Something like category theory but for physics

[Geofleur]

I was impressed how in R. Geroch's book, Mathematical Physics, category theory is used to unify so many different branches of mathematics. Is there a single framework that, in a similar way, unifies many or all branches of physics? If so, what are some good resources for learning it?

So far, the closest I have seen is perhaps the mathematics of linearity, of multi-linear maps and objects on which such maps act. Is there something better? It would be great to be able to see each area of physics as instantiating some one thing.

 

[Andy Resnick]

Classical field theory is one very general framework: classical mechanics, E&M, and thermodynamics can all be treated this way.

 

[Geofleur]

Is there a good book that presents it in this way?

 

[Fredrik]

I think the most general framework is the one that views all theories of physics (including the classical ones) as probability theories. This topic is unfortunately extremely difficult. I don't think there's a good book on the topic, at least not at a level that can be understood by someone below the level of a graduate student in mathematics.

I only understand bits and pieces of it. I think I understand enough to say that a really good book on this topic should identify mathematical structures (lattices, algebras) that can be associated with every set of statements that can be considered a "theory of physics", and then classify theories by additional conditions satisfied by these structures. Then the book should describe the most interesting classes of theories. There is however no such book. The closest thing I can think of is the article "Quantum probability theory" by Redei and Summers, which contains a classification of (generalized) probability theories defined by von Neumann algebras.

http://arxiv.org/abs/quant-ph/0601158

 

[Geofleur]

I only understand bits and pieces of it. I think I understand enough to say that a really good book on this topic should identify mathematical structures (lattices, algebras) that can be associated with every set of statements that can be considered a "theory of physics", and then classify theories by additional conditions satisfied by these structures. Then the book should describe the most interesting classes of theories.

That's exactly the sort of thing I had in mind with my question! If there are no books on the subject, and if the only paper is about the quantum aspects, perhaps this would be a good research topic?

 

[Fredrik]

The paper mentions that non-commutative von Neumann algebras lead to classical theories, so the classical theories are included in the framework defined by the paper. Unfortunately I don't understand this well enough to explain the details.

 

[Geofleur]

Thanks for the information, I appreciate it!

 

[Andy Resnick]

Is there a good book that presents it in this way?

The best source (that I have found so far) is here:

http://link.springer.com/book/10.1007/978-3-642-45943-6

and here:

http://link.springer.com/book/10.1007/978-3-642-46015-9