Manuscript X-H-ST@1 · X-H-ST.tex · section · 943aa27e

The natural reference action

Stage B: an explicit relative classical master action › The natural reference action · lines 621–693

X-H-ST@1#eq:Iinv, X-H-ST@1#eq:Igf, X-H-ST@1#eq:gfexplicit

Fix the curvature convention by

Γ^{ρ}_{μν} [q] R^{ρ}_{σ μν} R_{σ ν} = \frac{1}{2} q^{ρ λ} (\partial_{μ} q_{λ ν} + \partial_{ν} q_{λ μ} - \partial_{λ} q_{μν}), = \partial_{μ} Γ^{ρ}_{ν σ} - \partial_{ν} Γ^{ρ}_{μ σ} + Γ^{ρ}_{μ λ} Γ^{λ}_{ν σ} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ σ}, = R^{ρ}_{σ ρ ν} .

Write $I [q] = J_{q} R [q]$ , $J_{q} = - det q$ , and

L_{q} [ϕ] = \frac{1}{2} J_{q} (q^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}) .

The local reference invariant action is the class

I_{inv} = - \frac{1}{2 κ ^{2}} \int (I [q] - I [q ˉ]) d^{4} x + \int L_{q} [ϕ] d^{4} x .

These integrals denote variational classes, not an assumption of globally finite action.

The coefficient of $κ^{- 1}$ is

- \frac{1}{2} (\partial_{μ} \partial_{ν} r^{μν} - □ tr_{η} r),

a divergence. It is removed before the action is viewed as a nonnegative formal series. Thus no manipulation of $κ^{- 1}$ in a nilpotent coupling ring is involved.

The background de Donder vector and fermion are

C_{q ˉ}^{μ} [r] Ψ_{q ˉ} I = \nabla ˉ_{ν} (q ˉ^{ν α} q ˉ^{μ β} r_{α β} - \frac{1}{2} q ˉ^{ν μ} q ˉ^{α β} r_{α β}), = \int J_{q ˉ} c ˉ_{μ} (C_{q ˉ}^{μ} [r] - q ˉ^{μν} b_{ν}) d^{4} x, = I_{inv} + s Ψ_{q ˉ} .

In particular,

s Ψ_{q ˉ} = \int J_{q ˉ} (b_{μ} C_{q ˉ}^{μ} [r] - b_{μ} q ˉ^{μν} b_{ν} - c ˉ_{μ} C_{q ˉ}^{μ} [L_{c} q ˉ + κ L_{c} r]) d^{4} x .

The independent $b$ has not been eliminated.

Displayed formulas in this passage

unlabeled · line 624
$Γ^{ρ}_{μν} [q] R^{ρ}_{σ μν} R_{σ ν} = \frac{1}{2} q^{ρ λ} (\partial_{μ} q_{λ ν} + \partial_{ν} q_{λ μ} - \partial_{λ} q_{μν}), = \partial_{μ} Γ^{ρ}_{ν σ} - \partial_{ν} Γ^{ρ}_{μ σ} + Γ^{ρ}_{μ λ} Γ^{λ}_{ν σ} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ σ}, = R^{ρ}_{σ ρ ν} .$
X-H-ST@1#eq:Iinv · line 646
$I_{inv} = - \frac{1}{2 κ ^{2}} \int (I [q] - I [q ˉ]) d^{4} x + \int L_{q} [ϕ] d^{4} x .$
X-H-ST@1#eq:Igf · line 668
$C_{q ˉ}^{μ} [r] Ψ_{q ˉ} I = \nabla ˉ_{ν} (q ˉ^{ν α} q ˉ^{μ β} r_{α β} - \frac{1}{2} q ˉ^{ν μ} q ˉ^{α β} r_{α β}), = \int J_{q ˉ} c ˉ_{μ} (C_{q ˉ}^{μ} [r] - q ˉ^{μν} b_{ν}) d^{4} x, = I_{inv} + s Ψ_{q ˉ} .$
X-H-ST@1#eq:gfexplicit · line 684
$s Ψ_{q ˉ} = \int J_{q ˉ} (b_{μ} C_{q ˉ}^{μ} [r] - b_{μ} q ˉ^{μν} b_{ν} - c ˉ_{μ} C_{q ˉ}^{μ} [L_{c} q ˉ + κ L_{c} r]) d^{4} x .$
unlabeled · line 639
$L_{q} [ϕ] = \frac{1}{2} J_{q} (q^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}) .$
unlabeled · line 657
$- \frac{1}{2} (\partial_{μ} \partial_{ν} r^{μν} - □ tr_{η} r),$

The natural reference action

Displayed formulas in this passage

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