The

Weyl algebra is a

ring that arose out of work in

quantum theory in the 1920s by

Heisenberg,

Dirac
and others.

Here's a very natural description of the Weyl algebra in terms of differential operators. Start with a base field *k*; you can
take *k* to be the real or complex numbers **R** or **C**.

The Weyl algebra, normally written *A*_{1}(k) consists of all differential operators in *y*
with polynomial coefficients.
A typical element of the Weyl algebra is a sum of terms
of the form
*ay*^{i}(d/dy)^{j}
for non-negative integers *i* and *j* and *a* in *k*.

To keep the notation simpler, let's write *x=d/dy*
So *y* and *x* are elements
of *A*_{1} and so are *yx* and
*xy*.

What should the difference
*xy-yx*
be?

Well just think about applying this operator to
some power of *y*, say *y*^{n}.

*yx* applied to
*y*^{n} is *ny*^{n}. On the other hand,
*xy* applied to *y*^{n} is clearly *(n+1)y*^{n}. So we see that *xy-yx*
applied to *y*^{n} is *y*^{n}
again. That is *xy-yx* is the identity operator or

*xy-yx=1*

This tells us that the Weyl algebra is noncommutative.

In fact the Heisenberg Uncertainty Principle is a consequence of
this relation.