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 A1(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
ayi(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 A1 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 yn.
yx applied to
yn is nyn. On the other hand,
xy applied to yn is clearly (n+1)yn. So we see that xy-yx
applied to yn is yn
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.