Uniqueness of representation One would like to classify representations of the canonical commutation relation by two self-adjoint operators acting on separable Hilbert spaces, up to unitary equivalence. By Stone's theorem, there is a one-to-one correspondence between self-adjoint operators and (strongly continuous) one-parameter unitary groups. Let and be two self-adjoint operators satisfying the canonical commutation relation, , and and two real parameters. Introduce and , the corresponding unitary groups given by functional calculus. (For the explicit operators and defined above, these are multiplication by and pullback by translation .) A formal computation (using a special case of the Baker–Campbell–Hausdorff formula) readily yields Conversely, given two one-parameter unitary groups and satisfying the braiding relation formally differentiating at 0 shows that the two infinitesimal generators satisfy the above canonical commutation relation. This braiding formulation of the canonical commutation relations (CCR) for one-parameter unitary groups is called the Weyl form of the CCR. It is important to note that the preceding derivation is purely formal. Since the operators involved are unbounded, technical issues prevent application of the Baker–Campbell–Hausdorff formula without additional domain assumptions. Indeed, there exist operators satisfying the canonical commutation relation but not the Weyl relations (). Nevertheless, in "good" cases, we expect that operators satisfying the canonical commutation relation will also satisfy the Weyl relations. The problem thus becomes classifying two jointly irreducible one-parameter unitary groups and which satisfy the Weyl relation on separable Hilbert spaces. The answer is the content of the Stone–von Neumann theorem: all such pairs of one-parameter unitary groups are unitarily equivalent. In other words, for any two such and acting jointly irreducibly on a Hilbert space , there is a unitary operator so that where and are the explicit position and momentum operators from earlier. When is in this equation, so, then, in the -representation, it is evident that is unitarily equivalent to , and the spectrum of must range along the entire real line. The analog argument holds for . There is also a straightforward extension of the Stone–von Neumann theorem to degrees of freedom. Historically, this result was significant, because it was a key step in proving that Heisenberg's matrix mechanics, which presents quantum mechanical observables and dynamics in terms of infinite matrices, is unitarily equivalent to Schrödinger's wave mechanical formulation (see Schrödinger picture), Representation theory formulation In terms of representation theory, the Stone–von Neumann theorem classifies certain unitary representations of the Heisenberg group. This