Kiefer C. Quantum gravity

THE GEOMETRODYNAMICAL WAVE FUNCTION 155
Another instructive example for the discussion of anomalies is dilaton gravity
in 1+1 dimensions. It is well known that GR in 1+1 dimensions possesses no
dynamics (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 characteristic
of the two-dimensional manifold; if the manifold were a closed compact Riemann
surface with genus g, one would have χ =2(1−g). Although (5.59) plays a role in
string perturbation theory (see Chapter 9), it is of no use in a direct quantization
of GR. One can, however, construct non-trivial models in two dimensions if there
are degrees of freedom in the gravitational sector in addition to the metric. A
particular example is the presence of a dilaton field. Such a field occurs, for
example, in the ‘CGHS model’ presented in Callan et al. (1992). This model is
defined 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’) with
dimension L −2 . 11 Note that the gravitational constant G is dimensionless in
two dimensions. The name ‘dilaton’ comes from the fact that φ occurs in the
combination d 2 x √ −g e −2φ and can thus be interpreted as describing an effective
change of integration measure (‘change of volume’). It is commonly found in
string perturbation theory (Chapter 9), and its value there determines the string
coupling constant.
The simplest choice for the matter action S m is an ordinary scalar-field action,
∫
S m = 1 2
d 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 providing
the starting point for quantization (independent of whether equivalence to the old
variables holds or not). In the Hamiltonian version, one finds again constraints:
one Hamiltonian constraint and one momentum constraint. They read (after a
rescaling 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 0
11 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 spherically
symmetric gravity to two dimensions; see Grumiller et al. (2002) for a general review of
dilaton gravity in two dimensions.