Search results for "Path Integral"

QUANTUM YANG-MILLS THEORY ON ARBITRARY SURFACES

1992

We study quantum Maxwell and Yang-Mills theory on orientable two-dimensional surfaces with an arbitrary number of handles and boundaries. Using path integral methods we derive general and explicit expressions for the partition function and expectation values of contractible and noncontractible Wilson loops on closed surfaces of any genus, as well as for the kernels on manifolds with handles and boundaries. In the Abelian case we also compute correlation functions of intersecting and self-intersecting loops on closed surfaces, and discuss the role of large gauge transformations and topologically nontrivial bundles.

Fundamental Principles of Quantum Mechanics

2001

There are two alternative methods of quantizing a system: a) quantization via the Feynman Path Integral (equivalent to Schwinger’s Action Principle); b) canonical quantization.

Comparison of two non-primitive methods for path integral simulations: Higher-order corrections vs. an effective propagator approach

2002

Two methods are compared that are used in path integral simulations. Both methods aim to achieve faster convergence to the quantum limit than the so-called primitive algorithm (PA). One method, originally proposed by Takahashi and Imada, is based on a higher-order approximation (HOA) of the quantum mechanical density operator. The other method is based upon an effective propagator (EPr). This propagator is constructed such that it produces correctly one and two-particle imaginary time correlation functions in the limit of small densities even for finite Trotter numbers P. We discuss the conceptual differences between both methods and compare the convergence rate of both approaches. While th…

Glueball masses from ratios of path integrals

2011

By generalizing our previous work on the parity symmetry, the partition function of a Yang-Mills theory is decomposed into a sum of path integrals each giving the contribution from multiplets of states with fixed quantum numbers associated to parity, charge conjugation, translations, rotations and central conjugations. Ratios of path integrals and correlation functions can then be computed with a multi-level Monte Carlo integration scheme whose numerical cost, at a fixed statistical precision and at asymptotically large times, increases power-like with the time extent of the lattice. The strategy is implemented for the SU(3) Yang-Mills theory, and a full-fledged computation of the mass and …

Lattice QCD: A Brief Introduction

2014

A general introduction to lattice QCD is given. The reader is assumed to have some basic familiarity with the path integral representation of quantum field theory. Emphasis is placed on showing that the lattice regularization provides a robust conceptual and computational framework within quantum field theory. The goal is to provide a useful overview, with many references pointing to the following chapters and to freely available lecture series for more in-depth treatments of specifics topics.

Examples for Calculating Path Integrals

2001

We now want to compute the kernel K(b, a) for a few simple Lagrangians. We have already found for the one-dimensional case that $$\displaystyle{ K{\bigl (x_{2},t_{2};x_{1},t_{1}\bigr )} =\int _{ x(t_{1})=x_{1}}^{x(t_{2})=x_{2} }[dx(t)]\,\text{e}^{(\mathrm{i}/\hslash )S} }$$ (19.1) with $$\displaystyle{ S =\int _{ t_{1}}^{t_{2} }dt\,L(x,\dot{x};t)\;. }$$ First we consider a free particle, $$\displaystyle{ L = m\dot{x}^{2}/2\;, }$$ (19.2) and represent an arbitrary path in the form, $$\displaystyle{ x(t) =\bar{ x}(t) + y(t)\;. }$$ (19.3) Here, \(\bar{x}(t)\) is the actual classical path, i.e., solution to the Euler–Lagrange equation: $$\displaystyle{ \frac{\partial L} {\partial x}\Big\vert _{…

A diffusion Monte Carlo study of small para-Hydrogen clusters

2007

Abstract An improved Monte Carlo diffusion model is used to calculate the ground state energies and chemical potentials of parahydrogen clusters of three to forty molecules, using two different p-H2-p-H2 interactions. The improvement is due to three-body correlations in the importance sampling, to the time step adjustment and to a better estimation of statistical errors. In contrast to path-integral Monte Carlo results, this method predicts no magic clusters other than that with thirteen molecules.

Mean field methods in large amplitude nuclear collective motion

1984

The time dependent Hartree-Fock method (TDHF) is reviewed and its success and failure are discussed. It is demonstrated that TDHF is a semiclassical theory which is basically able to describe the time evolution of one-body operators, the energy loss in inclusive deep inelastic collisions, and fusion reactions above the Coulomb barrier. For genuine quantum mechanical processes as e.g. spontaneous fission, subbarrier fusion, phase shifts and the description of bound vibrations, the quantized adiabatic time dependent Hartree-Fock theory (quantized ATDHF) is suggested and reviewed. Realistic three-dimensional calculations for heavy ion systems of A1+A2<32 are presented. Applications to various …

Erratum to: Classical and Quantum Dynamics: From Classical Paths to Path Integrals

2017

Propagators for Particles in an External Magnetic Field

2001

In order to describe the propagation of a scalar particle in an external potential, we begin again with the path integral $$ K(r',t';r,0) = \int_{r,(0)}^{r',(t')} {[dr(t)]} \exp \left\{ {\frac{{\text{i}}} {\hbar }S[r(t)]} \right\} $$ (1) with $$ S[r(t)] = \int_0^{t'} {dt} L(r,\dot r). $$