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Stone's Theorem On One-parameter Unitary Groups
theorem relating unitary operators to one-parameter Lie groups
Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups.
The theorem was proved by Marshall Stone (1930, 1932), and Von Neumann (1932) harvtxt error: no target: CITEREFVon_Neumann1932 (help) showed that the requirement that be strongly continuous can be relaxed to say that it is merely weakly measurable, at least when the Hilbert space is separable.
This is an impressive result, as it allows to define the derivative of the mapping which is only supposed to be continuous. It is also related to the theory of Lie groups and Lie algebras.
The operator is called the infinitesimal generator of Furthermore, will be a bounded operator if and only if the operator-valued mapping is norm-continuous.
The infinitesimal generator of a strongly continuous unitary group may be computed as
with the domain of consisting of those vectors for which the limit exists in the norm topology. That is to say, is equal to times the derivative of with respect to at . Part of the statement of the theorem is that this derivative exists--i.e., that is a densely defined self-adjoint operator. The result is not obvious even in the finite-dimensional case, since is only assumed (ahead of time) to be continuous, and not differentiable.
The family of translation operators
is a one-parameter unitary group of unitary operators; the infinitesimal generator of this family is an extension of the differential operator
defined on the space of continuously differentiable complex-valued functions with compact support on Thus
Stone's theorem has numerous applications in quantum mechanics. For instance, given an isolated quantum mechanical system, with Hilbert space of states H, time evolution 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 .
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