Manuscript XX-C@1 · XX-C.tex · section · 7a8cab37

Causal and Hadamard data

Identical frozen background, operator, and rods › Causal and Hadamard data · lines 455–527

XX-C@1#eq:Green, XX-C@1#eq:intertwine, XX-C@1#eq:Wdata, XX-C@1#eq:Wwavefront, XX-C@1#eq:realpart, XX-C@1#eq:fullW

Let $G_{L}^{ret, adv}$ and $G_{Q}^{ret, adv}$ be the causal inverses supplied by the normally hyperbolic Green theorem BGP, and set

G_{B}^{ret, adv} = G_{L}^{ret, adv} A^{- 1}, E_{B} = G_{B}^{ret} - G_{B}^{adv}, E_{Q} = G_{Q}^{ret} - G_{Q}^{adv} .

The actual inverse equations use

δ_{g ˉ} (x, y) = a (t_{y})^{- 3} δ (t_{x} - t_{y}) δ^{3} (x - y) .

The exact intertwining identities are

B K E_{B} F^{†} = - r F^{†} Q, = - r^{- 1} K E_{Q}, K^{†} B F E_{B} = - r Q F, = - r^{- 1} E_{Q} K^{†} .

For example, the retarded version of the first Green identity follows because its two sides solve the same inhomogeneous equation and have the same causal support; uniqueness then applies.

The seed contractions obey

B W_{B} = W_{B} B W_{P}^{*} WF^{'} (W_{P}) = 0, = W_{P}, \subset N_{+} . Q W_{Q} = W_{Q} Q W_{P} - W_{P}^{T} = 0, = i E_{P}, P = B, Q,

Here $WF^{'}$ reverses the second covector, and $N_{+}$ is the positive-frequency null bicharacteristic relation in the XX-B convention.

In a real field trivialization, (eq:Wdata) is equivalently

W_{P} = μ_{P} + \frac{i}{2} E_{P}, μ_{P} real and symmetric .

No sign condition on $μ_{P}$ is imposed.

The auxiliary and ghost contractions are reconstructed by

W_{ev} = (W_{B} - r F W_{B} - r W_{B} F^{†} r^{2} F W_{B} F^{†}), W_{odd} = (0 W_{Q} - W_{Q} 0) .

Thus a stated bosonic normalization change also changes the auxiliary blocks; they are not silently kept fixed.

Displayed formulas in this passage

XX-C@1#eq:Green · line 460
$G_{B}^{ret, adv} = G_{L}^{ret, adv} A^{- 1}, E_{B} = G_{B}^{ret} - G_{B}^{adv}, E_{Q} = G_{Q}^{ret} - G_{Q}^{adv} .$
XX-C@1#eq:intertwine · line 474
$B K E_{B} F^{†} = - r F^{†} Q, = - r^{- 1} K E_{Q}, K^{†} B F E_{B} = - r Q F, = - r^{- 1} E_{Q} K^{†} .$
XX-C@1#eq:Wdata · line 489
$B W_{B} = W_{B} B W_{P}^{*} WF^{'} (W_{P}) = 0, = W_{P}, \subset N_{+} . Q W_{Q} = W_{Q} Q W_{P} - W_{P}^{T} = 0, = i E_{P}, P = B, Q,$
XX-C@1#eq:Wwavefront · line 489
$B W_{B} = W_{B} B W_{P}^{*} WF^{'} (W_{P}) = 0, = W_{P}, \subset N_{+} . Q W_{Q} = W_{Q} Q W_{P} - W_{P}^{T} = 0, = i E_{P}, P = B, Q,$
XX-C@1#eq:realpart · line 506
$W_{P} = μ_{P} + \frac{i}{2} E_{P}, μ_{P} real and symmetric .$
XX-C@1#eq:fullW · line 514
$W_{ev} = (W_{B} - r F W_{B} - r W_{B} F^{†} r^{2} F W_{B} F^{†}), W_{odd} = (0 W_{Q} - W_{Q} 0) .$
unlabeled · line 468
$δ_{g ˉ} (x, y) = a (t_{y})^{- 3} δ (t_{x} - t_{y}) δ^{3} (x - y) .$