arXiv subjects

R. Rosenfelder

Publications and source records attributed to R. Rosenfelder.

At least 19 recordsLinked to original sources

arXiv2021

On the numerical evaluation of real-time path integrals: Double exponential integration and the Maslov correction

Ooura's double exponential integration formula for Fourier transforms is applied to the oscillatory integrals occuring in the path-integral description of real-time Quantum Mechanics. Due to an inherent, implicit regularization multi-dimensional Gauss-Fresnel integrals are obtained numerically with high precision but modest number of function calls. In addition, the Maslov correction for the harmonic oscillator is evaluated numerically with an increasing number of time slices in the path integral thereby clearly demonstrating that the real-time propagator acquires an additional phase $ - π/2 $ each time the particle passes through a focal point. However, in the vicinity of these singularities an overall small damping factor is required. Prospects of evaluating scattering amplitudes of finite-range potentials by direct numerical evaluation of a real-time path integral are discussed.

physics.comp-ph↗

arXiv2020

On the Numerical Evaluation of a Class of Oscillatory Integrals in Worldline Variational Calculations

Filon-Simpson quadrature rules are derived for integrals of the type \int_a^b dx f(x) sin(xy)/(xy) and \int_a^b dx f(x) 4 sin^2(xy/2)/(xy)^2 which are needed in applications of the worldline variational approach to Quantum Field Theory. These new integration rules reduce to the standard Simpson rule for y = 0 and are exact for y \to \infty when a = 0 and f(0) \ne 0.The subleading term in the asymptotic expansion is also reproduced more and more precisely when the number of integration points is increased. Tests show that the numerical results are indeed stable over a wide range of y-values whereas usual Gauss-Legendre quadrature rules are more precise at low y but fail completely for large values of y. The associated Filon-Simpson weights are given in terms of sine and cosine integrals and have to be evaluated for each value of y. A Fortran program to calculate them in a fast and accurate manner is available. A detailed comparison with the double exponential method of Ooura and Mori is made.

hep-ph↗

arXiv2017

Path Integrals in Quantum Physics

These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, many-body physics and field theory and contain standard examples (quadratic Lagrangians, tunneling, description of bosons and fermions, quantization of gauge theories etc.) as well as specialized topics (scattering, dissipative systems, spin \& color in the path integral, lattice methods etc.). In each part simple Fortran programs which can be run on a PC, illustrate the numerical evaluation of (Euclidean) path integrals by Monte-Carlo or variational methods. Also included are the set of problems which accompanied the lectures and their solutions.

nucl-th↗

arXiv2013

Scattering Theory with Path Integrals

Starting from well-known expressions for the $T$-matrix and its derivative in standard nonrelativistic potential scattering I rederive recent path-integral formulations due to Efimov and Barbashov et al. Some new relations follow immediately.

nucl-th↗

arXiv2011

A New Path-Integral Representation of the $T$-Matrix in Potential Scattering

We employ the method used by Barbashov and collaborators in Quantum Field Theory to derive a path-integral representation of the $T$-matrix in nonrelativistic potential scattering which is free of functional integration over fictitious variables as was necessary before. The resulting expression serves as a starting point for a variational approximation applied to high-energy scattering from a Gaussian potential. Good agreement with exact partial-wave calculations is found even at large scattering angles. A novel path-integral representation of the scattering length is obtained in the low-energy limit.

nucl-th↗

arXiv2011

Exact Path-Integral Representations for the $T$-Matrix in Nonrelativistic Potential Scattering

Several path integral representations for the $T$-matrix in nonrelativistic potential scattering are given which produce the complete Born series when expanded to all orders and the eikonal approximation if the quantum fluctuations are suppressed. They are obtained with the help of "phantom" degrees of freedom which take away explicit phases that diverge for asymptotic times. Energy conservation is enforced by imposing a Faddeev-Popov-like constraint in the velocity path integral. An attempt is made to evaluate stochastically the real-time path integral for potential scattering and generalizations to relativistic scattering are discussed.

nucl-th↗

arXiv2010

Variational Approximations in a Path-Integral Description of Potential Scattering

