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

The natural background fermion

The nonlinear kinematics and background ghost operator › The natural background fermion · lines 845–872

X-G@1#eq:PsiB, X-G@1#eq:sPsiB, X-G@1#eq:Pgh

For the background de Donder vector of X-F, set

Ψ_{q ˉ} = \int - q ˉ c ˉ_{μ} (C^{μ}_{q ˉ} [h] - b^{μ}_{q ˉ}) d^{4} x .

The background is $s$ -inert, so

s Ψ_{q ˉ} = \int - q ˉ [b_{μ} C^{μ}_{q ˉ} [h] - b_{μ} b^{μ}_{q ˉ} - c ˉ_{μ} C^{μ}_{q ˉ} [L_{c} (q ˉ + κh)]] d^{4} x .

Eliminating $b$ gives exactly the bosonic gauge fixing of X-F.

Define the background ghost kinetic operator, with its Euler type, by

(P_{gh} [q ˉ] c)^{μ} = - - q ˉ C^{μ}_{q ˉ} [L_{c} q ˉ] .

Displayed formulas in this passage

X-G@1#eq:PsiB · line 848
$Ψ_{q ˉ} = \int - q ˉ c ˉ_{μ} (C^{μ}_{q ˉ} [h] - b^{μ}_{q ˉ}) d^{4} x .$
X-G@1#eq:sPsiB · line 855
$s Ψ_{q ˉ} = \int - q ˉ [b_{μ} C^{μ}_{q ˉ} [h] - b_{μ} b^{μ}_{q ˉ} - c ˉ_{μ} C^{μ}_{q ˉ} [L_{c} (q ˉ + κh)]] d^{4} x .$
X-G@1#eq:Pgh · line 867
$(P_{gh} [q ˉ] c)^{μ} = - - q ˉ C^{μ}_{q ˉ} [L_{c} q ˉ] .$