Local variational differential operators in field theory

The geometry of variations in Batalin–Vilkovisky formalism

“…Yet the misconception is still present in active research, e.g., see [4,27,39,44]. The fact of incompleteness of such heuristic analogies from usual geometry of manifolds is signalled in [51]. Paradoxically, it is simultaneously not true that a solution of the regularisation problem for BV-Laplacian has no analogues in the case of ODE dynamics on manifolds.…”

“…From section 1 below it is readily seen that good old techniques persist in the finite-dimensional ODE geometry at the level of standard linear algebra of dual vector spaces. 3 The article [51] is a considerable step towards a solution of the regularisation problem in BV-formalism. A weighted, critical overview of various inconsistencies, ad hoc practices, and roundabouts is summed up there.…”

“…A weighted, critical overview of various inconsistencies, ad hoc practices, and roundabouts is summed up there. The object of [51] was to formulate a self-contained analytic concept which would make the variational calculus of functionals free from divergencies and infinities. Still it remained unclear from [51] what the generality of underlying geometry is and why such self-consistent formalism should actually exist at the level of objects, i.e., beyond a mere ability to write formulas.…”

“…The object of [51] was to formulate a self-contained analytic concept which would make the variational calculus of functionals free from divergencies and infinities. Still it remained unclear from [51] what the generality of underlying geometry is and why such self-consistent formalism should actually exist at the level of objects, i.e., beyond a mere ability to write formulas. In particular, it remained unnoticed that the main motivating example -namely, the canonical BV-setup-itself is the only class of geometries in which the technique is grounded.…”

“…In particular, it remained unnoticed that the main motivating example -namely, the canonical BV-setup-itself is the only class of geometries in which the technique is grounded. 4 A correctness but incompleteness of the approach in [51] means the following in practice. Whenever a theorist refers to the formalism of loc.…”