Theorem relating unitary operators to one-parameter Lie groups
In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space formula_1 and one-parameter families
formula_2
of unitary operators that are strongly continuous, i.e.,
formula_3
and are homomorphisms, i.e.,
formula_4
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 John von Neumann (1932) showed that the requirement that formula_5 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 one to define the derivative of the mapping formula_6 which is only supposed to be continuous. It is also related to the theory of Lie groups and Lie algebras.
Formal statement.
The statement of the theorem is as follows. 
Theorem. Let formula_5 be a strongly continuous one-parameter unitary group. Then there exists a unique (possibly unbounded) operator formula_8, that is self-adjoint on formula_9 and such that
formula_10 
The domain of formula_11 is defined by 
formula_12
Conversely, let formula_8 be a (possibly unbounded) self-adjoint operator on formula_14 Then the one-parameter family formula_2 of unitary operators defined by 
formula_16
is a strongly continuous one-parameter group.
In both parts of the theorem, the expression formula_17 is defined by means of the functional calculus, which uses the spectral theorem for unbounded self-adjoint operators. 
The operator formula_11 is called the infinitesimal generator of formula_19 Furthermore, formula_11 will be a bounded operator if and only if the operator-valued mapping formula_21 is norm-continuous.
The infinitesimal generator formula_11 of a strongly continuous unitary group formula_2 may be computed as
formula_24
with the domain of formula_11 consisting of those vectors formula_26 for which the limit exists in the norm topology. That is to say, formula_11 is equal to formula_28 times the derivative of formula_29 with respect to formula_30 at formula_31. Part of the statement of the theorem is that this derivative exists—i.e., that formula_11 is a densely defined self-adjoint operator. The result is not obvious even in the finite-dimensional case, since formula_29 is only assumed (ahead of time) to be continuous, and not differentiable.
Example.
The family of translation operators
formula_34
is a one-parameter unitary group of unitary operators; the infinitesimal generator of this family is an extension of the differential operator
formula_35
defined on the space of continuously differentiable complex-valued functions with compact support on formula_36 Thus
formula_37
In other words, motion on the line is generated by the momentum operator.
Applications.
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 formula_1. 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 formula_39 is a locally compact abelian group. Non-degenerate *-representations of the group C*-algebra formula_40 are in one-to-one correspondence with strongly continuous unitary representations of formula_41 i.e., strongly continuous one-parameter unitary groups. On the other hand, the Fourier transform is a *-isomorphism from formula_40 to formula_43 the formula_44-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 formula_45 As every *-representation of formula_46 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:
The precise definition of formula_40 is as follows. Consider the *-algebra formula_65 the continuous complex-valued functions on formula_39 with compact support, where the multiplication is given by convolution. The completion of this *-algebra with respect to the formula_67-norm is a Banach *-algebra, denoted by formula_68 Then formula_40 is defined to be the enveloping formula_44-algebra of formula_71, i.e., its completion with respect to the largest possible formula_44-norm. It is a non-trivial fact that, via the Fourier transform, formula_40 is isomorphic to formula_45 A result in this direction is the Riemann-Lebesgue Lemma, which says that the Fourier transform maps formula_75 to formula_45
Generalizations.
The Stone–von Neumann theorem generalizes Stone's theorem to a "pair" of self-adjoint operators, formula_77, satisfying the canonical commutation relation, and shows that these are all unitarily equivalent to the position operator and momentum operator on formula_78
The Hille–Yosida theorem generalizes Stone's theorem to strongly continuous one-parameter semigroups of contractions on Banach spaces.
References.
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