Theorem: Relative classical BV construction

Let

be its nonnegative formal expansion through second order. Then on $U$,

as a local variational class.

The Hamiltonian operator

is an odd differential on the classical multilocal BV algebra modulo ${\kappa }^3$. Its restriction to fields is (eq:nonlinear) through that order.

Proof. The antifield-independent functional $I$ satisfies $(I,I)=0$. By Lemma lem:sign, $H_s$ acts on antifield-independent functionals as $s$. Since the bracket of two even arguments is symmetric,

The last equality is Lemma lem:action. The action has no negative formal coefficients after removal of the specified divergence. Thus truncation preserves this equality through order two. Explicitly,

The graded Jacobi identity yields

The generator action follows from Lemma lem:sign.