Manuscript X-A-ST@1 · X-A-ST.tex · section · d9ccbde1

Varied-action derivation: Route II

Varied-action derivation: Route II · lines 1303–1336

X-A-ST@1#eq:flatvariation, X-A-ST@1#eq:currentvariation

For completeness, define the symmetric bilinear expression

B_{m} (u, v) = \partial^{μ} u \partial_{μ} v - m^{2} uv, T_{2} = X_{ζ} + \frac{1}{2} X_{ξ}^{2} .

The two terms in (eq:varied) are explicitly

L_{η} [F^{*} ϕ] - L_{η} [ϕ] = Z_{F}^{μ} [L_{q} [ϕ]] = L^{(1)} [h, ϕ] = κ B_{m} (ϕ, X_{ξ} ϕ) + κ^{2} (B_{m} (ϕ, T_{2} ϕ) + \frac{1}{2} B_{m} (X_{ξ} ϕ, X_{ξ} ϕ)), κ ξ^{μ} L_{η} [ϕ] + κ^{2} (ξ^{μ} L^{(1)} [h, ϕ] + ζ^{μ} L_{η} [ϕ] + \frac{1}{2} ξ^{μ} D_{ξ} L_{η} [ϕ]), \frac{1}{2} (a_{1}^{μν} [h] \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} b_{1} [h] ϕ^{2}) .

For a $κ$ -dependent $ϕ$ , all products on the right are themselves expanded and truncated. No coefficient has been suppressed by treating that $ϕ$ as independent of $κ$ .

Displayed formulas in this passage

X-A-ST@1#eq:flatvariation · line 1312
$L_{η} [F^{*} ϕ] - L_{η} [ϕ] = Z_{F}^{μ} [L_{q} [ϕ]] = L^{(1)} [h, ϕ] = κ B_{m} (ϕ, X_{ξ} ϕ) + κ^{2} (B_{m} (ϕ, T_{2} ϕ) + \frac{1}{2} B_{m} (X_{ξ} ϕ, X_{ξ} ϕ)), κ ξ^{μ} L_{η} [ϕ] + κ^{2} (ξ^{μ} L^{(1)} [h, ϕ] + ζ^{μ} L_{η} [ϕ] + \frac{1}{2} ξ^{μ} D_{ξ} L_{η} [ϕ]), \frac{1}{2} (a_{1}^{μν} [h] \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} b_{1} [h] ϕ^{2}) .$
X-A-ST@1#eq:currentvariation · line 1312
$L_{η} [F^{*} ϕ] - L_{η} [ϕ] = Z_{F}^{μ} [L_{q} [ϕ]] = L^{(1)} [h, ϕ] = κ B_{m} (ϕ, X_{ξ} ϕ) + κ^{2} (B_{m} (ϕ, T_{2} ϕ) + \frac{1}{2} B_{m} (X_{ξ} ϕ, X_{ξ} ϕ)), κ ξ^{μ} L_{η} [ϕ] + κ^{2} (ξ^{μ} L^{(1)} [h, ϕ] + ζ^{μ} L_{η} [ϕ] + \frac{1}{2} ξ^{μ} D_{ξ} L_{η} [ϕ]), \frac{1}{2} (a_{1}^{μν} [h] \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} b_{1} [h] ϕ^{2}) .$
unlabeled · line 1306
$B_{m} (u, v) = \partial^{μ} u \partial_{μ} v - m^{2} uv, T_{2} = X_{ζ} + \frac{1}{2} X_{ξ}^{2} .$

Varied-action derivation: Route II

Displayed formulas in this passage

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