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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Stone's theorem on one-parameter
unitary group Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semi ...
s is a basic theorem of
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
that establishes a one-to-one correspondence between
self-adjoint operator In mathematics, a self-adjoint operator on a complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. That is, \langle Ax,y \rangle = \langle x,Ay \rangle for al ...
s on a
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
\mathcal and one-parameter families :(U_)_ of
unitary operator In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Non-trivial examples include rotations, reflections, and the Fourier operator. Unitary operators generalize unitar ...
s that are strongly continuous, i.e., :\forall t_0 \in \R, \psi \in \mathcal: \qquad \lim_ U_t(\psi) = U_(\psi), and are homomorphisms, i.e., :\forall s,t \in \R : \qquad U_ = U_t U_s. Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups. The theorem was proved by , and showed that the requirement that (U_t)_ 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 In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
of the mapping t \mapsto U_t, which is only supposed to be continuous. It is also related to the theory of
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
s and
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
s.


Formal statement

The statement of the theorem is as follows. Theorem 10.15 :Theorem. Let (U_t)_ be a strongly continuous one-parameter unitary group. Then there exists a unique (possibly unbounded) operator A: \mathcal_A \to \mathcal, that is self-adjoint on \mathcal_A and such that ::\forall t \in \R : \qquad U_t = e^. :The domain of A is defined by ::\mathcal_A = \left \. :Conversely, let A: \mathcal_A \to \mathcal be a (possibly unbounded) self-adjoint operator on \mathcal_A \subseteq \mathcal. Then the one-parameter family (U_)_ of unitary operators defined by ::\forall t \in \R : \qquad U_ := e^ :is a strongly continuous one-parameter group. In both parts of the theorem, the expression e^ is defined by means of the
functional calculus In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral theo ...
, which uses the
spectral theorem In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involvin ...
for unbounded
self-adjoint operator In mathematics, a self-adjoint operator on a complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. That is, \langle Ax,y \rangle = \langle x,Ay \rangle for al ...
s. The operator A is called the infinitesimal generator of (U_)_. Furthermore, A will be a bounded operator if and only if the operator-valued mapping t \mapsto U_ is norm-continuous. The infinitesimal generator A of a strongly continuous unitary group (U_)_ may be computed as :A\psi = -i\lim_\frac, with the domain of A consisting of those vectors \psi for which the limit exists in the norm topology. That is to say, A is equal to -i times the derivative of U_t with respect to t at t=0. Part of the statement of the theorem is that this derivative exists—i.e., that A is a densely defined self-adjoint operator. The result is not obvious even in the finite-dimensional case, since U_t is only assumed (ahead of time) to be continuous, and not differentiable.


Example

The family of translation operators :\left T_t(\psi) \rightx) = \psi(x + t) is a one-parameter unitary group of unitary operators; the infinitesimal generator of this family is an extension of the differential operator :-i \frac defined on the space of continuously differentiable complex-valued functions with
compact support In mathematics, the support of a real-valued function f is the subset of the function domain of elements that are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed ...
on \R. Thus :T_ = e^. In other words, motion on the line is generated by the momentum operator.


Applications

Stone's theorem has numerous applications in
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
. For instance, given an isolated quantum mechanical system, with Hilbert space of states , time evolution is a strongly continuous one-parameter unitary group on \mathcal. The infinitesimal generator of this group is the system
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
.


Using Fourier transform

Stone's Theorem can be recast using the language of the
Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
. The real line \R is a locally compact abelian group. Non-degenerate *-representations of the group C*-algebra C^*(\R) are in one-to-one correspondence with strongly continuous unitary representations of \R, i.e., strongly continuous one-parameter unitary groups. On the other hand, the Fourier transform is a *-isomorphism from C^*(\R) to C_0(\R), the C^*-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 C_0(\R). As every *-representation of C_0(\R) 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 (U_)_ be a strongly continuous unitary representation of \R on a
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
\mathcal. * Integrate this unitary representation to yield a non-degenerate *-representation \rho of C^*(\R) on \mathcal by first defining \forall f \in C_c(\R): \quad \rho(f) := \int_ f(t) ~ U_ dt, and then extending \rho to all of C^*(\R) by continuity. * Use the Fourier transform to obtain a non-degenerate *-representation \tau of C_0(\R ) on \mathcal. * By the Riesz-Markov Theorem, \tau gives rise to a
projection-valued measure In mathematics, particularly in functional analysis, a projection-valued measure, or spectral measure, is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. A projection-va ...
on \R that is the resolution of the identity of a unique
self-adjoint operator In mathematics, a self-adjoint operator on a complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. That is, \langle Ax,y \rangle = \langle x,Ay \rangle for al ...
A, which may be unbounded. * Then A is the infinitesimal generator of (U_)_. The precise definition of C^*(\R) is as follows. Consider the *-algebra C_c(\R), the continuous complex-valued functions on \R with compact support, where the multiplication is given by
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
. The completion of this *-algebra with respect to the L^1-norm is a Banach *-algebra, denoted by (L^1(\R),\star). Then C^*(\R) is defined to be the enveloping C^*-algebra of (L^1(\R),\star), i.e., its completion with respect to the largest possible C^*-norm. It is a non-trivial fact that, via the Fourier transform, C^*(\R) is isomorphic to C_0(\R). A result in this direction is the Riemann-Lebesgue Lemma, which says that the Fourier transform maps L^1(\R) to C_0(\R).


Generalizations

The Stone–von Neumann theorem generalizes Stone's theorem to a ''pair'' of self-adjoint operators, (P,Q), satisfying the
canonical commutation relation In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, hat x,\hat p ...
, and shows that these are all unitarily equivalent to the position operator and momentum operator on L^2(\R). The Hille–Yosida theorem generalizes Stone's theorem to strongly continuous one-parameter semigroups of contractions on
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
s.


References


Bibliography

* * * * * K. Yosida, ''Functional Analysis'', Springer-Verlag, (1968) {{DEFAULTSORT:Stone's Theorem On One-Parameter Unitary Groups Theorems in functional analysis