I know they have their adherents, but do more or less esoteric branches of mathematics such as Category Theory and/or Quantum Logic provide powerful tools for new theory development or are they just occasionally-useful presentational frameworks?
More particularly, is it worth investing the time to study these formalisms?
Recently, it is realized that quantum many-body states can be divided into short-range entangled states and long-range entangled states.
The quantum phases with long-range entanglements correspond to topologically ordered phases, which, in two spatial dimensions, can be described by tensor category theory (see cond-mat/0404617). Topological order in higher dimensions may need higher category to describe them.
One can also show that the quantum phases with short-range entanglements and symmetry $G$ in any dimensions can be "classified" by Borel group cohomology theory of the symmetry group. (Those phases are called symmetry protected trivial (SPT) phases.)
The quantum phases with short-range entanglements that break the symmetry are the familar Landau symmetry breaking states, which can be described by group theory.
So, to understand the symmetry breaking states, physicists have already been forced to learn group theory. It looks like to understand patterns of many-body entanglements that correspond to topological order and SPT order, we will be forced to learn tensor category theory and group cohomology theory. In modern quantum many-body physics and in modern condensed matter physics, tensor category theory and group cohomology theory will be as useful as group theory. The days when physics students need to learn tensor category theory and group cohomology theory are coming, may be soon.
The reason that people get away with ignoring category theory and homotopy theory in physics so much is that physics is already so rich locally and in perturbative approximations. But a general fact is that all global and non-perturbative effects, hence everything that concerns the full story, is fairly intractable without the toolbox of higher categroy/higher homotopy theory.
This begins with comparatively simple issues such as
but it doesn't stop there. It is remarkable that if one doesn't cheat all over the place, then many familiar types of systems in physics need tools from homotopy theory and category theory for the full description. This starts with archetypical examples such as charged and spinning particles and it becomes more and more true as one passes from there to charged and spinning strings, then charged and spinning membranes, ect.
A gentle exposition of how higher stuff is all over the place in physics is in this talk
For more on higher stuff in string theory see
My short answer is No, they're not too useful, but let me discuss some details, including positive ones.
Categories, especially derived categories, have been appropriate to describe D-brane charges - and not only charges - beyond the level accessible by homology and K-theory. See e.g.
However, I feel it is correct to say that the string theorists who approached D-branes in this way did so because they first learned lots of category theory - in mathematics courses - and then they tried to apply their knowledge.
I am not sure that a physicist would "naturally" discover the categories - or even formulated them in the very framework how they're usually defined and studied in mathematics. And on the contrary, I guess that the important qualitative as well as quantitative insights about the D-branes - including the complicated situations where category theory has been relevant - could have been obtained without any category theory, too.
But of course, people have different reactions to these issues and these reactions reflect their background. And I - a non-expert in category theory - could very well be missing something important that the category theory experts appreciate while others don't.
Most famously, Joe Polchinski - the very father of the D-branes - reacted wittily to the notion that the D-branes should have been rephrased in terms of category theory. In a talk, he spoke about an analogy with a dog named Ginger. We tell Ginger not to do many things and do others, Ginger. What Ginger hears is "blah blah blah blah Ginger blah blah blah".
In a similar way, Polchinski reprinted "what mathematicians say". It was a complicated paragraph about derived categories and their advanced methodology applied to D-branes. What Joe hears is "blah blah blah blah D-branes blah blah blah blah T-duality blah blah D-branes blah."
Some physicists also try to generalize gauge theory to some "higher gauge theory" using category theory but I don't think that there are any consistent and important theories of this kind. What they're doing is similar to the theories with $p$-forms and extended objects except that they don't do it right.
As always, category theory may offer one a rigorous language to talk about analogies etc. - but I don't think that physicists need anything beyond the common-sense understanding how the method of analogies works. So if you learn category theory - which is pretty tough - I think you should have better reasons than a hope that the theory could be useful for physics.
Nigel,
A few comments. Firstly I think we should separate the questions for Category Theory and Quantum Logic here, as I think they are rather different in a sense I will explain below.
Category Theory: My view is that most physicists look for insights and theories from a Geometric viewpoint. This somewhat unites Relativity Theorists and String Theorists. Also the Fibre Bundle model of particle physics (Yang-Mills) is largely a geometric structure too. So mainstream physicists are likely to continue to look to geometric models for answers and insights. This doesnt have to be "narrowly geometric" however, as there is some interesting work on Topological insights into physics: solitons, etc. However the most interesting material to me has been the topological discoveries which have a geometric form. Even Statistical Mechanics has some topological theorems of this sort.
