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CCCC – p. 14
Effective constraint algebras
Systems with several classical constraints C I :
→ Effective constraints C I,pol = 〈̂polĈI〉(〈·〉,∆(·)) with ̂pol
polynomial in basic operators.
→ Compute effective constraint algebra by Poisson brackets.
Easier than commutators of operators. Parameterizable.
→ Check off-shell consistency of first-class constraints.
Example: Hamiltonian and diffeomorphism constraints obey
hypersurface-deformation algebra of classical space-time.
{D[w a 1],D[w a 2]} = −D[L w2 w a 1]
{H[N],D[w a ]} = −H[w a ∇ a N]
{H[N 1 ],H[N 2 ]} = D[h ab (N 1 ∇ b N 2 −N 2 ∇ b N 1 )]