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

Quadratic closed-time-path bookkeeping

Quadratic closed-time-path bookkeeping · lines 1198–1246

XVIII-B@2#eq:Dpp, XVIII-B@2#eq:ctp

The closed-time-path construction concerns the same vacuum and two-stress insertion. It is not an all-orders source-evolution theorem.

Use the zero-momentum contacts, or transport a common contact $C$ consistently as described below. Define

D^{>} = \frac{i}{ℏ} W^{>}, D^{<} = \frac{i}{ℏ} W^{<},

and in Fourier space

D^{++} (p) D^{--} (p) D^{+-} = ℏ [Q_{2} (p) F_{2} (p^{2} + i 0) + Q_{0} (p) F_{0} (p^{2} + i 0)] + C (p), = - ℏ [Q_{2} (p) F_{2} (p^{2} - i 0) + Q_{0} (p) F_{0} (p^{2} - i 0)] - C (p), = D^{<}, D^{-+} = D^{>} .

Away from coincident points, these are respectively $i /ℏ$ times the time-ordered, anti-time-ordered, lesser, and greater correlators.

The minus sign in $D^{--}$ follows from its anti-time-ordering boundary value after multiplication by $i /ℏ$ ; it is not optional. The branch signs in the quadratic functional are separate from these kernel definitions.

For sources $H_{+}, H_{-}$ , let

H_{c} = \frac{H _{+} + H _{-}}{2}, H_{d} = H_{+} - H_{-}, N = \frac{1}{2} (W^{>} + W^{<}) .

All contractions below include spacetime integration and the two tensor slots.

Displayed formulas in this passage

XVIII-B@2#eq:Dpp · line 1214
$D^{++} (p) D^{--} (p) D^{+-} = ℏ [Q_{2} (p) F_{2} (p^{2} + i 0) + Q_{0} (p) F_{0} (p^{2} + i 0)] + C (p), = - ℏ [Q_{2} (p) F_{2} (p^{2} - i 0) + Q_{0} (p) F_{0} (p^{2} - i 0)] - C (p), = D^{<}, D^{-+} = D^{>} .$
XVIII-B@2#eq:ctp · line 1214
$D^{++} (p) D^{--} (p) D^{+-} = ℏ [Q_{2} (p) F_{2} (p^{2} + i 0) + Q_{0} (p) F_{0} (p^{2} + i 0)] + C (p), = - ℏ [Q_{2} (p) F_{2} (p^{2} - i 0) + Q_{0} (p) F_{0} (p^{2} - i 0)] - C (p), = D^{<}, D^{-+} = D^{>} .$
unlabeled · line 1208
$D^{>} = \frac{i}{ℏ} W^{>}, D^{<} = \frac{i}{ℏ} W^{<},$
unlabeled · line 1238
$H_{c} = \frac{H _{+} + H _{-}}{2}, H_{d} = H_{+} - H_{-}, N = \frac{1}{2} (W^{>} + W^{<}) .$

Quadratic closed-time-path bookkeeping

Displayed formulas in this passage

Nearby in this section