Manuscript XVIII-B@2 · XVIII-B-v2.tex · theorem · 27ed4f4b

Theorem: Evaluated stress spectral densities

The exact two-particle spectral calculation › Field and state normalization · lines 515–582 · proof included or linked

XVIII-B@2#thm:spectral, XVIII-B@2#eq:r2, XVIII-B@2#eq:r0, XVIII-B@2#eq:Wspec

Set $a = 4 m^{2}$ and, for $u \geq a$ , define

r_{2} (u) r_{0} (u) = \frac{1}{960 π ^{2}} (1 - \frac{a}{u})^{5/2}, = \frac{1}{96 π ^{2}} (1 - \frac{a}{u})^{1/2} (1 + \frac{2 m ^{2}}{u})^{2} .

Extend them by zero below $a$ . Then

W_{ab c d}^{>} (p) = 2 π ℏ^{2} Θ (p^{0}) [(Q_{2})_{ab c d} (p) r_{2} (p^{2}) + (Q_{0})_{ab c d} (p) r_{0} (p^{2})] .

Both densities are strictly positive above threshold. The corresponding projector densities are

ρ_{2} (u) = u^{2} r_{2} (u), ρ_{0} (u) = u^{2} r_{0} (u) .

The formula defines a positive, tempered vacuum two-point distribution.

Proof. In the rest frame $p = (s, 0)$ , $τ_{00} = τ_{0 i} = 0$ and

τ_{ij} = 2 K^{2} n_{i} n_{j} - \frac{s}{2} δ_{ij} .

Its spatial trace is $- (s + 2 m^{2})$ . The traceless part is $2 K^{2} (n_{i} n_{j} - δ_{ij} /3)$ . Lemma lem:phase therefore gives

⟨ τ_{ij} τ_{k l} ⟩ = \frac{8 K ^{4}}{15} (P_{2})_{ij k l} + \frac{( s + 2 m ^{2} ) ^{2}}{3} (P_{0})_{ij k l} .

Multiplying by $ℏ^{2} v / (16 π)$ from (eq:phaseformula) gives

ℏ^{2} \frac{s ^{2} v ^{5}}{480 π} P_{2} + ℏ^{2} \frac{v ( s + 2 m ^{2} ) ^{2}}{48 π} P_{0} .

This is (eq:Wspec). Lorentz covariance and transversality extend the rest-frame calculation to the future two-particle cone.

Positivity also follows directly from the squared matrix element in (eq:phaseformula). The polynomial tensor growth is at most quartic, while $r_{2}, r_{0}$ are bounded. Hence the spectral distribution is tempered.

Displayed formulas in this passage

XVIII-B@2#eq:r2 · line 518
$r_{2} (u) r_{0} (u) = \frac{1}{960 π ^{2}} (1 - \frac{a}{u})^{5/2}, = \frac{1}{96 π ^{2}} (1 - \frac{a}{u})^{1/2} (1 + \frac{2 m ^{2}}{u})^{2} .$
XVIII-B@2#eq:r0 · line 518
$r_{2} (u) r_{0} (u) = \frac{1}{960 π ^{2}} (1 - \frac{a}{u})^{5/2}, = \frac{1}{96 π ^{2}} (1 - \frac{a}{u})^{1/2} (1 + \frac{2 m ^{2}}{u})^{2} .$
XVIII-B@2#eq:Wspec · line 531
$W_{ab c d}^{>} (p) = 2 π ℏ^{2} Θ (p^{0}) [(Q_{2})_{ab c d} (p) r_{2} (p^{2}) + (Q_{0})_{ab c d} (p) r_{0} (p^{2})] .$
unlabeled · line 543
$ρ_{2} (u) = u^{2} r_{2} (u), ρ_{0} (u) = u^{2} r_{0} (u) .$
unlabeled · line 553
$τ_{ij} = 2 K^{2} n_{i} n_{j} - \frac{s}{2} δ_{ij} .$
unlabeled · line 560
$⟨ τ_{ij} τ_{k l} ⟩ = \frac{8 K ^{4}}{15} (P_{2})_{ij k l} + \frac{( s + 2 m ^{2} ) ^{2}}{3} (P_{0})_{ij k l} .$
unlabeled · line 569
$ℏ^{2} \frac{s ^{2} v ^{5}}{480 π} P_{2} + ℏ^{2} \frac{v ( s + 2 m ^{2} ) ^{2}}{48 π} P_{0} .$

Theorem: Evaluated stress spectral densities

Displayed formulas in this passage

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