Perturbatively Defined Effective Classical Potential in Curved Space

The partition function of a quantum statistical system in flat space can always be written as an integral over a classical Boltzmann factor \exp[ -\beta V^{\rm eff cl({\bf x}_0)], where V^{\rm eff cl({\bf x}_0) is the so-called effective classical potential containing the effects of all quantum fluctuations. The variable of integration is the temporal path average {\bf x_0\equiv \beta ^{-1}\int_0^ \beta d\tau {\bf x}(\tau). We show how to generalize this concept to paths $q^\mu(\tau)$ in curved space with metric g_{\mu \nu (q), and calculate perturbatively the high-temperature expansion of V^{\rm eff cl(q_0). The requirement of independence under coordinate transformations $q^\mu(\tau)\to q'^\mu(\tau)$ introduces subtleties in the definition and treatment of the path average $q_0^\mu$, and covariance is achieved only with the help of a suitable Faddeev-Popov procedure.Comment: Author Information under http://www.physik.fu-berlin.de/~kleinert/institution.html . Latest update of paper (including all PS fonts) at http://www.physik.fu-berlin.de/~kleinert/33

Perturbation Theory for Particle in a Box

Recently developed strong-coupling theory open up the possibility of treating quantum-mechanical systems with hard-wall potentials via perturbation theory. To test the power of this theory we study here the exactly solvable quantum mechanics of a point particle in a one-dimensional box. Introducing an auxiliary harmonic mass term $m$, the ground-state energy E^{(0) can be expanded perturbatively in powers of $1/md$, where $d$ is the box size. The removal of the infrared cutoff $m$ requires the resummation of the series at infinitely strong coupling. We show that strong-coupling theory yields a fast-convergent sequence of approximations to the well-known quantum-mechanical energy E^{(0)= \pi ^2/2d^2.Comment: Author Information under http://www.physik.fu-berlin.de/~kleinert/institution.html . Latest update of paper also at http://www.physik.fu-berlin.de/~kleinert/28

Coordinate Independence of of Quantum-Mechanical Path Integrals

We develop simple rules for performing integrals over products of distributions in coordinate space. Such products occur in perturbation expansions of path integrals in curvilinear coordinates, where the interactions contain terms of the form dot q^2 q^n, which give rise to highly singular Feynman integrals. The new rules ensure the invariance of perturbatively defined path integrals under coordinate transformations.Comment: Author Information under http://www.physik.fu-berlin.de/~kleinert/institution.html . Latest update of paper also at http://www.physik.fu-berlin.de/~kleinert/305

Reparametrization Invariance of Path Integrals

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 epsilon-> 0 -limit of 1+ epsilon -dimensional functional integrals.Comment: Author Information under http://www.physik.fu-berlin.de/~kleinert/institution.html . Latest update of paper also at http://www.physik.fu-berlin.de/~kleinert/kleiner_re289/preprint.htm