How essential is non-commutative algebra in theoretical physics?

[Grassman1]

For next term, I was wondering if I should take a course in non-commutative algebra. As of now, I'm focusing on mathematical physics, specifically on the mathematical side of string theory; mirror symmetry and field theories. I know that commutative algebra as well as algebraic geometry are definitely essential to my interests, but I'm not sure if non-commutative algebra shows up a lot. The course mainly covers semisimple rings and modules, module categories and Morita theory, Noetherian rings, basic dimension theory, and projective modules as an introduction to algebraic k-theory. So would these topics be vital to my interests?

Thanks!

 

[chiro]

Hey Grassman1.

If you are going to study any exotic systems like Spinor's and Clifford-based algebra approaches or others (even those involving something like Quaternions which are non-commutative) then I think it will be useful.

Non-commutative geometry also comes up a bit in theoretical physics where the mathematics put theories in the framework of high-dimensional groups and algebras. I guess one of the uses is that if you have a geometric intuition in higher-dimensions then putting the physics in that form will give you that particular insight.

Just remember that the whole Grassmann approach to geometry was that they were trying to find a way where division on vectors made sense: in other words, you could take vectors and find a way so that a*b*b^(-1) = a and this idea was pivotal to the work that in my opinion started it all - the exterior algebra's and the interior algebras which lead to the bi-vector approach offered by Grassmann.

It got extended with Hamilton (quaternions) and later people like Clifford tried to look at going even higher. On top of this some other extensions which take the way complex numbers work (you have i^2 = -1 instead of using sines and cosines) with multiplication tables and developing theories of high dimensional algebras where you would use multiplication tables instead of some kind of generator for a particular class of algebra's in some dimension with some property.

If the geometric insight exists for any algebraic and geometric (by this I mean the pure mathematics versions) constructions, then by all means use it as this is going to be the real benefit for using these complicated constructions.

Apart from looking at the high level theoretical physics with exotic algebra's and geometries, I would also recommend looking at works which take say a quaternion based approach to existing physics because you can understand quaternions quite well if you visualize their operations in terms of rotations and moving around on a sphere. This is what I am getting at with the geometric intuition. The algebraic intuition is a lot different and its geometric properties may not be as easy to identify or translate to.

 

[mathwonk]

I am not knowledgeable about physics, but I have been invited to lecture about mathematics to physicists, and my physics acquaintances once told me several decades ago, that theoretical physics at that time was essentially equivalent to group representations, the fundamental subject in non commutative algebra.

I.e. the basic object in algebra is the ring of matrices, a semi simple non commutative ring. These arise even if you start from a commutative ring. I.e. if R is a commutative ring, and M is an R module, then the set of R module maps from M to itself is a non commutative ring, an abstract version of the matrix ring. I.e. This ring of maps is a matrix ring precisely when M is a "free" R module, isomorphic to a direct sum of copies of R itself. Projective modules are "locally free" modules, the algebraic version of vector bundles, very basic in differential geometry and physics.

From another viewpoint, a very basic object is a vector space, and the ring of linear maps from a vector space to itself is a fundamentally important non commutative ring. Given any physical object, another fundamental invariant associated to it is its (usually non abelian) group of symmetries, e.g. its rigid motions.

So one cannot really avoid non commutative algebra. As you know, the vector cross product is non commutative, as is the wedge product of differential forms, both of which are used in basic physics.