### Other axiomatic approaches to the foundations of quantum mechanics

Pioneers in this area were Garreth Birkhoff & John von Neumann, George Mackey, Joseph Jauch & Constantin Piron, Dave Foulis & Charles Randall and Gunther Ludwig. Survey's are:

- Bob Coecke, David Moore and Alex Wilce's introduction to the volume with the same title, Operational quantum logic: An overview.
- Alex Wilce's entry in Stanford's Philosophy Encyclopedia, Quantum Logic and Probability Theory.

The founding paper was:

- Birkhoff, G. and von Neumann, J. (1936),
*The Logic of Quantum Mechanics*, Annals of Mathematics 37, 823—843.

There are also the tracts:

- Mackey, G.W. (1963),
*Mathematical Foundations of Quantum Mechanics*, W.A. Benjamin Inc. - Piron, C. (1976),
*Foundations of Quantum Physics*, W.A. Benjamin Inc. - Ludwig, G. (1985; 1987), An Axiomatic Basis of Quantum Mechanics:
*1. Derivation of Hilbert Space*; 2.*Quantum Mechanics and Macrosystems*, Springer-Verlag.

Some current developments are:

- Giacomo Mauro D'Ariano Operational Axioms for Quantum Mechanics
- Howard Barnum, Jonathan Barrett, Matthew Leifer and Alexander Wilce, Cloning and Broadcasting in Generic Probabilistic Theories
- Lucien Hardy Quantum Theory From Five Reasonable Axioms
- Hesteness and Hiley
- David Hestenes, Primer for Geometric Algebra
- (need reference for Hiley)

- Robert W. Spekkens In defense of the epistemic view of quantum states: a toy theory
- Dynamic operational quantum logic: Daniel-MFP-bxl-Baltag-Smets

### Categories in logic and foundations of computing

A Milestone:

- Jean-Yves Girard's Linear Logic (1987; orriginal paper).

Formally, this logical development stands orthogonal to Birkhoff/von Neumann quantum logic. It is rather this logic and not Birkhoff/von Neumann quantum logic which provides the logical foundation for the Monoidal approach, making the ability to copy and delete premisses explicit. Although not yet as articulated and exploited as Girrard's Linear Logic, one could argue that Linear Logic was already present in Jim Lambek's earlier work on mathematical linguistics (1956) and categorical logic (1970's). The categorical semantics of Linear Logic based on Mike Barr's *-autonomous categories is in:

- Robert Seely's Linear logic, *-autonomous categories and cofree coalgebras.

More recent computer science motivated developments are:

- Samson Abramsky's papers Interaction categories and Semantics of interaction.
- Marcelo Fiore's paper Differential structure in models of multiplicative biadditive intuitionistic linear logic and talks An axiomatics and a combinatorial model of creation/annihilation operators and differential structure & Adjoints and Fock space in the context of profunctors.

### Earlier work on categories in foundations of physics

- David A. Edwards' paper The Mathematical Foundations of Quantum Mechanics introduces category theory within quantum logic.

### Categories in mathematical physics

- Categories in topological quantum field theory: see Joachim Kock's book Frobenius algebras and 2D topological quantum field theories.
- Categories in quantum groups: see Ross Street's book Quantum Groups. A Path to Current Algebra.
- Categories in knot theory: see David Yetter's book Functorial knot theory. Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants.
- Categories in non-commutative geometry: see the paper Non-commutative geometry, categories and quantum physics by Bertozzini, Conti and Lewkeeratiyutkul for a "panoramic view".
- Functorial quantum field theory

## General category theory resources

- Categories mailing list.
- Jake's list of some Category theory resources.
- Visit the popular blog,
*The n-Category Café*. - Learn Category Theory via the Catsters youtube channel.
- Theory and Applications of Categories (TAC) journal.

## General physics resources

- Loads of quantum related stuff is available from quantalk.
- arXiv/quant-ph paper archive.