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Has category theory been used to explore relational physics in the extensive way that Rosen feels is warranted?
No… but it’s starting
… and even a 71 year old can get in on the ground floor.
(PS: read a little up on Rosen his book. Written in 2005, it lacks any mention (that I can see) to category theory. Shame: he is inventing terms and concepts that have already been invented and studied throughly in category theory.)
(PPS: there was a discussion on him in the N-cafe blog in 2007. They call him the “grandfather of categorical biology”… even though they haven’t read his paper. So maybe there is category theory in there….)
(PPPS: (face palm) look up anything by John Baez, category theory, and physics; if not the grandfather than he is certainly the first evangelizer of a categorical viewpoint to physics.)
I believe Rosen’s work on “categorical biology” has been extended by Andrée Ehresmann; see for instance, here.
Building on Thomas’ excellent suggestions, also check out categories for the practicing physicist written by the same person that wrote Kindergarten QM.
Just to highlight that the role of symmtric monoidal categories in quantum physics – which is the content of Baez’s “Quantum quandaries” and “Rosetta stone” and of Coecke’s topic finite quantum mechanics in terms of dagger-compact categories as well as of a plethora on texts on topological fieldtheories – captures only the kinematics (the spaces of states), not the dynamics (the evolution laws). Correspondingly Coecke’s “categories for the practicing physicist” introduces symmetric monoidal categories motivated by the example of quantum kinematics, but does not touch on the many other roles that category theory plays in the discussion of dynamics, where one is concerned with phase spaces, differential equations and other aspects of more geometric nature.
There is for instance Frederic Paugam’s book “Towards the Mathematics of Quantum Field Theory” which follows the more ambitious project of uncovering the correct category-theoretic structures in a more comprehensive formulation of physics, say the kind of physics that the historical figures mentioned in #1 cared about.
Specifically the entry general covariance mentioned in #1 refers to formulation of comprehensive physics in higher topos theory, which around here we have thought about a bit.
The introduction to Paugam’s “Towards a Math of QFT makes it very clear that the purpose of this book is to teach math, not find a categorical physics:
This book is an applied pure mathematics textbook on quantum field theory. Its aim is to introduce mathematicians (and, in particular, graduate students) to the mathematical methods of theoretical and experimental quantum field theory, with an emphasis on coordinate free presentations of the mathematical objects in play, but also of the mathematical theories underlying those mathematical objects.
In fact the first mention of CT is on page 35 (!!!) and most of it reads to me like every other set-based mathematical physics book read during my PhD but with a “dash” of category theory to make talking about coordinate-free systems easier.
Maybe I just can’t see it… but compare this introduction with Coecke’s “Kinder QM” or Caramello’s “Unifying theory” who present category theory (topos) on page one as the central aim as their research program.
Maybe I just can’t see it…
There is certainly some distance between Kindergarten and the real thing.
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