Quantum Gravity

DECOHERENCE AND THE QUANTUM UNIVERSE 311the universe. It has been suggested by Zeh (1986) that the irrelevant degreesof freedom are the variables describing density fluctuations and gravitationalwaves. Their interaction with the scale factor and other homogeneous degreesof freedom (such as an inflaton field) can render the latter classical. In a sense,then, a classical space–time arises from a ‘self-measurement’ of the universe.The following discussion will roughly follow Joos et al. (2003). Calculationsfor decoherence in quantum cosmology can be performed in the framework ofquantum geometrodynamics (Kiefer 1987), using the formalism of the Wheeler–DeWitt equation presented in Chapters 5 and 8. A prerequisite is the validity ofthe semiclassical approximation (Section 5.4) for the global variables. This bringsan approximate time parameter t into play. The irrelevant degrees of freedom(density fluctuations, gravitational waves) are described by the inhomogeneousvariables of Section 8.2. In Kiefer (1987), the relevant system was taken to be thescale factor (‘radius’) a of the universe together with a homogeneous scalar fieldφ (the ‘inflaton’); cf. the model discussed in Section 8.1.2. The inhomogeneousmodes of Section 8.2 can then be shown to decohere the global variables a andϕ.An open problem in Kiefer (1987) was the issue of regularization; the numberof fluctuations is infinite and would cause divergences, so an ad hoc cut-off wassuggested to consider only modes with wavelength bigger than the Planck length.The problem was again addressed in Barvinsky et al. (1999a) where a physicallymotivated regularization scheme was introduced. In the following, we shall brieflyreview this approach.As a (semi)classical solution for a and φ, onemayuseφ(t) ≈ φ, (10.6)a(t) ≈ 1 cosh H(φ)t , (10.7)H(φ)where H 2 (φ) =4πV (φ)/3m 2 P is the Hubble parameter generated by the inflatonpotential V (φ), cf. Section 8.3.2. It is approximately constant during the inflationaryphase in which φ slowly ‘rolls down’ the potential. We take into accountfluctuations of a field f(t, x) which can be a field of any spin (not necessarilya scalar field Φ). Space is assumed to be a closed three-sphere, so f(t, x) canbe expanded into a discrete series of spatial orthonormal harmonics Q n (x); cf.Section 8.2,f(t, x) = ∑ f n (t)Q n (x) . (10.8){n}One can thus represent the fluctuations by the degrees of freedom f n (in Section8.2, f n were the modes of a scalar field Φ).The aim is now to solve the Wheeler–DeWitt equation in the semiclassicalapproximation. This leads to the following solution:1Ψ(t|φ, f) = √ e ∏ −I(φ)/2+iS0(t,φ) ψ n (t, φ|f n ) . (10.9)vφ ∗(t) n