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The Fock number operator has its sector counting PVM

Statement

Assume Countable Choice The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $H$ be a complex Hilbert space, and let $N$ , $E_{N} (B)$ , and $P_{n}$ be as in Bosonic Fock number operator and counting PVM. Then $N$ is self-adjoint on its stated domain and $E_{N}$ is a projection-valued measure on the Borel subsets of $R$ . It is the sector counting resolution: $E_{N} (B)$ keeps exactly the sectors whose integer label lies in $B$ , and $N$ multiplies sector $n$ by $n$ . For any finite Borel partition $(B_{a})_{a = 1}^{m}$ of $R$ , the projections $E_{N} (B_{a})$ are mutually orthogonal and sum to the identity.

Facts & Assumptions

Given: Countable Choice, the square-summable symmetric Fock direct sum, the number-operator definition, and the standard definitions of adjoint, self-adjoint operator, and projection-valued measure.

[A1]

Countable Choice The Axiom of Countable Choice ( ${A C}_{ω}$ ) is the choice assumption of the Fock construction used here.

[F1]

Schwartz/distribution spaces and Hilbert Fock conventions: $F_{s} (H)$ is the Hilbert direct sum of the symmetric sectors $H_{s}^{(n)}$ , $n \geq 0$ , with $\sum_{n} ∥ Ψ_{n} ∥^{2} < \infty$ ; finite-sector vectors form a dense subspace.

[F2]

Bosonic Fock number operator and counting PVM: $D (N) = {Ψ : \sum_{n} n^{2} ∥ Ψ_{n} ∥^{2} < \infty}$ , $(N Ψ)_{n} = n Ψ_{n}$ , and $(E_{N} (B) Ψ)_{n} = 1_{B} (n) Ψ_{n}$ .

[F3]

Projection valued measure: every value must be an orthogonal projection, $E_{N} (\emptyset) = 0$ , $E_{N} (R) = I$ , $E_{N} (B \cap C) = E_{N} (B) E_{N} (C)$ , and disjoint countable unions must converge strongly.

[F4]

Under the second-variable-linear Fock pairing of Schwartz/distribution spaces and Hilbert Fock conventions, conjugating the first-variable-linear definition in Adjoint of a densely defined operator gives: $y \in D (N^{*})$ exactly when there is $z$ such that $⟨ N x, y ⟩_{2} = ⟨ x, z ⟩_{2}$ for every $x \in D (N)$ ; then $N^{*} y = z$ .

[F5]

Symmetric, self-adjoint and essentially self-adjoint operators: self-adjointness means equality of the operator and its adjoint, including equality of their domains.

Proof

Given: [A1], [F1]–[F5], and the number-operator and sector-projection formulas in [F2].

1.1A1F1F2algebra

Finite-sector vectors lie in $D (N)$ and are dense by [F1]. For $Ψ, Φ \in D (N)$ , the Cauchy–Schwarz inequality gives absolute convergence of $\sum_{n} n ⟨ Ψ_{n}, Φ_{n} ⟩_{2}$ , and because each $n$ is real, $⟨ N Ψ, Φ ⟩_{2} = \sum_{n} n ⟨ Ψ_{n}, Φ_{n} ⟩_{2} = ⟨ Ψ, N Φ ⟩_{2} .$ Thus $N$ is densely defined and symmetric.

1.2F2F3algebra

For each Borel $B$ , [F2] makes $E_{N} (B)$ the diagonal multiplier by the real indicator $1_{B} (n)$ . It is bounded with norm at most one, self-adjoint, and idempotent. The same coordinate formula gives $E_{N} (\emptyset) = 0$ , $E_{N} (R) = I$ , and $E_{N} (B) E_{N} (C) = E_{N} (B \cap C)$ .

2.1F1F2F4F5step 1.1algebra

Suppose $y \in D (N^{*})$ and write $z = N^{*} y$ . Test [F4] on vectors supported in only sector $n$ . Since $N$ acts there as $n I$ , the identity for every $x_{n} \in H_{s}^{(n)}$ gives $z_{n} = n y_{n}$ . Hence $\sum_{n} n^{2} ∥ y_{n} ∥^{2} = \sum_{n} ∥ z_{n} ∥^{2} < \infty$ , so $y \in D (N)$ and $N^{*} y = N y$ . Conversely, if $y \in D (N)$ , the convergent sector sum in step 1.1 gives $⟨ N x, y ⟩_{2} = ⟨ x, N y ⟩_{2}$ for every $x \in D (N)$ ; [F4] then gives $y \in D (N^{*})$ and $N^{*} y = N y$ . Thus $D (N^{*}) = D (N)$ and $N = N^{*}$ , which proves self-adjointness by [F5].

2.2F1F2F3step 1.2construct

Let $(B_{j})_{j \geq 0}$ be pairwise disjoint Borel sets with union $B$ , and fix $Ψ \in F_{s} (H)$ . Given $ϵ > 0$ , choose $M$ so that $\sum_{n > M} ∥ Ψ_{n} ∥^{2} < ϵ$ . Among the finitely many labels $0, \dots, M$ , each label in $B$ belongs to one $B_{j}$ ; choose $J$ at least as large as all those finitely many indices. Then $E_{N} (B) Ψ - \sum_{j = 0}^{J} E_{N} (B_{j}) Ψ$ has zero components through sector $M$ , and its squared norm is at most $\sum_{n > M} ∥ Ψ_{n} ∥^{2} < ϵ$ . This proves strong countable additivity. Together with step 1.2 it proves the PVM clauses in [F3]. The coordinate formula in [F2] shows this PVM retains exactly the sectors on which $N$ acts by their labels.

3.1F3step 1.2step 2.2∎

In a finite Borel partition $(B_{a})_{a = 1}^{m}$ of $R$ , distinct sets are disjoint, so step 1.2 gives $E_{N} (B_{a}) E_{N} (B_{b}) = 0$ for $a \neq b$ . Append empty sets to this finite list to apply the countable-additivity result of step 2.2; since the union is $R$ and $E_{N} (R) = I$ , the finite projections sum to $I$ .

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