Spectral Measures and Observables

Spectral measures replace the finite-dimensional instruction “diagonalize the matrix” by a construction that includes continuous spectrum, spectral multiplicity, measurable functions, and unbounded operator domains.

Reading sequence

On-ramp. State Vectors supplies the Hilbert-space language. Then read Self-Adjoint Operators so that the operator domain is part of the observable. The theorem also assumes Borel sets, countably additive measures, measurable functions, and Lebesgue integration. Projectors supplies the sharp-event geometry: it is helpful in step 1 and a hard prerequisite for step 2.

1. The Spectral Theorem for Unbounded Self-Adjoint Operators states

   $$A=\int_{\mathbb R}\lambda\,dE_A(\lambda)$$

   together with the exact domain

   $$D(A)= \left\{ \psi: \int\lambda^2\,d\langle\psi,E_A(\lambda)\psi\rangle<\infty \right\}.$$

2. Projection-Valued Measures then develops strong countable additivity, scalar state measures, discrete and continuous examples, coarse graining, and the PVM–POVM–instrument boundary.

The theorem page introduces the PVM definition compactly because the governed authoring sequence places the theorem first. The second page owns the measure-theoretic details, so the derivation is not duplicated.

What to carry forward

• A spectral value need not be an eigenvalue.
• An unbounded spectral integral has a domain fixed by square integrability.
• $E_A(\Delta)$ is a sharp event projector for a measurable set, not a state update map.
• Countable additivity is strong, not generally operator-norm, convergence.
• A POVM is broader than a PVM, and a Naimark dilation changes the Hilbert space.

For the dynamics branch, add Strongly Continuous Unitary Groups before Stone’s Theorem; Stone’s result needs both the spectral calculus and the unitary-group derivative domain. With Projectors and both pages in this chapter complete, continue instead to Gleason’s Theorem for consistent probabilities on projections.