Weyl algebra

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In abstract algebra, the Weyl algebras are abstracted from the ring of differential operators with polynomial coefficients. They are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics. In the simplest case, these are differential operators. Let F {\displaystyle F} be a field, and let F [ x ] {\displaystyle F[x]} be the ring of polynomials in one variable with coefficients in F {\displaystyle F} . Then the corresponding Weyl algebra consists of differential operators of the form f m ( x ) ∂ x m + f m − 1 ( x ) ∂ x m − 1 + ⋯ + f 1 ( x ) ∂ x + f 0 ( x ) {\displaystyle f_{m}(x)\partial _{x}^{m}+f_{m-1}(x)\partial _{x}^{m-1}+\cdots +f_{1}(x)\partial _{x}+f_{0}(x)} where f i ( x ) ∈ F [ x ] {\displaystyle f_{i}(x)\in F[x]} . This is the first Weyl algebra A 1 {\displaystyle A_{1}} . The n-th Weyl algebra A n {\displaystyle A_{n}} is constructed similarly. Alternatively, A 1 {\displaystyle A_{1}} can be constructed as the quotient of the free algebra on two generators, q and p, by the ideal generated by ( [ p , q ] − 1 ) {\displaystyle ([p,q]-1)} . Similarly, A n {\displaystyle A_{n}} is obtained by quotienting the free algebra on 2n generators by the ideal generated by ( [ p i , q j ] − δ i , j ) , ∀ i , j = 1 , … , n {\displaystyle ([p_{i},q_{j}]-\delta _{i,j}),\quad \forall i,j=1,\dots ,n} where δ i , j {\displaystyle \delta _{i,j}} is the Kronecker delta. More generally, let ( R , Δ ) {\displaystyle (R,\Delta )} be a partial differential ring with commuting derivatives Δ = { ∂ 1 , … , ∂ m } {\displaystyle \Delta =\lbrace \partial...