Institut für Theoretische Physik,
Freie Universität Berlin,
Arnimallee 14, D - 14195 Berlin
H. Kleinert
There
exists a simple
rule by which path integrals for the motion of a point particle
in a flat space can be
transformed correctly
into
those in curved space.
This rule arose
from
well-established methods
in the theory of plastic deformations,
where crystals with defects are described
mathematically by
applying nonholonomic coordinate
transformations
to ideal crystals. In the context of
time-sliced path integrals,
this has given rise to a quantum equivalence principle
which determines the measure of fluctating orbits
in spaces with curvature and torsion.
The nonholonomic
transformations
are accompanied by a nontrivial
Jacobian which
in curved spaces
produces
an additional
energy
proportional to the curvature scalar,
thereby canceling an equal term found earlier
by DeWitt from a naive formulation of Feynman's
time-sliced path
integral in curved space.
The importance of this cancelation has been documented
in various systems (H-atom, particle on the surface of a sphere, spinning top).
Here we point out its relevance
in the process of
bosonizing a nonabelian
one-dimensional quantum field theory,
whose fields live in a flat field space.
Its bosonized version
is a
quantum-mechanical path integral of
a point particle moving in a
space with constant curvature.
The additional term introduced by
the Jacobian is crucial
for
the identity between original and bosonized theory.
A useful bozonization tool is the so-called Hubbard-Stratonovich formula for which we find a nonabelian version.