Approaches to Quantum Gravity

238 T. ThiemannNotice that in general the functions f will be unbounded on M and thus will bepromoted to unbounded operators later on, which will raise inconvenient domain questions.One could use bounded functions instead but this usually comes at the price ofcomplicating the Poisson relations and thus the representation theory of A. An exceptionis when P is a real Poisson Lie algebra in which case we can pass to the uniqueassociated Weyl C ∗ -algebra generated by unitary operators in any representation.III. Representation theory of A.The next step is the choice of a representation π of the elements a of A by (unbounded)operators π(a) on a Hilbert space H. We will not enter into the discussion of themost general representation but describe an important and large subclass arising frompositive linear functionals ω on the ∗ −algebra A, that is, ω(a ∗ a) ≥ 0 for all a ∈ A.The associated, so-called GNS representation is constructed as follows. Consider theset I ω := {a ∈ A; ω(a ∗ a) = 0}. One can show that this defines a left ideal andtherefore the natural operations [a]+[b] :=[a + b], [a]·[b] :=[a · b] on the classes[a] :={a + b, b ∈ I ω } are well defined. We define ω := [1] and π ω (a) := [a].TheHilbert space H ω is the completion of the vector space A/I ω in the following innerproduct< [a], [b] >=< π ω (a) ω ,π ω (b) ω >:= ω(a ∗ b). (13.3)Notice the double role of A as a Hilbert space and as a space of operators. The vector ω is automatically cyclic in this representation and obviously there are no domainquestions: all operators are densely defined on the dense subspace A/I ω . The GNSdata (π ω , H ω , ω ) are uniquely determined by ω up to unitary equivalence.The choice of ω is again far from unique and will be guided by physical input. Forinstance, it may be true that a subset of the constraints generates a Poisson Lie groupG. One then has a natural action of G on P via f ↦→ α g ( f ) where α g denotesthe Hamiltonian flow corresponding to g ∈ G. For instance in the case of the freeMaxwell field∫α g ( f ) = exp({ d 3 x∂ a E a ,.}) · f, (13.4)where g = exp(i) is a local U(1) gauge transformation and E denotes the electricfield. The action α is obviously a Poisson automorphism and extends to Avia α g (( f 1 , .., f n )) := (α g ( f 1 ), .., α g ( f n )). In this situation it is natural to lookfor states ω which are G-invariant, that is ω ◦ α g = ω for all g ∈ G becausethe following representation of the gauge group G on H ω is automatically unitary:U ω (g)π ω (a) ω := π ω (a) ω .Further criteria are the irreducibility of the representation. All we know is that thevector ω is cyclic. Irreducibility means that all vectors are cyclic. If a representationis not irreducible then the Hilbert space is a direct sum of irreducible subspaces andno observables exist which map between these sectors, they are superselected. Hencethe physically interesting information is realised already in one of those sectors.