Ring theory in physics

Question

Surely group theory is a very handy tool in the problems dealing with symmetry. But is there any application for ring theory in physics? If not, what's this that makes rings not applicable in physics problems?

Answer 1

This really comes down to the question of how broadly you define ring theory. Special types of rings appear all over the place in physics, but often their focused study is given a more specialized name. The term "ring theory" is sometimes used to indicate the specific study of rings as a general class, and under that interpretation, the discipline seems to be closer to logic and set theory than questions of current physical relevance.

In any case, rings show up in the following contexts (this list is not comprehensive):

1. The representation theory of a finite or compact group $G$ can be studied from the optic of ring theory, since representations are modules over a (suitably topologized in the infinite case) group ring $\mathbb{C}[G]$.

2. Algebras of operators, such as C* algebras and von Neumann algebras, are rings. The ring of functions on a manifold is a commutative C* algebra, and geometric quantization is done by deforming the product structure on the ring of functions on a symplectic manifold (e.g., taking the ring of functions on the cotangent bundle of a configuration space to the ring of differential operators on the space).

3. Algebraic varieties such as Calabi-Yau varieties and Riemann surfaces show up in some string theory and conformal field theory papers. They are patched together using commutative rings of functions on open sets.

4. Cohomology and K-theory of a topological space form (graded-)commutative rings. I am told that string theorists sometimes view these rings as places where certain charges live.

5. If you like vertex algebras, their representation theory is captured by the module theory of a current algebra, which is a big ring.

6. I'm told that perturbative renormalization can be placed into Wightman's axiomatic framework in a mathematically rigorous fashion, if the framework is suitably generalized by replacing $\mathbb{C}$ with a formal power series ring in one ore more variables, such as $\mathbb{C}[[\lambda]]$, where the variables are the coupling constants, assumed to be infinitesimal. Objects like the Hilbert space of states are replaced with modules over this ring equipped with a sesquilinear form. Borcherds has a recent preprint establishing this.

Answer 2

I don't know any important applications of rings in undergraduate level physics, but there are plenty of applications in the study of supersymmetric field theories. For example, take an $N=2$ supersymmetric $\sigma$-model with whose target space is a Kahler manifold $X$. One can define a set of twisted supercharges $Q$ (the exact form depends on the twisting) and there is a ring structure on the cohomology of $Q$. Classically this is the same as the cohomology ring of $X$, but quantum mechanically there are corrections from instantons. The chiral ring of the Q's is often called the quantum cohomology of $X$. The chiral ring structure also played an important role in the discovery of mirror symmetry. The lectures by E. Witten in "Quantum fields and strings: a course for mathematicians, Volume 2" are a good reference on this for the mathematically inclined. If there are some more down to earth examples where rings are important in physics I'd also be interested to hear about them.

Answer 3

In geometry, spaces are often characterized by their ring of continuous (or smooth) R or C valued functions. This is the basic philosophy of algebraic geometry, noncommutative geometry, and deformation theory, which all have applications to physics (the latter two being almost exclusively motivated by quantum mechanics). For example any compact Hausdorff space can be constructed just from its ring of continuous functions so one could say that a space is a ring. This ring is always commutative since we multiply pointwise and R and C are commutative. The viewpoint of noncommutative geometry is to study noncommutative rings as if they were the ring of continuous functions of some "noncommutative" space. This is motivated by quantum mechanics where the algebra of observables is noncommutative.