Andreas Doring

A topos foundation for theories of physics: I. Formal languages for physics

This paper is the first 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 discuss two different types of language that can be attached to a system S. The first is a propositional language PL(S); the second is a higher-order, typed language L(S). Both languages provide deductive systems with an intuitionistic logic. The reason for introducing PL(S) is that, as shown in Paper II of the series, it is the easiest way of understanding, and expanding on, the earlier work on topos theory and quantum physics. However, the main thrust o...

A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory

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 ...

A Topos Foundation for Theories of Physics: IV. Categories of Systems

This paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics. The motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity. 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. The previous papers in this series are concerned with implementing this program for a single system. In the present paper, we turn to considering a collection of systems; in particular, we are interested in the relation between the topos representation for a composite system and the representations for its constituents. We also study this problem for the disjoint sum of two systems. Our approach to these matters is to construct a category of systems and to find a topos representation of the entir...

A topos foundation for theories of physics: III. The representation of physical quantities with arrows δ̆o(A):Σ̱→R≽̱

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 δo(A):Σ→R≽, where R≽ is the quantity-value object for this theory. The construction of δo(A) 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 a...

Kochen-Specker Theorem for von Neumann Algebras

The Kochen–Specker theorem has been discussed intensely ever since its original proof in 1967. It is one of the central no-go theorems of quantum theory, showing the non-existence of a certain kind of hidden states models. In this paper, we first offer a new, non-combinatorial proof for quantum systems with a type In factor as algebra of observables, including I∞. Afterwards, we give a proof of the Kochen–Specker theorem for an arbitrary von Neumann algebra

Topos Theory and ‘Neo-Realist’ Quantum Theory

{\cal R}

Observables as functions: Antonymous functions

without summands of types I1 and I2, using a known result on two-valued measures on the projection lattice