Using a recent path integral representation for the T-matrix in nonrelativistic potential scattering we investigate new variational approximations in this framework. By means of the Feynman-Jensen variational principle and the most general ansatz quadratic in the velocity variables -- over which one has to integrate functionally -- we obtain variational equations which contain classical elements (trajectories) as well as quantum-mechanical ones (wave spreading).We analyse these equations and solve them numerically by iteration, a procedure best suited at high energy. The first correction to the variational result arising from a cumulant expansion is also evaluated. Comparison is made with exact partial-wave results for scattering from a Gaussian potential and better agreement is found at large scattering angles where the standard eikonal-type approximations fail.

nucl-th↗

arXiv2009

Path Integrals for Potential Scattering

Two path integral representations for the $T$-matrix in nonrelativistic potential scattering are derived and proved to produce the complete Born series when expanded to all orders. They are obtained with the help of "phantom" degrees of freedom which take away explicit phases that diverge for asymptotic times. In addition, energy conservation is enforced by imposing a Faddeev-Popov-like constraint in the velocity path integral. These expressions may be useful for attempts to evaluate the path integral in real time and for alternative multiple scattering expansions. Standard and novel eikonal-type high-energy approximations and systematic expansions immediately follow.

nucl-th↗

arXiv2009

Perturbation Theory Without Diagrams: The Polaron Case

Higher-order perturbative calculations in Quantum (Field) Theory suffer from the factorial increase of the number of individual diagrams. Here I describe an approach which evaluates the total contribution numerically for finite temperature from the cumulant expansion of the corresponding observable followed by an extrapolation to zero temperature. This method (originally proposed by Bogolyubov and Plechko) is applied to the calculation of higher-order terms for the ground-state energy of the polaron. Using state-of-the-art multidimensional integration routines two new coefficients are obtained corresponding to a four- and five-loop calculation. Several analytical and numerical procedures have been implemented which were crucial for obtaining reliable results.

hep-th↗

arXiv2007

Perturbative Results Without Diagrams

Higher-order perturbative calculations in Quantum (Field) Theory suffer from the factorial increase of the number of individual diagrams. Here I describe an approach which evaluates the total contribution numerically for finite temperature from the cumulant expansion of the corresponding observable followed by an extrapolation to zero temperature. This method (originally proposed by Bogolyubov and Plechko) is applied to the calculation of higher-order terms for the ground-state energy of the polaron. Using state-of-the-art multidimensional integration routines 2 new coefficients are obtained corresponding to a 4- and 5-loop calculation.

hep-th↗

arXiv2006

Variational Worldline Approximation for the Relativistic Two-Body Bound State in a Scalar Model

We use the worldline representation of field theory together with a variational approximation to determine the lowest bound state in the scalar Wick-Cutkosky model where two equal-mass constituents interact via the exchange of mesons. Self-energy and vertex corrections are included approximately in a consistent way as well as crossed diagrams. Only vacuum-polarization effects of the heavy particles are neglected. In a path integral description of an appropriate current-current correlator an effective, retarded action is obtained by integrating out the meson field. As in the polaron problem we employ a quadratic trial action with variational functions to describe retardation and binding effects through multiple meson exchange.The variational equations for these functions are derived, discussed qualitatively and solved numerically. We compare our results with the ones from traditional approaches based on the Bethe-Salpeter equation and find an enhanced binding contrary to some claims in the literature. For weak coupling this is worked out analytically and compared with results from effective field theories. However, the well-known instability of the model, which usually is ignored, now appears at smaller coupling constants than in the one-body case and even when self-energy and vertex corrections are turned off. This induced instability is investigated analytically and the width of the bound state above the critical coupling is estimated.

hep-ph↗

arXiv2006

Vacuum Polarization Effects in the Worldline Variational Approach to Quantum Field Theory

The worldline variational approach is extended beyond the quenched approximation, i.e. to include virtual pair production of heavy particles. This is achieved either by an expansion of the functional determinant to second order or by an hybrid ansatz for the quadratic trial action consisting of fields for the light particles and worldlines for the heavy ones as in the linear polaron model. Numerical results and analytic approximations show a reduction of radiative effects with increasing number of flavors.

