Editing Weyl algebra (section)

=== General Leibniz rule ===
{{Main|General Leibniz rule}}

{{Math theorem
| name = Theorem
| note = general Leibniz rule
| math_statement = <math display="block">
p^k q^m = \sum_{l=0}^k \binom{k}{l} \frac{m!}{(m-l)!} q^{m-l} p^{k-l} = q^mp^k + mk q^{m-1}p^{k-1} + \cdots 
</math>
}}

{{Math proof|title=Proof|proof=
 Under the <math>
p \mapsto x, q \mapsto \partial_x
</math> representation, this equation is obtained by the general Leibniz rule. Since the general Leibniz rule is provable by algebraic manipulation, it holds for <math>
A_1
</math> as well.
}}In particular, <math display="inline">[q, q^m p^n] = -nq^mp^{n-1}</math> and <math display="inline">[p, q^mp^n] = mq^{m-1}p^n</math>.
{{Math theorem
| math_statement = The [[Center (ring theory)|center]] of Weyl algebra <math>A_n</math> is the underlying field of constants <math>F</math>.
| name = Corollary
}}

{{Math proof|title=Proof|proof= 
If the commutator of <math>f</math> with either of <math>p, q</math> is zero, then by the previous statement, <math>f</math> has no monomial <math>p^nq^m</math> with <math>n > 0</math> or <math>m > 0</math>.
}}