Republication of: 5. Comment on the axially symmetric solutions to Einstein’s equations of gravitation

This is the English translation of the second of a series of 3 papers by Hermann Weyl (the third one jointly with Rudolf Bach), first published in 1917–1922, in which the authors derived and discussed the now-famous Weyl two-body static axially symmetric vacuum solution of Einstein’s equations. The English translations of the other two papers are published alongside this one. The papers have been selected by the Editors of General Relativity and Gravitation for re-publication in the Golden Oldies series of the journal. This republication is accompanied by an editorial note written by Gernot Neugebauer, David Petroff and Bahram Mashhoon, and by a brief biography of R. Bach, written by H. Goenner.     

Republication of: New solutions to Einstein’s equations of gravitation. B. Explicit determination of static,axially symmetric fields. By Rudolf Bach. With a supplement on the static two-body problem. By H. Weyl.

This is the English translation of the third of a series of 3 papers by Hermann Weyl (the third one jointly with Rudolf Bach), first published in 1917–1922, in which the authors derived and discussed the now-famous Weyl two-body static axially symmetric vacuum solution of Einstein’s equations. The English translations of the other two papers are published alongside this one. The papers have been selected by the Editors of General Relativity and Gravitation for re-publication in the Golden Oldies series of the journal. This republication is accompanied by an editorial note written by Gernot Neugebauer, David Petroff and Bahram Mashhoon, and by a brief biography of R. Bach, written by H. Goenner.     

Republication of: 3. On the theory of gravitation

This is the English translation of the first of a series of 3 papers by Hermann Weyl (the third one jointly with Rudolf Bach), first published in 1917–1922, in which the authors derived and discussed the now-famous Weyl two-body static axially symmetric vacuum solution of Einstein’s equations. The English translations of the other two papers are published alongside this one. The papers have been selected by the Editors of General Relativity and Gravitation for re-publication in the Golden Oldies series of the journal. This republication is accompanied by an editorial note written by Gernot Neugebauer, David Petroff and Bahram Mashhoon, and by a brief biography of R. Bach, written by H. Goenner.     

A unitary model for three-body collisions

A connected 3 → 3 formalism for three-body collision processes is reduced to a hierarchy of three on-energy-shell integral equations and one off-energy-shell integral equation. Only the on-energy-shell equations, which involve only on-energy-shell three-body and two-body amplitudes, need be solved exactly in order to obtain elastic and break-up amplitudes satisfying the unitarity constraints exactly. Applied to n-d break-up, the on-energy-shell equations ensure that the n-d initial-state interaction, the nucleon-nucleon final-state interactions, and more complicated 3 → 3 processes are correctly described. After angular momentum analysis the on-energy-shell equations are one-dimensional integral equations, even in the case of local two-body potentials. This unitary model provides a practical scheme for calculating approximate three-body elastic and break-up amplitudes when two-body local potentials are used to describe the two-body subsystems.     

Structure and Occurrence of Three-Body Halos in Two Dimensions

 The quantum-mechanical three-body problem is reformulated in two dimensions by use of hyperspherical coordinates and an adiabatic expansion of the Faddeev equations. The effective radial potentials are calculated and their large-distance asymptotic behavior is derived analytically for short-range two-body interactions. Energies and wave functions are computed numerically for various potentials. An infinite series of Efimov states does not exist in two dimensions. Borromean systems, i.e. bound three-body systems without bound binary subsystems, can only appear when a short-range repulsive barrier at finite distance is present in the two-body interaction. The corresponding Borromean state is never spatially extended. For a system of three weakly interacting identical bosons we find two bound states with both binding energies proportional to the two-body binding energy. In the limit of small binding these states are spatially located at the very large distances characterized by the scattering length. Their properties are universal and independent of the details of the potential. We compare throughout with the corresponding properties in three dimensions. Received September 25, 1998; accepted for publication January 30, 1999     

Integral equations for N-particle scattering

Integral equations are derived for the N-particle transition operators. The equations couple together only transition operators between two-body channels. The kernel of the equations becomes connected after a single iteration. Transition operators involving channels with three or more particles can be obtained by quadratures from the solution of the equations. It is also shown that the N-particle equations can be reduced to multichannel two-body equations by the use of the quasiparticle method.     

正则形式的QED约束关联动力学（Ⅱ）等时关联格林函数的运动方程与二体关联近似

本文把多时关联格林函数的运动方程转变成等时关联格林函数的运动方程，包括电子和光子的密度矩阵的运动方程以及等时顶点的运动方程，在二体关联截断近似下，给出运动方程，高斯定律及Ｗａｒｄ恒等式的明显表达式．     

Constrained Correlation Dynamics of QED in Canonical Form(Ⅱ)Equal Time Limit and Two-Body Correlation Dynamics

The equations of motion for multi-time correlation Green's functions are transformed into those for equal-time correlation Green's functions,which include the equations of motion for electron's and photon's density matrices as well as vertex functions.In two-body correlation truncation approximation,we present the explicit expressions for the equations of motion,Gauss law and Ward identities explicitly.     

Generalised Scalar-tensor Theory in the Bianchi Type I Model

We use a conformal transformation to find solutions to the generalised scalar-tensor theory, with a coupling constant dependent on a scalar field, in an empty Bianchi type I model. We describe the dynamical behaviour of the metric functions for three different couplings: two exact solutions to the field equations and a qualitative one are found. They exhibit non-singular behaviours and kinetic inflation. Two of them admit both General Relativity and string theory in the low-energy limit as asymptotic cases.     

