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Logarithmic coherent infrared tail: finite energy, no Fock limit

Statement

Use the transverse one-particle space $H_{T}$ and multiplier $ω (k)$ of Free transverse-photon energy multiplier, and form its symmetric Fock space $F_{s} (H_{T})$ . Let $H_{p h} : = ℏ c d Γ (ω)$ , with $ℏ, c > 0$ and $d Γ (ω)$ acting on the $n$ -particle sector as multiplication by $\sum_{j = 1}^{n} ∣ k_{j} ∣$ , as in Second-quantized multipliers, joint spectral measure and exact domains. For a coherent vector use $ϵ (f) : = ⨁_{n = 0}^{\infty} \frac{f^{\otimes n}}{\sqrt{n!}}, C (f) : = e^{- ∥ f ∥^{2} / 2} ϵ (f),$ using the tensor and direct-sum conventions of Schwartz/distribution spaces and Hilbert Fock conventions.

Let $κ > 0$ . Let $b : S^{2} \to C^{3}$ be measurable with $b \in L^{2} (S^{2}, d Ω; C^{3})$ , $n \cdot b (n) = 0$ for a.e. $n$ , and $C_{b} : = \int_{S^{2}} ∣ b (n) ∣^{2} d Ω > 0.$ Take $k, κ, ϵ$ in inverse metres and $b$ dimensionless; then $f_{ϵ}$ has momentum-wavefunction units $m^{3 / 2}$ , $C_{b}$ is dimensionless, and $ℏ c (κ - ϵ)$ has energy units. For $0 < ϵ < κ$ , define $f_{ϵ} (0) = 0$ and $f_{ϵ} (k) : = 1_{{ϵ < ∣ k ∣ < κ}} ∣ k ∣^{- 3 / 2} b ⁣ (\frac{k}{∣ k ∣}) .$ Then $∥ f_{ϵ} ∥^{2} = C_{b} \log \frac{κ}{ϵ}, ⟨ C (f_{ϵ}), H_{p h} C (f_{ϵ}) ⟩ = ℏ c C_{b} (κ - ϵ) .$ Each $C (f_{ϵ})$ is a unit vector in $D (H_{p h})$ . As $ϵ ↓ 0$ , $C (f_{ϵ}) ⇀ 0$ weakly, but the family has no strong Fock-space limit. The statement is an exact free transverse-mode construction; it asserts no interacting charged sector, continuum QED limit, or empirical observation.

Facts & Assumptions

Given: Countable Choice; the transverse-space and multiplier definition cited below; and the coherent-vector and second-quantized multiplier interfaces stated below.

[A1]

Countable Choice, written ${A C}_{ω}$ , is assumed as in the Fock and multiplier suppliers. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[A2]

$ℏ, c > 0$ , $κ > 0$ , $b \in L^{2} (S^{2}, d Ω; C^{3})$ , $n \cdot b (n) = 0$ a.e., and $C_{b} = \int_{S^{2}} ∣ b ∣^{2} d Ω > 0$ ; each cutoff satisfies $0 < ϵ < κ$ . In the QFT length convention $k, κ, ϵ$ have units m $^{- 1}$ and $b$ is dimensionless. These are the explicit data hypotheses of the Statement.

[F1]

Free transverse-photon energy multiplier: $H_{T} = Ran M_{P_{T}} \subset L^{2} (R^{3}, d^{3} k; C^{3})$ , $P_{T} (k) = I - n n^{T}$ , and the nonnegative maximal multiplication operator $M_{ω}$ with $ω (k) = ∣ k ∣$ . This local definition uses the exact QFT-MG01 transverse-space draft registered in the completed batch-44 manifest; the definition and this theorem remain conditional on that mathematical supplier, with native readiness and independent review separate.

[F2]

Schwartz/distribution spaces and Hilbert Fock conventions: symmetric tensor sectors have the product inner pairing on elementary tensors, and bosonic Fock space is their square-summable direct sum with componentwise pairing.

