John Baez

Quantum Physics and Logic 7, Oxford University

May 29, 2010

Duality in Logic and Physics

Duality has many manifestations in logic and physics. In classical logic, propositions form a partially ordered set and negation is an order-reversing involution which switches "true" and "false". The same holds in quantum logic, with propositions corresponding to closed subspaces of a Hilbert space. But the full structure of quantum physics involves more: at the very least, the category of Hilbert spaces and bounded linear operators. This category has another kind of duality, a contravariant involution that switches "preparation" and "observation". Other closely related dualities in quantum physics include "charge conjugation" (switching matter and antimatter), "parity" (switching left and right), and "time reversal" (switching future and past). The quest to find a unified mathematical framework for dualities leads to a fascinating variety of structures: star-autonomous categories, n-categories with duals, and more. We give a tour of these, with an effort to focus on conceptual rather than technical issues. A few key points:

You can see slides from the talk and also a video of the talk.

For more on this subject, try these introductory papers:

Also try these somewhat more technical ones:

© 2010 John Baez