arXiv subjects

C. J. Isham

Publications and source records attributed to C. J. Isham.

25 records · Page 2Linked to original sources

arXiv1996

Information-entropy and the space of decoherence functions in generalised quantum theory

In standard quantum theory, the ideas of information-entropy and of pure states are closely linked. States are represented by density matrices $ρ$ on a Hilbert space and the information-entropy $-tr(ρ\logρ)$ is minimised on pure states (pure states are the vertices of the boundary of the convex set of states). The space of decoherence functions in the consistent histories approach to generalised quantum theory is also a convex set. However, by showing that every decoherence function can be written as a convex combination of two other decoherence functions we demonstrate that there are no `pure' decoherence functions. The main content of the paper is a new notion of information-entropy in generalised quantum mechanics which is applicable in contexts in which there is no a priori notion of time. Information-entropy is defined first on consistent sets and then we show that it decreases upon refinement of the consistent set. This information-entropy suggests an intrinsic way of giving a consistent set selection criterion.

quant-ph↗

arXiv1996

Topos Theory and Consistent Histories: The Internal Logic of the Set of all Consistent Sets

A major problem in the consistent-histories approach to quantum theory is contending with the potentially large number of consistent sets of history propositions. One possibility is to find a scheme in which a unique set is selected in some way. However, in this paper we consider the alternative approach in which all consistent sets are kept, leading to a type of `many world-views' picture of the quantum theory. It is shown that a natural way of handling this situation is to employ the theory of varying sets (presheafs) on the space $\B$ of all Boolean subalgebras of the orthoalgebra $\UP$ of history propositions. This approach automatically includes the feature whereby probabilistic predictions are meaningful only in the context of a consistent set of history propositions. More strikingly, it leads to a picture in which the `truth values', or `semantic values' of such contextual predictions are not just two-valued (\ie true and false) but instead lie in a larger logical algebra---a Heyting algebra---whose structure is determined by the space $\B$ of Boolean subalgebras of $\UP$.

gr-qc↗

arXiv1995

Perennials and the Group-Theoretical Quantization of a Parametrized Scalar Field on a Curved Background

The perennial formalism is applied to the real, massive Klein-Gordon field on a globally-hyperbolic background space-time with compact Cauchy hypersurfaces. The parametrized form of this system is taken over from the accompanying paper. Two different algebras ${\cal S}_{\text{can}}$ and ${\cal S}_{\text{loc}}$ of elementary perennials are constructed. The elements of ${\cal S}_{\text{can}}$ correspond to the usual creation and annihilation operators for particle modes of the quantum field theory, whereas those of ${\cal S}_{\text{loc}}$ are the smeared fields. Both are shown to have the structure of a Heisenberg algebra, and the corresponding Heisenberg groups are described. Time evolution is constructed using transversal surfaces and time shifts in the phase space. Important roles are played by the transversal surfaces associated with embeddings of the Cauchy hypersurface in the space-time, and by the time shifts that are generated by space-time isometries. The automorphisms of the algebras generated by this particular type of time shift are calculated explicitly.

gr-qc↗

arXiv1995

The symplectic geometry of a parametrized scalar field on a curved background

We study the real, massive Klein-Gordon field on a $C^\infty$ globally-hyperbolic background space-time with compact Cauchy hypersurfaces. In particular, the parametrization of this system as initiated by Dirac and Kuchař is put on a rigorous basis. The discussion is focussed on the structure of the set of spacelike embeddings of the Cauchy manifold into the space-time, and on the associated $e$-tensor density bundles and their tangent and cotangent bundles. The dynamics of the field is expressed as a set of automorphisms of the space of initial data in which each pair of embeddings defines one such automorphism. Using these results, the extended phase space of the system is shown to be a weak-symplectic manifold, and the Kuchař constraint is shown to define a smooth constraint submanifold which is foliated smoothly by the constraint orbits. The pull-back of the symplectic form to the constraint surface is a presymplectic form which is singular on the tangent spaces to the constraint orbits. Thus, the geometric structure of this infinite-dimensional system is analogous to that of a finite-dimensional, first-class parametrized system, and hence many of the results for the latter can be transferred to the infinite-dimensional case without difficulty.

gr-qc↗

arXiv1993

Prima Facie Questions in Quantum Gravity

The long history of the study of quantum gravity has thrown up a complex web of ideas and approaches. The aim of this article is to unravel this web a little by analysing some of the {\em prima facie\/} questions that can be asked of almost any approach to quantum gravity and whose answers assist in classifying the different schemes. Particular emphasis is placed on (i) the role of background conceptual and technical structure; (ii) the role of spacetime diffeomorphisms; and (iii) the problem of time.

gr-qc↗

arXiv1993

Quantum Logic and the Histories Approach to Quantum Theory

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.

gr-qc↗

arXiv1992

Representations of the holonomy algebras of gravity and non-Abelian gauge theories

Holonomy algebras arise naturally in the classical description of Yang-Mills fields and gravity, and it has been suggested, at a heuristic level, that they may also play an important role in a non-perturbative treatment of the quantum theory. The aim of this paper is to provide a mathematical basis for this proposal. The quantum holonomy algebra is constructed, and, in the case of real connections, given the structure of a certain C-star algebra. A proper representation theory is then provided using the Gel'fand spectral theory. A corollory of these general results is a precise formulation of the ``loop transform'' proposed by Rovelli and Smolin. Several explicit representations of the holonomy algebra are constructed. The general theory developed here implies that the domain space of quantum states can always be taken to be the space of maximal ideals of the C-star algebra. The structure of this space is investigated and it is shown how observables labelled by ``strips'' arise naturally.

hep-th↗