Manuscript VII-B@1 · VII-B.tex · section · be632835

Gauge fixing, ghosts, and auxiliary field

An explicit action and vertex ledger › Gauge fixing, ghosts, and auxiliary field · lines 1470–1519

VII-B@1#eq:gauge-generator, VII-B@1#eq:gf-ghost, VII-B@1#eq:shifted-gf

For an odd vector field $c^{μ}$ , define

(D_{κ} (h) c)_{μν} = \partial_{μ} c_{ν} + \partial_{ν} c_{μ} + κ (c^{ρ} \partial_{ρ} h_{μν} + h_{ρ ν} \partial_{μ} c^{ρ} + h_{μ ρ} \partial_{ν} c^{ρ}) .

The notation $D_{κ} (h)$ here denotes the infinitesimal metric gauge generator, not an Euler operator.

With an unshifted auxiliary field $B_{μ}$ , choose

L_{gf + gh} = B_{μ} C^{μ} [h] - \frac{1}{4} B_{μ} B^{μ} - c ˉ_{μ} C^{μ} [D_{κ} (h) c] .

The identity

C^{μ} [\partial c + \partial c] = □ c^{μ}

gives

L_{gf + gh} = C_{μ} [h] C^{μ} [h] - \frac{1}{4} β_{μ} β^{μ} - c ˉ_{μ} □ c^{μ} - κ c ˉ_{μ} C^{μ} [L_{c} h], β_{μ} = B_{μ} - 2 C_{μ} [h] .

Adding $C_{μ} C^{μ}$ to (eq:FP) gives the graviton quadratic density in (eq:free-action), modulo an unswitched divergence. The shifted auxiliary field has no interaction vertices in this parametrization.

The chosen action is sufficient to define the gauge-fixed perturbative field content. No switched BRST or Ward identity is inferred from these formulas.

Displayed formulas in this passage

VII-B@1#eq:gauge-generator · line 1473
$(D_{κ} (h) c)_{μν} = \partial_{μ} c_{ν} + \partial_{ν} c_{μ} + κ (c^{ρ} \partial_{ρ} h_{μν} + h_{ρ ν} \partial_{μ} c^{ρ} + h_{μ ρ} \partial_{ν} c^{ρ}) .$
VII-B@1#eq:gf-ghost · line 1489
$L_{gf + gh} = B_{μ} C^{μ} [h] - \frac{1}{4} B_{μ} B^{μ} - c ˉ_{μ} C^{μ} [D_{κ} (h) c] .$
VII-B@1#eq:shifted-gf · line 1501
$L_{gf + gh} = C_{μ} [h] C^{μ} [h] - \frac{1}{4} β_{μ} β^{μ} - c ˉ_{μ} □ c^{μ} - κ c ˉ_{μ} C^{μ} [L_{c} h], β_{μ} = B_{μ} - 2 C_{μ} [h] .$
unlabeled · line 1497
$C^{μ} [\partial c + \partial c] = □ c^{μ}$