Complex Numbers, One-Parameter of Unitary Transformations and Stone's Theorem in Topos Quantum Theory

Abstract

Topos theory has been suggested first by Isham and Butterfield, and then by Isham and Doering, as an alternative mathematical structure within which to formulate physical theories. In particular, it has been used to reformulate standard quantum mechanics in such a way that a novel type of logic is used to represent propositions. In recent years the topic has been considerably progressing with the introduction of probabilities, group and group transformations. In the present paper we will introduce a candidate for the complex quantity value object and analyse its relation to the real quantity value object. By defining the Grothendieck k-extension of these two objects, so as to turn them into abelian groups, it is possible to define internal one parameter groups in a topos. We then use this new definition to construct the topos analogue of the Stone's theorem.

References in corpus (5)

• A Topos Foundation for Theories of Physics: I. Formal Languages for Physics
• A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory
• A Topos Foundation for Theories of Physics: IV. Categories of Systems
• A Topos Foundation for Theories of Physics: III. The Representation of Physical Quantities With Arrows
• Observables II : Quantum Observables