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

The frozen action

Switched scalar action and coefficient equations › The frozen action · lines 1102–1148

X-A@1#eq:Lq, X-A@1#eq:Ag, X-A@1#eq:Lg, X-A@1#eq:action, X-A@1#eq:Euler

Let

L_{q} [ϕ] = \frac{1}{2} (A_{q}^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} - q ϕ^{2}), L_{η} [ϕ] = \frac{1}{2} (η^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}),

both scalar densities relative to $d^{4} x$ . Define the switched density coefficients

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

and the switched density

L_{g} [q, ϕ] = \frac{1}{2} (A_{g} [q]^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϱ_{g} [q] ϕ^{2}) = (1 - g) L_{η} [ϕ] + g L_{q} [ϕ] .

The switched scalar action is

S_{g}^{sc} [q, ϕ] = \int_{M_{T}} L_{g} [q, ϕ] d^{4} x,

understood variationally: through its Euler–Lagrange density and through differences of integrands with compact support. Its numerical value need not be finite. This is exactly the prescription “one switch multiplies the complete curved-minus-flat density before variation”.

The Euler operator, with the sign convention $δ S_{g}^{sc} / δ ϕ = - E_{g} [q] ϕ$ , is

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

Its output is a scalar density.

Displayed formulas in this passage

X-A@1#eq:Lq · line 1105
$L_{q} [ϕ] = \frac{1}{2} (A_{q}^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} - q ϕ^{2}), L_{η} [ϕ] = \frac{1}{2} (η^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}),$
X-A@1#eq:Ag · line 1115
$A_{g} [q]^{μν} = η^{μν} + g (A_{q}^{μν} - η^{μν}), ϱ_{g} [q] = 1 + g (- q - 1),$
X-A@1#eq:Lg · line 1122
$L_{g} [q, ϕ] = \frac{1}{2} (A_{g} [q]^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϱ_{g} [q] ϕ^{2}) = (1 - g) L_{η} [ϕ] + g L_{q} [ϕ] .$
X-A@1#eq:action · line 1130
$S_{g}^{sc} [q, ϕ] = \int_{M_{T}} L_{g} [q, ϕ] d^{4} x,$
X-A@1#eq:Euler · line 1142
$E_{g} [q] ϕ = \partial_{μ} (A_{g} [q]^{μν} \partial_{ν} ϕ) + m^{2} ϱ_{g} [q] ϕ .$