Quantum logic and decohering histories

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'.Comment: 14 pages LaTeX; Writeup of lecture given at conference ``Theories of fundamental interactions'', Maynooth Eire 24--26 May 1995

Quantising on a category

We review the problem of finding a general framework within which one can construct quantum theories of non-standard models for space, or space-time. The starting point is the observation that entities of this type can typically be regarded as objects in a category whose arrows are structure-preserving maps. This motivates investigating the general problem of quantising a system whose `configuration space' (or history-theory analogue) is the set of objects \Ob\Q in a category \Q. We develop a scheme based on constructing an analogue of the group that is used in the canonical quantisation of a system whose configuration space is a manifold $Q\simeq G/H$, where $G$ and $H$ are Lie groups. In particular, we choose as the analogue of $G$ the monoid of `arrow fields' on \Q. Physically, this means that an arrow between two objects in the category is viewed as an analogue of momentum. After finding the `category quantisation monoid', we show how suitable representations can be constructed using a bundle (or, more precisely, presheaf) of Hilbert spaces over \Ob\Q. For the example of a category of finite sets, we construct an explicit representation structure of this type.Comment: To appear in a volume dedicated to the memory of James Cushin

A Topos Perspective on State-Vector Reduction

A preliminary investigation is made of possible applications in quantum theory of the topos formed by the collection of all $M$-sets, where $M$ is a monoid. Earlier results on topos aspects of quantum theory can be rederived in this way. However, the formalism also suggests a new way of constructing a `neo-realist' interpretation of quantum theory in which the truth values of propositions are determined by the actions of the monoid of strings of finite projection operators. By these means, a novel topos perspective is gained on the concept of state-vector reduction

A topos perspective on the Kochen-Specker theorem: II. Conceptual Aspects, and Classical Analogues:

In a previous paper, we have proposed assigning as the value of a physical quantity in quantum theory, a certain kind of set (a sieve) of quantities that are functions of the given quantity. The motivation was in part physical---such a valuation illuminates the Kochen-Specker theorem; and in part mathematical---the valuation arises naturally in the topos theory of presheaves. This paper discusses the conceptual aspects of this proposal. We also undertake two other tasks. First, we explain how the proposed valuations could arise much more generally than just in quantum physics; in particular, they arise as naturally in classical physics. Second, we give another motivation for such valuations (that applies equally to classical and quantum physics). This arises from applying to propositions about the values of physical quantities some general axioms governing partial truth for any kind of proposition.Comment: Small changes and correction

On the Emergence of Time in Quantum Gravity

We discuss from a philosophical perspective the way in which the normal concept of time might be said to `emerge' in a quantum theory of gravity. After an introduction, we briefly discuss the notion of emergence, without regard to time (Section 2). We then introduce the search for a quantum theory of gravity (Section 3); and review some general interpretative issues about space, time and matter Section 4). We then discuss the emergence of time in simple quantum geometrodynamics, and in the Euclidean approach (Section 5). Section 6 concludes.Comment: To appear in ``The Arguments of Time'', ed. J. Butterfield, Oxford University Press, 199

A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations

The Kochen-Specker theorem asserts the impossibility of assigning values to quantum quantities in a way that preserves functional relations between them. We construct a new type of valuation which is defined on all operators, and which respects an appropriate version of the functional composition principle. The truth-values assigned to propositions are (i) contextual; and (ii) multi-valued, where the space of contexts and the multi-valued logic for each context come naturally from the topos theory of presheaves. The first step in our theory is to demonstrate that the Kochen-Specker theorem is equivalent to the statement that a certain presheaf defined on the category of self-adjoint operators has no global elements. We then show how the use of ideas drawn from the theory of presheaves leads to the definition of a generalized valuation in quantum theory whose values are sieves of operators. In particular, we show how each quantum state leads to such a generalized valuation.Comment: Clarification of situation for situation for operators with continuous spectr