Infrared properties of the gauge theory coupling constant III

In previous papers we have outlined a program for deriving the infrared behavior of the axial gauge gluon propagator in a pure Yang-Mills theory. The program is based on an integral equation for the gluon propagator derived from the Dyson equation and the Ward identities. Here we present a solution to this equation, obtained numerically. The solution exhibits a Singularity in the infrared, and therefore presumably predicts confinement of color. The method is supposed to be exact in the infrared. Away from the infrared, therefore, our solution is only approximate. Nevertheless, even in the ultraviolet, our solution for the propagator is not very different from the known asymptotic freedom result, so it may be that it is a reasonable approximation over the entire range of momentum.     

Infrared gluon and ghost propagators from lattice QCD

We report on the infrared limit of the quenched lattice Landau gauge gluon and ghost propagators as well as the strong-coupling constant computed from large asymmetric lattices. The infrared lattice propagators are compared with the pure power law solutions from Dyson-Schwinger equations (DSE). For the gluon propagator, the lattice data is compatible with the DSE solution. The preferred measured gluon exponent being ∼0.52, favouring a vanishing propagator at zero momentum. The lattice ghost propagator shows finite-volume effects and, for the volumes considered, the propagator does not follow a pure power law. Furthermore, the strong-coupling constant is computed and its infrared behaviour investigated.     

A non-perturbative calculation of the infrared limit of the axial gauge gluon propagator (II)

We show, numerically, that the integral equation for the axial gauge gluon propagator developed in the preceding paper has an explicit solution with the features outlined there. This solution is expected to be exact in the infrared limit. We find, however, that even in the ultraviolet limit it does not differ greatly from the known asymptotic freedom behavior of the propagator. It may, therefore, be a reasonable approximation over the whole range.     

A microscopic derivation of the critical magnetic field in a superconductor

The propagator for a noninteracting many electron system in a constant magnetic field in three space time dimensions is computed. This formula and the results of [FT1,2] are used to give a microscopic derivation of a BCS-equation with magnetic field. It is shown that this equation has no solution if the magnetic field is sufficiently large. Perturbation theory in the interaction around the magnetic field propagator is discussed.     

Theory of Spin Echo in Restricted Geometries under a Step-wise Gradient Pulse Sequence

A closed matrix form solution of the Bloch-Torrey equation is presented for the magnetization density of spins diffusing in a bounded region under a steady gradient field and for the Stejskal-Tanner gradient pulse sequence, assuming straightforward generalization to any step-wise gradient profile. The solution is expressed in terms of the eigenmodes of the diffusion propagator in a given geometry with appropriate boundary conditions (perfectly reflecting or relaxing walls). Applications to rectangular, cylindrical, and spherical geometries are discussed. The relationship with the multiple propagator approach is established and an alternative step-wise gradient discretization procedure is suggested to handle arbitrary gradient waveforms.     

Study of the gluon propagator in the axial gauge

It is shown that a non-linear integral equation for the gluon propagator in the axial gauge (Baker et al.) can be simplified considerably. A comparison is made with an approximate equation for the gluon propagator in the Landau gauge (Mandelstam). Both equations have polynomial kernels where the argument is the divisor of the internal and external momenta. A solution which behaves as a double pole for low momenta remains consistent.     

Evolution of weakly correlated systems

A propagator with an effective action is constructed for a quantum many-particle system in the approximation taking into account only pair correlations between particles. The theory can easily be generalized to the case of two species of particles. In the case of a rarefied plasma and a gas, this propagator for a one-particle density matrix provides a solution to the kinetic problem over time intervals longer than the relaxation time.     

The Epstein–Glaser causal approach to the light-front QED4. II: Vacuum polarization tensor

In this work we show how to construct the one-loop vacuum polarization for light-front QED₄ in the framework of the perturbative causal theory. Usually, in the canonical approach, it is considered for the fermionic propagator the so-called instantaneous term, but it is known in the literature that this term is controversial because it can be omitted by computational reasons; for instance, by compensation or vanishing by dimensional regularization. In this work we propose a solution to this paradox. First, in the Epstein–Glaser causal theory, it is shown that the fermionic propagator does not have instantaneous term, and with this propagator we calculate the one-loop vacuum polarization, from this calculation it follows the same result as those obtained by the standard approach, but without reclaiming any extra assumptions. Moreover, since the perturbative causal theory is defined in the distributional framework, we can also show the reason behind our obtaining the same result whether we consider or not the instantaneous fermionic propagator term.     

自旋为任意整数的传播子

以自旋为任意整数的自由粒子的波函数(Bargmann-Wigner方程的解)为基础，进一步研究了 自旋为任意整数的投影算符和传播子.证明了Behrends和Fronsdal所构造的投影算符是正确 的.导出了自旋为任意整数的场的一般对易规则和费恩曼传播子的一般表达式. 关键词： 整数自旋 投影算符 对易规则 费恩曼传播子     

Renormalization of the Mass Gap

We have explicitly shown that QCD is the color gauge invariant theory at non-zero mass gap as well. It has been defined as the value of the regularized full gluon self-energy at some finite point. The mass gap is mainly generated by the nonlinear interaction of massless gluon modes. All this allows one to establish the structure of the full gluon propagator in the explicit presence of the mass gap. In this case, the two independent general types of formal solutions for the full gluon propagator as a function of the regularized mass gap have been found. The nonlinear iteration solution at which the gluons remain massless is explicitly present. The existence of the solution with an effective gluon mass is also demonstrated.     

Solving the BSE with an Integral Representation in Minkowski Space

Special method of solution of BSE in Minkowski space is presented. Perturbation theory integral representation is introduced for BSE and the applicability is discussed beyond the simplest kernel and constituent propagator approximation.     

On the Summation of Non-leading Logarithms in QCD

Exploiting the relations between the expansion coefficients of Green's functions and those of β and γ functions in massless QFT, the summation up to the n-th non-leading logarithms is reduced to the solution of a system of linear differential equations, which in general differ from renormalization group equations. Applying this procedure to the gluon propagator the leading log approximation is modified near the mass shell by a constant factor only. Furthermore for the gluon propagator exponentiation in terms of the running coupling constant is argued to be restricted to leading logarithms.     

Infrared gluon and ghost propagator exponents from lattice QCD

The compatibility of the pure power law infrared solution of QCD and lattice data for the gluon and ghost propagators in Landau gauge is discussed. For the gluon propagator, the lattice data are well described by a pure power law with an infrared exponent κ∼0.53, in the Dyson–Schwinger notation. κ is measured using a technique that suppresses finite volume effects. This value is consistent with a vanishing zero momentum gluon propagator, in agreement with the Gribov–Zwanziger confinement scenario. For the ghost propagator, the lattice data seem not to follow a pure power law, at least for the range of momenta accessed in our simulation.     

Projection Operator and Feynman Propagator for a Free MassiVe Particle of Arbitrary Spin

Based on the solution to the Bargmann-Wigner equations, a direct derivation of the projection operator and Feynman propagator for a free massive particle of arbitrary spin is worked out. The projection operator constructed by Behrends and Fronsdal is re-deduced and confirmed, and simplified in the case of half-integral spin, the general commutation rules and Feynman propagator with additional non-covariant terms for a free massive particle with any spin are derived, and explicit expressions for the propagators for spins 3/2, 2, 5/2, 3, 7/2, and 4 are provided.     

Feynman propagator for an arbitrary half—integral spin

Based on the solution to Bargmann－Wigner equation for a particle with arbitrary half-integral spin, a direct derivation of the projection operator and propagator for a particle with arbitrary half-integral spin is worked out. The projection operator constructed by Behrends and Fronsdal is re-deduced and confirmed and simplified, the general commutation rules and Feynman propagator with additional non-covariant terms for a free particle with arbitrary half-integral spin are derived, and explicit expressions for the propagators for spins 3/2, 5/2 and 7/2 are provided.     

Evaluation of the angular spectrum approach for simulations of near-field pressures

The implementation of the angular spectrum approach based on the two-dimensional fast Fourier transform is evaluated for near-field pressure simulations of square ultrasound transducers, where the three-dimensional pressure field is calculated from the normal velocity distribution on the transducer surface. The pressure field is propagated in the spatial frequency domain with the spatial propagator or the spectral propagator. The spatial propagator yields accurate results in the central portion of the computational grid while significant errors are produced near the edge due to the finite extent of the window applied to the spatial propagator. Likewise, the spectral propagator is inherently undersampled in the spatial frequency domain, and this causes high frequency errors in the computed pressure field. This aliasing problem is alleviated with angular restriction. The results show that, in nonattenuating media, the spatial propagator achieves smaller errors than the spectral propagator after the region of interest is truncated to exclude the windowing error. For pressure calculations in attenuating media or with apodized pistons as sources, the spatial and spectral propagator achieve similar accuracies. In all simulations, the angular spectrum calculations with the spatial propagator take more time than calculations with the spectral propagator.     

Kugo–Ojima color confinement criterion and Gribov–Zwanziger horizon condition

We rewrite the Zwanziger horizon condition in terms of the Kugo–Ojima parameter for color confinement. This enables one to explain which value of the Kugo–Ojima parameter is allowed if the horizon condition is imposed. Although all the calculations are performed in the limit of vanishing Gribov parameter for simplicity, the obtained value is consistent with the result of numerical simulations. Consequently, the ghost propagator behaves like free and the gluon propagator is non-vanishing at low momenta, in harmony with recent lattice results and decoupling solution of the Schwinger–Dyson equation. The Kugo–Ojima criterion is realized only when the restriction is removed.     

Feynman propagator for a particle with arbitrary spin

Based on the solution to the Rarita-Schwinger equations, a direct derivation of the projection operator and propagator for a particle with arbitrary spin is worked out. The projection operator constructed by Behrends and Fronsdal is re-deduced and confirmed, and simplified in the case of half-integral spin; the general commutation rules and Feynman propagator for a free particle of any spin are derived, and explicit expressions for the propagators for spins 3/2, 2, 5/2, 3, 7/2, 4 are provided.Received: 13 March 2003, Revised: 24 April 2005, Published online: 6 July 2005     

On the theory of X-ray refractive optics: Exact solution for a parabolic medium

The exact solution is obtained for a propagator describing x-ray propagation through a refractive parabolic medium. Such a medium arises in compound many-element refractive x-ray lenses that are used in synchrotron radiation sources. The solution obtained allows one to analyze such lenses in detail to predict their operation in particular applications (beam focusing, microobject imaging, and Fourier transform).