Stone's theorem on one-parameter unitary 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 and one-parameter families
and are homomorphisms, i.e.,
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) 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.
The statement of the theorem is as follows.
- Theorem: Let be a strongly continuous one-parameter unitary group. Then there exists a unique (possibly unbounded) operator , that is self-adjoint on and such that
The domain of is defined by .
- Conversely, let be a (possibly unbounded) self-adjoint operator on . Then the one-parameter family of unitary operators defined by
- is a strongly continuous one-parameter group.
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
In other words, motion on the line is generated by the momentum operator.
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 .
- 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 .
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 .
- Hall 2013 Theorem 10.15
- Hall, B.C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, 267, Springer, ISBN 978-1461471158
- Neumann, J. von (1932), "Über einen Satz von Herrn M. H. Stone", Annals of Mathematics, Second Series (in German), Annals of Mathematics, 33 (3): 567–573, doi:10.2307/1968535, ISSN 0003-486X, JSTOR 1968535
- Stone, M. H. (1930), "Linear Transformations in Hilbert Space. III. Operational Methods and Group Theory", Proceedings of the National Academy of Sciences of the United States of America, National Academy of Sciences, 16 (2): 172–175, doi:10.1073/pnas.16.2.172, ISSN 0027-8424, JSTOR 85485
- Stone, M. H. (1932), "On one-parameter unitary groups in Hilbert Space", Annals of Mathematics, 33 (3): 643–648, doi:10.2307/1968538, JSTOR 1968538
- K. Yosida, Functional Analysis, Springer-Verlag, (1968)