Manuscript X-J-ST@1 · X-J-ST.tex · proposition · 8f0d1a36

Proposition: Charge identity

Section A: two action-derived compact charged coefficients › The charges and the formal-order correction · lines 760–792 · proof included or linked

X-J-ST@1#prop:charge, X-J-ST@1#eq:qi, X-J-ST@1#eq:BRSTcharge, X-J-ST@1#eq:weightedcharge

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For every $ξ \in g_{U}$ ,

q_{i} (ϕ; ξ) := q_{C_{i}} (ϕ; ξ) = \int w_{i} ξ^{μ} \partial_{μ} ϕ d^{4} x .

On scalar functionals the retained BV differential gives

Q_{λ}^{[3]} C_{i} = κ J_{i} [ϕ, c], J_{i} [ϕ, c] = \int w_{i} c^{μ} \partial_{μ} ϕ d^{4} x .

The polynomial field rule also gives exactly

s O_{i} = κ^{4} J_{i} .

Proof. Differentiate the scalar pullback and use (eq:moment). For the BV action, only the scalar antifield coupling acts on $C_{i}$ ; its rule is (eq:sscalar). The calibration scalar is not varied. Multiplication by $κ^{3}$ gives (eq:weightedcharge).

Displayed formulas in this passage

X-J-ST@1#eq:qi · line 763
$q_{i} (ϕ; ξ) := q_{C_{i}} (ϕ; ξ) = \int w_{i} ξ^{μ} \partial_{μ} ϕ d^{4} x .$
X-J-ST@1#eq:BRSTcharge · line 770
$Q_{λ}^{[3]} C_{i} = κ J_{i} [ϕ, c], J_{i} [ϕ, c] = \int w_{i} c^{μ} \partial_{μ} ϕ d^{4} x .$
X-J-ST@1#eq:weightedcharge · line 778
$s O_{i} = κ^{4} J_{i} .$

Nearby in this section

scope: Charge versus retained formal order