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CCR/CAR and particle occupations in adopted free models

Physical scope

Conditional algebraic consequences of the adopted bosonic scalar symmetric-Fock and Dirac CAR-Fock free models. The result concerns normalizable one-particle wave-packet modes; it is not a general spin-statistics theorem, an interacting-field result, or an empirical claim.

Statement

Assume ${A C}_{ω}$ . In the selected model of Specified bosonic scalar and CAR Dirac model adoption, let $H_{μ}$ be the scalar one-particle Hilbert space and let $K = H_{+} \oplus \overline{H_{-}}$ be the Dirac particle/antiparticle one-particle space. Let $Ω_{s}$ and $Ω_{D}$ be their zero-particle Fock vectors. The assignments of $H_{μ}$ to bosonic Fock space and $K$ to CAR Fock space are the physical assumptions of that postulate.

For every unit scalar wave-packet mode $h \in H_{μ}$ , define $∣ n ⟩_{h} = \frac{(a^{†} (h))^{n}}{\sqrt{n!}} Ω_{s}, n \geq 0.$ These vectors are normalized and mutually orthogonal, belong to the finite-particle domain, and $a (h) ∣ n ⟩_{h} = \sqrt{n} ∣ n - 1 ⟩_{h}, a^{†} (h) ∣ n ⟩_{h} = \sqrt{n + 1} ∣ n + 1 ⟩_{h},$ where $a (h) ∣ 0 ⟩_{h} = 0$ . The maximal operator domains supplied by Bosonic and fermionic Fock operators with exact domains are $D (a (h)) = {Ψ = (Ψ_{n}) \in F_{s} (H_{μ}) : \sum_{n \geq 1} ∥ a_{n} (h) Ψ_{n} ∥^{2} < \infty}, D (a^{†} (h)) = {Ψ = (Ψ_{n}) \in F_{s} (H_{μ}) : \sum_{n \geq 0} ∥ a_{n}^{†} (h) Ψ_{n} ∥^{2} < \infty} .$ Define the one-mode number operator as the composition $N_{h} : = a^{†} (h) a (h), D (N_{h}) : = {Ψ \in D (a (h)) : a (h) Ψ \in D (a^{†} (h))}, N_{h} Ψ : = a^{†} (h) (a (h) Ψ) .$ The displayed finite-particle vectors $∣ n ⟩_{h}$ belong to $D (N_{h})$ and satisfy $N_{h} ∣ n ⟩_{h} = n ∣ n ⟩_{h}$ .

For every unit Dirac particle mode $u \in H_{+}$ or antiparticle mode $u \in \overline{H_{-}}$ , regarded as a vector of $K$ , the bounded operator $N_{u} = c^{†} (u) c (u)$ is an orthogonal projection. The one-mode subspace generated by $u$ is the closed span of ${(c^{†} (u))^{n} Ω_{D} : n \geq 0}$ . It is spanned by $Ω_{D}$ and $c^{†} (u) Ω_{D}$ ; the latter is normalized and $c^{†} (u)^{2} = 0$ . Thus a single Dirac CAR mode has occupation $0$ or $1$ . All statements are conditional on the adopted representations and ${A C}_{ω}$ . They do not derive which statistics nature assigns to all fields and do not assert an experimental observation.

Facts & Assumptions

Given: The physical species assignments in [A1], Countable Choice in [A2], and the mathematical Fock operator construction in [F1].

[A1]

Specified bosonic scalar and CAR Dirac model adoption adopts the real scalar bosonic symmetric-Fock model and the Dirac particle/antiparticle CAR-Fock model. This is a physical model assumption, not a universal statistics theorem.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) states ${A C}_{ω}$ : every sequence of nonempty sets has a choice function. The Fock supplier assumes this principle, which is carried into this consumer.

[F1]

Bosonic and fermionic Fock operators with exact domains assumes ${A C}_{ω}$ and a separable complex Hilbert space. It constructs the symmetric and antisymmetric Fock spaces and proves that $D_{f i n}$ is dense; the bosonic operators are mutual adjoints on their displayed domains and on $D_{f i n}$ satisfy $[a (h), a^{†} (k)] = ⟨ h, k ⟩_{2} I$ , while the fermionic operators are bounded on the whole Fock space and satisfy ${c (h), c^{†} (k)} = ⟨ h, k ⟩_{2} I$ and ${c^{†} (h), c^{†} (k)} = 0$ .

[F2]

Bosonic and fermionic Fock operators with exact domains gives the exact bosonic maximal domains displayed in the Statement.

[F3]

Schwartz/distribution spaces and Hilbert Fock conventions defines each Fock vacuum as $Ω = (1, 0, 0, \dots)$ , defines creation and annihilation by sector insertion and contraction, and states $a (h) Ω = c (h) Ω = 0$ .

