Manuscript X-G@1 · X-G.tex · section · 01566e14

The ghost Green operators

The normalized free BRST multiplet › The ghost Green operators · lines 655–690

X-G@1#eq:ghostGreen, X-G@1#eq:causal

Use left Grassmann variations. In the order $(c, c ˉ)$ , the ghost kinetic matrix and its inverses are

L_{gh} = (0 - □ □ 0), G_{gh}^{\pm} = (0 E_{0}^{\pm} - E_{0}^{\pm} 0) .

The flat vector–covector fibre pairing is implicit. Matrix multiplication verifies both inverse identities.

Let

δ_{μ} = E_{μ}^{R} - E_{μ}^{A} .

The nonzero causal blocks are

Δ_{hh} Δ_{hb} Δ_{c c ˉ} = - 4 P δ_{0}, = K δ_{0}, = - δ_{0}, Δ_{ϕϕ} Δ_{bh} Δ_{c ˉ c} = - δ_{m}, = - 2 C P δ_{0}, = δ_{0} .

The ghost block has the required graded symmetry because $δ_{0}^{T} = - δ_{0}$ .

Displayed formulas in this passage

X-G@1#eq:ghostGreen · line 659
$L_{gh} = (0 - □ □ 0), G_{gh}^{\pm} = (0 E_{0}^{\pm} - E_{0}^{\pm} 0) .$
X-G@1#eq:causal · line 677
$Δ_{hh} Δ_{hb} Δ_{c c ˉ} = - 4 P δ_{0}, = K δ_{0}, = - δ_{0}, Δ_{ϕϕ} Δ_{bh} Δ_{c ˉ c} = - δ_{m}, = - 2 C P δ_{0}, = δ_{0} .$
unlabeled · line 672
$δ_{μ} = E_{μ}^{R} - E_{μ}^{A} .$