Do Category Theory and/or Quantum Logic add value in physics?
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
- Why are bundles so important in physics
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
- Higher Structures in Mathematics and Physics
For more on higher stuff in string theory see
- PhysicsForums Insights -- Why Higher Category Theory in Physics
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.
http://arxiv.org/abs/hep-th/0104200
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.