a choice of scalar matrix – a choice of scale. Given such a choice of scale, a central representation of the Heisenberg group is a map of -algebras , which is the formal way of saying that it sends the center to a chosen scale. Then the Stone–von Neumann theorem is that, given the standard quantum mechanical scale (effectively, the value of ħ), every strongly continuous unitary representation is unitarily equivalent to the standard representation with position and momentum. Reformulation via Fourier transform Let be a locally compact abelian group and be the Pontryagin dual of . The Fourier–Plancherel transform defined by extends to a C*-isomorphism from the group C*-algebra of and , i.e. the spectrum of is precisely . When is the real line , this is Stone's theorem characterizing one-parameter unitary groups. The theorem of Stone–von Neumann can also be restated using similar language. The group acts on the *-algebra by right translation : for in and in , Under the isomorphism given above, this action becomes the natural action of on : So a covariant representation corresponding to the *-crossed product is a unitary representation of and of such that It is a general fact that covariant representations are in one-to-one correspondence with *-representation of the corresponding crossed product. On the other hand, all irreducible representations of are unitarily equivalent to the , the compact operators on . Therefore, all pairs are unitarily equivalent. Specializing to the case where yields the Stone–von Neumann theorem. The Heisenberg group The above canonical commutation relations for , are identical to the commutation relations that specify the Lie algebra of the general Heisenberg group for a positive integer. This is the Lie group of square matrices of the form In fact, using the Heisenberg group, one can reformulate the Stone von Neumann theorem in the language of representation theory. Note that the center of consists of matrices . However, this center is not the identity operator in Heisenberg's original CCRs. The Heisenberg group Lie algebra generators, e.g. for , are and the central generator is not the identity. All these representations are unitarily inequivalent; and any irreducible representation which is not trivial on the center of is unitarily equivalent to exactly one of these. Note that is a unitary operator because it is the composition of two operators which are easily seen to be unitary: the translation to the left by