Editing Stone–von Neumann theorem (section)

== Example: Segal–Bargmann space ==
The [[Segal–Bargmann space]] is the space of holomorphic functions on {{math|'''C'''<sup>''n''</sup>}} that are square-integrable with respect to a Gaussian measure. Fock observed in 1920s that the operators
<math display="block"> a_j = \frac{\partial}{\partial z_j}, \qquad a_j^* = z_j, </math>
acting on holomorphic functions, satisfy the same commutation relations as the usual annihilation and creation operators, namely,
<math display="block"> \left [a_j,a_k^* \right ] = \delta_{j,k}. </math>

In 1961, Bargmann showed that {{math|''a''{{su|b=''j''|p=&lowast;}}}} is actually the adjoint of {{math|''a<sub>j</sub>''}} with respect to the inner product coming from the Gaussian measure. By taking appropriate linear combinations of {{math|''a<sub>j</sub>''}} and {{math|''a''{{su|b=''j''|p=&lowast;}}}}, one can then obtain "position" and "momentum" operators satisfying the canonical commutation relations. It is not hard to show that the exponentials of these operators satisfy the Weyl relations and that the exponentiated operators act irreducibly.{{r|Hall 2013|p=Section 14.4}} The Stone–von Neumann theorem therefore applies and implies the existence of a unitary map from {{math|''L''<sup>2</sup>('''R'''<sup>''n''</sup>)}} to the Segal–Bargmann space that intertwines the usual annihilation and creation operators with the operators {{math|''a<sub>j</sub>''}} and {{math|''a''{{su|b=''j''|p=&lowast;}}}}. This unitary map is the [[Segal–Bargmann space#The Segal–Bargmann transform|Segal–Bargmann transform]].