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.
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.
We provide 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, we will show all constructions explicitly for a three-dimensional Hilbert space and the spin-z operator of a spin-1 particle. This article is a companion to the article by Isham in the same volume.
The basic notion of how topoi can be utilized in physics is presented here. Topos and category theory serve as valuable tools which extend our ordinary set-theoretical conceptions, can further the study of quantum logic and give rise to new and 'neo-realistic' descriptions of quantum physics, i.e. make possible the construction of a general scheme for quantum physics, which 'looks like' the classical one. A novel way of writing theories of physics, by representing a formal language of a physical system, into an appropriate topos, is analyzed here. This novel approach is initiated by C.J. Isham and A. Doering and the major part of this paper comes from there; however there has been an effort in simplifying the basic ideas and do this work accessible to 'less experts' of the field. A minor, but not less important, part of this paper is devoted to resuming, in a concise way the background knowledge of logic, quantum logic and category theory, that is necessary for anyone who wishes to follow the whole topos approach.
This paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper, we study in depth the topos representation of the propositional language, PL(S), for the case of quantum theory. In doing so, we make a direct link with, and clarify, the earlier work on applying topos theory to quantum physics. The key step is a process we term “daseinisation” by which a projection operator is mapped to a subobject of the spectral presheaf—the topos quantum analog of a classical state space. In the second part of the paper, we change gear with the introduction of the more sophisticated local language L(S). From this point forward, throughout the rest of the series of papers, our attention will be devoted almost entirely to this language. In the present paper, we use L(S) to study “truth objects” in the topos. These are objects in the topos that play the role of states: a necessary development as the spectral presheaf has no global elements, and hence, there are no micro-states in the sense of classical physics. Truth objects therefore play a crucial role in our formalism.
This paper is the third in a series whose goal is to develop a fundamentally new way of viewing theories of physics. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. In Paper II, we studied the topos representations of the propositional language PL(S) for the case of quantum theory, and in the present paper we do the same thing for the, more extensive, local language L(S). One of the main achievements is to find a topos representation for self-adjoint operators. This involves showing that, for any physical quantity A, there is an arrow δ̆o(A):Σ̱→R≽̱, where R≽̱ is the quantity-value object for this theory. The construction of δ̆o(Â) is an extension of the daseinisation of projection operators that was discussed in Paper II. The object R≽̱ is a monoid object only in the topos, τϕ=SetsV(H)op, of the theory, and to enhance the applicability of the formalism, we apply to R≽̱ a topos analog of the Grothendieck extension of a monoid to a group. The resulting object, k(R≽̱), is an abelian group object in τϕ. We also discuss another candidate, R↔̱, for the quantity-value object. In this presheaf, both inner and outer daseinisations are used in a symmetric way. Finally, there is a brief discussion of the role of unitary operators in the quantum topos scheme.
What is the role of topos theory in the topos models for quantum theory as used by Isham, Butterfield, Doring, Heunen, Landsman, Spitters and others? In other words, what is the interplay between physical motivation for the models and the mathematical framework used in these models? Concretely, we show that the presheaf topos model of Butterfield, Isham and Doring resembles classical physics when viewed from the internal language of the presheaf topos, similar to the copresheaf topos model of Heunen, Landsman and Spitters. Both the presheaf and copresheaf models provide a `quantum logic' in the form of a complete Heyting algebra. Although these algebras are natural from a topos theoretic stance, we seek a physical interpretation for the logical operations. Finally, we investigate dynamics. In particular we describe how an automorphism on the operator algebra induces a homeomorphism (or isomorphism of locales) on the associated state spaces of the topos models, and how elementary propositions and truth values transform under the action of this homeomorphism. Also with dynamics the focus is on the internal perspective of the topos.
What is the role of topos theory in the topos models for quantum theory as used by Isham, Butterfield, Döring, Heunen, Landsman, Spitters, and others? In other words, what is the interplay between physical motivation for the models and the mathematical framework used in these models? Concretely, we show that the presheaf topos model of Butterfield, Isham, and Döring resembles classical physics when viewed from the internal language of the presheaf topos, similar to the copresheaf topos model of Heunen, Landsman, and Spitters. Both the presheaf and copresheaf models provide a “quantum logic” in the form of a complete Heyting algebra. Although these algebras are natural from a topos theoretic stance, we seek a physical interpretation for the logical operations. Finally, we investigate dynamics. In particular, we describe how an automorphism on the operator algebra induces a homeomorphism (or isomorphism of locales) on the associated state spaces of the topos models, and how elementary propositions and truth values transform under the action of this homeomorphism. Also with dynamics the focus is on the internal perspective of the topos.