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H. Kleinert

Publications and source records attributed to H. Kleinert.

At least 109 records · Page 6Linked to original sources
arXiv1997

Functional determinants via Wronski construction of Green functions

A general technique is developed for calculating functional determinants of second-order differential operators with Dirichlet, periodic, and antiperiodic boundary conditions. As an example, we give simple formulas for a harmonic oscillator with an arbitrary time-dependent frequency. Here our result is a generalization of Gel'fand-Yaglom's famous formula which was restricted to Dirichlet boundary conditions. Apart from the generalization, our derivation is more transparent than theirs, the determinants requiring only knowledge of the classical trajectories. Special properties of operators with a zero mode are exhibited. Our technique does not require the calculation of the spectrum and is as simple as Wronski's method for Green functions.

math-ph↗
arXiv1997

Spaces with torsion from embedding and the special role of autoparallel trajectories

As a contribution to the ongoing discussion of trajectories of spinless particles in spaces with torsion we show that the geometry of such spaces can be induced by embedding their curves in a euclidean space without torsion. Technically speaking, we define the tangent (velocity) space of the embedded space imposing non-holonomic constraints upon the tangent space of the embedding space. Parallel transport in the embedded space is determined as an induced parallel transport on the surface of constraints. Gauss' principle of least constraint is used to show that autoparallels realize a constrained motion that has a minimal deviation from the free, unconstrained motion, this being a mathematical expression of the principle of inertia.

gr-qc↗
arXiv1997

Quantum Equivalence Principle

A simple mapping procedure is presented by which classical orbits and path integrals for the motion of a point particle in flat space can be transformed directly into those in curved space with torsion. Our procedure evolved from well-established methods in the theory of plastic deformations, where crystals with defects are described mathematically as images of ideal crystals under active nonholonomic coordinate transformations. Our mapping procedure may be viewed as a natural extension of Einstein's famous equivalence principle. When applied to time-sliced path integrals, it gives rise to a new "quantum equivalence principle" which determines short-time action and measure of fluctuating orbits in spaces with curvature and torsion. The nonholonomic transformations possesses a nontrivial Jacobian in the path measure which produces in a curved space an additional term proportional to the curvature scalar R, thus canceling a similar term found earlier by DeWitt. This cancelation is important for correctly describing semiclassically and quantum mechanically various systems such as the hydrogen atom, a particle on the surface of a sphere, and a spinning top. It is also indispensable for the process of bosonization, by which Fermi particles are redescribed by those fields.

quant-ph↗
arXiv1996

Relations Between Markov Processes Via Local Time and Coordinate Transformations

The Duru-Kleinert (DK) method of solving unknown path integrals of quantum mechanical systems by relating them to known ones does not apply to Markov processes since the DK-transform of a Fokker-Planck equation is in general not a Fokker-Planck equation. In this note, we present a significant modification of the method, based again on a combination of path-dependent time and coordinate transformations, to obtain such relations after all. As an application we express unknown Green functions for a one-parameter family of Markov processes in terms of the known one for the Schenzle-Brandt process.

cond-mat↗
arXiv1996

Stability of 3D Cubic Fixed Point in Two-Coupling-Constant ϕ^4-Theory

For an anisotropic euclidean $ϕ^4$-theory with two interactions $[u (\sum_{i=1^M ϕ_i^2)^2+v \sum_{i=1}^M ϕ_i^4]$ the $β$-functions are calculated from five-loop perturbation expansions in $d=4-\varepsilon$ dimensions, using the knowledge of the large-order behavior and Borel transformations. For $\varepsilon=1$, an infrared stable cubic fixed point for $M \geq 3$ is found, implying that the critical exponents in the magnetic phase transition of real crystals are of the cubic universality class. There were previous indications of the stability based either on lower-loop expansions or on less reliable Pad\'{e approximations, but only the evidence presented in this work seems to be sufficently convincing to draw this conclusion.

quant-ph↗
arXiv1996

Transformation Properties of Classical and Quantum Laws under Some Nonholonomic Spacetime Transformations

Nonrelativistic Newton and Schroedinger equations remain correct not only under holonomic but also under nonholonomic transformations of the spacetime coordinates. Here we study the properties of transformations which are holonomic in the space coordinates while additionally tranforming the time in a path-dependent way. This makes them nonholonomic in spacetime. The resulting transformation formulas of physical quantities establish relations between different physical systems. Furthermore we point out certain differential-geometric features of these relations.

quant-ph↗
arXiv1996

Classical and Fluctuating Paths in Spaces with Curvature and Torsion

This conference talk elaborates on a recently discovered mapping procedure by which classical orbits and path integrals for the motion of a point particle in flat space can be transformed correctly into those in curved space. This procedure evolved 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, there seems to exists a quantum equivalence principle which determines the measures of fluctating orbits in spaces with curvature and torsion. The nonholonomic transformations produce a nontrivial Jacobian in the path measure which in a curved space produces an additional term proportional to the curvature scalar canceling a similar term found earlier by DeWitt from a naive formulation of Feynman's time-sliced path integral. This cancelation is important in correctly describing semiclassically and quantum mechanically various systems such as the hydrogen atom, a particle on the surface of a sphere, and a spinning top. It is also indispensible for the process of bosonization, by which Fermi particles are redescribed in terms of Bose fields.

quant-ph↗
arXiv1996

Nonabelian Bosonization as a Nonholonomic Transformations from Flat to Curved Field Space

