As in a classical theory, the admissible eventualities in a quantum theory for the description of a system A will be identifiable with subsets of a corresponding set I {A}. The essential new feature distinguishing a quantum theory is that I is endowed with a Hilbert space structure, and that the admissible eventualities are identifiable not with arbitrary subsets, but only with those that are Hilbert subspaces.

If e1 and e2 are the Hilbert subspaces representing a pair of admissible eventualities, their intersection e1 n e2 will also be a Hilbert subspace, representing the corresponding conjoint eventuality, but their union e1 U e2 will in general not have the structure of a Hilbert subspace and thus (unlike the classical case) will not represent an admissible eventuality. The eventualities of a quantum theory do, nevertheless, have an additive structure that is naturally induced by the Hilbert space structure: the sum e1 © e2 is defined to be the Hilbert subspace that is spanned by the separate Hilbert subspaces e1 and e2. What this means, using the standard notation scheme originally developed by Dirac [7], is that the state £ e1 © e2 if and only if is a Hilbert space vector having the form = |^1) + |^2) for some pair of

Hilbert space vectors such that E e1 and |^2) E e2. In the particular case for which every such pair of vectors satisfies the orthogonality condition = 0, the corresponding subspaces e1 C I and e2 C I will be describable as mutually orthogonal.

Orthogonality in this sense is what characterizes the kind of exclusivity required for the definition of what is generally known as an observable in the context of quantum theory. Thus an observable in a quantum theory for the system A (i.e. a qualitative observable, as distinct from a quantitative observable of the related kind to be discussed below) can be formally defined to consist of a complete set {e} of mutually orthogonal Hilbert subspaces e1,..., e_N, where the condition of completeness means that they span the entire Hilbert space I {A}, i.e. that e1 © ••• © en = I.

For any particular eventuality, the corresponding subspace e C I will determine and be determined by an associated Hilbert space projection operator e = e2, defined so as to be automatically Hermitian by the conditions e = whenever lies in e and e = 0 whenever is orthogonal to the subspace e. The condition for a set {e} of eventualities {ei} (i = 1,...,n) to constitute an observable is thus expressible as the condition that the corresponding operators should satisfy the orthogonality requirement ei ej = 0 for i = j and the completeness condition ^i ei = I, where I is the unit operator on I.

In the earliest versions of quantum theory, it was postulated that the relevant probabilities would be given just by the specification of a single state vector G I {A}, subject to the normalization condition = 1, according to a prescription expressible in the following familiar form:

This is just a conditional probability, subject to the requirement that the relevant observation, Oe, be actually carried out.

Soon after the original development of this Dirac-Heisenberg paradigm, it was recognized that a prescription of the simple form in Eq. (19.1) is too restrictive for typical cases in which the system A may interact with another (internal or external) system B. The extended system A, consisting of the combination of A and B, will be characterized by a Hilbert space I = I {a!} which is constructed as the tensor product of I {A} and I {B}. This means that a state vector I1 ) E 1 for the extended system will be expressible in terms of a basis of vectors |$a) E I{B} satisfying the orthonormality condition ($a|$b) = ôab. It must therefore have the form

a for some corresponding set of vectors |^a) £ I {A} that will not in general be orthonormal but must satisfy the condition ^a(^a|^a) = 1 in order for the unit normalization condition (^) = 1 to be satisfied. If ei is a subspace of dimension Ri within the original Hilbert space I {A} of dimension N{A}, then it will determine a corresponding subspace ei of dimension RiN{B} in the tensor product Hilbert space I, where N{B} is the dimension of I {B}. Within the original Hilbert space I = I {A}, the corresponding projection operator will have rank given by its trace, Ri = tr{ei}, while the corresponding operator ei of projection onto ei in / will have rank Ri N{B}. According to the natural extension of Eq. (19.1), a unit state vector I1) in / will specify a (conditional) probability distribution given by

