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

Topos theory and `neo-realist' quantum theory

Topos theory, a branch of category theory, has been proposed as mathematical basis for the formulation of physical theories. In this article, we give a brief introduction to this approach, emphasising the logical aspects. Each topos serves as a `mathematical universe' with an internal logic, which is used to assign truth-values to all propositions about a physical system. We show in detail how this works for (algebraic) quantum theory.Comment: 22 pages, no figures; contribution for Proceedings of workshop "Recent Developments in Quantum Field Theory", MPI MIS Leipzig, July 200

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