Reparametrization invariance of path integrals

Abstract

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...

6D trace anomalies from quantum mechanical path integrals

“…Recently, following the suggestion in ref. [9] of using dimensional regularization a third way of properly defining the path integrals has been developed in [10]: the dimensional regularization scheme (DR). While the counterterms required in MR and TS are noncovariant, they happen to be covariant in the DR scheme [10,11,9].…”

Dimensional regularization of nonlinear sigma models on a finite time interval

“…This problem has a simple solution [15] which we use below: one starts with the action ∂ µ φ∂ µ φ in n dimensions and then all Lorentz indices µ, ν in all contractions are unambiguous.…”

“…for by previous authors" [15]. Of course, this refers to possible noncovariant counterterms since a one-dimensional model cannot test counterterms proportional to R. However, the interpretation "...artificial potential term..." of the results of [1][2][3][4][5][6][8][9][10][11][12][13][14] may be misleading.…”

“…One can now regulate nonlinear sigma models on a finite time interval by using dimensional regularization. We were inspired by recent papers [15] which applied standard dimensional regularization on an infinite time interval to a model with a mass term. Such a mass term is also needed in more general models on an infinite time interval [16] to regularize infrared divergences, but it breaks general covariance in target space.…”

On the simplified path integral on spheres

“…The nonlinearities present in the kinetic term make the definition of the path integral rather subtle, carrying the necessity of specifying a regularization scheme together with the fixing of corresponding finite counterterms. The latter are needed for specifying a well-defined quantum theory; see [2][3][4][5] for the known regularization schemes. The development of those regularization schemes was prompted by the a e-mail: fiorenzo.bastianelli@bo.infn.it b e-mail: olindo.corradini@unimore.it desire of extending the quantum mechanical method of computing chiral anomalies [6][7][8] to trace anomalies [9,10].…”