# 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 Template:Mvar and one-parameter families

of unitary operators 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 Template:Harvs, and Template:Harvtxt 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 a very stunning theorem, as it allows to define the derivative of the mapping *t* ↦ *U _{t}*, which is only supposed to be continuous. It is also related to the theory of Lie groups and Lie algebras.

## Formal statement

Let be a strongly continuous one-parameter unitary group. Then there exists a unique (not necessarily bounded) self-adjoint operator Template:Mvar such that

Conversely, let Template:Mvar be a (not necessarily bounded) self-adjoint operator on a Hilbert space Template:Mvar. Then the one-parameter family of unitary operators defined by (using the Spectral Theorem for Self-Adjoint Operators)

is a strongly continuous one-parameter group.

The infinitesimal generator of is defined to be the operator *iA*. This mapping is a bijective correspondence. Furthermore, Template:Mvar will be a bounded operator if and only if the operator-valued mapping *t* ↦ *U _{t}* is norm-continuous.

Stone's Theorem can be recast using the language of the Fourier transform. 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 complex-valued continuous 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 be a strongly continuous unitary representation of
**R**on a Hilbert space Template:Mvar. - Integrate this unitary representation to yield a non-degenerate *-representation Template:Mvar of
*C*^{∗}(**R**) on Template:Mvar by first defining

- and then extending Template:Mvar to all of
*C*^{∗}(**R**) by continuity.

- Use the Fourier transform to obtain a non-degenerate *-representation Template:Mvar of
*C*_{0}(**R**) on Template:Mvar. - By the Riesz-Markov Theorem, Template:Mvar gives rise to a projection-valued measure on
**R**that is the resolution of the identity of a unique self-adjoint operator Template:Mvar, which may be unbounded. - Then
*iA*is the infinitesimal generator of .

The precise definition of *C*^{∗}(**R**) is as follows. Consider the *-algebra *C _{c}*(

**R**), the complex-valued continuous functions on

**R**with compact support, where the multiplication is given by convolution. The completion of this *-algebra with respect to the

*L*

^{1}-norm is a Banach *-algebra, denoted by . Then

*C*

^{∗}(

**R**) is defined to be the

**enveloping C*-algebra**of , 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**).

## 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 complex-valued continuously differentiable functions of compact support on **R**. Thus

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 Template:Mvar, time evolution is a strongly continuous one-parameter unitary group on Template:Mvar. The infinitesimal generator of this group is the system Hamiltonian.

## Generalizations

The Stone–von Neumann theorem generalizes Stone's theorem to a *pair* of self-adjoint operators, *Q*, *P* satisfying the canonical commutation relation, 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 spaces.

## References

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- K. Yosida,
*Functional Analysis*, Springer-Verlag, (1968)