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:

• There are versions of matrix mechanics describing both quantum and classical physics. Both involve dagger-compact categories. There is a no-cloning theorem in classical mechanics.

• The study of duality unifies real, complex and quaternionic quantum mechanics into a single theory which is already implicit in standard physics.

• Dagger-compact categories are the n = 1, k = 3 example of k-monoidal n-categories with duals — the case most relevant to particles in 4d spacetime, but just one of many.

• Treating profunctors as categorified linear operators relates propositional logic to categorified 2d topological quantum field theories in a somewhat mysterious way.

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

For more on this subject, try these introductory papers:

• John Baez, Quantum Quandaries: A Category-Theoretic Perspective.

• John Baez and Mike Stay, Physics, Topology, Logic and Computation: A Rosetta Stone.

• John Baez and Aaron Lauda, A Prehistory of n-Categorical Physics.

• Bob Coecke, Kindergarten Quantum Mechanics.

• Samson Abramsky and Bob Coecke, A Categorical Semantics of Quantum Protocols.

• John Baez and James Dolan, Higher-Dimensional Algebra and Topological Quantum Field Theory.

• John Baez, Higher-Dimensional Algebra II: 2-Hilbert Spaces.

• Dion Coumans and Bart Jacobs, Scalars, Monads, and Categories.

• Freeman J. Dyson, The Threefold way: Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics, Jour. Math. Phys. 3 (1962), 1199-1215.

• Aaron Fenyes, There's No Cloning in Symplectic Mechanics.

• Grigori Litvinov, The Maslov Dequantization, Idempotent and Tropical Mathematics: a Brief Introduction.

• Jacob Lurie, On the Classification of Topological Quantum Field Theories.

• Jeffrey Morton, Categorified Algebra and Quantum Mechanics.

• Peter Selinger, Dagger Compact Closed Categories and Completely Positive Maps.

• Ross Street, Frobenius Monads and Pseudomonoids.

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