The Spectral Theorem

The spectral theorem is the crown of operator theory: it says every self-adjoint (more generally normal) operator is "diagonal" in a precise sense, even when its spectrum is continuous and it has no eigenvectors in the Hilbert space. It is the mathematical content of the measurement postulate and, via Stone's theorem, of unitary time evolution. This page builds on bounded-operators.md; the unbounded case it also covers is detailed in unbounded-operators.md.

From eigenvalues to spectral measures

For a Hermitian matrix, $A={\sum }_n{\lambda }_n{P}_n$ with $P_n$ the projection onto the ${\lambda }_n$-eigenspace. In infinite dimensions with continuous spectrum this sum must become an integral against a projection-valued measure. A projection-valued measure (PVM) $E$ assigns to each (Borel) subset $\Omega \subseteq \mathbb{R}$ an orthogonal projection $E(\Omega )$, with $E(\mathbb{R})=1$, $E(\varnothing )=0$, and countable additivity on disjoint sets. It is the operator analogue of a probability distribution — indeed $⟨\psi ,E(\Omega )\psi ⟩$ is the probability distribution of the observable in state $\psi$.

The spectral theorem

Spectral theorem (self-adjoint form). For every self-adjoint operator $A$ there is a unique projection-valued measure $E$ supported on $\sigma (A)>\subseteq \mathbb{R}$ such that $$A={\int }_{\sigma (A)}\lambda dE(\lambda ).$$ Equivalently, $A$ is unitarily equivalent to multiplication by the real variable $\lambda$ on some $L^2(\sigma (A),d\mu )$ (multiplication-operator form).

So every self-adjoint operator "is" multiplication by a real coordinate in a suitable representation. Discrete eigenvalues contribute atoms of $E$ (rank of the projection = degeneracy); continuous spectrum contributes a continuous part with no eigenvectors in $\mathcal{H}$ — the position operator $\hat{x}$ on $L^2(\mathbb{R})$ is already the multiplication operator, its "eigenkets" $|x⟩$ living in the rigged space.

Functional calculus

The spectral measure lets one apply functions to operators: for any (bounded Borel) $f$, $$f(A)=\int f(\lambda )dE(\lambda ),$$ a homomorphism $f\mapsto f(A)$ from functions to operators with $\overline{f}(A)=f(A{)}^{†}$. This functional calculus is ubiquitous in physics: it defines $e^{-iHt}$ (dynamics), $e^{-\beta H}$ (thermal density matrices), projections $1_{[a,b]}(A)$ onto spectral ranges (measurement outcomes), and resolvents $(A-\lambda {)}^{-1}$ (Green's functions). It is how a Hamiltonian's spectrum determines everything dynamical.

Stone's theorem

Specializing the functional calculus to $f_t(\lambda )=e^{-i\lambda t}$ ties self-adjoint operators to unitary dynamics:

Stone's theorem. Strongly continuous one-parameter unitary groups $\{U(t){\}}_{t\in \mathbb{R}}$ correspond bijectively to self-adjoint operators $A$ via $U(t)=e^{-iAt}$, with $A=i\frac{d}{dt}U(t){|}_{t=0}$ (the generator).

This is the rigorous statement that observables generate symmetries and, applied to the Hamiltonian, that $U(t)=e^{-iHt}$ is the unique unitary evolution — the Schrödinger dynamics of QM. Self-adjointness (not mere symmetry) is exactly the condition for $e^{-iAt}$ to be unitary for all $t$, which is why the domain subtleties of unbounded-operators.md are physically essential: a merely symmetric Hamiltonian does not generate a well-defined dynamics.

References

• Reed & Simon, Methods of Modern Mathematical Physics I, Ch. VII–VIII.
• Hall, Quantum Theory for Mathematicians, Ch. 7–10.
• von Neumann, Mathematical Foundations of Quantum Mechanics.