Changing dimension and time: two well-founded and practical techniques for path integration in quantum physics

Abstract

In a reasonably self-contained presentation mathematical rigour is supplied to the important ideas of solving certain non-Gaussian path integrals by changes of dimension and/or path-dependent time transformations. The resulting genuine path-integral calculus neither requires discretization prescriptions nor sophisticated methods from the theory of stochastic differential equations. The power of the calculus is illustrated by two standard quantum-physics applications. First, the calculation of the time-dependent propagator corresponding to a particle on the half-line in a harmonic plus inverse-square potential is shown to be a simple exercise. Second, the first rigorous derivation of the energy-dependent Green function of the one-dimensional Morse system is given.