Approaches to Quantum Gravity

368 R. WilliamsDropping the 1/d correction one obtains to leading order√d 2d/22 ( λ − kd 3) ( 1 − 1 ∑)ɛ2d!8 ij + ··· . (19.24)The partition function can be formally computed viaZ =∫N ∏i=1dɛ i e −ɛ M ɛ = π N/2√detM, (19.25)with N = 2d(d + 1). Convergence of the Gaussian integral then requires kd 3 >λ,and one has√ dd 2 2 +1 (log Z =kd 3 − λ ) [√ dd 2 2 +1 (− d (d + 1) log kd 3 − λ ) ]/8πd!d!with the first term arising from the constant term in the action, and the secondterm from the ɛ-field Gaussian integral. Therefore the general structure, to leadingorder in the weak field expansion at large d, islogZ = c 1 (kd 3 − λ) −d(d + 1) log(kd 3 − λ) + c 2 with c 1 and c 2 d-dependent constants, and therefore∂ 2 log Z/∂k 2 ∼ 1/(kd 3 − λ) 2 with divergent curvature fluctuations in the vicinityof the critical point at kd 3 = λ.If we apply the ideas of mean field theory, we need to keep the terms of order1/d in Eq. (19.23). In the ɛ ij ɛ ik term, we assume that the fluctuations are smalland replace ɛ ik by its average ¯ɛ. Each ɛ ij has 4d − 2 neighbours (edges with onevertex in common with it); this has to be divided by 2 to avoid double counting inthe sum, so the contribution is (2d − 1) ¯ɛ. Thentolowestorderin1/d, the actionis proportional to( ) [ λ − kd31 − 1 ∑ɛ28 ij + 1 4 ¯ɛ ∑ ]ɛ ij . (19.26)This gives rise to the same partition function as obtained earlier, and using it tocalculate the average value of ɛ ij gives ¯ɛ, as required for consistency.19.4 Regge calculus in quantum cosmologyIn quantum cosmology, interest is focused on calculations of the wave functionof the universe. According to the Hartle–Hawking prescription [41], the wavefunction for a given 3-geometry is obtained from a path integral over all 4-geometries which have the given 3-geometry as a boundary. To calculate such anobject in all its glorious generality is impossible, but one can hope to capture theessential features by integrating over those 4-geometries which might, for whatever