Quantum Gravity

THE GEOMETRODYNAMICAL WAVE FUNCTION 155Another instructive example for the discussion of anomalies is dilaton gravityin 1+1 dimensions. It is well known that GR in 1+1 dimensions possesses nodynamics (see e.g. Brown (1988) for a review), since∫d 2 x √ −g (2) R =4πχ , (5.59)where (2) R is the two-dimensional Ricci scalar, and χ is the Euler characteristicof the two-dimensional manifold; if the manifold were a closed compact Riemannsurface with genus g, one would have χ =2(1−g). Although (5.59) plays a role instring perturbation theory (see Chapter 9), it is of no use in a direct quantizationof GR. One can, however, construct non-trivial models in two dimensions if thereare degrees of freedom in the gravitational sector in addition to the metric. Aparticular example is the presence of a dilaton field. Such a field occurs, forexample, in the ‘CGHS model’ presented in Callan et al. (1992). This model isdefined by the action∫4πGS CGHS = d 2 x √ ( )−g e −2φ (2) R +4g µν ∂ µ φ∂ ν φ − λ + S m , (5.60)where φ is the dilaton field, and λ is a parameter (‘cosmological constant’) withdimension L −2 . 11 Note that the gravitational constant G is dimensionless intwo dimensions. The name ‘dilaton’ comes from the fact that φ occurs in thecombination d 2 x √ −g e −2φ and can thus be interpreted as describing an effectivechange of integration measure (‘change of volume’). It is commonly found instring perturbation theory (Chapter 9), and its value there determines the stringcoupling constant.The simplest choice for the matter action S m is an ordinary scalar-field action,∫S m = 1 2d 2 x √ −gg µν ∂ µ ϕ∂ ν ϕ. (5.61)Cangemi et al. (1996) make a series of redefinitions and canonical transformations(partly non-local) to simplify this action. The result is then defined as providingthe starting point for quantization (independent of whether equivalence to the oldvariables holds or not). In the Hamiltonian version, one finds again constraints:one Hamiltonian constraint and one momentum constraint. They read (after arescaling λ → λ/8πG)H ⊥ = (π 1) 2 − (π 0 ) 2− λ 2λ 2 ([r0 ] ′ ) 2 + λ 2 ([r1 ] ′ ) 2 + 1 2 (π2 ϕ +[ϕ ′ ] 2 ) , (5.62)H 1 = −[r 0 ] ′ π 0 − [r 1 ] ′ π 1 − ϕ ′ π ϕ , (5.63)where r 0 and r 1 denote the new gravitational variables (found from the metric—the only dynamical part being its conformal part—and the dilaton), and π 011 One can exhaust all dilaton models by choosing instead of λ any potential V (φ); cf. Louis-Martinez and Kunstatter (1994). A particular example is the dimensional reduction of sphericallysymmetric gravity to two dimensions; see Grumiller et al. (2002) for a general review ofdilaton gravity in two dimensions.