Kiefer C. Quantum gravity

MINISUPERSPACE MODELS 253
( )
∂
2
Ĥψ(a, χ) ≡ (−H a + H χ )ψ ≡
∂a 2 − ∂2
∂χ 2 − a2 + χ 2 ψ =0. (8.28)
This has the form of an ‘indefinite oscillator’—two harmonic oscillators distinguished
by a relative sign in the Hamiltonian. The model defined by (8.28) is
the simplest non-trivial model in quantum cosmology. (It also arises from scalar–
tensor theories of gravity; cf. Lidsey 1995.) Its classical solutions are represented
by Lissajou ellipses confined to a rectangle in configuration space. The corresponding
wave packets are obtained if a normalization condition is imposed with
respect to a and χ. In this way, one obtains an ordinary quantum-mechanical
Hilbert space. The wave packets follow here the classical Lissajous ellipses without
dispersion (Kiefer 1990; Gousheh and Sepangi 2000). From (8.28), one finds
H a ϕ n (a)ϕ n (χ) =H χ ϕ n (a)ϕ n (χ) ,
where ϕ n denote again the usual harmonic-oscillator eigenfunctions. A wavepacket
solution can then be constructed according to
ψ(a, χ) = ∑ n
A n ϕ n (a)ϕ n (χ) = ∑ n
H n (a)H n (χ)
A n
2 n e −a2 /2−χ 2 /2 , (8.29)
n!
with suitable coefficients A n . From the properties of the Hermite polynomials,
it is evident that the wave packet has to satisfy the ‘initial condition’ ψ(0,χ)=
ψ(0, −χ). The requirement of normalizability thus gives a restriction on possible
initial conditions. A particular example of such a wave-packet solution is depicted
in Fig. 8.2. One recognizes a superposition of (half of) two Lissajous ellipses.
In a more general oscillator model one would expect to find two different
frequencies for a and χ, that is, instead of the potential −a 2 + χ 2 , one would
have −ωa 2a2 + ωχ 2χ2 . The demand for normalizability would then entail the commensurability
condition
ω χ
= 2n a +1
ω a 2n χ +1 , (8.30)
where n a and n χ are integer numbers. Thus, one gets from normalizability a
restriction on the ‘coupling constants’ of this model. It is imaginable that such
conditions may also hold in the full theory, for example, for the cosmological
constant. 3
As Page (1991) has demonstrated, one can map various minisuperspace models
into each other. He also presents a plenty of classical and quantum solutions
for these models. For example, one can find a map between the cases of minimally
and non-minimally coupled scalar fields. The fields have to be rescaled and,
what is most important, they differ in the range of allowed values. As mentioned
above, this has consequences for the initial value problem in quantum gravity
3 Quantization conditions for the cosmological constant may also arise in string theory; cf.
Bousso and Polchinski (2000) and Feng et al. (2001).