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

Lemma: Pure-gauge variation of the scalar operator

Switched scalar action and coefficient equations › Coefficient operators with switch derivatives displayed · lines 1181–1202 · proof included or linked

X-A@1#lem:QK, X-A@1#eq:QK

Back to manuscript

For every smooth $ψ$ and every smooth vector field $ξ$ ,

Q^{(m)}_{K ξ} ψ = D_{ξ} (□ + m^{2}) ψ - (□ + m^{2}) X_{ξ} ψ .

Proof. Direct expansion. With $a_{1} [K ξ] = (\partial_{ρ} ξ^{ρ}) η - \partial^{μ} ξ^{ν} - \partial^{ν} ξ^{μ}$ and $b_{1} [K ξ] = \partial_{ρ} ξ^{ρ}$ ,

Q^{(m)}_{K ξ} ψ = (\partial_{ρ} ξ^{ρ}) (□ + m^{2}) ψ - (□ ξ^{ν}) \partial_{ν} ψ - 2 \partial^{μ} ξ^{ν} \partial_{μ} \partial_{ν} ψ,

because $\partial_{μ} \partial^{ν} ξ^{μ} \partial_{ν} ψ$ cancels against $\partial_{μ} (\partial_{ρ} ξ^{ρ}) \partial^{μ} ψ$ . The right-hand side of (eq:QK) expands to the same expression, since $X_{ξ}$ commutes with the constant mass term.

Displayed formulas in this passage

X-A@1#eq:QK · line 1184
$Q^{(m)}_{K ξ} ψ = D_{ξ} (□ + m^{2}) ψ - (□ + m^{2}) X_{ξ} ψ .$
unlabeled · line 1193
$Q^{(m)}_{K ξ} ψ = (\partial_{ρ} ξ^{ρ}) (□ + m^{2}) ψ - (□ ξ^{ν}) \partial_{ν} ψ - 2 \partial^{μ} ξ^{ν} \partial_{μ} \partial_{ν} ψ,$

Nearby in this section

lemma: Lemma: Expansion of the switched Euler operator