28 research outputs found
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
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 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 introduces subtleties in the
definition and treatment of the path average , 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 , the ground-state energy E^{(0) can be
expanded perturbatively in powers of , where is the box size. The
removal of the infrared cutoff 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
Integrals over Products of Distributions and Coordinate Independence of Zero-Temperature Path Integrals
In perturbative calculations of quantum-statistical zero-temperature path
integrals in curvilinear coordinates one encounters Feynman diagrams involving
multiple temporal integrals over products of distributions, which are
mathematically undefined. In addition, there are terms proportional to powers
of Dirac delta-functions at the origin coming from the measure of path
integration. We give simple rules for integrating products of distributions in
such a way that the results ensure coordinate independence of the path
integrals. The rules are derived by using equations of motion and partial
integration, while keeping track of certain minimal features originating in the
unique definition of all singular integrals in dimensions. Our
rules yield the same results as the much more cumbersome calculations in 1-
epsilon dimensions where the limit epsilon --> 0 is taken at the end. They also
agree with the rules found in an independent treatment on a finite time
interval.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