The Weyl algebra
is a ring
that arose out of work in quantum theory
in the 1920s by Heisenberg
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
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
What should the difference
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
This tells us that the Weyl algebra is noncommutative.
In fact the Heisenberg Uncertainty Principle is a consequence of