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 ${𝔓}_{\mathrm{BV}}$ has a Chevalley-Eilenberg algebra $\mathrm{CE}({𝔓}_{\mathrm{BV}})$ – 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 $d_{\mathrm{BV}}$ on $\mathrm{CE}({𝔓}_{\mathrm{BV}})$

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

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

$$\{S,S\}=0.$$\{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 $\mathrm{CE}({𝔓}_{\mathrm{BV}})$ 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 $\hslash$.

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

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