Quantum Gravity

84 PARAMETRIZED AND RELATIONAL SYSTEMSintrinsic physical meaning. In addition, there is the Poincaré symmetry of thebackground Minkowski space–time, which is of minor interest here.Defining the two-dimensional energy–momentum tensor according to 3T αβ = − 4πα′ √hδS Pδh αβ = ∂ αX µ ∂ β X µ − 1 2 h αβh γδ ∂ γ X µ ∂ δ X µ , (3.53)one findsh αβ T αβ =0. (3.54)This tracelessness of the energy–momentum tensor is a consequence of Weylinvariance. This can be easily seen as follows. Consider a general variation of theaction,∫δS = d 2 σδS ∫δh αβ δhαβ ∝ d 2 σ √ hT αβ δh αβ .Under (3.52) we haveδh αβ = −2(δω)h αβ .Therefore, the demand that δS = 0 under (3.52) leads to (3.54). In the quantumtheory, a ‘Weyl anomaly’ may occur in which the trace of the energy–momentumtensor is proportional to times the two-dimensional Ricci scalar; see (2.124).The demand for this anomaly to vanish leads to restrictions on the parametersof the theory; see below.Using the field equations δS P /δh αβ = 0, one findsT αβ =0. (3.55)In a sense, these are the Einstein equations with the left-hand side missing, sincethe Einstein–Hilbert action is a topological invariant. As (3.55) has no secondtime derivatives, it is in fact a constraint—a consequence of diffeomorphisminvariance. From (3.55), one can easily derive thatdetG αβ = h 4 (hαβ G αβ ) 2 , (3.56)where G αβ is the induced metric (3.39). Inserting this into (3.51) gives backthe action (3.37). Therefore, ‘on-shell’ (i.e., using the classical equations) bothactions are equivalent.The constraints (3.42) and (3.43) can also be found directly from (3.51)—defining the momenta conjugate to X µ in the usual manner—after use has beenmade of (3.55). One can thus formulate instead of (3.51) an alternative canonicalaction principle,∫S = d 2 σ (P µ Ẋ µ − NH ⊥ − N 1 H 1 ) , (3.57)Mwhere H ⊥ and H 1 are given by (3.45) and (3.46), respectively.3 Compared to GR there is an additional factor −2π here, which is introduced for convenience.