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 H and one-parameter families

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

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

 \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.,

 \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 (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 tUt, which is only supposed to be continuous. It is also related to the theory of Lie groups and Lie algebras.

Formal statement[edit]

Let (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 A such that

 \forall t \in \mathbf{R}: \qquad U_{t} = e^{i t A}.

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

 \forall t \in \mathbf{R}: \qquad U_{t} := e^{i t A}

is a strongly continuous one-parameter group.

The infinitesimal generator of (U_{t})_{t \in \mathbf{R}} is defined to be the operator iA. This mapping is a bijective correspondence. Furthermore, A will be a bounded operator if and only if the operator-valued mapping tUt 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 C0(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 C0(R). As every *-representation of C0(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_{t})_{t \in \mathbf{R}} be a strongly continuous unitary representation of R on a Hilbert space H.
  • Integrate this unitary representation to yield a non-degenerate *-representation ρ of C(R) on H by first defining
\forall f \in C_c(\mathbf{R}): \qquad \rho(f) := \int_{\mathbf{R}} f(t) U_{t} \, dt,
and then extending ρ to all of C(R) by continuity.

The precise definition of C(R) is as follows. Consider the *-algebra Cc(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 L1-norm is a Banach *-algebra, denoted by (L^1(\mathbf{R}),\star) . Then C(R) is defined to be the enveloping C*-algebra of (L^1(\mathbf{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 C0(R). A result in this direction is the Riemann-Lebesgue Lemma, which says that the Fourier transform maps L1(R) to C0(R).

Example[edit]

The family of translation operators

 \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

 \frac{d}{dx} = i \frac{1}{i} \frac{d}{dx}

defined on the space of complex-valued continuously differentiable functions of compact support on R. Thus

 T_t = e^{t \, \frac{d}{dx}}.

In other words, motion on the line is generated by the momentum operator.

Applications[edit]

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 H. The infinitesimal generator of this group is the system Hamiltonian.

Generalizations[edit]

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 L2(R).

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

References[edit]