A First Course in Topos Quantum Theory, 2013 Lecture Notes in Physics Series, Vol. 868

In the last five decades various attempts to formulate theories of quantum gravity have been made, but none has fully succeeded in becoming the quantum theory of gravity. One possible explanation for this failure might be the unresolved fundamental issues in quantum theory as it stands now. Indeed, most approaches to quantum gravity adopt standard quantum theory as their starting point, with the hope that the theory?s unresolved issues will get solved along the way. However, these fundamental issues may need to be solved before attempting to define a quantum theory of gravity.

The present text adopts this point of view, addressing the following basic questions:  What are the main conceptual issues in quantum theory? How can these issues be solved within a new theoretical framework of quantum theory?

A possible way to overcome critical issues in present-day quantum physics ? such as a priori assumptions about space and time that are not compatible with a theory of quantum gravity, and the impossibility of talking about systems without reference to an external observer ? is through a reformulation of quantum theory in terms of a different mathematical framework called topos theory.

This course-tested primer sets out to explain to graduate students and newcomers to the field alike, the reasons for choosing topos theory to resolve the above-mentioned issues and how it brings quantum physics back to looking more like a ?neo-realist? classical physics theory again.

Introduction.- Philosophical Motivations.- Kochen-Specker Theorem.- Introducing Category Theory.- Functors.- The Category of Functors.- Topos.- Topos of Presheaves.- Topos Analogue of the State Space.- Topos Analogue of Propositions.- Topos Analogues of States.- Truth Values.- Quantity Value Object and Physical Quantities.- Sheaves.- Probabilities in Topos Quantum theory.- Group Action in Topos Quantum Theory.- Topos History Quantum Theory.- Normal Operators.- KMS States.- Future Research.- Topos and Logic.- Worked out Examples.