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 {\em 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.

hep-th↗
arXiv1996

Resummation of anisotropic quartic oscillator. Crossover from anisotropic to isotropic large-order behavior

We present an approximative calculation of the ground-state energy for the anisotropic anharmonic oscillator Using an instanton solution of the isotropic action $δ= 0$, we obtain the imaginary part of the ground-state energy for small negative $g$ as a series expansion in the anisotropy parameter $δ$. From this, the large-order behavior of the $g$-expansions accompanying each power of $δ$ are obtained by means of a dispersion relation in $g$. These $g$-expansions are summed by a Borel transformation, yielding an approximation to the ground-state energy for the region near the isotropic limit. This approximation is found to be excellent in a rather wide region of $δ$ around $δ= 0$. Special attention is devoted to the immediate vicinity of the isotropic point. Using a simple model integral we show that the large-order behavior of an $δ$-dependent series expansion in $g$ undergoes a crossover from an isotropic to an anisotropic regime as the order $k$ of the expansion coefficients passes the value $k_{{\rm cross} \sim 1/ |δ|$.

quant-ph↗
arXiv1996

Anholonomic Transformations of Mechanical Action Principle

We exhibit the transformation properties of the mechanical action principle under anholonomic transformations. Using the fact that spaces with torsion can be produced by anholonomic transformations we derive the correct action principle in these spaces which is quite different from the conventional action principle.

gr-qc↗
arXiv1996

Evidence for Negative Stiffness of QCD Strings

QCD strings are color-electric flux tubes between quarks with a finite thickness and thus a finite curvature stiffness. Contrary to an earlier rigid-string by Polyakov and Kleinert, and motivated by the properties of a magnetic flux tubes in type-II superconductors we put forward the hypothesis that QCD strings have a {\em negative\/ stiffness. We set up a new string model with this property and show that it is free of the three principal problems of rigid-strings --- particle states with negative norm, nonexistence of a lowest-energy state, and wrong high-temperature behavior of string tension --- thus making it a better candidate for a string description of quark forces than previous models.

hep-th↗
arXiv1995

Quantum Equivalence Principle for Path Integrals in Spaces with Curvature and Torsion

We formulate a new quantum equivalence principle by which a path integral for a particle in a general metric-affine space is obtained from that in a flat space by a non-holonomic coordinate transformation. The new path integral is free of the ambiguities of earlier proposals and the ensuing Schrödinger equation does not contain the often-found but physically false terms proportional to the scalar curvature. There is no more quantum ordering problem. For a particle on the surface of a sphere in $D$ dimensions, the new path integral gives the correct energy $\propto \hat L^2$ where $\hat L$ are the generators of the rotation group in ${\bf x}$-space. For the transformation of the Coulomb path integral to a harmonic oscillator, which passes at an intermediate stage a space with torsion, the new path integral renders the correct energy spectrum with no unwanted time-slicing corrections.

quant-ph↗
arXiv1995

Convergence behavior of variational perturbation expansion-- A method for locating Bender-Wu singularities

Variational perturbation expansions have recently been used to calculate directly the strong-coupling expansion coefficients of the anharmonic oscillator. The convergence is exponentially fast with superimposed oscillations, as recently observed empirically by the authors. In this note, the observed behavior is explained and used to determine accurately the magnitude and phase of the leading Bender-Wu singularity which is responsible for the finite convergence radius in the complex coupling constant plane.

quant-ph↗
arXiv1995

Dual description of the superconducting phase transition

The dual approach to the Ginzburg-Landau theory of a Bardeen-Cooper-Schrieffer superconductor is reviewed. The dual theory describes a grand canonical ensemble of fluctuating closed magnetic vortices, of arbitrary length and shape, which interact with a massive vector field representing the local magnetic induction. When the critical temperature is approached from below, the magnetic vortices proliferate. This is signaled by the disorder field, which describes the loop gas, developing a non-zero expectation value in the normal conducting phase. It thereby breaks a {\it global} U(1) symmetry. The ensuing Goldstone field is the magnetic scalar potential. The superconducting-to-normal phase transition is studied by applying renormalization group theory to the dual formulation. In the regime of a second-order transition, the critical exponents are given by those of a superfluid with a reversed temperature axis.

cond-mat↗
arXiv1995

Large-Order Behavior of Two-coupling Constant $ϕ^4$-Theory with Cubic Anisotropy

For the anisotropic $[u (\sum_{i=1^N ϕ_i^2)^2+v \sum_{i=1^N ϕ_i^4]$-theory with {$N=2,3$} we calculate the imaginary parts of the renormalization-group functions in the form of a series expansion in $v$, i.e., around the isotropic case. Dimensional regularization is used to evaluate the fluctuation determinants for the isotropic instanton near the space dimension 4. The vertex functions in the presence of instantons are renormalized with the help of a nonperturbative procedure introduced for the simple $g{ϕ^4$-theory by McKane et al.

hep-th↗
arXiv1995

Variational Resummation of Divergent Series with known Large-Order Behavior

Recently-developed variational perturbation expansions converge exponentially fast for positive coupling constants. They do not, however, possess the correct left-hand cut in the complex coupling constant plane, implying a wrong large-order behavior of their Taylor expansion coefficients. We correct this deficiency and present a method of resumming divergent series with their proper large-order behavior. For a given set of expansion coefficients, knowledge of the large-order behavior considerably improves the quality of the approximation.

quant-ph↗