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.

*The more precisely the position is determined,*

*the less precisely the momentum is known in this instant, and vice versa*.”

*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.

*PM*≠

*MP*, 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.

**Question**. Given an algebra generated in variables

*x*

_{1},…,

*x*, subject to relations

_{n}*f*

_{1}(

__x__)=…=

*f*(

_{m}__x__) = 0,

*A*

_{1},…,

*A*to this system of equations?

_{n}*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.

*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:

*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.

*A*and

*B*, we have that

*AB*) = tr(

*BA*) and tr(

*A*-

*B*) = tr(

*A*) - tr(

*B*).

*PM*-

*MP*=

*I*, then

*PM*) - tr(

*MP*) = tr(

*PM*-

*MP*) = tr(

*I*) =

*d*.

*d*= 0, which is impossible.

*d*finite, there is no matrix solution to the equation,

*xy*-

*yx*= 1.

*xy*-

*yx*= 1. Take for instance,

*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.