Manuscript VII-A@1 · VII-A.tex · section · 11da3fe6

Action-level switching

The specified switched scalar action › Action-level switching · lines 543–592

VII-A@1#eq:Ag, VII-A@1#eq:rhog, VII-A@1#eq:switchedaction, VII-A@1#eq:Dg

Define

A_{g} [g]^{μν} ρ_{g} [g] = η^{μν} + g (A_{g}^{μν} - η^{μν}), = 1 + g (ρ_{g} - 1) .

The switched scalar action is

S_{g}^{sc} [g, ϕ] = - \frac{1}{2} \int_{M_{+}} (A_{g} [g]^{μν} \partial_{μ} ϕ \partial_{ν} ϕ + m^{2} ρ_{g} [g] ϕ^{2}) d^{4} x .

The action is understood variationally, using compactly supported variations or variations respecting the prescribed boundary data; its numerical value need not be finite on every smooth configuration.

This prescription is equivalent to multiplying the difference between the curved and flat scalar Lagrangian densities by $g$ . In particular, the quadratic metric vertex carries one factor of $g$ , not a factor of $g^{2}$ . Two first-order insertions will nevertheless produce two switch factors in the iterated response.

The corresponding Euler operator is

D_{g} [g] ϕ = \partial_{μ} (A_{g} [g]^{μν} \partial_{ν} ϕ) - m^{2} ρ_{g} [g] ϕ .

Its output represents a scalar density relative to $d^{4} x$ .

Displayed formulas in this passage

VII-A@1#eq:Ag · line 546
$A_{g} [g]^{μν} ρ_{g} [g] = η^{μν} + g (A_{g}^{μν} - η^{μν}), = 1 + g (ρ_{g} - 1) .$
VII-A@1#eq:rhog · line 546
$A_{g} [g]^{μν} ρ_{g} [g] = η^{μν} + g (A_{g}^{μν} - η^{μν}), = 1 + g (ρ_{g} - 1) .$
VII-A@1#eq:switchedaction · line 559
$S_{g}^{sc} [g, ϕ] = - \frac{1}{2} \int_{M_{+}} (A_{g} [g]^{μν} \partial_{μ} ϕ \partial_{ν} ϕ + m^{2} ρ_{g} [g] ϕ^{2}) d^{4} x .$
VII-A@1#eq:Dg · line 582
$D_{g} [g] ϕ = \partial_{μ} (A_{g} [g]^{μν} \partial_{ν} ϕ) - m^{2} ρ_{g} [g] ϕ .$