Manuscript XI-A-ST@1 · XI-A-ST.tex · section · c3f0929d

The quadratic normalization

The signed action and its Euler derivatives › The quadratic normalization · lines 881–992

XI-A-ST@1#eq:N, XI-A-ST@1#eq:B, XI-A-ST@1#rest:noether, XI-A-ST@1#eq:noether, XI-A-ST@1#eq:noethergf, XI-A-ST@1#eq:N0, XI-A-ST@1#eq:N1, XI-A-ST@1#eq:N2

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For a symmetric contravariant density $u^{μν}$ , define

N_{ν} (u) = 2 \partial_{μ} (u^{μ ρ} q_{ρ ν}) - u^{μ ρ} \partial_{ν} q_{μ ρ} .

The scaled active variations are

δ_{ξ} q = κ L_{ξ} q, δ_{ξ} γ = L_{ξ} q = K ξ + κ L_{ξ} γ, δ_{ξ} ϕ = κ ξ^{ρ} \partial_{ρ} ϕ,

where

(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ} .

The backgrounds $η$ , $g$ , and $σ_{ε}$ are held fixed in this active transformation.

Define

B_{ν} = (\partial_{μ} g) T_{0}^{μ}_{ν} - (1 - g) (P_{m} ϕ) \partial_{ν} ϕ + (\partial_{ν} g) (L_{η} - L_{q}) .

The exact invariant identity is

N_{ν} (e) = κ (E_{ϕ} \partial_{ν} ϕ - B_{ν}) .

The gauge-fixed identity is instead

N_{ν} (e + e_{gf}) = κ (E_{ϕ} \partial_{ν} ϕ - B_{ν}) + N_{ν} (e_{gf}) .

On $U$ , $B_{ν} = 0$ , but the gauge-fixing contribution is not removed.

Proof. These are XI-A :noether, :noether, and :noether-gf CheungXIA. For a compact interior parameter, direct integration by parts gives

δ_{ξ} S_{inv} = \int ξ^{ν} [- N_{ν} (e) + κ E_{ϕ} \partial_{ν} ϕ] d^{4} x .

The inherited fixed-background action variation has coefficient $κ \int ξ^{ν} B_{ν} d^{4} x$ . Comparison gives (eq:noether). The variation of the independent term (eq:gf) gives (eq:noethergf). The pairing transfers by Lemma lem:transfer. No new proof of the relational Route II theorem is asserted.

The raw divergence and the adjoint Noether operator are different:

\partial_{μ} e_{gf}^{μν} = - □ D^{ν}, N_{ν} (e_{gf})_{κ = 0} = - 2 □ D_{ν} .

The factor two comes from the leading flat part of (eq:N).

For a $κ$ -independent scalar, write $e = e_{0} + κ e_{1} + κ^{2} e_{2} + \dots$ , and put

Q_{ν} (u) = 2 \partial_{μ} (u^{μ ρ} γ_{ρ ν}) - u^{μ ρ} \partial_{ν} γ_{μ ρ} .

The retained coefficient audit is

2 \partial_{μ} (e_{0}^{μ ρ} η_{ρ ν}) 2 \partial_{μ} (e_{1}^{μ ρ} η_{ρ ν}) + Q_{ν} (e_{0}) 2 \partial_{μ} (e_{2}^{μ ρ} η_{ρ ν}) + Q_{ν} (e_{1}) = 0, = - g (P_{m} ϕ) \partial_{ν} ϕ - (\partial_{μ} g) T_{0}^{μ}_{ν}, = - (V_{1, g} ϕ) \partial_{ν} ϕ + (\partial_{ν} g) L_{(1)},

where

L_{(1)} = \frac{1}{2} (a_{1}^{α β} p_{α} p_{β} - m^{2} b_{1} ϕ^{2}) .

These formulas retain the nonlinear adjoint contribution; a bare partial divergence is not the complete identity.

