Manuscript X-G-ST@1 · X-G-ST.tex · proposition · 3c97567f

Proposition: Full switched-scalar breaking

Stage 2A: the switched scalar has a classical gauge breaking › The reference action being varied · lines 1111–1162 · proof included or linked

X-G-ST@1#prop:breaking, X-G-ST@1#eq:breaking, X-G-ST@1#eq:B

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For compact ghost coefficient functions, modulo integrated divergences,

s_{kin} S_{ref} = s_{kin} S_{sc, g} = κ \int ((1 - g) T_{0}^{μν} \partial_{μ} c_{ν} + (\partial_{μ} g) c^{μ} (L_{η} - L_{q})) d^{4} x = κ \int c^{ν} B_{ν} d^{4} x,

where

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 full off-shell breaking is not supported only on $supp d g$ .

Proof. The curved scalar density is natural: $s_{kin} L_{q} = κ \partial_{μ} (c^{μ} L_{q})$ . For the flat density, $η$ is fixed, so

s_{kin} L_{η} = κ (\partial_{μ} (c^{μ} L_{η}) + T_{0}^{μν} \partial_{μ} c_{ν}) .

Multiplying by $g, 1 - g$ and integrating gives the first integral in (eq:breaking). Integrating its stress term once more and using the stress-divergence identity gives (eq:B).

The Einstein–Hilbert variation is a compact divergence. Lemma lem:kin gives $s_{kin}^{2} Ψ_{q ˉ} = 0$ . Neither contributes a breaking here.

Where $g = 0$ and $d g = 0$ , the off-shell term $- (P_{m} ϕ) \partial_{ν} ϕ$ can still be nonzero.

Displayed formulas in this passage

X-G-ST@1#eq:breaking · line 1115
$s_{kin} S_{ref} = s_{kin} S_{sc, g} = κ \int ((1 - g) T_{0}^{μν} \partial_{μ} c_{ν} + (\partial_{μ} g) c^{μ} (L_{η} - L_{q})) d^{4} x = κ \int c^{ν} B_{ν} d^{4} x,$
X-G-ST@1#eq:B · line 1128
$B_{ν} = (\partial_{μ} g) T_{0}^{μ}_{ν} - (1 - g) (P_{m} ϕ) \partial_{ν} ϕ + (\partial_{ν} g) (L_{η} - L_{q}) .$
unlabeled · line 1144
$s_{kin} L_{η} = κ (\partial_{μ} (c^{μ} L_{η}) + T_{0}^{μν} \partial_{μ} c_{ν}) .$

Proposition: Full switched-scalar breaking

Displayed formulas in this passage

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