正则形式的SU（N）规范理论的约束关联动力学（Ⅲ）等时关联Green函数的运动方程与二体关联近似

把多时关联Ｇｒｅｅｎ函数的运动方程转变成等时关联Ｇｒｅｅｎ函数的运动方程，其中包括夸克和胶子的密度矩阵的运动方程以及顶角函数的运动方程．在二体关联截断近似下，给出运动方程、高斯定律和Ｗａｒｄ恒等式的明显表达式．     

Eine nichtlineare lorentzinvariante Gravitationstheorie

Nonlinear Lorentz-Invariant Theory of Gravitation A nonlinear Lorentz-invariant theory of gravitation and a Lorentz-invariant Hamiltonian for a particle with spin in the gravitational field are developed. The equations of motions are studied. The theory is applied to the three well known tests of General Relativity. In the special case of the red shift of spectral lines and of the deflection of light, the theory gives the same results as the General Theory of Relativity, whereas in the case of the perihelion of the Mercury, the theory gives 40,3″, in good agreement with experimental results of DICKE .     

Behaviour of anisotropic ten-dimensional cosmologies

A class of ten-dimensional anisotropic cosmologies, with three independent spatial scale factors, is explored. Extensions of the Bianchi types I and IX space-times are analysed, in both the case of the field equations of General Relativity and the case of generalized equations of relativity altered by low-energy superstring correction terms.     

Differential equations in momentum space for the three-body problem in the case of pointlike pair interactions

Corrects schemes for solving equations of three-body dynamics for systems governed by a zerorange two-body interaction are considered. Correlations between spectral features of a three-boson system are obtained. The results are compared with the results obtained by calculating the spectra and scattering lengths in the system of three helium atoms with realistic two-body interaction potentials.     

The general class of the vacuum spherically symmetric equations of the general relativity theory

The system of the spherical-symmetric vacuum equations of the General Relativity Theory is considered. The general solution to a problem representing two classes of line elements with arbitrary functions g ₀₀ and g ₂₂ is obtained. The properties of the found solutions are analyzed.     

An extended time-dependent mean-field theory for strongly correlated nuclear systems

We perform an extension of the time-dependent mean-field theory by an explicit inclusion of strong two-body correlations of short range on a level of microscopic reversibility relating them to realistic nucleon-nucleon forces. Invoking a least action principle for correlated basis functions, equations of motion for the correlation functions and the single-particle model wave function are derived to the lowest order of the FAHT cluster expansion. Higher order effects as well as longrange correlations we consider only to the extent to which they contribute to the mean field via a readjusted phenomenological effective two-body interaction. The corresponding correlated stationary problem is investigated and appropriate initial conditions to describe a heavy ion reaction are proposed. The single-particle density matrix is evaluated. Norm, energy and particle number conservation are proved. Possible simplifications are discussed. Standard TDHF appears as a limiting case if the range of the explicitly considered part of the bare nucleon-nucleon interaction goes to zero.     

A Solution to Einstein's Equations for the Mixmaster Universe in Complex General Relativity

Starting from Einstein's equations of the Classical General Relativity, new kinds of solutions for the Mixmaster model are explored. By dispensing with the extension to the complex variable field, which is usual in problems such as the Laplace equation or the harmonic oscillator, in a similar manner to that of Quantum Mechanics, the equations appear to have solutions that belong to the complex General Relativity. A first integral is performed by establishing a separation of the first derivatives. Then a second integral is obtained once the respective equations with separate variables are found and whose integrals provide a family of complex solutions. However, reality conditions do not seem to be easily imposed at this stage. Above all, it is significant that the classical Einstein's equations for the debatably integrable Mixmaster model present complex solutions.     

Physical theories in Galilean space-time and the origin of Schrödinger-like equations

A method to develop physical theories of free particles in space-time with the Galilean metric is presented. The method is based on a Principle of Analyticity and a Principle of Relativity, and uses the Galilei group of the metric. The first principle requires that state functions describing the particles are analytic and the second principle demands that dynamical equations for these functions are Galilean invariant. It is shown that the method can be used to formally derive Schrödinger-like equations and to determine modifications of the Galilei group of the metric that are necessary to fullfil the requirements of analyticity and Galilean invariance. The obtained results shed a new light on the origin of Schrödinger’s equation of non-relativistic quantum mechanics.     

Is the Regge Calculus a Consistent Approximation to General Relativity?

We will ask the question of whether or not the Regge calculus (and two related simplicial formulations) is a consistent approximation to General Relativity. Our criteria will be based on the behaviour of residual errors in the discrete equations when evaluated on solutions of the Einstein equations. We will show that for generic simplicial lattices the residual errors cannot be used to distinguish metrics which are solutions of Einstein's equations from those that are not. We will conclude that either the Regge calculus is an inconsistent approximation to General Relativity or that it is incorrect to use residual errors in the discrete equations as a criteria to judge the discrete equations.     

Three-Body Systems with Square-Well Potentials in L = 0 States

The angular part of the Faddeev equations is solved analytically for s-states in case of two-body square-well potentials. The results are, still analytically, generalized to arbitrary short-range potentials for both small and large distances. We consider systems with three identical bosons, three non-identical particles, and two identical spin- fermions plus a third particle with arbitrary spin. The angular wave functions are in general linear combinations of trigonometric and exponential functions. The Efimov conditions are obtained at large distances. General properties and applications to short-range potentials are discussed. Gaussian potentials are used for illustrations. The results are useful for numerical calculations, where, for example, large distances can be treated analytically and matched to the numerical solutions at smaller distances. The saving in computational efforts could be substantial. Received May 20, 1996; revised January 24, 1997; accepted for publication January 27, 1997