2,259 research outputs found
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
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
Variational Perturbation Theory for Fokker-Planck Equation with Nonlinear Drift
We develop a recursive method for perturbative solutions of the Fokker-Planck
equation with nonlinear drift. The series expansion of the time-dependent
probability density in terms of powers of the coupling constant is obtained by
solving a set of first-order linear ordinary differential equations. Resumming
the series in the spirit of variational perturbation theory we are able to
determine the probability density for all values of the coupling constant.
Comparison with numerical results shows exponential convergence with increasing
order.Comment: Author Information under
http://www.theo-phys.uni-essen.de/tp/ags/pelster_dir
Nonholonomic Mapping Principle for Classical Mechanics in Spaces with Curvature and Torsion. New Covariant Conservation Law for Energy-Momentum Tensor
The lecture explains the geometric basis for the recently-discovered
nonholonomic mapping principle which specifies certain laws of nature in
spacetimes with curvature and torsion from those in flat spacetime, thus
replacing and extending Einstein's equivalence principle. An important
consequence is a new action principle for determining the equation of motion of
a free spinless point particle in such spacetimes. Surprisingly, this equation
contains a torsion force, although the action involves only the metric. This
force changes geodesic into autoparallel trajectories, which are a direct
manifestation of inertia. The geometric origin of the torsion force is a
closure failure of parallelograms. The torsion force changes the covariant
conservation law of the energy-momentum tensor whose new form is derived.Comment: Corrected typos. Author Information under
http://www.physik.fu-berlin.de/~kleinert/institution.html . Paper also at
http://www.physik.fu-berlin.de/~kleinert/kleiner_re261/preprint.htm
Brownian motion of Massive Particle in a Space with Curvature and Torsion and Crystals with Defects
We develop a theory of Brownian motion of a massive particle, including the
effects of inertia (Kramers' problem), in spaces with curvature and torsion.
This is done by invoking the recently discovered generalized equivalence
principle, according to which the equations of motion of a point particle in
such spaces can be obtained from the Newton equation in euclidean space by
means of a nonholonomic mapping. By this principle, the known Langevin equation
in euclidean space goes over into the correct Langevin equation in the Cartan
space. This, in turn, serves to derive the Kubo and Fokker-Planck equations
satisfied by the particle distribution as a function of time in such a space.
The theory can be applied to classical diffusion processes in crystals with
defects.Comment: LaTeX, http://www.physik.fu-berlin.de/kleinert.htm
Autoparallels From a New Action Principle
We present a simpler and more powerful version of the recently-discovered
action principle for the motion of a spinless point particle in spacetimes with
curvature and torsion. The surprising feature of the new principle is that an
action involving only the metric can produce an equation of motion with a
torsion force, thus changing geodesics to autoparallels. This additional
torsion force arises from a noncommutativity of variations with parameter
derivatives of the paths due to the closure failure of parallelograms in the
presence of torsionComment: Paper in src. Author Information under
http://www.physik.fu-berlin.de/~kleinert/institution.html Read paper directly
with Netscape under
http://www.physik.fu-berlin.de/~kleinert/kleiner_re243/preprint.htm
Smearing Formula for Higher-Order Effective Classical Potentials
In the variational approach to quantum statistics, a smearing formula
describes efficiently the consequences of quantum fluctuations upon an
interaction potential. The result is an effective classical potential from
which the partition function can be obtained by a simple integral. In this
work, the smearing formula is extended to higher orders in the variational
perturbation theory. An application to the singular Coulomb potential exhibits
the same fast convergence with increasing orders that has been observed in
previous variational perturbation expansions of the anharmonic oscillator with
quartic potential.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_re267/preprint.htm
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