Manuscript I-B-VP@3 · I-B-VP-v3.tex · section · 93c04eb4

Renormalization, complex first jets, and Ward identities

Renormalization, complex first jets, and Ward identities · lines 959–997

I-B-VP@3#eq:pointsplit, I-B-VP@3#eq:current-prescription

This is the local framework of Wald1977,HollandsWald2001,HollandsWald2005,DecaniniFolacci. Its local formulas are applied on regular charts, not as a globally hyperbolic existence theorem for all of $M$ .

Let brackets denote coincidence and let $H$ be the regular complex two-point kernel in a parallel frame. Put

F = [H] .

For one real field use $h = Re H$ , and for complex stress use $h = 2 Re H$ . For a smooth symmetric real bisolution define

[D_{μν} h] = (1 - 2 ξ) [\nabla_{(μ} \nabla_{ν)^{'}} h] + (2 ξ - \frac{1}{2}) g_{μν} [\nabla_{ρ} \nabla^{ρ^{'}} h] - ξ [(\nabla_{μ} \nabla_{ν} + \nabla_{μ^{'}} \nabla_{ν^{'}}) h] .

Indices are parallel transported before coincidence.

For the charged field,

⟨ j_{μ} ⟩_{ren} = i e [(D_{μ} - D_{μ^{'}}^{*}) H], j_{μ} = i e [Φ^{†} D_{μ} Φ - (D_{μ} Φ)^{†} Φ] .

The two slots carry opposite connection characters.

Displayed formulas in this passage

I-B-VP@3#eq:pointsplit · line 975
$[D_{μν} h] = (1 - 2 ξ) [\nabla_{(μ} \nabla_{ν)^{'}} h] + (2 ξ - \frac{1}{2}) g_{μν} [\nabla_{ρ} \nabla^{ρ^{'}} h] - ξ [(\nabla_{μ} \nabla_{ν} + \nabla_{μ^{'}} \nabla_{ν^{'}}) h] .$
I-B-VP@3#eq:current-prescription · line 988
$⟨ j_{μ} ⟩_{ren} = i e [(D_{μ} - D_{μ^{'}}^{*}) H], j_{μ} = i e [Φ^{†} D_{μ} Φ - (D_{μ} Φ)^{†} Φ] .$
unlabeled · line 969
$F = [H] .$

Renormalization, complex first jets, and Ward identities

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