Noncommutative Algebra

I am a Noncommutative Algebraist, which means that I study objects in algebra. For instance, integers have commutative multiplication (e.g., 3x5 always equals 5x3), but matrices do not (e.g., given matrices A and B, we have that A x B usually does not equal B x A). If we think more abstractly, then we could construct more interesting examples of noncommutative systems. Consider doing laundry, for example. The order in which one washes clothes and dries clothes matters; one could end up with clean and dry laundry, or clean and wet laundry! In fact, “washing” and “drying” are functions with inputs and outputs both being clothes. The act of washing then drying, or drying then washing, is the multiplication (or more technically, the composition) of such functions, which we have shown is noncommutative.

There is an abundance of examples of noncommutative functions (besides doing laundry) that arise in our everyday lives and in nature. Noncommutative algebras can be used to model their behavior. One important illustration of this arose in the first developments of quantum mechanics in the mid-1920’s. While modeling the behavior of subatomic particles, Werner Heisenberg noticed that their dynamical variables (e.g., momentum, position) should be represented by matrices. A surprising outcome of this observation is summarized in Heisenberg’s Uncertainty Principle:

The more precisely the position is determined,
the less precisely the momentum is known in this instant, and vice versa.”
–Werner Heisenberg, 1927

More explicitly, if P and M represent the position and the momentum of a particle, respectively, then their relationship is determined by the fundamental equation of quantum mechanics: PM - MP ih. Here, i is the square root of -1 and h is Planck’s constant. Oftentimes, one normalizes this equation to yield:
(†)        PM - MP = 1.

Since PMMP, one concludes that the algebraic behavior of a particle is indeed noncommutative. The algebraic model for this phenomenon is the first Weyl algebra W, which is a structure generated by two variables x and y, subject to the relation:
xy - yx = 1.

Many mathematicians are interested in studying algebraic properties of structures like Weyl algebras, particularly algebras that arise in physics. One natural question is:

Question. Given an algebra generated in variables x1,…,xn, subject to relations
f1(x)=…=fm(x) = 0,
what are the matrix solutions A1,…,An to this system of equations?

Only square matrices of a prescribed dimension are under consideration. Of course, explicit matrix solutions are desired, especially for physical purposes, but oftentimes this is difficult to compute. Instead, one may ask: Do d x d matrix solutions exist? If so, for which d? Is d finite? In addition, can such solutions be parametrized by a geometric object, like a curve or a surface? All of these questions lie in the rapidly expanding area of representation theory, which is a subfield of noncommutative algebra.

Returning to the example of the Weyl algebra W, recall that we are interested in matrix solutions to the equation: xy - yx = 1. In other words, are there d x d matrices P and M so that PM - MP = I, where I is the d x d identity matrix? If d is a finite number, then the answer is “no". The proof is this fact is quite short:

• For a d x d matrix A with d a *positive* finite number, the trace of A, denoted by tr(A), is the sum of its diagonal elements.

• For square matrices A and B, we have that
tr(AB) = tr(BA) and tr(A - B) = tr(A) - tr(B).

• Hence, if PM - MP = I, then
0 = tr(PM) - tr(MP) = tr(PM - MP) = tr(I) = d.

• So, d = 0, which is impossible.

• We conclude that for d finite, there is no matrix solution to the equation, xy - yx = 1.

On the other hand, there are infinite dimensional matrix solutions to xy - yx = 1. Take for instance,

## matrices.png

In this case, we say that W has no finite dimensional representations, yet has an infinite dimensional representation. Thus, given Heisenberg’s Uncertainty Principle, or the normalized fundamental equation (†), we conclude that the position and the momentum of a subatomic particle at an instant must be modeled by infinite dimensional matrices.

Since then, noncommutative algebra has played a vital role in the understanding of quantum phenomena. The philosophy is that one should not try to understand the quantum objects themselves, but rather the functions of them. Moreover, as functions are part of our everyday lives, the implications of noncommutative algebra are endless.

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