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

Abstract

AbstractThe 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 arrow \(\breve{A}_\phi:\varSigma_\phi\rightarrow{\cal R}_\phi\) where Σ φ and \({\cal R}_\phi\) 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, \(\mathcal{PL}(S)\), is a propositional language; the second, \(\mathcal{L}({S})\), is a higher-order, typed language. Both languages provide deductive systems with an intuitionistic logic. With the aid of \(\mathcal{PL}(S)\) we expand and develop some of the earlier work1 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 \({\underline{\varSigma}}\)—the topos quantum analogue of a classical state space. The topos concerned is \(\textbf{Sets}{\cal V}({\cal H})^{\textrm{op}}\): the category of contravariant set-valued functors on the category (partially ordered set) \({\cal V}({\cal H})\) of commutative sub-algebras of the algebra of bounded operators on the quantum Hilbert space \({\cal H}\).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 \({\cal H}\).We then introduce the more sophisticated language \(\mathcal{L}({S})\) 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 operator \({\hat A}\), there is a corresponding arrow \({\breve{\delta}^o(\hat{A})}:{\underline{\varSigma}}\rightarrow{\underline{{\mathbb{R}}^{\succeq}}}\) where \({\underline{{\mathbb{R}}^{\succeq}}}\) is the quantity-value object for this theory. The construction of \({\breve{\delta}^o(\hat{A})}\) is an extension of the daseinisation of projection operators.The object \({\underline{{\mathbb{R}}^{\succeq}}}\) can serve as the quantity-value object if only outer daseinisation of self-adjoint operators is used in the construction of arrows \({\breve{\delta}^o(\hat{A})}:{\underline{\varSigma}}\rightarrow{\underline{{\mathbb{R}}^{\succeq}}}\). If both inner and outer daseinisation are used, then a related presheaf \({\underline{\mathbb{R}^{\leftrightarrow}}}\) 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 \({\underline{{\mathbb{R}}^{\succeq}}}\) (resp. \({\underline{\mathbb{R}^{\leftrightarrow}}}\)). The resulting object, \({k({\underline{{\mathbb{R}}^{\succeq}}})}\) (resp. \(k({\underline{\mathbb{R}^{\leftrightarrow}}})\)), is an abelian group-object in τ φ .Finally we turn to considering a collection of systems: in particular, we are interested in the relation between the topos representation of a composite system, and the representations of its constituents. Our approach to these matters is to construct a category of systems and to find coherent topos representations of the entire category.