In order to express such a prescription within the simpler framework of the original Hilbert space I{A} of the subsystem A with which we are particularly concerned, it is necessary to use a prescription of the kind whose development was attributed by Dirac to von Neumann. In the Dirac-von Neumann paradigm, instead of being specified by just a single state vector the (conditional) probability distribution (for the outcome of an observation Oe if actually performed) is specified by a Hermitian probability density operator P with unit trace on I according to the prescription

This prescription is compatible with the original pure state paradigm, as specified by a single vector satisfying the unit normalization = 1, according to Eq. (19.1). The effect of this can be seen to be the same as taking P = in the general formula Eq. (19.4). The advantage of the von Neumann type formulation in Eq. (19.4) is that it can also express the result of the more general prescription given in Eq. (19.3), whose effect can be seen to be the same as that of taking

a where the (generally non-orthonormal) set of vectors |^a) is specified by the decomposition in Eq. (19.2).

Many authors - particularly those influenced by the Everett doctrine [4] -have continued to hanker after the original Heisenberg-type paradigm, meaning the supposition that the probabilities should ultimately be determined by a pure state in a very large all-embracing Hilbert space, characterizing the universe as a whole. Such authors - including Hawking [8] - have been inclined to regard the use of a von Neumann operator as a rather unsatisfactory approximation device that may be made necessary by our ignorance due to the regrettable loss of some of the relevant information in, for example, a black hole. However, my own attitude is like that of the distrustful insurance agent, who doubts whether what was alleged to have been lost was ever actually possessed. I personally see no reason why - to encompass more and more detailed microstructure and more and more extended macrostructure - the process of construction of successively larger and larger Hilbert spaces should ever come to an end. In other words, the search [9] for a single, ultimate, all-embracing 'wave-function of the universe', or even an ultimate, all-embracing von Neumann operator, may be like the pursuit of the proverbially elusive 'Will o' the wisp'. It seems more reasonable to accept that any system sufficiently simple to be amenable to our mathematical analysis can only be a model of an incomplete subcomponent of something larger, and that it is therefore unreasonable to demand that it be describable by a pure state rather than a more general von Neumann operator. However that may be, advocates of the Everett doctrine would agree that there can in any case be no harm in working throughout in terms of the von Neumann paradigm, as will be done here, because it includes the more restricted Heisenberg-type pure state paradigm as a special case.

Before continuing, it should be remarked that the term observable has been used here to designate what - in a more pedantically explicit terminology -would be called a qualitative observable, in order to distinguish it from the quantitative observables that are definable as functions thereof. Thus, any qualitative observable {e} determines and is determined by a corresponding equivalence class of quantitative variables, in which any particular member E is determined by a corresponding non-degenerate real-valued function Ei with the index labelling the admissible alternatives ei for {e}. The condition for non-degeneracy of the function is to be understood as meaning that Ei = Ej whenever i = j. In a quantum theory for a system characterized by a Hilbert space I {A}, such a quantitative observable will be identifiable with a corresponding Hermitian operator E whose eigenspaces are the Hilbert subspaces ei C I {A}, while the corresponding eigenvalues are the real numbers Ei. One therefore has

Such a quantitative variable E will have a mean (expectation) value {E) given by

in which the operator E will be expressible in terms of the relevant projection operators ei in the explicit form

The simplest illustration is provided by the familiar Stern-Gerlach example, for which the observable E represents the spin energy of an electron (with respect to its own rest frame) in a uniform magnetic field. For this application, the relevant Hilbert space I has only two (complex) dimensions, being spanned by a subspace e1 representing the eventuality that the spin be aligned with the magnetic field, and a subspace e2 representing the eventuality that it be aligned in the opposite direction. Other eventualities, corresponding to alignment in other directions, will not be characterized by well defined energy values. The quantum analogue of the unbiased coin toss theory (considered in Section 19.3) is the unbiased spin theory, specified by adopting the isotropic probability distribution given (as the high temperature limit of an ordinary thermal distribution) by P = I/2.

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