(higher) category theory and physics
state, configuration space;
propagation
Lagrangian mechanics?
Axiomatizations
Tools
Models
Phenomena
Types of quantum field thories
Which mathematical structures are natural in physics?
The history of theoretical physics is not the least the story of a search process for suitably notions: while mathematical theories – such as symplectic geometry, group theory, differential geometry, etc – are a priori “just languages” – and each of these languages was upon its introduction to theoretical physics originally met with some hostility (just compare the Gruppenpest (plague of group theory) complaint by nobody less than Wolfgang Pauli) – we do know in retrospect that the modern insights and theorems of, respectively, classical mechanics, quantum mechanics and general relativity would have been literally unthinkable without usage of these languages.
In pure mathematics the result of such a search process for natural notions has been fully established in the last century: category theory – direct generalization in particular of group theory – does not only help to put into order existing mathematical knowledge, it has proven to be necessary for even making thinkable certain insights. Such as set theory in 19th century math does category theory appear today as the natural language of mathematics. This goes as far as making rather philosophical sounding questions such as “What does ‘natural’ mean in mathematics?” attain a formal and useful meaning: indeed category theory was discovered by Eilenberg and MacLane in the process of formalizing this term.
The inclined theoretical-mathematical physicist can hardly regard this development without wondering about concept formation in physics. In view of the in part deeply ontologically satisfactory category theoretic explanations and discoveries in mathematics, the justifiable question is how this convergency to a natural mathematical reality which is clearly being achieved here should reach over to the natural physical reality.
And indeed: quite a few fundamental insights in category theory originated from theoretical physicists who took the time to follow the structures they had to deal with to their proper nature. In the realm of classical physics this notably is true for Bill Lawvere, whose study of continuum mechanics led him to profound categorical formulations of basic physical and ontological concepts (see for instance space and quantity). In the realm of quantum mechanics there is John Roberts, whose work on quantum field theory and in particular on algebraic quantum field theory led , among other things, to formulations of strict omega-category and nonabelian cohomology. Generally, the quest for an understanding of quantum field theory keeps driving category theoretic developments, more recently in the context of the generalized tangle hypothesis.
Lawvere was led from the plain study of differential equations in classical continuum mechanics, via the old question of Newton and Leibniz concerning the indispensable notion of the infinitesimal, to an impressively powerful categorical concept formation which today is well established as topos theory. That Lawevere’s clear-sighted new viewpoint on classical mechanics is only beginning to be appreciated among a few mathematical physicists is less a sign for a justification of the maybe expected allegation of a Topospest than a sign for the distance which can be traversed in the space of categorically formulated physics which is to be explored now: Lawvere goes as far as not only identifying categorical structures in physical phenomena, but pointing out that many fundamental ontological notions do find a useful formalization this way. Such as for instance the duality between space and quantity – a fundamental theme in the development of theoretical physics, back then for the development of general relativity and quantum mechanics just as well as today for string theory and its generalized geometries – which Lawvere identifies with the elementary categorical duality between sheaves and co-sheaves, thereby laying the conceptual groundwork for currently active research on generalized – to a large extent quantum mechanically motivated – spaces, for instance in noncommutative and derived geometry.
Still, the Lawvere-ification of the proper center of modern theoretical physics is still to be done: that of quantum field theory. Established for decades, it is just too easy for the practicing physicist to forget that this current culmination point of our understanding of fundamental physical reality is to a large extent still nothing but a mystery that is waiting to be solved. As in Kepler’s times with the description of the orbits of planets by a handful of ad-hoc postulated rules there is a wealth of cooking recipes which describe many aspects of quantum field theory, but the discovery of the analog of Newton’s explanation of Kepler’s laws as a consequence of a closed theory behind it is still amiis in quantum field theory.
One good approximation to this far goal is certainly the system of algebraic quantum field theory as developed by Haag and Kastler: this axiomatization builds in a remarkably crucial way on the categorical notion of a co-presheaf (the net of local observables of a quantum field theory). More remarkable is maybe only that this fact is hardly mentioned in much of the existing literature:
Even thought its full range for our conceptual understanding of quantum field theory has still barely unraveled, it was early on recognized and followed by John Roberts. Higher sheaves and co-sheaves form the ground stone of a generalized cohomology theory? which only in recent times is being studied more intensively under the headline of infinity-stacks, which in turn are finding their application in quantum field theory (see for geometric function theory).
John Roberts already saw in the 1960s from his quantum field theoretic considerations the necessity of higher category theory. Indeed, Ross Street’s notion of strict omega-category established shortly afterwards builds on Roberts’ early work. Not only that, but Roberts also saw that a general theory of cohomology – which he saw as the home of certain quantum field theoretic invariants – was to be described in terms of (co)sheaves with coefficients in such higher categories, so in terms of infinity pre-stacks as one would say today.
This remarkable amount of fundamental category theoretic concept formation which has been extracted out of theoretical physics in particular by Lawvere and Roberts should probably make both theoretical physicists as well as mathematicians and philosophers interested in natural sciences think.
In parts this is already the case. For instance the second existing proposal for an axiomatization of quantum field theory – Atiyah and Segal's definition of a quantum field theory as a functorial representation of cobordism categories – this, too, an fundamental concept which is hardly thinkable without category theoretic language – has led to remarkable activity. A couple of comparatively simple special cases of quantum field theories on topologically nontrivial spaces have been made accessible to a rigorous mathematical classification: besides topological quantum field theories for exmaple rational conformal field theories as well as cohomological field theories. Only recently did Lurie and Hopkins, building on Joyal’s notion of (infinity,1)-category, formalize and prove the by now ten year old Baez-Dolan hypothesis on extended topological quantum field theory in the language of (infinity,n)-categories.
Also on the closely related area of the theory of open topological strings and string-theoretical mirror symmetry has categorical language – not the least through work of Kontsevich and collaborators on generalized geometries – has categorical language become indispensable. In these areas of mathematica-theoretical physics categroy theoretic language is so well established that its use is not much noticed anymore among practitioners. This is the clearest sign of a natural language.
…
…
…
John Baez and Aaron Lauda, A prehistory of -categorical physics (pdf)
John Baez, Quantum Quandaries: a Category-Theoretic Perspective (arXiv)
John Baez and Mike Stay, Physics, topology, logic and computation: a Rosetta Stone (arXiv)
Daniel S. Freed, Higher Algebraic Structures and Quantization (arXiv)
But for one the most important points there is to date no good comprehensive survey: lots of well-known structures in physics are secretly higher categorical structures. See BV theory.