[F3]

Second-quantized multipliers, joint spectral measure and exact domains: for a real measurable one-particle multiplier $ω$ , its second quantization acts on sector $n$ by $\sum_{j = 1}^{n} ω (k_{j})$ with the maximal square-integrability domain. The draft statement and full proof were inspected; it assumes Countable Choice. For the vector-valued transverse space, identify $L^{2} (R^{3}; C^{3})$ with scalar $L^{2} (R^{3} \times {1, 2, 3})$ before restricting to the transverse range.

[F4]

Surface integration on compact C1 hypersurfaces: under ${A C}_{ω}$ , surface integration on $S^{2}$ is the Euclidean chart integral using the induced metric Jacobian.

[F5]

Agreement with the existing polar sphere measure: under ${A C}_{ω}$ , this chart surface measure on $S^{2}$ equals the polar measure $σ$ .

[F6]

Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma: under ${A C}_{ω}$ , every nonnegative Borel function on $R^{3}$ integrates as $\int_{0}^{\infty} \int_{S^{2}} F (r n) r^{2} d σ (n) d r$ .

[F7]

The complex exponential by its power series defines $\exp z = \sum_{n \geq 0} z^{n} / n!$ ; The complex exponential series converges absolutely for every complex argument supplies absolute convergence for every $z \in C$ .

[F8]

$\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential: $\exp (z + w) = \exp z \exp w$ , and $\exp (x + 0 i) = e^{x}$ for real $x$ .

[F9]

$\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ : $∣ \exp z ∣ = e^{Re z}$ .

[F10]

For $x > 0$ , $\log x = \int_{1}^{x} d u / u$ and $\log^{'} (x) = 1 / x$ . The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t

[F11]

In a real or complex inner-product space, $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ . Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors

Proof

1.1A1A2F1F4F5F6algebra

For $n \in S^{2}$ , $P_{T} (n) = I - n n^{⊤}$ is an orthogonal projection and [A2] gives $n \cdot b (n) = 0$ ; hence $k \cdot f_{ϵ} (k) = 0$ a.e. Since $ϵ < ∣ k ∣ < κ$ on the support, the radial norm integrals in [F4]–[F6] are finite on these strictly positive finite bounds, so both $f_{ϵ}$ and $ω f_{ϵ}$ are square-integrable.

1.2A1A2F1F4F5F6F10algebra

By [F4]–[F6], spherical coordinates give $d^{3} k = r^{2} d r d Ω$ . Substituting $r = ϵ u$ in the radial integral and using [F10] gives $\int_{ϵ}^{κ} d r / r = \log (κ / ϵ)$ , so $∥ f_{ϵ} ∥^{2} = C_{b} \int_{ϵ}^{κ} r^{- 1} d r = C_{b} \log (κ / ϵ)$ , $⟨ f_{ϵ}, ω f_{ϵ} ⟩ = C_{b} (κ - ϵ)$ , and $∥ ω f_{ϵ} ∥^{2} = C_{b} (κ^{2} - ϵ^{2}) / 2$ .

2.1F2F3F7F8F9step 1.1step 1.2algebra

For any $u, v \in H_{T}$ , [F2] gives $⟨ u^{\otimes n}, v^{\otimes n} ⟩ = ⟨ u, v ⟩^{n}$ and $∥ u^{\otimes n} ∥^{2} = ∥ u ∥^{2 n}$ . By [F7]–[F8], $\sum_{n \geq 0} ∥ u ∥^{2 n} / n! = e^{∥ u ∥^{2}} < \infty$ , so the displayed $ϵ (u)$ belongs to symmetric Fock space. Its direct-sum pairing with $ϵ (v)$ is the absolutely convergent series $\sum_{n \geq 0} ⟨ u, v ⟩^{n} / n! = e^{⟨ u, v ⟩}$ , by [F7]. The addition and modulus laws [F8]–[F9] therefore give that $C (u)$ has norm one and $∣ ⟨ C (u), C (v) ⟩ ∣ = e^{- ∥ u - v ∥^{2} / 2}$ . This direct tensor calculation requires no separability hypothesis on $H_{T}$ . On the full vector space $L^{2} (R^{3}; C^{3})$ , multiplication by $ω$ commutes with the multiplier $P_{T}$ . Its tensor-power projections therefore commute with the sector multiplier $\sum_{j} ∣ k_{j} ∣$ ; the transverse symmetric Fock space is a reducing subspace for the second-quantized multiplier of [F3]. If $r = ∥ f_{ϵ} ∥^{2}$ , $q = ⟨ f_{ϵ}, ω f_{ϵ} ⟩$ , and $s = ∥ ω f_{ϵ} ∥^{2}$ , the sector formulas give $∥ d Γ (ω) C (f_{ϵ}) ∥^{2} = e^{- r} \sum_{n \geq 1} \frac{n s r^{n - 1} + n (n - 1) q^{2} r^{n - 2}}{n!} = s + q^{2} < \infty .$ The same sector sum with one multiplier gives $⟨ C (f_{ϵ}), d Γ (ω) C (f_{ϵ}) ⟩ = q$ ; [F2] gives $∥ C (f_{ϵ}) ∥ = 1$ . Multiplying by $ℏ c$ and substituting [step 1.2] proves the stated domain and energy claims.

