Kiefer C. Quantum gravity

156 QUANTUM GEOMETRODYNAMICS
and π 1 are their respective momenta. The form of H 1 is similar to the case
of parametrized field theory and string theory; cf. (3.80) and (3.49). One recognizes
explicitly that the kinetic term in H ⊥ is indefinite. In fact, the Hamiltonian
constraint describes an ‘indefinite harmonic oscillator’ (Zeh 1988)—the sum of
two ordinary oscillators where one comes with the opposite sign (see also Section
8.1.2).
According to our general prescription one has in the quantum theory
Ĥ ⊥ 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 expect
to find a product state as a common solution (‘correlation interaction’). All
physical states are probably entangled among all degrees of freedom. The algebra
of constraints (3.90)–(3.92) then reads in the quantum theory, 12
i[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) − c2
12π 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. This
is different from the (3+1)-dimensional case. The reason is that hh ab =1inone
spatial dimension and that the constraint generators have been rescaled by a
factor √ 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 must
necessarily be a Schwinger term in the commutator
[Ĥ⊥(x), Ĥ1(y)] .
This theorem was proven, however, within standard Poincaré-invariant local field
theory, with the additional assumption that there be a ground state of the Hamiltonian.
This is certainly not a framework that is applicable in a gravitational
context. But since the equations (5.62) and (5.63) have the form of equations
in flat space–time, one can tentatively apply this theorem. The central charge is
then 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 variables
r 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). Standard
methods (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 ).