Proof

Given: [A1], [A2], and [F1]–[F3].

1.1givenA1A2F1F2F3algebra

Fix a unit $h \in H_{μ}$ and put $v_{n} = (a^{†} (h))^{n} Ω_{s}$ . The vacuum relation is $a (h) Ω_{s} = 0$ and the CCR give $a (h) a^{†} (h) = a^{†} (h) a (h) + I$ . The base case is $a (h) v_{1} = Ω_{s}$ . If $a (h) v_{n} = n v_{n - 1}$ , then $a (h) v_{n + 1} = a (h) a^{†} (h) v_{n} = a^{†} (h) a (h) v_{n} + v_{n} = n a^{†} (h) v_{n - 1} + v_{n} = (n + 1) v_{n}$ . Thus $a (h) v_{n} = n v_{n - 1}$ for every $n \geq 1$ .

1.2A1A2F1F3algebra

Fix a unit $u \in K$ . By [F1], $c (u) c^{†} (u) + c^{†} (u) c (u) = I$ and $c^{†} (u)^{2} = 0$ . On finite-particle vectors, the insertion and contraction formulas of [F3] give $⟨ c^{†} (u) ξ, η ⟩_{2} = ⟨ ξ, c (u) η ⟩_{2}$ : the antisymmetric projection can be removed against $η$ , after which the tensor pairing contracts the inserted $u$ with the first factor. Since both operators are bounded by [F1] and $D_{f i n}$ is dense by [F1], this identity extends to the whole Fock space and gives $c^{†} (u) = c (u)^{*}$ . Also $c (u) Ω_{D} = 0$ by [F3]. Hence $∥ c^{†} (u) Ω_{D} ∥^{2} = ⟨ Ω_{D}, c (u) c^{†} (u) Ω_{D} ⟩_{2} = ⟨ Ω_{D}, (I - c^{†} (u) c (u)) Ω_{D} ⟩_{2} = 1$ . A second creation vanishes. Since vacuum and one-particle sectors are orthogonal, the cyclic subspace generated by $u$ is exactly $span {Ω_{D}, c^{†} (u) Ω_{D}}$ . Moreover, $N_{u}^{*} = N_{u}$ and $N_{u}^{2} = c^{†} (u) (I - c^{†} (u) c (u)) c (u) = N_{u}$ because $c^{†} (u)^{2} = 0$ . Thus $N_{u}$ is a projection, so its only possible eigenvalues are $0$ and $1$ ; it acts as $0$ on $Ω_{D}$ and as $1$ on $c^{†} (u) Ω_{D}$ .

2.1A2F1F2F3step 1.1algebra

Since $a^{†} (h)$ and $a (h)$ are mutual adjoints on the finite-sector vectors, $∥ v_{n + 1} ∥^{2} = ⟨ v_{n}, a (h) a^{†} (h) v_{n} ⟩_{2} = (n + 1) ∥ v_{n} ∥^{2}$ . Starting from $∥ v_{0} ∥ = 1$ gives $∥ v_{n} ∥^{2} = n!$ . The vectors $v_{n}$ lie in distinct Fock sectors and are orthogonal. Consequently the displayed $∣ n ⟩_{h}$ are orthonormal, and dividing $a (h) v_{n} = n v_{n - 1}$ by the factorial normalizations gives $a (h) ∣ n ⟩_{h} = \sqrt{n} ∣ n - 1 ⟩_{h}$ . The creation formula gives $a^{†} (h) ∣ n ⟩_{h} = \sqrt{n + 1} ∣ n + 1 ⟩_{h}$ . The exact domains [F2] show that every finite-particle vector lies in both $D (a (h))$ and $D (a^{†} (h))$ , since each defining sum then has only finitely many finite sector terms. For $n \geq 1$ , $a (h) ∣ n ⟩_{h} = \sqrt{n} ∣ n - 1 ⟩_{h}$ is finite-particle, so $∣ n ⟩_{h} \in D (a (h))$ and $a (h) ∣ n ⟩_{h} \in D (a^{†} (h))$ ; hence $∣ n ⟩_{h} \in D (N_{h})$ and $N_{h} ∣ n ⟩_{h} = a^{†} (h) (\sqrt{n} ∣ n - 1 ⟩_{h}) = n ∣ n ⟩_{h}$ . For $n = 0$ , $a (h) ∣ 0 ⟩_{h} = 0 \in D (a^{†} (h))$ , so $∣ 0 ⟩_{h} \in D (N_{h})$ and $N_{h} ∣ 0 ⟩_{h} = 0$ .

3.1A1step 2.1step 1.2given∎

The scalar ladder and Dirac one-mode exclusion above follow only after the species-to-Fock assignments in [A1] are made. The argument uses the specified representations and proves no relation assigning statistics to every relativistic field.

CCR/CAR and particle occupations in adopted free models

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources