Quantum Gravity

LOOP QUANTUM COSMOLOGY 275obeying〈µ 1 |µ 2 〉 = δ µ1,µ 2. (8.85)In the standard Schrödinger representation one would here have the delta functionδ(µ 1 − µ 2 ) on the right-hand side. The occurrence of the Kronecker symbolexpresses the fact that the Hilbert space is non-separable.How is p being quantized? Since according to (8.80) it is conjugated to c, werepresent it by a derivative operator,ˆp = −i 8πβl2 P3ddc , (8.86)leading to 8ˆp|µ〉 = 4πβl2 Pµ|µ〉 ≡p µ |µ〉 . (8.87)3The basis states are thus eigenstates of the ‘flux operator’ ˆp. This is possiblebecause of the orientation freedom for the triads, allowing µ ∈ R. Still, thespectrum of ˆp is ‘discrete’ in the sense that its eigenstates |µ〉 are normalizable(this is possible because the Hilbert space is non-separable). The discreteness ofthe full theory thus survives in the truncated version. We note also the relationfrom which one getsê iµ′ c/2|µ〉 = |µ + µ ′ 〉 , (8.88)〈µ|êiµ′ c/2|µ〉 = δ 0,µ ′ .From here one recognizes that the operator êiµ′ c/2is not continuous in µ, whichis why the Stone–von Neumann theorem of quantum mechanics does not applyand one indeed obtains a representation which is inequivalent to the standardSchrödinger representation.We shall now turn to a discussion of the quantum Hamiltonian constraint forthe Friedmann universe with a scalar field. Consider first the matter Hamiltonian(8.82). The major problem is to construct a well-defined operator for theclassically diverging (in the limit a → 0) expression |p| −3/2 ∝ a −3 .Ascanbeseen from (8.87), the operator ˆp possesses a discrete spectrum containing zeroand is thus not invertible. In order to deal with this situation one makes us of a‘Poisson-bracket trick’ similarly to the full theory in which one uses the identity(4.135) and transforms the Poisson bracket occurring therein into a commutator.Here instead of |p| −3/2 one directly uses the functiond(p) = 13πβG3∑i=1tr(τ i U i {U −1i , √ V }) 6 , (8.89)where U i denotes the holonomies of the isotropic connections, and V = |p| 3/2is the volume. For large p one has d(p) ∼ |p| −3/2 as required. Turning the8 The difference to the numerical factors appearing in some of the cited references is due tothe fact that in much of the literature on loop quantum cosmology 8πl 2 P is used instead of l2 P .