Manuscript X-A@1 · X-A.tex · lemma · dc7f5c0a

Lemma: Expansion of the switched Euler operator

Switched scalar action and coefficient equations › Coefficient operators with switch derivatives displayed · lines 1151–1179 · proof included or linked

X-A@1#lem:Eexp, X-A@1#eq:V1g, X-A@1#eq:V2g

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$E_{g} [q] = (□ + m^{2}) + κ V_{1, g} [h] + κ^{2} V_{2, g} [h, k] (mod κ^{3})$ , where

V_{1, g} [h] ψ V_{2, g} [h, k] ψ = \partial_{μ} (g a_{1}^{μν} [h] \partial_{ν} ψ) + g m^{2} b_{1} [h] ψ = g [\partial_{μ} (a_{1}^{μν} [h] \partial_{ν} ψ) + m^{2} b_{1} [h] ψ] + (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} ψ, = \partial_{μ} (g (a_{1}^{μν} [k] + a_{2}^{μν} [h]) \partial_{ν} ψ) + g m^{2} (b_{1} [k] + b_{2} [h]) ψ = g [\partial_{μ} ((a_{1}^{μν} [k] + a_{2}^{μν} [h]) \partial_{ν} ψ) + m^{2} (b_{1} [k] + b_{2} [h]) ψ] + (\partial_{μ} g) (a_{1}^{μν} [k] + a_{2}^{μν} [h]) \partial_{ν} ψ .

In particular $V_{1, g} [h] = Q^{(m)}_{g h}$ , where $Q^{(m)}_{h} ψ = \partial_{μ} (a_{1}^{μν} [h] \partial_{ν} ψ) + m^{2} b_{1} [h] ψ$ is the unswitched first variation.

Proof. Insert Lemma lem:density into (eq:Ag)–(eq:Euler) and expand. Linearity of $a_{1}, b_{1}$ gives the last statement.

Displayed formulas in this passage

X-A@1#eq:V1g · line 1155
$V_{1, g} [h] ψ V_{2, g} [h, k] ψ = \partial_{μ} (g a_{1}^{μν} [h] \partial_{ν} ψ) + g m^{2} b_{1} [h] ψ = g [\partial_{μ} (a_{1}^{μν} [h] \partial_{ν} ψ) + m^{2} b_{1} [h] ψ] + (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} ψ, = \partial_{μ} (g (a_{1}^{μν} [k] + a_{2}^{μν} [h]) \partial_{ν} ψ) + g m^{2} (b_{1} [k] + b_{2} [h]) ψ = g [\partial_{μ} ((a_{1}^{μν} [k] + a_{2}^{μν} [h]) \partial_{ν} ψ) + m^{2} (b_{1} [k] + b_{2} [h]) ψ] + (\partial_{μ} g) (a_{1}^{μν} [k] + a_{2}^{μν} [h]) \partial_{ν} ψ .$
X-A@1#eq:V2g · line 1155
$V_{1, g} [h] ψ V_{2, g} [h, k] ψ = \partial_{μ} (g a_{1}^{μν} [h] \partial_{ν} ψ) + g m^{2} b_{1} [h] ψ = g [\partial_{μ} (a_{1}^{μν} [h] \partial_{ν} ψ) + m^{2} b_{1} [h] ψ] + (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} ψ, = \partial_{μ} (g (a_{1}^{μν} [k] + a_{2}^{μν} [h]) \partial_{ν} ψ) + g m^{2} (b_{1} [k] + b_{2} [h]) ψ = g [\partial_{μ} ((a_{1}^{μν} [k] + a_{2}^{μν} [h]) \partial_{ν} ψ) + m^{2} (b_{1} [k] + b_{2} [h]) ψ] + (\partial_{μ} g) (a_{1}^{μν} [k] + a_{2}^{μν} [h]) \partial_{ν} ψ .$

Nearby in this section

lemma: Lemma: Pure-gauge variation of the scalar operator