Displayed formulas in this passage

XI-A-ST@1#eq:N · line 884
$N_{ν} (u) = 2 \partial_{μ} (u^{μ ρ} q_{ρ ν}) - u^{μ ρ} \partial_{ν} q_{μ ρ} .$
XI-A-ST@1#eq:B · line 905
$B_{ν} = (\partial_{μ} g) T_{0}^{μ}_{ν} - (1 - g) (P_{m} ϕ) \partial_{ν} ϕ + (\partial_{ν} g) (L_{η} - L_{q}) .$
XI-A-ST@1#eq:noether · line 916
$N_{ν} (e) = κ (E_{ϕ} \partial_{ν} ϕ - B_{ν}) .$
XI-A-ST@1#eq:noethergf · line 922
$N_{ν} (e + e_{gf}) = κ (E_{ϕ} \partial_{ν} ϕ - B_{ν}) + N_{ν} (e_{gf}) .$
XI-A-ST@1#eq:N0 · line 972
$2 \partial_{μ} (e_{0}^{μ ρ} η_{ρ ν}) 2 \partial_{μ} (e_{1}^{μ ρ} η_{ρ ν}) + Q_{ν} (e_{0}) 2 \partial_{μ} (e_{2}^{μ ρ} η_{ρ ν}) + Q_{ν} (e_{1}) = 0, = - g (P_{m} ϕ) \partial_{ν} ϕ - (\partial_{μ} g) T_{0}^{μ}_{ν}, = - (V_{1, g} ϕ) \partial_{ν} ϕ + (\partial_{ν} g) L_{(1)},$
XI-A-ST@1#eq:N1 · line 972
$2 \partial_{μ} (e_{0}^{μ ρ} η_{ρ ν}) 2 \partial_{μ} (e_{1}^{μ ρ} η_{ρ ν}) + Q_{ν} (e_{0}) 2 \partial_{μ} (e_{2}^{μ ρ} η_{ρ ν}) + Q_{ν} (e_{1}) = 0, = - g (P_{m} ϕ) \partial_{ν} ϕ - (\partial_{μ} g) T_{0}^{μ}_{ν}, = - (V_{1, g} ϕ) \partial_{ν} ϕ + (\partial_{ν} g) L_{(1)},$
XI-A-ST@1#eq:N2 · line 972
$2 \partial_{μ} (e_{0}^{μ ρ} η_{ρ ν}) 2 \partial_{μ} (e_{1}^{μ ρ} η_{ρ ν}) + Q_{ν} (e_{0}) 2 \partial_{μ} (e_{2}^{μ ρ} η_{ρ ν}) + Q_{ν} (e_{1}) = 0, = - g (P_{m} ϕ) \partial_{ν} ϕ - (\partial_{μ} g) T_{0}^{μ}_{ν}, = - (V_{1, g} ϕ) \partial_{ν} ϕ + (\partial_{ν} g) L_{(1)},$
unlabeled · line 891
$δ_{ξ} q = κ L_{ξ} q, δ_{ξ} γ = L_{ξ} q = K ξ + κ L_{ξ} γ, δ_{ξ} ϕ = κ ξ^{ρ} \partial_{ρ} ϕ,$
unlabeled · line 897
$(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ} .$
unlabeled · line 938
$δ_{ξ} S_{inv} = \int ξ^{ν} [- N_{ν} (e) + κ E_{ϕ} \partial_{ν} ϕ] d^{4} x .$
unlabeled · line 955
$\partial_{μ} e_{gf}^{μν} = - □ D^{ν}, N_{ν} (e_{gf})_{κ = 0} = - 2 □ D_{ν} .$
unlabeled · line 966
$Q_{ν} (u) = 2 \partial_{μ} (u^{μ ρ} γ_{ρ ν}) - u^{μ ρ} \partial_{ν} γ_{μ ρ} .$
unlabeled · line 987
$L_{(1)} = \frac{1}{2} (a_{1}^{α β} p_{α} p_{β} - m^{2} b_{1} ϕ^{2}) .$

The quadratic normalization

Displayed formulas in this passage

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