3.1A1A2F2F4F5F6F10step 1.2step 2.1algebra

Put $ϵ_{j} = κ 2^{- j}$ for $j \geq 1$ . The difference $f_{ϵ_{j + 1}} - f_{ϵ_{j}}$ is supported on the shell $(ϵ_{j + 1}, ϵ_{j})$ , so its squared norm is $C_{b} \int_{ϵ_{j + 1}}^{ϵ_{j}} r^{- 1} d r = C_{b} \log 2 > 0$ , using $r = ϵ_{j + 1} u$ and [F10]. From the pairing identity in step 2.1, $∣ ⟨ C (f_{ϵ_{j}}), C (f_{ϵ_{j + 1}}) ⟩ ∣ = e^{- ∥ f_{ϵ_{j + 1}} - f_{ϵ_{j}} ∥^{2} / 2} = e^{- C_{b} \log 2 / 2} < 1.$ Thus $∥ C (f_{ϵ_{j}}) - C (f_{ϵ_{j + 1}}) ∥^{2} \geq 2 (1 - e^{- C_{b} \log 2 / 2}) > 0$ , and this sequence is not Cauchy.

4.1A2F2F7F8F11step 1.2step 2.1step 3.1algebra∎

Let $Ψ$ have only sectors $0, \dots, N$ . By sectorwise Cauchy–Schwarz [F11], $∣ ⟨ Ψ, C (f_{ϵ}) ⟩ ∣ \leq e^{- ∥ f_{ϵ} ∥^{2} / 2} \sum_{n = 0}^{N} \frac{∥ Ψ_{n} ∥ ∥ f_{ϵ} ∥^{n}}{\sqrt{n!}} ⟶ 0$ as $ϵ ↓ 0$ , as follows explicitly. If $ϵ < κ 2^{- m}$ , the radial integral in step 1.2 contains $m$ disjoint dyadic intervals; on each $(a, 2 a)$ , $\int_{a}^{2 a} d r / r \geq 1 / 2$ . Thus $∥ f_{ϵ} ∥^{2} \geq C_{b} m / 2 \to \infty$ . Write $a = ∥ f_{ϵ} ∥$ and choose an integer $M > N / 2$ . The nonnegative real exponential series [F7]–[F8] gives $e^{a^{2} / 2} \geq (a^{2} / 2)^{M} / M!$ . Hence for $0 \leq n \leq N$ , $a^{n} e^{- a^{2} / 2} \leq 2^{M} M! a^{n - 2 M} \to 0$ , proving the displayed finite-sum limit. Finite-sector vectors are dense: for any Fock vector $Ψ = (Ψ_{n})_{n \geq 0}$ , the truncations $\sum_{n = 0}^{N} Ψ_{n}$ converge in norm because $\sum_{n} ∥ Ψ_{n} ∥^{2} < \infty$ . Since $∥ C (f_{ϵ}) ∥ = 1$ , [F11] bounds the pairing of each truncation error by its norm, uniformly in $ϵ$ . First taking a sufficiently accurate truncation and then the limit just proved gives weak convergence to zero for every Fock vector. A norm limit would have to equal the weak limit zero, contradicting the unit norms.

Logarithmic coherent infrared tail: finite energy, no Fock limit

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources