and multiplication by a function of absolute value 1. To show is multiplicative is a straightforward calculation. The hard part of the theorem is showing the uniqueness; this claim, nevertheless, follows easily from the Stone–von Neumann theorem as stated above. We will sketch below a proof of the corresponding Stone–von Neumann theorem for certain finite Heisenberg groups. In particular, irreducible representations , of the Heisenberg group which are non-trivial on the center of are unitarily equivalent if and only if for any in the center of . One representation of the Heisenberg group which is important in number theory and the theory of modular forms is the theta representation, so named because the Jacobi theta function is invariant under the action of the discrete subgroup of the Heisenberg group. Relation to the Fourier transform For any non-zero , the mapping is an automorphism of which is the identity on the center of . In particular, the representations and are unitarily equivalent. This means that there is a unitary operator on such that, for any in , Moreover, by irreducibility of the representations , it follows that up to a scalar, such an operator is unique (cf. Schur's lemma). Since is unitary, this scalar multiple is uniquely determined and hence such an operator is unique. This means that, ignoring the factor of in the definition of the Fourier transform, This theorem has the immediate implication that the Fourier transform is unitary, also known as the Plancherel theorem. Moreover, From this fact the Fourier inversion formula easily follows. Example: The Segal–Bargmann space The Segal–Bargmann space is the space of holomorphic functions on that are square-integrable with respect to a Gaussian measure. Fock observed in 1920s that the operators acting on holomorphic functions, satisfy the same commutation relations as the usual annihilation and creation operators, namely, In 1961, Bargmann showed that is actually the adjoint of with respect to the inner product coming from the Gaussian measure. By taking appropriate linear combinations of and , 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. The Stone–von Neumann theorem therefore applies and implies the existence of a unitary map from to the Segal–Bargmann space that intertwines the usual annihilation and creation operators with the operators and .