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 orderreversing 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: starautonomous
categories, ncategories 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 daggercompact categories. There is a
nocloning 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.

Daggercompact categories are the n = 1, k = 3 example of
kmonoidal ncategories 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:
Also try these somewhat more technical ones:

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

John Baez and James Dolan,
HigherDimensional
Algebra and Topological Quantum Field Theory.

John Baez,
HigherDimensional
Algebra II: 2Hilbert 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), 11991215.

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.
© 2010 John Baez
baez@math.removethis.ucr.andthis.edu