Abstract
We explain why no sources of divergence are built into the Batalin–Vilkovisky (BV) Laplacian, whence there is no need to postulate any ad hoc conventions such as "δ(0) = 0" and "log δ(0) = 0" within BV-approach to quantisation of gauge systems. Remarkably, the geometry of iterated variations does not refer at all to the construction of Dirac's δ-function as a limit of smooth kernels. We illustrate the reasoning by re-deriving –but not just 'formally postulating' – the standard properties of BV-Laplacian and Schouten bracket and by verifying their basic inter-relations (e.g., cohomology preservation by gauge symmetries of the quantum master-equation).
Highlights
This is a paper about geometry of variations
1 Another convention is log δ(0) = 0; we show that natural counterparts of the true geometry of variations lead to this intuitive convention and simultaneously to (2) — none of the two being required
We argue that a logical complexity of geometric objects grows while they accumulate the variations ; a conversion of such composite-structure objects into maps which take physical field configurations to numbers entails a decrease of the complexity via a loss of information
Summary
This is a paper about geometry of variations. We formulate definitions of the objects and structures which are cornerstones of Batalin–Vilkovisky formalism [5, 7, 20, 22, 55]. We tacitly describe a geometric mechanism which is responsible for the anti-commutation of differential one-forms ; such mechanism ensures that the results of calculations match empiric data even if the exterior algebras of forms are introduced by hand 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 exist at the level of objects, i.e., beyond a mere ability to write formulas. We properly enlarge the space of functionals and adjust a description of the geometry for the functionals’ variations: each variation brings its own copy of the base M n into the picture (see Fig. 3 on p. 14)
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