Action input and signed Hamiltonian lift

The field rules do not determine an action. Let $S_{\mathrm{ret}}(\kappa )$ be the separately specified, even, ghost-number-zero classical action of the retained relative problem, including its actual switches, backgrounds, and declared classical deformation terms.

For the free-recovery statement its quadratic, zero-coupling part is required to be (eq:freeaction). An equality modulo a relative support convention is used only on the support class where it has been established. Switch derivatives are not deleted by calling a region a plateau.

Variational action notation is used in its usual local sense: compact variations determine the Euler derivatives. It is not an assertion that an integral of a noncompact configuration is a finite number FRclassical.

Introduce antifields ${{\Phi }_A}^{‡}$ of parity $|{\Phi }_A|+1$ and ghost number $-1-\mathrm{gh}({\Phi }_A)$. Use the odd canonical bracket convention

The bracket extends by the odd biderivation rules. Antifields are written on the right in

Consequently

Reordering (eq:kinham) with antifields on the left introduces the factor $(-1{)}^{|{\Phi }_A|+1}$. A different canonical bracket orientation requires its corresponding overall convention change.

Let $\mathcal{B}$ be the coefficientwise classical local variational BV algebra, together with finite products of its compactly supported functionals, on which these brackets are defined. No distributional quantum contraction is used.