Simple explicit formulas for Gaussian path integrals with time-dependent frequencies

Abstract

Quadratic fluctuations require an evaluation of ratios of functional determinants of second-order differential operators. We relate these ratios to the Green functions of the operators for Dirichlet, periodic and antiperiodic boundary conditions on a line segment. This permits us to take advantage of Wronski's construction method for Green functions without knowledge of eigenvalues. Our final formula expresses the ratios of functional determinants in terms of an ordinary 2 × 2 -determinant of a constant matrix constructed from two linearly independent solutions of a the homogeneous differential equations associated with the second-order differential operators. For ratios of determinants encountered in semiclassical fluctuations around a classical solution, the result can further be expressed in terms of this classical solution.In the presence of a zero mode, our method allows for a simple universal regularization of the functional determinants. For Dirichlet's boundary condition, our result is equivalent to Gelfand-Yaglom's.Explicit formulas are given for a harmonic oscillator with an arbitrary time-dependent frequency.

Generating functionals for harmonic expectation values of paths with fixed end points: Feynman diagrams for nonpolynomial interactions

“…Note that similar results can also be derived for periodic and antiperiodic boundary conditions [3,4].…”

“…In order to calculate the operator determinant (3.31) it is advantageous to introduce a one-parameter family of operators [3,4] …”

“…the correlation functions (4.1) may be reexpressed as 4) where the generating functional is given by (3.42). The currents j(t) and k(t) are specialized to…”

Beyond the thin-wall approximation: Precise numerical computation of prefactors in false vacuum decay

“…We adapt a method of McKane and Tarlie [39] (see also [28,29,40]). First, add a small quantity, k 2 , to the operator possessing the zero mode, and to the corresponding free operator (although this latter addition is not important in the end).…”

Two-loop corrections to false vacuum decay in scalar field theory

“…The one-loop computations are the most developed. To evaluate functional determinants we may choose from direct evaluation of the spectrum for solvable potentials 2 , heat kernel methods [34][35][36][37], Green function methods [38][39][40][41][42][43] or Gel'fand-Yaglom method [44] and its generalizations [45,46]. Beyond one loop the corresponding techniques were mostly developed in the study of instantons [47][48][49] in quantum mechanics [50][51][52][53][54][55], investigation of effective Euler-Heisenberg lagrangian [56][57][58] and computation of quantum corrections to classical string solutions [59,60].…”