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CCCC – p. 5
Hypersurface-deformation algebra
Diffeomorphism constraint D[w a ] and Hamiltonian constraint H[N].
Classical algebra
{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 )]
ensures that theory respects
→ gauge transformations for coordinate changes: space-time
Lie derivative {f,H[ǫ]+D[ξ a ]} = L (ǫ/N,ξa +ǫN a /N)f if
constraints satisfied and time direction t a = Nn a +N a . [Dirac]
→ slicing independence: algebra amounts to deformations of
hypersurfaces
[Kuchar, Teiltelboim]
Anomaly-free quantum-corrected constraints generate modified
gauge transformations. Space-time?