Stone–von Neumann theorem

In mathematics and in theoretical physics, the Stone–von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. The name is for Marshall Stone and John von Neumann (1931). U ( t ) V ( s ) = e − i s t V ( s ) U ( t ) ∀ s , t , {displaystyle U(t)V(s)=e^{-ist}V(s)U(t)qquad forall s,t,}    (E1) In mathematics and in theoretical physics, the Stone–von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. The name is for Marshall Stone and John von Neumann (1931). In quantum mechanics, physical observables are represented mathematically by linear operators on Hilbert spaces. For a single particle moving on the real line R, there are two important observables: position and momentum. In the Schrödinger representation quantum description of such a particle, the position operator x and momentum operator p are respectively given by on the domain V of infinitely differentiable functions of compact support on R. Assume ℏ to be a fixed non-zero real number — in quantum theory ℏ is (up to a factor of 2π) Planck's constant, which is not dimensionless; it takes a small numerical value in terms of (action) units of the macroscopic world. The operators x, p satisfy the canonical commutation relation Lie algebra, Already in his classic book, Hermann Weyl observed that this commutation law was impossible to satisfy for linear operators p, x acting on finite-dimensional spaces unless ℏ vanishes. This is apparent from taking the trace over both sides of the latter equation and using the relation Trace(AB) = Trace(BA); the left-hand side is zero, the right-hand side is non-zero. Further analysis shows that, in fact, any two self-adjoint operators satisfying the above commutation relation cannot be both bounded. For notational convenience, the nonvanishing square root of ℏ may be absorbed into the normalization of p and x, so that, effectively, it is replaced by 1. We assume this normalization in what follows. The idea of the Stone—von Neumann theorem is that any two irreducible representations of the canonical commutation relations are unitarily equivalent. Since, however, the operators involved are necessarily unbounded (as noted above), there are tricky domain issues that allow for counter-examples. To obtain a rigorous result, one must require that the operators satisfy the exponentiated form of the canonical commutation relations, known as the Weyl relations. Although, as noted below, these relations are formally equivalent to the standard canonical commutation relations, this equivalence is not rigorous, because (again) of the unbounded nature of the operators. There is also a discrete analog of the Weyl relations, which can hold in a finite-dimensional space, namely Sylvester's clock and shift matrices in the finite Heisenberg group, discussed below. 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 Q and P be two self-adjoint operators satisfying the canonical commutation relation, = i, and s and t two real parameters. Introduce eitQ and eisP, the corresponding unitary groups given by functional calculus. (For the explicit operators x and p defined above, these are multiplication by exp(itx) and pullback by translation x → x+s.) A formal computation (using a special case of the Baker–Campbell–Hausdorff formula) readily yields