Stone's Theorem on One-Parameter Unitary Groups

Given a densely defined, closed, self-adjoint operator A on a Hilbert space H, Stone's Theorem states that there exists a unique one-parameter unitary group $U(t) = e^{itA}$, such that $U(t)$ is strongly continuous, and

$$\frac{d}{dt} U(t)|_{t=0} = iAU(0)$$.

Here's a brief explanation of each component of the theorem:

• Densely defined: The domain of the operator is a dense subset of the Hilbert space.
• Closed operator: The operator for which the graph (a subset of H×H) is a closed set in the topology induced by H×H.
• Self-adjoint: The operator equals its own adjoint, i.e.,

$$\langle\psi|A\phi\rangle = \langle A\psi|\phi\rangle \forall \psi, \phi \in \text{domain}(A)$$

• One-parameter unitary group: A family of operators $U(t)$ indexed by a real number t, with the properties that $U(t)$ is a unitary operator for each t, $U(0)$ is the identity operator, and

$$U(s + t) = U(s)U(t) \forall s,t \in \mathbb{R}$$

• Strongly continuous: The map $t \to U(t)\psi$ is continuous for each fixed $\psi$ in H, in the strong operator topology.

The proof outline of the theorem is as follows:

1. Define a new operator B by setting

$$B\psi = -i \frac{d}{dt} U(t)\psi(t)|_{t=0}$$

for each $\psi$ in the domain of A. This operator is initially only densely defined, but we can extend it to the entire Hilbert space by continuity.

2. Show that B is a self-adjoint extension of A. This involves showing that $B\psi = A\psi$ for all $\psi$ in the domain of A, and that the domain of B includes the entire Hilbert space. The first part requires using the strong continuity of $U(t)$, while the second part uses the fact that A is densely defined.
3. Show that

$$U(t) = e^{itB} \forall t$$

This requires using the functional calculus for self-adjoint operators, which allows us to define the exponential of an operator.

4. Show that the operator B is unique. This follows from the spectral theorem for self-adjoint operators, which tells us that two self-adjoint operators are equal if they have the same spectral projections.

Thus, the theorem provides a profound connection between the abstract properties of self-adjoint operators and the concrete representations of time evolution in quantum mechanics.

References:

• Reed, M. & Simon, B. (1972) "Methods of Modern Mathematical Physics I: Functional Analysis". Academic Press. ISBN 0-12-585050-6.
• Hall, B.C. (2013) "Quantum Theory for Mathematicians". Graduate Texts in Mathematics 267. Springer. ISBN 978-1-4614-7116-5.