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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

This tells us that the Weyl algebra is noncommutative.

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

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