A Second Course in Topos Quantum Theory

Description

This advanced course, a sequel to the first volume of this lecture series on topos quantum theory, delves deeper into the theory, addressing further technical aspects and recent advances. These include, but are not limited to, the development of physical quantities and self-adjoint operators; insights into the quantization process; the description of an alternative, covariant version of topos quantum theory; and last but not least, the development of a new concept of spacetime. The book builds on the concepts introduced in the first volume (published as Lect. Notes Phys. 868), which presents the main building blocks of the theory and how it could provide solutions to interpretational problems in quantum theory, such as: What are the main conceptual issues in quantum theory? And how can these issues be solved within a new theoretical framework of quantum theory? These two volumes together provide a complete, basic course on topos quantum theory, offering a set of mathematical tools to readers interested in tackling fundamental issues in quantum theory in general, and in quantum gravity in particular. From the reviews of the first volume: The book is self-contained and can be used as a textbook or self-study manual teaching the usage of category theory and topos theory, in particular in theoretical physics or in investigating the foundations of quantum theory in mathematically rigorous terms. [The] book is a very welcome contribution. Frank Antonsen, Mathematical Reviews, December, 2013 Front Matter ....Pages i-xiv Introduction (Cecilia Flori)....Pages 1-7 Logic of Propositions in Topos Quantum Theory (Cecilia Flori)....Pages 9-24 Alternative Group Action in Topos Quantum Theory (Cecilia Flori)....Pages 25-63 Observables in Terms of Antonymous and Observable Functions (Cecilia Flori)....Pages 65-78 Interpreting Self-Adjoint Operators as q-Functions (Cecilia Flori)....Pages 79-105 What Information Can Be Recovered from the Abelian Subalgebras of a von Neumann Algebra (Cecilia Flori)....Pages 107-113 Grothendieck Topoi (Cecilia Flori)....Pages 115-148 Locales (Cecilia Flori)....Pages 149-169 Internalizing Objects in Topos Theory (Cecilia Flori)....Pages 171-188 Geometric Logic (Cecilia Flori)....Pages 189-199 Brief Introduction to Covariant Topos Quantum Theory (Cecilia Flori)....Pages 201-224 Space Time in Topos Quantum Theory (Cecilia Flori)....Pages 225-246 Extending the Topos Quantum Theory Approach (Cecilia Flori)....Pages 247-289 Quantization in Topos Quantum Theory: An Open Problem (Cecilia Flori)....Pages 291-313 Back Matter ....Pages 315-342