Manuscript XX-B@1 · XX-B.tex · section · 56d708a5

Gauge condition and auxiliary normalization

Gauge fixing and the normally hyperbolic realization › Gauge condition and auxiliary normalization · lines 720–787

XX-B@1#eq:F, XX-B@1#eq:beta, XX-B@1#eq:BRST, XX-B@1#eq:Psi, XX-B@1#eq:Sgf, XX-B@1#eq:Qdef, XX-B@1#eq:Fadj, XX-B@1#eq:Q

Retain XX-A's gauge-condition candidate:

F_{ν} (h, φ) = \nabla^{μ} h_{μν} - \frac{1}{2} \nabla_{ν} h^{μ}_{μ} - M^{- 2} v_{ν} φ .

Fix

r = \frac{M ^{2}}{2}, β = r^{- 1} = \frac{2}{M ^{2}} .

Thus a previously unspecified positive gauge parameter is now fixed; the rod prescription is not changed.

The free graded fields are

z = (h, φ), B_{ν}, c_{ν}, c ˉ_{ν},

with $z, B$ even and $c, c ˉ$ odd. Set

s_{0} z = K c, s_{0} c = 0, s_{0} c ˉ = B, s_{0} B = 0.

The gauge fermion and quadratic action are

Ψ S_{2, gf} Q = (c ˉ, F z + \frac{1}{2 r} B)_{1}, = \frac{1}{2} (z, H z)_{0} + (B, F z)_{1} + \frac{1}{2 r} (B, B)_{1} - (c ˉ, Q c)_{1}, = F K .

The odd Leibniz rule gives the ghost sign in (eq:Sgf). Since $H K = 0$ , $s_{0} S_{2, gf} = 0$ and $s_{0}^{2} = 0$ on the off-shell free multiplet.

Let

D_{ν} h = \nabla^{μ} h_{μν} - \frac{1}{2} \nabla_{ν} h^{μ}_{μ} .

Its adjoint and the adjoint of $F$ are

(D^{†} a)_{μν} F^{†} a = - \nabla_{(μ} a_{ν)} + \frac{1}{2} g ˉ_{μν} \nabla^{ρ} a_{ρ}, = (D^{†} a, - M^{- 2} v^{ν} a_{ν}) .

The ghost operator is explicitly

(Q ξ)_{ν} = □_{g ˉ} ξ_{ν} + R ˉ_{ν ρ} ξ^{ρ} - M^{- 2} v_{ν} v_{ρ} ξ^{ρ} .

It is normally hyperbolic and formally self-adjoint for $(\cdot, \cdot)_{1}$ .

Displayed formulas in this passage

XX-B@1#eq:F · line 723
$F_{ν} (h, φ) = \nabla^{μ} h_{μν} - \frac{1}{2} \nabla_{ν} h^{μ}_{μ} - M^{- 2} v_{ν} φ .$
XX-B@1#eq:beta · line 731
$r = \frac{M ^{2}}{2}, β = r^{- 1} = \frac{2}{M ^{2}} .$
XX-B@1#eq:BRST · line 743
$s_{0} z = K c, s_{0} c = 0, s_{0} c ˉ = B, s_{0} B = 0.$
XX-B@1#eq:Psi · line 749
$Ψ S_{2, gf} Q = (c ˉ, F z + \frac{1}{2 r} B)_{1}, = \frac{1}{2} (z, H z)_{0} + (B, F z)_{1} + \frac{1}{2 r} (B, B)_{1} - (c ˉ, Q c)_{1}, = F K .$
XX-B@1#eq:Sgf · line 749
$Ψ S_{2, gf} Q = (c ˉ, F z + \frac{1}{2 r} B)_{1}, = \frac{1}{2} (z, H z)_{0} + (B, F z)_{1} + \frac{1}{2 r} (B, B)_{1} - (c ˉ, Q c)_{1}, = F K .$
XX-B@1#eq:Qdef · line 749
$Ψ S_{2, gf} Q = (c ˉ, F z + \frac{1}{2 r} B)_{1}, = \frac{1}{2} (z, H z)_{0} + (B, F z)_{1} + \frac{1}{2 r} (B, B)_{1} - (c ˉ, Q c)_{1}, = F K .$
XX-B@1#eq:Fadj · line 770
$(D^{†} a)_{μν} F^{†} a = - \nabla_{(μ} a_{ν)} + \frac{1}{2} g ˉ_{μν} \nabla^{ρ} a_{ρ}, = (D^{†} a, - M^{- 2} v^{ν} a_{ν}) .$
XX-B@1#eq:Q · line 779
$(Q ξ)_{ν} = □_{g ˉ} ξ_{ν} + R ˉ_{ν ρ} ξ^{ρ} - M^{- 2} v_{ν} v_{ρ} ξ^{ρ} .$
unlabeled · line 739
$z = (h, φ), B_{ν}, c_{ν}, c ˉ_{ν},$
unlabeled · line 765
$D_{ν} h = \nabla^{μ} h_{μν} - \frac{1}{2} \nabla_{ν} h^{μ}_{μ} .$