Manuscript X-G-ST@1 · X-G-ST.tex · section · 3c97567f

Ghost matrix and full causal form

Stage 1: the normalized free auxiliary–ghost multiplet › Ghost matrix and full causal form · lines 794–849

X-G-ST@1#eq:ghostGreen, X-G-ST@1#eq:causalblocks

Use left Grassmann variations, placing variations on the left. The ghost Euler derivatives of $- \int c ˉ □ c$ are

E_{c} = □ c ˉ, E_{c ˉ} = - □ c .

Consequently, in the order $(c, c ˉ)$ ,

L_{gh} = (0 - □ □ 0), G_{gh}^{R / A} = (0 E_{0}^{R / A} - E_{0}^{R / A} 0) .

Ordinary matrix multiplication verifies the two inverse identities with the indicated fibre pairings.

For the complete order $(r, b, ϕ, c, c ˉ)$ , use

G^{R / A} = diag (G_{r b}^{R / A}, - E_{m}^{R / A}, G_{gh}^{R / A}) .

Put

δ_{μ} = E_{μ}^{R} - E_{μ}^{A}, Δ = G^{R} - G^{A} .

Its nonzero blocks are

\begin{array}{c|c@{\qquad}c|c}
 AB&\Delta_{AB}&AB&\Delta_{AB}\\
 \hline
 rr&-4\mathsf P_{\mathrm{tr}}\delta_0&\phi\phi&-\delta_m\\
 rb&K\delta_0&br&-2\mathsf C\mathsf P_{\mathrm{tr}}\delta_0\\
 c\bar c&-\delta_0&\bar c c&\delta_0
\end{array}
 \label{eq:causalblocks}

and $Δ_{bb} = 0$ . The scalar sign comes from its action Hessian. The ghost form is graded antisymmetric: its odd–odd blocks are symmetric under simultaneous exchange of species and spacetime arguments, because $δ_{0}^{T} = - δ_{0}$ .

Displayed formulas in this passage

X-G-ST@1#eq:ghostGreen · line 803
$L_{gh} = (0 - □ □ 0), G_{gh}^{R / A} = (0 E_{0}^{R / A} - E_{0}^{R / A} 0) .$

X-G-ST@1#eq:causalblocks · line 833

\begin{array}{c|c@{\qquad}c|c}
 AB&\Delta_{AB}&AB&\Delta_{AB}\\
 \hline
 rr&-4\mathsf P_{\mathrm{tr}}\delta_0&\phi\phi&-\delta_m\\
 rb&K\delta_0&br&-2\mathsf C\mathsf P_{\mathrm{tr}}\delta_0\\
 c\bar c&-\delta_0&\bar c c&\delta_0
\end{array}
 \label{eq:causalblocks}

unlabeled · line 799
$E_{c} = □ c ˉ, E_{c ˉ} = - □ c .$
unlabeled · line 818
$G^{R / A} = diag (G_{r b}^{R / A}, - E_{m}^{R / A}, G_{gh}^{R / A}) .$
unlabeled · line 826
$δ_{μ} = E_{μ}^{R} - E_{μ}^{A}, Δ = G^{R} - G^{A} .$