Assumptions behind the Quantum Master Equation derived using Batalin-Vilkovisky Formalism

Question

Is there any underlying assumption(s) behind the Batalin-Vilkovisky Quantum Master Equation: $$\frac{1}{2}(S,S) = i\hbar\Delta(S)~?$$

As an example, if we consider the Nakajima–Zwanzig Master Equation, it is assumed that the slow, collective 'relevant' part only contributes, rapidly fluctuating 'irrelevant' part doesn't. Similarly there are some clear-cut assumptions also in the famous Lindblad Equation, Redfield Equation, Markovian Master Equation and in many others.

Thus does there any similar assumptions in B-V Quantum Master Equation also?

Answer 1

The Batalin-Vilkovisky (BV) formalism is a formal theory of theories, i.e. it was heuristically developed for an arbitrary gauge theory [1-4] without rigorously addressing e.g. the issues of regularization, renormalization$^1$ and non-perturbative features. In practice, for a given specific gauge theory, there is obviously a long list of assumptions that need to be fulfilled in order to successfully quantize the theory; let us mention a few of them. E.g. the gauge theory should not be anomalous. Also the classical gauge theory should have a local action formulation to begin with. Moreover, the BV recipe may lead to an infinite tower of gauge-for-gauge symmetries and ghost-for-ghosts ad infinitum, which are often not useful.

Briefly, the quantum master equation (QME) encodes gauge symmetry (more precisely: BRST symmetry). It is related to the Maurer-Cartan equation. It is an off-shell equation for the quantum master action, which in turn is a cohomological deformation of the original classical action.

The other master equations that OP mentions are usually applied to QM rather than QFT, and the gauge symmetry no longer takes center stage.

References:

1. I.A. Batalin & G.A. Vilkovisky, Gauge Algebra and Quantization, Phys. Lett. B 102 (1981) 27–31.

2. M. Henneaux & C. Teitelboim, Quantization of Gauge Systems, 1994.

3. M. Henneaux, Lectures on the antifield-BRST formalism for gauge theories, Nucl. Phys. B Proc. Suppl. 18 (1990) 47.

4. J. Gomis, J. Paris & S. Samuel, Antibracket, Antifields and Gauge-Theory Quantization, arXiv:hep-th/9412228.

5. S. Weinberg, Quantum Theory of Fields, Vol. 2, 1996; chapter 17.

6. K.J. Costello, Renormalisation and the BV formalism, arXiv:0706.1533.

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$^1$ In practice the BV formalism works well with renormalization [5-6].