Manuscript VI-B@1 · VI-B-v1.tex · section · 33798723

Compact switching: exact defect and plateau obstruction

Compact switching: exact defect and plateau obstruction · lines 844–902

VI-B@1#eq:Q, VI-B@1#eq:QL, VI-B@1#eq:Qswitch, VI-B@1#eq:candidate, VI-B@1#eq:free-identification

Fix a real compact switch

g \in C_{c}^{\infty} (D_{R, b}^{\circ}) .

It is not transformed in this section. The compact minimal scalar vertex has classical density

V_{1}^{m} (g; h, φ) = \frac{1}{2} \int g h^{μν} T_{μν} (η, φ) d^{4} x .

For $H = g h$ , its scalar Hessian is

Q_{H} = - \partial_{μ} (H ˉ^{μν} \partial_{ν}) - \frac{1}{2} m^{2} tr_{η} H .

Integration by parts shows formal self-adjointness. Expanding the derivatives gives

Q_{H} = L_{H} + \frac{1}{2} (tr_{η} H) P .

Hence, for $P ϕ_{0} = 0$ ,

Q_{g h} ϕ_{0} = g L_{h} ϕ_{0} - (\partial_{μ} g) h ˉ^{μν} \partial_{ν} ϕ_{0} .

Define the classical zero-past response

ϕ_{1, g} [h, ϕ_{0}] = - G_{R} Q_{g h} ϕ_{0} .

It equals the zero-data solution of $P ψ = - L_{g h} ϕ_{0}$ , by (eq:QL) and uniqueness. In particular, multiplication by $g$ occurs inside the full metric operator, not merely in front of $L_{h}$ .

Consider the candidate

A_{f, g}^{(1)} (h, ϕ_{0}) = \int f (ϕ_{1, g} [h, ϕ_{0}] - X_{w [h]} ϕ_{0}) d^{4} x

under the unchanged-free-input identification

(h, ϕ_{0}) \sim (h + K ξ, ϕ_{0}), ξ \in g .

This is not the full action on $ϕ_{1}, ϕ_{2}$ in (eq:gauge-coefficients).

Displayed formulas in this passage

VI-B@1#eq:Q · line 858
$Q_{H} = - \partial_{μ} (H ˉ^{μν} \partial_{ν}) - \frac{1}{2} m^{2} tr_{η} H .$
VI-B@1#eq:QL · line 866
$Q_{H} = L_{H} + \frac{1}{2} (tr_{η} H) P .$
VI-B@1#eq:Qswitch · line 871
$Q_{g h} ϕ_{0} = g L_{h} ϕ_{0} - (\partial_{μ} g) h ˉ^{μν} \partial_{ν} ϕ_{0} .$
VI-B@1#eq:candidate · line 888
$A_{f, g}^{(1)} (h, ϕ_{0}) = \int f (ϕ_{1, g} [h, ϕ_{0}] - X_{w [h]} ϕ_{0}) d^{4} x$
VI-B@1#eq:free-identification · line 895
$(h, ϕ_{0}) \sim (h + K ξ, ϕ_{0}), ξ \in g .$
unlabeled · line 848
$g \in C_{c}^{\infty} (D_{R, b}^{\circ}) .$
unlabeled · line 853
$V_{1}^{m} (g; h, φ) = \frac{1}{2} \int g h^{μν} T_{μν} (η, φ) d^{4} x .$
unlabeled · line 879
$ϕ_{1, g} [h, ϕ_{0}] = - G_{R} Q_{g h} ϕ_{0} .$

Compact switching: exact defect and plateau obstruction

Displayed formulas in this passage

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