PhD thesis in English

2. Diagonalization of Transition AmplitudesA (p) (0, 0, t)21.51p = 1p = 10p = 20p = 30p = 400.500 0.5 1 1.5 2 2.5 3 3.5 4tFigure 2.7: Transition amplitude A (p) (0, 0; t) as a function of the time of propagationt, calculated analytically using different levels p of the effective action. The plot isfor the quartic anharmonic potential V (x) = k 22x 2 + k 424 x4 , with parameters k 2 = 1,k 4 = 10.its derivatives. The truncation of the series for the effective potential up to ordert p−1 , designated by W (p−1) (x, δ; t), gives the expansion of the transition amplitudeaccurate to t p ,A (p) (x, y; t) =1−tW√(2πt)d e−(x−y)2 2t(p−1) ( x+y ,x−y;t) 2 . (2.28)The analytic expressions for higher-order effective actions therefore yield analyticapproximations for amplitudes with the convergence behavior given by Eq. (2.25).We emphasize that although the structure of the effective action solution form (2.26)is motivated by the path integral formalism, the expression for amplitudes obtainedin the above approach contain no integrals and can be used straightforwardly aslong as the time of propagation is below the radius of convergence of the short-timeseries expansion.For the exactly solvable case of a harmonic oscillator one finds that the radiusof convergence is τ c = π. The radius of convergence is simply the distance in thecomplex time plain from the origin to the nearest singularity of the propagator.For the harmonic oscillator the singularities are located at ±ikπ, k ∈ N. Theconsequence of these singularities is that the power series for the effective potential35