# Batalin-Vilkovisky quantization

Batalin-Vilkovisky (BV) quantization is way of quantizing a theory, which is apparently more powerful than BRST quantization. It has been used, for example, for string field theory, in the closed string approach.

Weinberg's book (Vol 2, chapter 15.9) is quite hard to understand for me, as I don't get the physical motivation of such approach and most of the calculations. Wikipedia is even worse for me, as it just focuses on defining operators and their properties, without giving a hint on the physical meaning of these requirements.

Why is it different from BRST quantization?

• See the answers to this question very related to the one of Yours. – Ramiro Hum-Sah Jun 12 at 18:02

1. Gauge theory & BRST formulation originally referred to just Yang-Mills (YM) theory but the terms are nowadays applied to any gauge theory, cf. e.g. this Phys.SE post.

2. A gauge theory is a theory with a local (=$$x$$-dependent) symmetry. Note that the structure constants in YM are replaced by field-dependent structure functions in a generic gauge theory. Also the gauge algebra might be reducible and/or open, cf. e.g. this Phys.SE post and references therein.

3. To have a BRST formulation, the gauge transformations should be encoded via a nilpotent Grassmann-odd BRST differential of ghost number 1.

4. The Batalin-Vilkovisky (BV) formalism is a BRST formulation in the above sense. However it assumes more. In particular:

• The original gauge theory should have an action formulation.

• It introduces so-called non-minimal fields to enable gauge-fixing. For irreducible gauge algebra the non-minimal fields are just the Faddeev-Popov antighosts and the Lautrup-Nakanishi fields, but it becomes more complicated for reducible gauge algebras.

• It introduces an antifield of opposite statistics to each dynamical & auxiliary field. The (infinite-dimensional) supermanifold of fields and antifield is endowed with an Grassmann-odd Poisson structure known as an anti-bracket.

• A quantum master action that satisfies a so-called quantum master equation.

• The BRST differential is derived from the quantum master action in a specific way.