋ α and that all the maps {. j } j should be maps from all the projections in ℋ α to the values {0,1}. The content of requirement (B) is, however, less clear because there is consensus neither about how to define the negation, conjunction and disjunction of properties in quantum mechanics nor about how to impose the logical relations. In Section 5.1 I present my choice for the definitions of the negation, conjunction and disjunction.

DOI: 10.1017/cbo9780511608230.006

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The full property ascription

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The different modal interpretations all advance a core property ascription {〈P j , C α j 〉} j . This chapter is about how this core property ascription fixes the full property ascription. I start with some logic and algebra. I then present two existing proposals for determining the full property ascriptions as well as four conditions one can impose on them. Lastly I give my own proposal and end by discussing how the full property ascription leads to a value assignment to magnitudes. Some logic and algebra To prepare the ground for discussing the full property ascription, I firstly define the logical connectives (negation, conjunction and disjunction) for properties. Then because, as discussed in Section 4.1, all the modal interpretations which I consider give up on the idea that thefull property ascription assigns definite values to all properties, I secondly define two types of subsets of properties: Boolean algebras and faux-Boolean algebras. I have already assumed that every property Q α pertaining to a system α is represented by one and only one projection Q α defined on the Hilbert space ℋα associated with α. Each projection Q α in its turn corresponds one-to-one to the subspace of ℋα, denoted by 2α, onto which it projects. One thus has a bijective mapping from a property Q α to a projection Q α to a subspace 2α. For the set of subspaces of a Hilbert space one can in a natural way define an orthocomplement, a meet and a join. I now choose 45 to define the logical connectives for properties pertaining to a system by means of this orthocomplement, meet and join for subspaces and the bijective mapping between properties and subspaces (by means of an isomorphism, more shortly).

Cambridge University Press eBooksFeb 5, 20100 citations

Formalism

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In Chapter 4 the different modal interpretations are introduced. Their property ascription to a system α is characterised by a core property ascription {〈P j ,C α J 〉} j . Chapter 5 treats the question of how the core property ascription to a system determines the full property ascription to the system, and of how that full property ascription induces an assignment of values to the magnitudes of the system. In Chapter 6 it is determined whether the properties that modal interpretations simultaneously ascribe to different systems can be correlated. A no-go theorem is derived which restricts the possibility of giving such correlations. In Chapter 7 it is shown that the set of properties, which a system possibly possesses, evolves in a number of undesirable ways. This evolution is, for instance, discontinuous and unstable. Chapter 8 is concerned with the evolution of the actually possessed properties of a system. It is proved for the case of freely evolving systems that this evolution is deterministic and for the case of interacting systems it is argued that this evolution cannot be uniquely fixed. In Chapter 9 it is proved that the evolution of the actually possessed properties of systems violates a number of the Dynamical Autonomy assumptions presented in Section 3.3. It is shown that this allows the descriptions of reality by modal interpretations to be non-local in a quite explicit way.

Cambridge University Press eBooksFeb 5, 20100 citations

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Conclusions

Pieter E. Vermaas

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Cambridge University Press eBooksFeb 5, 20100 citations

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Cambridge University Press eBooksFeb 5, 20100 citations

Modal interpretations

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Cambridge University Press eBooksFeb 5, 20100 citations