Quantum Gravity

160 QUANTUM GEOMETRODYNAMICSn AA′ = e AA′µ n µ , (5.87)obeyingn AA ′e AA′a =0, n AA ′n AA′ =1. (5.88)Analogous to (4.40) one can expand the remaining components of the spinorialtetrad ase AA′0 = Nn AA′ + N a e AA′a , (5.89)with lapse function N and shift vector N a . The canonical formalism starts withthe definition of the momenta. The momenta conjugate to N, N a , ψ0 A ,and¯ψ 0 A′ are all zero, since these variables are Lagrange multipliers. The momentaconjugate to the gravitino fields are 14πA a = δSδ ˙ψ aA˜π A a = δS′ δ ˙¯ψA ′a= − 1 2 ɛabc ¯ψA ′b e AA ′ c , (5.90)= 1 2 ɛabc ψ A b e AA ′ c . (5.91)Since the action is linear in Dψ and D ¯ψ, the time derivatives ˙ψ and ˙¯ψ do notoccur on the right-hand sides. Therefore, these equations are in fact constraints.It turns out that these constraints are of second class, that is, the Poisson bracketsof the constraints do not close on the constraints again; cf. Section 3.1.2.As a consequence, one can eliminate the momenta πA a and ˜πa A from the canonicalaction by using these constraints. Finally, the momentum conjugate to the′spinorial tetrad can be found fromp a AA ′ =δSδė AA′ a, (5.92)from which the ordinary spatial components follow via p ab = −e AA′a p b AA ′.Thesymmetric part of p ab can be expressed exactly as in (4.63) in terms of the secondfundamental form K ab on t = constant,√h(p (ab) = K (ab) − Kh ab) . (5.93)16πGHowever, due to the presence of torsion, K ab now possesses also an antisymmetricpart,K [ab] = S 0ab = n µ S µab . (5.94)If second-class constraints are present, one has to use Dirac brackets insteadof Poisson brackets for the canonically conjugate variables (Dirac 1964; Sundermeyer1982; Henneaux and Teitelboim 1992). Dirac brackets coincide with14 The Grassmann-odd variables must be brought to the left before the functional differentiationis carried out. The momentum ˜π a A ′ is minus the Hermitian conjugate of π a A .