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Complete sets of compatible non selfadjoint observables

Giuseppina Epifanio, Todor S. Todorov, Camillo Trapanı

Published Feb 4, 1992• 4 citations
Publisher

Abstract

Compatibility for observables understood as symmetric but not selfadjoint operators is defined in terms of commutation of the isometry semigroups that they generate. Compatible observables are shown to admit a joint distribution. The completeness of a set of commuting observables is discussed by means of the generated von Neumann algebra
DOI
10.5169/seals-116383

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From here, we start relaxing the constraints that we had imposed in the earlier chapters for simplicity, and start moving towards more general scenarios. First, measurements with inadequate resolution, employing insufficiently selective measurement devices have been introduced. In this book they have been termed imprecise measurements. Here, we admit for the first time that all measurements do not prepare quantum states. We argue how the description of imprecise observables requires the use of Hermitian operators with degenerate eigenvalues: degeneracy encodes insufficient selectiveness. The general law of state preparation, the standard jargon for which is “collapse postulate”, is described in its full glory here. We then move on to introduce the idea of compatibility of observables and demonstrate how certain groups of imprecise measurements can collaborate to prepare states. This leads to the important idea of “complete set of compatible observables” and the related concepts of “complete and incomplete measurements”. Subsequently, the association of compatible observables and commuting operators is established which also connects complete set of compatible observables and complete set of commuting operators (both of which are accronymed CSCO, without ambiguity). We also give an explicit illustration of the algebraic construction of a CSCO.
This paper i a continuation of the first part under the same title. The author studies a joint distribution in σ\sigmaσ-finite measures for noncompatible observables of a quantum logic defined on some system of σ\sigmaσ-independent Boolean sub-σ\sigmaσ-algebras of a Boolean σ\sigmaσ-algebra. We present some necessary and sufficient conditions fot the existence of a joint distribution. In particular, it is shown that an arbitrary system of obsevables has a joint distribution in a measure iff it may be embedded into a system of compatible observables of some quantum logic. The methods used are different from those developed for finite measures. Finally, the author deals with the connection between the existence of a joint distribution and the existence of a commutator of observables, and the quantum logic of a nonseparable Hilbert space is mentioned.
We present in this paper a set of postulates for a physical system and deduce from these the main general features of the quantum theory of stationary states. Our theory is strictly operational in the sense that only the observables of the physical system are involved in the postulates. The collection of all bounded self-adj oint operators on a Hilbert space, which has previously been used as a mathematical model for the observables in quantum mechanics, satisfy the postulates, as do a variety of considerably more general mathematical structures. We show that for any two observables there exists a pure state of the system (this term being defined essentially as done by von Neumann and Weyl) in which they have different (expectation) values. Each observable has a spectral resolution, and a state of the system induces, in a natural way, a probability distribution on the range of spectral values of each observable. A number of observables are simultaneously observable if and only if they commute (although the product of two observables is not defined in general, a reasonable notion of commutativity can be introduced). Inasmuch as Hilbert space plays no role in our theory, our proofs are necessarily of a different character from the proofs of these results for the case of the system of all bounded selfadjoint operators. Actually, Hilbert space appears to be somewhat inadequate as a state space even for the latter system, in that there exist pure states of the system which cannot be represented in the usual way by rays of the Hilbert space. The postulates are partly algebraic and partly metric. The algebraic postulates require essentially that an observable can be multiplied by real numbers and raised to integral powers, and that any two observables can be added. It is assumed that the usual algebraic laws are satisfied so that (1) the observables can be treated like the elements of a linear space, and (2) the usual rules for dealing with polynomials in one variable with real coefficients remain valid when the variable is replaced by an observable. It is not assumed that two observables have a product. The metric postulates require that for each observable there be defined a kind of maximum numerical value, which plays the part of a norm, and has various properties in accord with its physical significance. While this norm is quite essential to the development of the theory, an interesting consequence of the theory is that the norm can be (uniquely) defined in a purely algebraic fashion. This shows that the objective features of a physical system,-the spectral values and probability distributions of the observables, and the pure states,-are completely determined by the algebra of observables, i.e., by the rules for addition, scalar multiplication, and powers, of observables.