In
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 ...
and one-parameter families
:
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.,
:
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 , and 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 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 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
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.
In both parts of the theorem, the expression
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
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.
Example
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
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
Thus
:
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
. 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
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
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 ...
.
* 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
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
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 ...
, 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
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
-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
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
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