is a strongly continuous one-parameter unitary group on . The infinitesimal generator of this group is the system Hamiltonian. Using Fourier transform Stone's Theorem can be recast using the language of the Fourier transform. The real line is a locally compact abelian group. Non-degenerate *-representations of the group C*-algebra are in one-to-one correspondence with strongly continuous unitary representations of i.e., strongly continuous one-parameter unitary groups. On the other hand, the Fourier transform is a *-isomorphism from to the -algebra of continuous complex-valued functions on the real line that vanish at infinity. Hence, there is a one-to-one correspondence between strongly continuous one-parameter unitary groups and *-representations of As every *-representation of corresponds uniquely to a self-adjoint operator, Stone's Theorem holds. Therefore, the procedure for obtaining the infinitesimal generator of a strongly continuous one-parameter unitary group is as follows: Let be a strongly continuous unitary representation of on a Hilbert space . Integrate this unitary representation to yield a non-degenerate *-representation of on by first defining and then extending to all of by continuity. Use the Fourier transform to obtain a non-degenerate *-representation of on . By the Riesz-Markov Theorem, gives rise to a projection-valued measure on that is the resolution of the identity of a unique self-adjoint operator , which may be unbounded. Then is the infinitesimal generator of The precise definition of is as follows. Consider the *-algebra the continuous complex-valued functions on with compact support, where the multiplication is given by convolution. The completion of this *-algebra with respect to the -norm is a Banach *-algebra, denoted by Then is defined to be the enveloping -algebra of , i.e., its completion with respect to the largest possible -norm. It is a non-trivial fact that, via the Fourier transform, is isomorphic to A result in this direction is the Riemann-Lebesgue Lemma, which says that the Fourier transform maps to Generalizations The Stone–von Neumann theorem generalizes Stone's theorem to a pair of self-adjoint operators, , satisfying the canonical commutation relation, and shows that these are all unitarily equivalent to the position operator and momentum operator on The Hille–Yosida theorem generalizes Stone's theorem to strongly continuous one-parameter semigroups of contractions on Banach spaces. References Bibliography K. Yosida, Functional Analysis, Springer-Verlag, (1968) Theorems in functional analysis