Introduction

This chapter provides a conceptual discussion and physical interpretation of some of the quite abstract constructions in the topos approach to physics. In particular, the daseinisation process for projection operators and for self-adjoint operators is motivated and explained from a physical point of view. Daseinisation provides the bridge between the standard Hilbert space formalism of quantum theory and the new topos-based approach to quantum theory. As an illustration, I show all constructions explicitly for a three-dimensional Hilbert space and the spin-z operator of a spin-1 particle. Throughout, I refer to joint work with Chris Isham, and this chapter is intended to serve as a companion to the one he contributed to this volume.

The Topos Approach

The topos approach to quantum theory was initiated by Isham [21] and Butterfield and Isham [19, 22–24]. It was developed and broadened into an approach to the formulation of physical theories in general by Isham and by this author [12–15]. The long article [16] gives a more or less exhaustive and coherent overview of the approach. More recent developments are the description of arbitrary states by probability measures [9] and further developments [10] concerning the new form of quantum logic that constitutes a central part of the topos approach. For background, motivation, and the main ideas, see also Isham's Chapter 3 in this volume.