We demonstrate the reparametrization invariance of perturbatively defined one-dimensional functional integrals up to the three-loop level for a path integral of a quantum-mechanical point particle in a box. We exhibit the origin of the failure of earlier authors to establish reparametrization invariance which led them to introduce, superfluously, a compensating potential depending on the connection of the coordinate system. We show that problems with invariance are absent by defining path integrals as the ǫ → 0 -limit of 1 + ǫ-dimensional functional integrals.1. The long-standing problem of the reparametrization invariance of perturbatively defined one-dimensional functional integrals, called path integrals, has recently become an important issue in the context of worldline formulations of quantum field theory [1][2][3][4]. This problem was apparently first encountered in two-loop perturbative calculations in Ref. [5], and a solution was attempted once more in Ref. [6]. The latter authors exhibited in great detail the change of results obtained after subjecting a path integral to a coordinate transformation. The results could only be forced to agree by adding to the tranformed action an artificial potential term, of the orderh 2 , depending on the connection of the coordinate transformation.Such a noninvariance under coordinate transformations, if unavoidable, would be extremely unpleasant for two reasons: First, the functional integral of the much more complicated quantum gravity is known to be invariant under coordinate transformations if the infinities are regularized a la t' Hooft and Veltman [7] in 4 − ǫ dimensions, and it would be surprising to see a failure in simple quantum mechanics. Second, a similar initially encountered problem in the time-sliced definition of path integrals has been solved elegantly in the textbook (11), by defining it as the image of a euclidean path integral (in curved space via a nonholonomic coordinate transformations). It would be embarrassing if a perturbative definition on a continuous time axis which is standard in quantum field theory would fail to match the invariance of the time-sliced definition.An brief inspection of the problems encountered in Refs. [5,6] reveals immediately the central unsatisfactory feature of their treatment. The perturbation expansion of the transformed path integral leads to Feynman integrals involving space-dependent kinetic terms whose result depends on the order of evaluation of the individual one-dimensional momentum integrals. As a typical example, take the Feynman integral Y = dk 2π dp 1 2π dp 2 2πIntegrating this first over k, then over p 1 and p 2 yields 3/64m. In the order first p 1 , then p 2 and k, we find −1/64m. As we shall see below, the correct result is the average og the two, 1/32m. The purpose of this note is to exhibit the origin of this ambiguity and to remove it by a proper definition of the path integral as the analytic continuation of a corresponding functional integral in D spacetime dimensions to D = 1. This leads to a uni...

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General solutions for the system of nonlinear equations in the second order partial derivatives with two independent variables are obtained. They determine the basic differential forms of the two-dimensional minimal surface embedded into n-dimensional pseudo-Euclidean space.

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.

We show how to perform integrals over products of distributions in coordinate space such as to reproduce the results of momentum space Feynman integrals in dimensional regularization. This ensures the invariance of path integrals under coordinate transformations. The integrals are performed by expressing the propagators in 1 − ε dimensions in terms of modified Bessel functions.