The recent suggestion that a temporal form of quantum logic provides the natural mathematical framework within which to discuss the proposal by Gell-Mann and Hartle for a generalized form of quantum theory based on the ideas of histories and decoherence functionals is analyzed and developed herein. Particular stress is placed on properties of the space of decoherence functionals, including one way in which certain global and topological properties of a classical system are reflected in a quantum history theory.
The recent suggestion that a temporal form of quantum logic provides the natural mathematical framework within which to discuss the proposal by Gell-Mann and Hartle for a generalized form of quantum theory based on the ideas of histories and decoherence functionals is analyzed and developed herein. Particular stress is placed on properties of the space of decoherence functionals, including one way in which certain global and topological properties of a classical system are reflected in a quantum history theory.
An extensive analysis is made of the Gell-Mann and Hartle axioms for a generalized ‘histories’ approach to quantum theory. Emphasis is placed on finding analogs of the lattice structure employed in standard quantum logic. Particular attention is given to ‘quasitemporal’ theories in which the notion of time-evolution in conventional Hamiltonian physics is replaced by something that is much broader; theories of this type are expected to arise naturally in the context of quantum gravity and quantum field theory in a curved space–time. The quasitemporal structure is coded in a partial semigroup of ‘temporal supports’ that underpins the lattice of history propositions. Nontrivial examples include quantum field theory on a non-globally-hyperbolic space–time, and a possible cobordism approach to a theory of quantum topology. A key result is the demonstration that the set of history propositions in standard quantum theory can be realized in such a way that each such proposition is represented by a genuine projection operator. This gives valuable insight into the possible lattice structure in general history theories.
An extended analysis is made of the Gell-Mann and Hartle axioms for a generalised `histories' approach to quantum theory. Emphasis is placed on finding equivalents of the lattice structure that is employed in standard quantum logic. Particular attention is given to `quasi-temporal' theories in which the notion of time-evolution is less rigid than in conventional Hamiltonian physics; theories of this type are expected to arise naturally in the context of quantum gravity and quantum field theory in a curved space-time. The quasi-temporal structure is coded in a partial semi-group of `temporal supports' that underpins the lattice of history propositions. Non-trivial examples include quantum field theory on a non globally-hyperbolic spacetime, and a simple cobordism approach to a theory of quantum topology. It is shown how the set of history propositions in standard quantum theory can be realised in such a way that each history proposition is represented by a genuine projection operator. This provides valuable insight into the possible lattice structure in general history theories, and also provides a number of potential models for theories of this type.
I review the decoherent (or consistent) histories approach to quantum mechanics, due to Griffiths, to Gell-Mann and Hartle, and to Omnes. This is an approach to standard quantum theory specifically designed to apply to genuinely closed systems, up to and including the entire universe. It does not depend on an assumed separation of classical and quantum domains, on notions of measurement, or on collapse of the wave function. Its primary aim is to find sets of histories for closed systems exhibiting negligble interference, and therefore, to which probabilities may be assigned. Such sets of histories are called consistent or decoherent, and may be manipulated according to the rules of ordinary (Boolean) logic. The approach provides a framework from which one may predict the emergence of an approximately classical domain for macroscopic systems, together with the conventional Copenhagen quantum mechanics for microscropic subsystems. In the special case in which the total closed system naturally separates into a distinguished subsystem coupled to an environment, the decoherent histories approach is closed related to the quantum state diffusion approach of Gisin and Percival.
Gell-Mann and Hartle have proposed a significant generalization of quantum theory with a scheme whose basic ingredients are ‘‘histories’’ and decoherence functionals. Within this scheme it is natural to identify the space 𝒰𝒫 of propositions about histories with an orthoalgebra or lattice. This raises the important problem of classifying the decoherence functionals in the case where 𝒰𝒫 is the lattice of projectors 𝒫(𝒱) in some Hilbert space 𝒱; in effect, one seeks the history analog of Gleason’s famous theorem in standard quantum theory. In the present article the solution to this problem for the case where 𝒱 is finite dimensional is presented. In particular, it is shown that every decoherence functional d(α,β), α,β∈𝒫(𝒱) can be written in the form d(α,β)=tr𝒱⊗𝒱(α⊗βX) for some operator X on the tensor-product space 𝒱⊗𝒱.
I give an informal overview of the decoherent histories approach to quantum mechanics, due to Griffiths, to Omnès, and to Gell-Mann and Hartle is given. Results on the connections between decoherence, records, correlation and entropy are described. The emphasis of the presentation is on understanding the broader meaning of the conditions of consistency and decoherence, and in particular, the extent to which they permit one to assign definite properties to the system. The quantum Brownian motion model is briefly discussed. (To appear in proceedings of the workshop, "Stochastic Evolution of Quantum States in Open Systems and Measurement Processes", Budapest, March, 1993, edited by L.Diosi).
Precise rules are developed in order to formalize the reasoning processes involved in standard non-relativistic quantum mechanics, with the help of analogies from classical physics. A classical or quantum description of a mechanical system involves a {\it framework}, often chosen implicitly, and a {\it statement} or assertion about the system which is either true or false within the framework with which it is associated. Quantum descriptions are no less ``objective'' than their classical counterparts, but differ from the latter in the following respects: (i) The framework employs a Hilbert space rather than a classical phase space. (ii) The rules for constructing meaningful statements require that the associated projectors commute with each other and, in the case of time-dependent quantum histories, that consistency conditions be satisfied. (iii) There are incompatible frameworks which cannot be combined, either in constructing descriptions or in making logical inferences about them, even though any one of these frameworks may be used separately for describing a particular physical system. A new type of ``generalized history'' is introduced which extends previous proposals by Omnès, and Gell-Mann and Hartle, and a corresponding consistency condition which does not involve density matrices or single out a direction of time. Applications which illustrate the formalism include: measurements of spin, two-slit diffraction, and the emergence of the classical world from a fully quantum description.