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arXiv:gr-qc/9308006 · Journal of Mathematical Physics

Quantum logic and the histories approach to quantum theory

C. J. Isham

Published May 1, 1994• 236 citations
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Abstract

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.

DOI
10.1063/1.530544

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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.
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 axiomatic development of dynamics of systems in the framework of histories is given which contains the history versions of classical and traditional quantum mechanics as special cases. We consider theories which admit a quasitemporal structure (a generalization of the concept of time using partial semigroups) and whose “single time” propositions have the mathematical structure of a logic; isomorphism of logics at different “instants of time” is not assumed. The concept of directed partial semigroup is introduced to incorporate the concept of direction of flow of time in quasitemporal theories. Starting with a few simple axioms, the space of history propositions is explicitly constructed and shown to be an orthoalgebra (as envisaged in the scheme of Isham and Linden [J. Math. Phys. 35, 5452–5476 (1994)]); its subspace consisting of history filters (“homogeneous histories”) is a meet semilattice. The partial semigroups employed allow semi-infinite irreducible decompositions.
An introduction is given to an algebraic formulation and generalisation of the consistent histories approach to quantum theory. The main technical tool in this theory is an orthoalgebra of history propositions that serves as a generalised temporal analogue of the lattice of propositions of standard quantum logic. Particular emphasis is placed on those cases in which the history propositions can be represented by projection operators in a Hilbert space, and on the associated concept of a `history group'.
These lecture notes cover the important developments in histories approach to quantum mechanics with overall content and emphasis somewhat different from other reviews and books on the subject.The idea of Houtappel, Van Dam and Wigner of employing objects based on primitive concepts of physical theories is discussed in some detail and the fact that histories are such objects is emphasized. Application of histories formalism to the problem of understanding the quasiclassical domain is treated in some detail. Other topics discussed include generalized histories-based quantum mechanics and its application to the quantum mechanics of space-time,generalization of the notion of time sequences employing partial semigroups,quasitemporal structures, history projection operator (HPO) formalism, the algebraic scheme of Isham and Linden, an axiomatic scheme for quasitemporal histories-based theories and symmetries and conservation laws in histories-based theories.
Topos theory has been suggested by Döring and Isham as an alternative mathematical structure with which to formulate physical theories. In particular, it has been used to reformulate standard quantum mechanics in such a way that a novel type of logic is used to represent propositions. In this paper we extend this formulation to include temporally ordered collections of propositions as opposed to single-time propositions. That is to say, we have developed a quantum history formalism in the language of topos theory where truth values can be assigned to temporal propositions. We analyze the extent to which such truth values can be derived from the truth values of the constituent, single-time propositions.