A topos foundation for theories of physics: III. The representation of physical quantities with arrows delta(o)(A) : (Sigma)under-bar ->(R->=)under-bar

摘要

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 delta(o)(A):(Sigma) under bar ->(R->=) under bar, where (R->=) under bar is the quantity-value object for this theory. The construction of delta(o)((A) over cap) is an extension of the daseinisation of projection operators that was discussed in Paper II. The object (R->=) under bar is a monoid object only in the topos, tau(phi) = SetsV(H)(op), of the theory, and to enhance the applicability of the formalism, we apply to (R->=) under bar a topos analog of the Grothendieck extension of a monoid to a group. The resulting object, k((R->=) under bar), is an abelian group object in tau(phi). We also discuss another candidate, (R-<->) under bar, 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. (C) 2008 American Institute of Physics.