Having said this there is a "school" of fundamental "pre-geometry" physicists (which can be found at another site that some of the Stack physics group use called the "Institute for Fundamental Questions"). This link is to an author there I have briefly interacted with:
http://www.fqxi.org/community/forum/topic/482
They may not use Category Theory itself however, but some are looking for a way to derive spacetime geometry and quantum theory (thereby providing a framework for Quantum Gravity) from something else: but what can that "something else" mathematically be? Chris Isham quoted in another Answer is motivated by this too.
Futhermore although Category Theory is billed as a "unifying" theory, physicists tend to get excited by more specific unifications between specific models: the String Theorists have shown us that too, although such unifications also (seem to) happen outside of both String Theory and Category Theory too.
EDIT: There is an online Category Theory, Philosophy and Physics Wiki: a bit technical in detail but some sections outline why the researchers believe that Category Theory is useful in their area e.g in Path Integral Modelling.
The unifying attraction of Category Theory in a less Geometric world, like Abstract Algebra or Software Engineering Logics is higher - it might have value there.
Quantum Logic: Unlike mathematical theories or Software Logics this arose as a result of a theorem from Von Neumann about what Quantum Mechanics was supposed to be telling us about physical reality. However this theorem has been disputed, see http://write4science.com/Assets/pdfs/Global%20logic.pdf (around page 61 of 74), and although controversy still exists quantum logic per se seems to dead. It has left us with related lattice structures though, and again these might prove useful to those interested in a novel "Quantum Gravity Foundation".
Roy.
Extra Note: Chris Isham has also managed to keep QL going with his Temporal Quantum Logic http://en.wikipedia.org/wiki/HPO_formalism.
John Baez, a well-known mathematical physicist presently (2014) in the Department of Mathemetics at the University of California, Riverside is trying to find more and more ways that category theory connects with physics, chemistry, electrical engineering, network theory, etc. You can read more about his efforts:
Category theory has some potential for physics. Quantum logic I am less sanguine about. It has always struck me as a way of expressing something we understand in set theoretic language. It has always struck me as a formalistic study that brings little additional content.
Philip Goyal demonstrated how the above summation over intermediate points is equivalent in a discrete setting to the concatenation of measurements, where the summation is over all possible outcomes [1]. The complex amplitudes are products of complex numbers, and in a discrete setting this is a multiplication rule which requires complex numbers. The result is that quantum mechanics is reduced to a simple system of associative and commutative mathematics of complex numbers with no reference to classical mechanics, or any notion of space or spacetime. However, the one requirement is that the points intermediate to the initial and final points be intermediate in time.
The Goyal logic is a summation of Stern-Gerlach experiments. The intermediate point corresponds to some intermediate measurement between the source of particles and the final Stern-Gerlach (SG) apparatus. If the outcome of the intermediate SG apparatus is ignored, or no measurement is performed of their outcomes, the split beams recombine as a discrete summation. So the intermediate SG apparatus represents a sum over elementary quantum events. This summation in the complex algebra corresponding to this logic recovers the quantum superposition.
These summations over SG experiments occur in a sequence. There is no ambiguity in the ordering of these events. Further, the process appears well defined in a discrete setting. The Goyal approach for discrete quantum mechanics, even if the number of elements is enormous, but not infinite, indicates some sort of quantization of time, and a discrete spacetime.
This has elements of Zariski topology. Consider the affine space $A_n$ as the n-dimensional space over a closed field $F$. The topology is constructed from closed sets defined by the polynomial set $S~\in~F$ by $$ V (S)~=~\{x~\in~A_n| f(x)~=~0;~\forall f~ \in~S\} $$ For two polynomials in the set S we have the following rules: $$ V (p_1)\cup V (p_2)~=~V (p_1\times p_2),~ V (p_1)\cap V (p_2)~=~ V (p_1~+~p_2) $$ which serve as the representation map between the logic of outcomes and the algebra of quantum operations demonstrated by Goyal. This closed set topology defines the Zariski topology on the affine set $A_n$. So a connection to quantum mechanics exists within this system with respect to Zariski topology. This is the topology of {\’E}tale and Grothendieck, or topos theory. An overview of topos theory in physics is in [2] by Isham.
[1] P. Goyal, K. H. Knuth, J. Skilling, "Origin of Complex Quantum Amplitudes and Feynman's Rules," {\it Phys. Rev.} {\bf A 81}, 022109 (2010) http://arxiv.org/abs/0907.0909
[2] C. J. Isham, "Topos Methods in the Foundations of Physics," {\it"Deep Beauty,} ed. Hans Halvorson, Cambridge University Press (2010) http://arxiv.org/abs/1004.3564
