DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Bosonic Fock number operator and counting PVM

Definition

Let $H$ be a complex Hilbert space and let $F_{s} (H) = ⨁_{n = 0}^{\infty} H_{s}^{(n)}$ be its bosonic Fock Hilbert space under Schwartz/distribution spaces and Hilbert Fock conventions. A vector is a square-summable sequence $Ψ = (Ψ_{n})_{n \geq 0}$ with $Ψ_{n} \in H_{s}^{(n)}$ .

Define the number operator $N$ by $D (N) = {Ψ \in F_{s} (H) : \sum_{n = 0}^{\infty} n^{2} ∥ Ψ_{n} ∥^{2} < \infty}, (N Ψ)_{n} = n Ψ_{n} .$

For every Borel set $B \subseteq R$ , define the sector projection $E_{N} (B) Ψ = (1_{B} (n) Ψ_{n})_{n \geq 0}, P_{n} : = E_{N} ({n}) .$ The definitions specify a candidate number observable and its candidate spectral projections. They assign no physical probability; well-definedness, self-adjointness, and the projection-valued-measure property are established by The Fock number operator has its sector counting PVM ↗.

Depends on

Dependency tree · 2 levels

Sources

Mark Srednicki, Quantum Field Theory, author-hosted 2006 prepublication book draft, §55 “Electrodynamics in Coulomb Gauge,” printed pp. 335–338