“What is a Thing?”: Topos Theory in the Foundations of Physics

Abstract

The goal of this article is to summarise the first steps in developing 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. In doing so we provide a new answer to Heidegger's timeless question "What is a thing?".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 uses the topos of sets. Other theories involve a different topos. For the types of theory discussed in this article, a key goal is to represent any physical quantity A with an arrowȂ φ : Σ φ → R φ where Σ φ and R φ are two special objects (the "state object" and "quantity-value object") in the appropriate topos, τ φ .We discuss two different types of language that can be attached to a system, S. The first, PL (S), is a propositional language; the second, L(S), is a higher-order, typed language. Both languages provide deductive systems with an intuitionistic logic. With the aid of PL(S) we expand and develop some of the earlier work 1 on topos theory and quantum physics. A key step is a process we term "daseinisation" by which a projection operator is mapped to a sub-object of the spectral presheaf Σ-the topos quantum analogue of a classical state space. The topos concerned is Sets V(H)op : the category of contravariant set-valued functors on the category (partially ordered set) V(H) of commutative sub-algebras of the algebra of bounded operators on the quantum Hilbert space H.There are two types of daseinisation, called "outer" and "inner": they involve approximating a projection operator by projectors that are, respectively, larger and smaller in the lattice of projectors on H.We then introduce the more sophisticated language L(S) and use it to study "truth objects" and "pseudo-states" in the topos. These objects play the role of states: a necessary development as the spectral presheaf has no global elements, and hence there are no microstates in the sense of classical physics.One of the main mathematical achievements is finding a topos representation for self-adjoint operators. This involves showing that, for any bounded, self-adjoint operatorÂ, there is a corresponding arrowδ o (Â) : Σ → R where R is the quantity-value object for this theory. The construction ofδ o (Â) is an extension of the daseinisation of projection operators.The object R can serve as the quantity-value object if only outer daseinisation of self-adjoint operators is used in the construction of arrowsδ o (Â) : Σ → R . If both inner and outer daseinisation are used, then a related presheaf R ↔ is the appropriate choice. Moreover, in order to enhance the applicability of the quantity-value object, one can consider a topos analogue of the Grothendieck extension of a monoid to a group, applied to R (resp. R ↔ ). The resulting object, k(R ) (resp. k(R ↔ )), is an abelian group-object in τ φ .Finally we...

The Space of Measurement Outcomes as a Spectral Invariant for Non-Commutative Algebras

“…The assignment p → (C(p), p), to be defined below in Theorem 7, injects traditional orthomodular quantum logic of projections into our intuitionistic quantum logic. A similar embedding can be found in [8,Sect. 5.5] and [14,Sect.…”

“…By Zorn, 8 each commutative subalgebra is contained in a maximal commutative one. Hence the collection of maximal commutative subalgebras is dense.…”

“…Proof The Kochen-Specker theorem can be reformulated as the non-existence of certain global sections [4,8,14]. This connection carries over essentially unchanged to the present situation.…”

Modality and Contextuality in Topos Quantum Theory

“…Over the past twenty years, the topos theoretic reformulation of quantum theory (TQT) has been developed by Isham, Butterfield and Döring (e.g [1,2,7]), amongst others. 3 TQT was originally motivated largely by the Kochen-Specker theorem, which states the impossibility of simultaneously assigning classical truth values to all of the projections onto a Hilbert space (of dimension greater than 2) in a way that respects the functional relations between those projections.…”

“…We will subsequently go on to study the implications of this relationship for the problem of defining modal operators in TQT. 1,2 In Section 2 we will give a short introductory account of some of the key concepts and results from the topos theoretic reformulation of quantum mechanics. In Section 3, we will describe the logical structure that arises naturally from this reformulation, paying special attention to the fact that the algebra of propositions associated with a quantum system is not an orthomodular lattice, but rather a complete bi-Heyting algebra.…”