Quantum Gravity

250 QUANTUM COSMOLOGYOne recognizes from (8.16) that α plays the role of an ‘intrinsic time’—the variable that comes with the opposite sign in the kinetic term. 2 Since thepotential (8.17) obeys V (α, φ) ≠ V (−α, φ), there is no invariance with respect toreversal of intrinsic time. This is of crucial importance to understand the originof irreversibility; cf. Section 10.2. Moreover, writing (8.16) in the form− 2 ∂2∂α 2 ψ ≡ h2 αψ, (8.18)the ‘reduced Hamiltonian’ h α is not self-adjoint, so there is no unitary evolutionwith respect to α. This is, however, not a problem since α is no external time.Unitarity with respect to an intrinsic time is not expected to hold.Models such as the one above can thus serve to illustrate the difficultieswhich arise in the approaches of ‘reduced quantization’ discussed in Section 5.2.Choosing classically a = t and solving H ≈ 0 with respect to p a leads to p a +h a ≈0 and, therefore, to a reduced Hamiltonian h a equivalent to the one in (8.18),h a = ±√p 2 φt 2 − t2 + Λt43 + m2 φ 2 t 4 . (8.19)One can recognize all the problems that are connected with such a formulation—explicit t-dependence of the Hamiltonian, no self-adjointness, complicated expression.One can make alternative choices for t—either p a = t (‘extrinsic time’),φ = t (‘matter time’), or a mixture of all these. This non-uniqueness is an expressionof the ‘multiple-choice problem’ mentioned in Section 5.2. Due to theseproblems, we shall restrict our attention to the discussion of the Wheeler–DeWittequation and do not follow the reduced approach any further.A simple special case of (8.16) is obtained if we set Λ = 0 and m = 0. Againalso setting =1,onegets( )∂2∂α 2 − ∂2∂φ 2 − e4α ψ(α, φ) =0. (8.20)A separation ansatz leads immediately to the special solution( ) eψ k (α, φ) =e −ikφ 2αK ik/2 , (8.21)2where K ik/2 denotes a Bessel function. This solution has been chosen in order tofulfil the boundary condition that ψ → 0forα →∞. A general solution with this2 In two-dimensional configuration spaces, only the relative sign seems to play a role. However,in higher-dimensional minisuperspaces, it becomes clear that the variable connected withthe volume of the universe is the time-like variable; see Section 5.2.2.