Approaches to Quantum Gravity

512 F. GirelliThe discussion can be made more precise when addressed in the Hamiltonianformalism. GR is an example of a constrained theory: there is a set of first classconstraints that encode the diffeomorphism symmetry. 1 Observable quantities arefunctions on phase space that commute with the constraints. It is pretty hard toconstruct a general complete set of observables. However, taking advantage of thefact that physics should be relational allowed us to construct a large set of suchobservables [13].To simplify the analysis let us consider the relativistic free particle: in this case,we have time reparametrization invariance, encoded in the mass shell constraintH = p 2 − m 2 = 0. It is easy to construct the algebra of observables: it is given bythe Poincaré algebra {J μν , p μ }. This set of observables does not include the importantnotion of position. To define this concept, we need to introduce the followingRovelli terminology of a partial observable 2 b(τ) as a clock. A natural observable 3is then the value of another partial observable a(τ) when b(τ) is T . If separately aand b are not observable, since not commuting with the mass shell constraint, thequantity∫a(b −1 (T )) = dτa(τ)ḃ(τ)δ (b(τ) − T ) (26.1)is clearly time reparametrization invariant, and therefore observable. For example,if we take b to be x 0 ,anda to be x i , we obtain the trajectories of the relativisticparticle in terms of the time x 0 :x μ (T ) = x μ + p μ(x 0 − T ).p 0Notice that of course this observable can be constructed from the Poincaréalgebra [14].In the language of constrained mechanics, b(τ) = T is a gauge fixing, or asecond class constraint. From the physics point of view, the degree of freedom b isthe reference frame. Obviously there is the issue of the invertibility of b. In generalthe choice of clock might not lead to a function which is invertible everywhere.This means that the clock ceased to be a good clock. This issue has to be studiedin a case per case analysis.Introducing second class constraints means that we can reduce the phase spaceto obtain the physical phase space. The reduced symplectic form is called the Diracbracket, and is not in general identical to the canonical Poisson bracket. This leadsto complications when one wants to quantize such system.1 There might be more constraints according to the choice of variables. For example, if using the pair (tetrad,connection), there is also the Gauss law.2 Let be f a function on phase space, not commuting with the first class constraint H, then we define f (τ) =e τ{H,.} f .3 That is the outcome of a measurement or, according to Rovelli, a complete observable [12].