Quantum Gravity

316 INTERPRETATIONFor a massive minimally coupled field, the new frequency function reads[ √n2( 1˜Ω n = + m 2 a 2 m 2 a 2 )]+isinhHt2 n 2 + m 2 a 2 + O(1/m) . (10.31)Note that, in contrast to (10.24), there is no factor of a 2 in front of this expression.Since (10.31) is valid in the large-mass limit, it corresponds to modes that evolveadiabatically on the gravitational background, the imaginary part in (10.31)describing particle creation.It turns out that the imaginary part of the decoherence factor has at most logarithmicdivergences and, therefore, affects only the phase of the density matrix.Moreover, these divergences decompose into an additive sum of one-argumentfunctions and can thus be cancelled by adding counterterms to the classicalaction S 0 (and S 0) ′ in (10.18) (Paz and Sinha 1992). The real part is simply convergentand gives a finite decoherence amplitude. This result is formally similarto the result for the decoherence factor in QED (Kiefer 1992).For a ≫ a ′ (far off-diagonal terms), one gets the expression| ˜D(t|φ, φ ′ )|≃exp[− (ma)324(π − 8 3)+ O(m 2 )]. (10.32)Compared with the naively regularized (and inconsistent) expression (10.26), πhas effectively been replaced by 8/3 − π. In the vicinity of the diagonal, oneobtainsln | ˜D(t|φ, φ ′ )| = − m3 πa(a − a ′ ) 2, (10.33)64a behaviour similar to (10.32).An interesting case is also provided by minimally coupled massless scalarfields and by gravitons. They share the basis- and frequency functions in theirrespective conformal parametrizations,ṽ ∗ n(t) =( 1+isinhHt1 − isinhHt) n/2 ( n − isinhHtn +1), (10.34)˜Ω n = n(n2 − 1)n 2 − 1+H 2 a 2 − a 2√ H 2 a 2 − 1iH2 n 2 − 1+H 2 a 2 . (10.35)They differ only by the range of the quantum number n (2 ≤ n for inhomogeneousscalar modes and 3 ≤ n for gravitons) and by the degeneracies of the ntheigenvalue of the Laplacian,dim(n) scal = n 2 , (10.36)dim(n) grav =2(n 2 − 4). (10.37)For far off-diagonal elements one obtains the decoherence factor| ˜D(t|φ, φ ′ )|∼e −C(Ha)3 , a ≫ a ′ , C > 0 , (10.38)