Approaches to Quantum Gravity

242 T. Thiemannorder to avoid anomalies, one should not order the constraints symmetrically unless{C K , f IJ K }=0, which is not the case in GR.VI. Dirac observables and the problem of time.Classically, (weak) Dirac observables are defined by {C I , O} M=0 = 0 for all I .It is easy to check that this system of conditions is equivalent to a single relation,namely {O, {O, M}} M=0 = 0. There is a formal but rather natural way to constructthem [15; 16]. Consider a system of phase space functions T I such that the matrixA IJ := {C I , T J } is at least locally invertible and define the equivalent set of constraintsCI′ := (A −1 ) IJ C J . Remarkably, these constraints have weakly commutingHamiltonian vector fields X I . It is then tedious but straightforward to check that forany function f on phase space the functionF τ f,T := ∑ {n J }∏ (τ I − T I ) n IIn I !∏IX n II· f (13.10)is a weak Dirac observable. Here the sum runs over all sequences {n I } of non-negativeintegers. The physical interpretation of (13.10) is as follows. The constraint surface Mof the unconstrained phase space can be thought of as a fibre bundle with base givenby the physical phase [M] ={[m]; m ∈ M}, where [m] :={m ′ ∈ M; m ′ ∼ m}denotes the gauge orbit through m while the fibre above [m] are the points of thesubset [m] ⊂M. By assumption, the functions T I are local coordinates in the fibresabove each point, that is, given m ∈ M we may coordinatise it by m ↦→ ([m], T (m)).Hence we have a local trivialisation of the bundle. The gauge condition T (m) = τfor a value τ in the range of T now fixes a unique point m T (τ, [m]) in the fibre above[m] and at that point F τ f,T obviously assumes the value f (m T (τ, [m])). Since F τ f,T isgauge invariant, we have F τ f,T (m) = f (m T (τ, [m])) for all m ∈ M. It follows thatF τ f,Tonly depends on [M] for all values of τ and its value at p ∈[M] is the value off at the point m ∈ M with local coordinates ([m] =p, T (m) = τ).The functional (13.10) is what one calls a relational observable: none of the functionsf, T I is gauge invariant and therefore not observable. Only F τ f,Tis observable. Thisis precisely what happens in physics: consider the example of a relativistic particle.Like GR, the relativistic particle has no Hamiltonian, only a Hamiltonian constraintwhich in this case is the mass shell constraint C = (p 2 + m 2 )/2 = 0. It arises becausethe classical action is reparameterisation invariant. None of the coordinates X μ of theparticle is gauge invariant and thus observable. What is observable is the trajectory ofthe particle, that is, its graph. It can be implicitly described by P 0 X a −P a X 0 = const.or explicitly by F τ = X a + (τ − X 0 )P a /P 0 .X a ,X 0Relational observables also solve the Problem of Time: since the vector fields X Iare weakly commuting it is easy to see that f ↦→ F τ f,Tis a Poisson automorphismamong f which satisfy { f, T I } = 0 for all I . Therefore, the multifingered physicaltime automorphism F.,T τ has canonical generators defined by {H I (τ), F τ f,T } =