hep-ph↗

arXiv2005

Worldline Variational Approximation: A New Approach to the Relativistic Binding Problem

We determine the lowest bound-state pole of the density-density correlator in the scalar Wick-Cutkosky model where two equal-mass constituents interact via the exchange of mesons. This is done by employing the worldline representation of field theory together with a variational approximation as in Feynman's treatment of the polaron. Unlike traditional methods based on the Bethe-Salpeter equation, self-energy and vertex corrections are (approximately) included as are crossed diagrams. Only vacuum-polarization effects of the heavy particles are neglected. The well-known instability of the model due to self-energy effects leads to large qualitative and quantitative changes compared to traditional approaches which neglect them. We determine numerically the critical coupling constant above which no real solutions of the variational equations exist anymore and show that it is smaller than in the one-body case due to an induced instability. The width of the bound state above the critical coupling is estimated analytically.

hep-ph↗

arXiv2004

Eikonal Amplitude in the Gravireggeon Model at Superplanckian Energies

The gravity effects in high-energy scattering, described by a four-dimensional eikonal amplitude related to gravireggeons induced by compact extra dimensions are studied. It is demonstrated that the real part of the eikonal (with a massless mode subtracted) dominates its imaginary part at both small and large impact parameters, in contrast to the usual case of hadronic high-energy behavior. The real part of the scattering amplitude exhibits an exponential falloff at large momentum transfer, similar to that of the imaginary part of the amplitude.

hep-ph↗

arXiv2003

Structure function of a damped harmonic oscillator

Following the Caldeira-Leggett approach to describe dissipative quantum systems the structure function for a harmonic oscillator with Ohmic dissipation is evaluated by an analytic continuation from euclidean to real time. The analytic properties of the Fourier transform of the structure function with respect to the energy transfer (the ``characteristic function'') are studied and utilized. In the one-parameter model of Ohmic dissipation we show explicitly that the broadening of excited states increases with the state number without violating sum rules. Analytic and numerical results suggest that this is a phenomenologically relevant, consistent model to include the coupling of a single (sub-)nuclear particle to unobserved and complex degrees of freedom.

nucl-th↗

arXiv2003

The W_t Transcendental Function and Quantum Mechanical Applications

We discuss the function wt(x) defined via the implicit equation wt(x)*tan[wt(x)]=x which appears in certain quantum mechanical and field theoretic applications. We investigate its analytic structure, develop series expansions for both small and large x, and provide various techniques for its numerical evaluation in the complex plane.

math-ph↗

arXiv2002

Improved variational description of the Wick-Cutkosky model with the most general quadratic trial action

We generalize the worldline variational approach to field theory by introducing a trial action which allows for anisotropic terms to be induced by external 4-momenta of Green's functions. By solving the ensuing variational equations numerically we demonstrate that within the (quenched) scalar Wick-Cutkosky model considerable improvement can be achieved over results obtained previously with isotropic actions. In particular, the critical coupling associated with the instability of the model is lowered, in accordance with expectations from Baym's proof of the instability in the unquenched theory. The physical picture associated with a different quantum mechanical motion of the dressed particle along and perpendicular to its classical momentum is discussed. Indeed, we find that for large couplings the dressed particle is strongly distorted in the direction of its four-momentum. In addition, we obtain an exact relation between the renormalized coupling of the theory and the propagator. Along the way we introduce new and efficient methods to evaluate the averages needed in the variational approach and apply them to the calculation of the 2-point function.

hep-th↗

arXiv2001

On the Best Quadratic Approximation in Feynman's Path Integral Treatment of the Polaron

The best quadratic approximation to the retarded polaron action due to Adamowski {\it et al.} and Saitoh is investigated numerically for a wide range of coupling constants. The non-linear variational equations are solved iteratively with an efficient method in order to obtain the ground state energy and the effective mass of the polaron. The virial theorem and expansions for small and large couplings are used to check the high accuracy of the numerical results. Only small improvements over Feynman's (non-optimal) results are observed. For a moving polaron it is shown that the most general quadratic trial action may contain anisotropic terms which, however, do not lead to improvements for the ground state energy and effective mass.

cond-mat↗