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 BV\mathfrak{P}_{BV} has a Chevalley-Eilenberg algebra CE(𝔓 BV)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 Δ\Delta, such that there is a “HamiltonianSS for the differential d BVd_{BV} on CE(𝔓 BV)CE(\mathfrak{P}_{BV})

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

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

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

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

Under quantization the graded-commutative dg-algebra CE(𝔓 BV)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\{S,S\} = 0 by terms of order O()O(\hbar). For a consistent BV-quantization (…) the condition is that after quantization the relation is

{S,S}+ΔS=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.

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