# Stone's theorem on one-parameter unitary groups

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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 ${\displaystyle {\mathcal {H}}}$ and one-parameter families

${\displaystyle (U_{t})_{t\in \mathbf {R} }}$

of unitary operators that are strongly continuous, i.e.,

${\displaystyle \forall t_{0}\in \mathbf {R} ,~\xi \in H:\qquad \lim _{t\to t_{0}}{U_{t}}(\xi )={U_{t_{0}}}(\xi ),}$

and are homomorphisms, i.e.,

${\displaystyle \forall s,t\in \mathbf {R} :\qquad U_{t+s}=U_{t}U_{s}.}$

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 ${\displaystyle (U_{t})_{t\in \mathbf {R} }}$ 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 ${\displaystyle t\mapsto U_{t}}$, which is only supposed to be continuous. It is also related to the theory of Lie groups and Lie algebras.

## Formal statement

Let ${\displaystyle (U_{t})_{t\in \mathbf {R} }}$ be a strongly continuous one-parameter unitary group. Then there exists a unique (not necessarily bounded) self-adjoint operator ${\displaystyle A}$ such that

${\displaystyle \forall t\in \mathbf {R} :\qquad U_{t}=e^{itA}.}$

Conversely, let ${\displaystyle A}$ be a (not necessarily bounded) self-adjoint operator on a Hilbert space ${\displaystyle {\mathcal {H}}}$. Then the one-parameter family ${\displaystyle (U_{t})_{t\in \mathbf {R} }}$ of unitary operators defined by (using the Spectral Theorem for Self-Adjoint Operators)

${\displaystyle \forall t\in \mathbf {R} :\qquad U_{t}:=e^{itA}}$

is a strongly continuous one-parameter group.[1]

The infinitesimal generator of ${\displaystyle (U_{t})_{t\in \mathbf {R} }}$ is defined to be the operator iA. This mapping is a bijective correspondence. Furthermore, ${\displaystyle A}$ will be a bounded operator if and only if the operator-valued mapping ${\displaystyle t\mapsto U_{t}}$ is norm-continuous.

Stone's Theorem can be recast using the language of the Fourier transform. The real line ${\displaystyle \mathbf {R} }$ is a locally compact abelian group. Non-degenerate *-representations of the group C*-algebra ${\displaystyle {C^{*}}(\mathbf {R} )}$ are in one-to-one correspondence with strongly continuous unitary representations of ${\displaystyle \mathbf {R} }$, i.e., strongly continuous one-parameter unitary groups. On the other hand, the Fourier transform is a *-isomorphism from ${\displaystyle {C^{*}}(\mathbf {R} )}$ to ${\displaystyle {C_{0}}(\mathbf {R} )}$, the ${\displaystyle C^{*}}$-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 ${\displaystyle {C_{0}}(\mathbf {R} )}$. As every *-representation of ${\displaystyle {C_{0}}(\mathbf {R} )}$ 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 ${\displaystyle (U_{t})_{t\in \mathbf {R} }}$ be a strongly continuous unitary representation of ${\displaystyle \mathbf {R} }$ on a Hilbert space ${\displaystyle {\mathcal {H}}}$.
• Integrate this unitary representation to yield a non-degenerate *-representation ${\displaystyle \rho }$ of ${\displaystyle {C^{*}}(\mathbf {R} )}$ on ${\displaystyle {\mathcal {H}}}$ by first defining
${\displaystyle \forall f\in {C_{c}}(\mathbf {R} ):\qquad \rho (f):=\int _{\mathbf {R} }f(t)~U_{t}~d{t},}$
and then extending ${\displaystyle \rho }$ to all of ${\displaystyle {C^{*}}(\mathbf {R} )}$ by continuity.
• Use the Fourier transform to obtain a non-degenerate *-representation ${\displaystyle \tau }$ of ${\displaystyle {C_{0}}(\mathbf {R} )}$ on ${\displaystyle {\mathcal {H}}}$.
• By the Riesz-Markov Theorem, ${\displaystyle \tau }$ gives rise to a projection-valued measure on ${\displaystyle \mathbf {R} }$ that is the resolution of the identity of a unique self-adjoint operator ${\displaystyle A}$, which may be unbounded.
• Then ${\displaystyle iA}$ is the infinitesimal generator of ${\displaystyle (U_{t})_{t\in \mathbf {R} }}$.

The precise definition of ${\displaystyle {C^{*}}(\mathbf {R} )}$ is as follows. Consider the *-algebra ${\displaystyle {C_{c}}(\mathbf {R} )}$, the continuous complex-valued functions on ${\displaystyle \mathbf {R} }$ with compact support, where the multiplication is given by convolution. The completion of this *-algebra with respect to the ${\displaystyle L^{1}}$-norm is a Banach *-algebra, denoted by ${\displaystyle ({L^{1}}(\mathbf {R} ),\star )}$. Then ${\displaystyle {C^{*}}(\mathbf {R} )}$ is defined to be the enveloping ${\displaystyle C^{*}}$-algebra of ${\displaystyle ({L^{1}}(\mathbf {R} ),\star )}$, i.e., its completion with respect to the largest possible ${\displaystyle C^{*}}$-norm. It is a non-trivial fact that, via the Fourier transform, ${\displaystyle {C^{*}}(\mathbf {R} )}$ is isomorphic to ${\displaystyle {C_{0}}(\mathbf {R} )}$. A result in this direction is the Riemann-Lebesgue Lemma, which says that the Fourier transform maps ${\displaystyle {L^{1}}(\mathbf {R} )}$ to ${\displaystyle {C_{0}}(\mathbf {R} )}$.

## Example

The family of translation operators

${\displaystyle \left[{T_{t}}(\psi )\right](x)=\psi (x+t)}$

is a one-parameter unitary group of unitary operators; the infinitesimal generator of this family is an extension of the differential operator

${\displaystyle {\frac {d}{dx}}=i\cdot {\frac {1}{i}}{\frac {d}{dx}}}$

defined on the space of continuously differentiable complex-valued functions with compact support on ${\displaystyle \mathbf {R} }$. Thus

${\displaystyle T_{t}=e^{t{\frac {d}{dx}}}.}$

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 ${\displaystyle {\mathcal {H}}}$. 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, ${\displaystyle (P,Q)}$, satisfying the canonical commutation relation, and shows that these are all unitarily equivalent to the position operator and momentum operator on ${\displaystyle {L^{2}}(\mathbf {R} )}$.

The Hille–Yosida theorem generalizes Stone's theorem to strongly continuous one-parameter semigroups of contractions on Banach spaces.