# nLab quantum master equation

## Idea

The quantum master equation is an condition on a consistent quantization of a derived phase space, given dually by a BV-BRST complex:

as explained there, this derived L-∞ algebroid $\mathfrak{P}_{BV}$ has a Chevalley-Eilenberg algebra $CE(\mathfrak{P}_{BV})$ – a graded-commutative dg-algebra – which is equipped with the structure of a BV-algebra, hence with a Poisson n-algebra bracket $\{-,-\}$ and a BV-operator $\Delta$, such that there is a “Hamiltonian$S$ for the differential $d_{BV}$ on $CE(\mathfrak{P}_{BV})$

$d_{BV} = \{S, -\} \,,$

so that the condition on the differential $(d_{BV})^2 = 0$ is equivalently

$\{S,S\} = 0 \,.$

This $S$ is an extension of the classical action: the BV-action . This equation $\{S,S\} = 0$ is called the classical master equation.

Under quantization the graded-commutative dg-algebra $CE(\mathfrak{P}_{BV})$ becomes a non-commutative dg-algebra. For instance in deformation quantization it becomes a non-commutative algebra over the power series ring in a formal parameter $\hbar$.

In general such a deformation breaks the condition $\{S,S\} = 0$ by terms of order $O(\hbar)$. For a consistent BV-quantization (…) the condition is that after quantization the relation is

$\{S,S\} + \hbar \Delta S = 0 \,.$

This equation is called the quantum master equation in the context of BV-quantization.

If after quantization this condition does not hold one says that the system has a gauge quantum anomaly . See there for more references on this point.

Created on September 20, 2011 15:42:45 by Urs Schreiber (82.113.99.58)