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

Lemma: Vacuum-to-two-particle stress matrix element

The exact two-particle spectral calculation › Field and state normalization · lines 443–475 · proof included or linked

XVIII-B@2#lem:matrixelement, XVIII-B@2#eq:phaseformula

For on-shell positive-energy $k, q$ , define

τ_{ab} (k, q) = - (k_{a} q_{b} + k_{b} q_{a}) + η_{ab} (k \cdot q + m^{2}) .

The vacuum matrix element is $ℏ τ_{ab}$ , and

W_{ab c d}^{>} (p) = \frac{ℏ ^{2}}{2} \int d μ_{m} (k) d μ_{m} (q) (2 π)^{4} δ^{4} (p - k - q) τ_{ab} (k, q) τ_{c d} (k, q) .

Moreover $(k + q)^{a} τ_{ab} = 0$ .

Proof. The two annihilation operators in (eq:stress) give the two derivative contractions $- (k_{a} q_{b} + k_{b} q_{a})$ . The trace term gives $η_{ab} (k \cdot q + m^{2})$ . The factor $ℏ$ comes from the two field factors. Inserting the two-particle identity gives (eq:phaseformula) and its factor $1/2$ .

Using $k^{2} = q^{2} = m^{2}$ , contraction with $k + q$ cancels the two terms.

Displayed formulas in this passage

XVIII-B@2#eq:phaseformula · line 451
$W_{ab c d}^{>} (p) = \frac{ℏ ^{2}}{2} \int d μ_{m} (k) d μ_{m} (q) (2 π)^{4} δ^{4} (p - k - q) τ_{ab} (k, q) τ_{c d} (k, q) .$
unlabeled · line 446
$τ_{ab} (k, q) = - (k_{a} q_{b} + k_{b} q_{a}) + η_{ab} (k \cdot q + m^{2}) .$

Lemma: Vacuum-to-two-particle stress matrix element

Displayed formulas in this passage

Nearby in this section