Quantum Gravity

156 QUANTUM GEOMETRODYNAMICSand π 1 are their respective momenta. The form of H 1 is similar to the caseof parametrized field theory and string theory; cf. (3.80) and (3.49). One recognizesexplicitly that the kinetic term in H ⊥ is indefinite. In fact, the Hamiltonianconstraint describes an ‘indefinite harmonic oscillator’ (Zeh 1988)—the sum oftwo ordinary oscillators where one comes with the opposite sign (see also Section8.1.2).According to our general prescription one has in the quantum theoryĤ ⊥ Ψ(r 0 ,r 1 ,ϕ)=0, Ĥ 1 Ψ(r 0 ,r 1 ,ϕ)=0. (5.64)Although both H ⊥ and H 1 are a sum of independent terms, one cannot expectto find a product state as a common solution (‘correlation interaction’). Allphysical states are probably entangled among all degrees of freedom. The algebraof constraints (3.90)–(3.92) then reads in the quantum theory, 12i[Ĥ⊥(x), Ĥ⊥(y)] = (Ĥ1(x)+Ĥ1(y))δ ′ (x − y) , (5.65)i[Ĥ⊥(x), Ĥ1(y)] = (Ĥ⊥(x)+Ĥ⊥(y))δ ′ (x − y) − c212π δ′′′ (x − y) , (5.66)i[Ĥ1(x), Ĥ1(y)] = (Ĥ1(x)+Ĥ1(y))δ ′ (x − y) . (5.67)Note the absence of the metric on the right-hand side of these equations. Thisis different from the (3+1)-dimensional case. The reason is that hh ab =1inonespatial dimension and that the constraint generators have been rescaled by afactor √ h.In (5.66) an additional ‘Schwinger term’ with central charge c has been added.The reason is a theorem by Boulware and Deser (1967) stating that there mustnecessarily be a Schwinger term in the commutator[Ĥ⊥(x), Ĥ1(y)] .This theorem was proven, however, within standard Poincaré-invariant local fieldtheory, with the additional assumption that there be a ground state of the Hamiltonian.This is certainly not a framework that is applicable in a gravitationalcontext. But since the equations (5.62) and (5.63) have the form of equationsin flat space–time, one can tentatively apply this theorem. The central charge isthen a sum of three contributions (Cangemi et al. 1996),c = c g + c m ≡ c g 0 + cg 1 + cm , (5.68)where c g 0 and cg 1 are the central charges connected with the gravitational variablesr 0 and r 1 , respectively, and c m is the central charge connected with the field ϕ.The result for c depends on the notion of vacuum (if there is one). Standardmethods (decomposition into creation and annihilation operators) yield c g 1 =1.What about c g 0 ? If the sign in front of the (π 0) 2 -term in (5.62) were positive,12 The Virasoro form (3.62) of the algebra follows for the combinations θ ± =(H ⊥ ∓H 1 ).