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A Finite BV Density Counted Once

Example

In BV Density Bridge and Pushforward Master Defect, fix a finite Darboux chart, even classical master action $S_{0}$ , even field-only $f$ , and even parameter $σ$ . Let $μ_{σ} = e^{σ f}$ and $ρ_{σ} = μ_{σ}^{2} ρ_{0}$ . Then the representations $I_{1}$ with $Δ_{ρ_{σ}}$ and $J_{1} = I_{1} - σ f$ with $Δ_{0}$ have the same one-loop defect. Pairing $J_{1}$ with $Δ_{ρ_{σ}}$ instead adds the spurious term $- σ (S_{0}, f)$ . Here $σ$ is a finite variation parameter, not a continuum Weyl anomaly, and no prior calculation is alleged to have double-counted a density.

Statement references

BV Density Bridge and Pushforward Master Defect

Facts & Assumptions

Given: A fixed finite Darboux chart, even field-only $f$ , and the finite density/action bridge of [F1].

[F1]

BV Density Bridge and Pushforward Master Defect: $Δ_{ρ} = Δ_{0} + (\log μ, -)$ , $J_{1} = I_{1} - \log μ$ , and $(S_{0}, I_{1}) - Δ_{ρ} S_{0} = (S_{0}, J_{1}) - Δ_{0} S_{0}$ .

Proof

1.1F1algebra

Since $\log μ_{σ} = σ f$ , $Δ_{ρ_{σ}} = Δ_{0} + σ (f, -)$ and $J_{1} = I_{1} - σ f$ . The even shifted-skew rule gives $(f, S_{0}) = (S_{0}, f)$ , so $(S_{0}, I_{1}) - Δ_{ρ_{σ}} S_{0} = (S_{0}, I_{1}) - Δ_{0} S_{0} - σ (f, S_{0}) = (S_{0}, I_{1} - σ f) - Δ_{0} S_{0}$ . This is equality of the two defect representatives.

2.1F1step 1.1algebra∎

If one instead retains $J_{1}$ and also uses $Δ_{ρ_{σ}}$ , the defect becomes $(S_{0}, J_{1}) - Δ_{0} S_{0} - σ (f, S_{0})$ , differing from the reference-density defect by $- σ (S_{0}, f)$ . The term is exactly the same density variation already used in defining $J_{1}$ , so the finite example identifies the algebraic double count without alleging that it occurred in a physical source calculation.

Depends on

BV Density Bridge and Pushforward Master Defect

Dependency tree · 2 levels

Sources

Joaquim Gomis, Jordi París, and Stuart Samuel, Antibracket, Antifields and Gauge-Theory Quantization