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

Two different density operators

Frozen classical prescription › Two different density operators · lines 610–642

X-A-ST@1#eq:switchedaction, X-A-ST@1#eq:Euler

For the scalar dynamics define instead

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

Thus

L_{g} = (1 - g) L_{η} + g L_{q} .

The action $S_{g}^{sc} = \int_{M_{T_{*}}} L_{g}$ is understood variationally; its numerical value on arbitrary smooth fields need not be finite. Compactly supported variations and the finite changes used below are well defined.

The sign convention is

\frac{δ S _{g} ^{sc}}{δ ϕ} = - E_{g} [q] ϕ, E_{g} [q] ϕ = \partial_{μ} (A_{g} [q]^{μν} \partial_{ν} ϕ) + m^{2} r_{g} [q] ϕ .

Displayed formulas in this passage

X-A-ST@1#eq:switchedaction · line 613
$A_{g} [q]^{μν} r_{g} [q] L_{g} [q, ϕ] = η^{μν} + g (A_{q}^{μν} - η^{μν}), = 1 + g (v_{q} - 1), = \frac{1}{2} (A_{g} [q]^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} r_{g} [q] ϕ^{2}) .$
X-A-ST@1#eq:Euler · line 633
$\frac{δ S _{g} ^{sc}}{δ ϕ} = - E_{g} [q] ϕ, E_{g} [q] ϕ = \partial_{μ} (A_{g} [q]^{μν} \partial_{ν} ϕ) + m^{2} r_{g} [q] ϕ .$
unlabeled · line 623
$L_{g} = (1 - g) L_{η} + g L_{q} .$

Two different density operators

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