Manuscript XV-F-patch@1 · XV-F-patch.tex · definition · 6d6ef16d

Definition: Polarized stress convention of the patch

Patch J: the target, blocks, stress, and map clauses › One stress polarization fixed by the displayed vertex · lines 1661–1682

XV-F-patch@1#def:polarization, XV-F-patch@1#eq:Tpolar

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For two scalar directions $ψ_{1}, ψ_{2}$ , define

T^{μν} (ψ_{1}, ψ_{2}) := \frac{1}{2} (\partial^{μ} ψ_{1} \partial^{ν} ψ_{2} + \partial^{μ} ψ_{2} \partial^{ν} ψ_{1}) - \frac{1}{2} η^{μν} (\partial_{ρ} ψ_{1} \partial^{ρ} ψ_{2} - m^{2} ψ_{1} ψ_{2}) .

This is the convention used in the patch's mixed Hessian. It satisfies $T (ϕ, ϕ) = T_{0} [ϕ]$ . It is introduced explicitly here from the displayed $T_{0}$ in (eq:vertex), not attributed to an expanded tensor formula in XIV-J.

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XV-F-patch@1#eq:Tpolar · line 1664
$T^{μν} (ψ_{1}, ψ_{2}) := \frac{1}{2} (\partial^{μ} ψ_{1} \partial^{ν} ψ_{2} + \partial^{μ} ψ_{2} \partial^{ν} ψ_{1}) - \frac{1}{2} η^{μν} (\partial_{ρ} ψ_{1} \partial^{ρ} ψ_{2} - m^{2} ψ_{1} ψ_{2}) .$

Definition: Polarized stress convention of the patch

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