Feynman Path Integral Formulation

160 5 Semiclassical Gravityof the time dependence of the metric (which can be visualized as a time dependentexternal quantum mechanical perturbation).After substituting Eq. (5.83) into Eq. (5.81) and equating it to the expansionin Eq. (5.78), one obtains a direct expression for the second set of creation anddestruction operators b † k , b k, c † k , c k in terms of the original set a † k , a k, with[b k = ∑ ᾱ kk ′ a k ′ − ¯β]kk ′ a † k ′ , (5.84)k ′and similarly for the c k operators.The last equality allows one to compute the expectation value of the numberoperator N (b)k= b † k b k, in a state which does not contain any incoming particles (andtherefore, as stated at the beginning, is |0 − 〉 with a k |0 − 〉 = 0), giving therefore forthe mode k〈0 − |b † k b k |0 − 〉 = ∑|β kk ′| 2 . (5.85)k ′Therefore the problem of computing the number of particles created and emitted toinfinity has been reduced to computing the complex expansion coefficients β kk ′ inEq. (5.83).To derive an expression for the coefficients β kk ′ it is sufficient to consider a metricwith spherical symmetry, where the solutions to the wave equation ∇ 2 f k = 0 can bewritten as a product of spherical harmonics Y lm ( theta,φ) times a time dependentradial wave function. The latter can be written in terms of advanced and retardedsolutions with frequency dependence1√2πωe iω u , (5.86)with u and v retarded and advanced coordinatesr∣ ∣∣u = t − r − 2MG log∣2MG − 1 r∣ ∣∣v = t + r + 2MG log∣2MG − 1 , (5.87)and invariant distanceds 2 = − 2MG e −r/2MG e (v−u)/4MG dudv + r 2 dΩ2 2 , (5.88)rwith r = r(u,v). Then the sum over modes ∑ k gets replaced by a sum over frequencies∑ ω lm , so that for each partial wave (l,m)f ω ′ lm =1√ f2πω ′ ω ′(r)Y lm (θ,φ) eiω′ vr(5.89)p ω lm =1√ p ω (r)Y lm (θ,φ) eiω u2πω r, (5.90)