Showing papers on "Configuration space published in 1975"
TL;DR: Wilson's lattice gauge model is presented as a canonical Hamiltonian theory and the structure of the model is reduced to the interactions of an infinite collection of coupled rigid rotators as mentioned in this paper.
Abstract: Wilson's lattice gauge model is presented as a canonical Hamiltonian theory. The structure of the model is reduced to the interactions of an infinite collection of coupled rigid rotators. The gauge-invariant configuration space consists of a collection of strings with quarks at their ends. The strings are lines of non-Abelian electric flux. In the strong-coupling limit the dynamics is best described in terms of these strings. Quark confinement is a result of the inability to break a string without producing a pair.
1,388 citations
TL;DR: In this article, the density of every infinite semitrajectory in configuration space is proved for a billiard trajectory in a polygon Q⊂R2 having all angles commensurate with π.
Abstract: Consider a billiard in a polygon Q⊂R2 having all angles commensurate with π. For the majority of initial directions, density of every infinite semitrajectory in configuration space is proved. Also proved is the typicality of polygons for which some billiard trajectory is dense in phase space.
300 citations
TL;DR: In this paper, it was shown that the conserved magnetic charge discovered by ’t Hooft in non−Abelian gauge theories with spontaneous symmetry breaking is not associated with the invariance of the action under a symmetry group.
Abstract: It is shown that the conserved magnetic charge discovered by ’t Hooft in non−Abelian gauge theories with spontaneous symmetry breaking is not associated with the invariance of the action under a symmetry group. Rather, it is a topological characteristic of an isotriplet of Higgs fields in a three−dimensional space: the Brouwer degree of the mapping between a large sphere in configuration space and the unit sphere in field space provided by the normalized Higgs field ? a = φ a (φ b φ b )−1/2. The use of topological methods in determining magnetic charge configurations is outlined. A peculiar interplay between Dirac strings and zeros of the Higgs field under gauge transformations is pointed out. The monopole−antimonopole system is studied.
285 citations
TL;DR: In this article, it was shown that symmetry, positivity, and normalization are sufficient and sufficient conditions for ρ to be a correlation measure, and the main operative condition is (P) which says that ξφ(x)dρ≧0 must hold for every function φ for which Sφ (ξ)≧ 0 identically for ξ e X, where φ → S φ is a certain linear operator whose properties we study.
Abstract: This paper is devoted to the solution of the problem of characterizing correlation measures arising naturally in classical statistical mechanics of point particles. A correlation measure ρ must be related to a (not necessarily unique) probability measure μ over an infinite particle configuration space X by the formula μ(H)=∝NH(ξ)dμ where {NH} is a certain family of integer valued random variables. We prove that there are three conditions, namely (S) symmetry, (P) positivity, and (N) normalization, which together are sufficient as well as necessary for ρ to be a correlation measure. The main operative condition is (P), which says that ξφ(x)dρ≧0 must hold for every function φ for which Sφ(ξ)≧0 identically for ξ e X, where φ → Sφ is a certain linear operator whose properties we study. Condition (P) gives rise to a large class of inequalities satisfied by the ρ-measures of certain sets. The theory is also generalized to the case when there is a group of translations acting in the one-particle space, the concern then being with measures ρ as well as μ that are invariant with respect to the group.
136 citations
TL;DR: In this paper, the Gibbs potential is assumed to be invariant under general assumption on the potential, and it is shown that each Gibbs random field with this potential is also G-invariant.
Abstract: Two-dimensional lattice model is considered. The connected Lie groupG acts on a configuration space. The Gibbs potential assumed to be invariant under this action. We prove, that under general assumption on the potential, each Gibbs random field with this potential is alsoG-invariant.
127 citations
TL;DR: In this paper, the Weyl transform is used to rigorously derive path integral forms for position and momentum transition amplitudes from the time-dependent Schrodinger equation for arbitrary Hermitian Hamiltonians.
Abstract: The method of Weyl transforms is used to rigorously derive path integral forms for position and momentum transition amplitudes from the time‐dependent Schrodinger equation for arbitrary Hermitian Hamiltonians. It is found that all paths in phase space contribute equally in magnitude, but that each path has a different phase, equal to 1/h/ times an ’’effective action’’ taken along it. The latter is the time integral of p⋅q−h (p,q), h (p,q) being the Weyl transform of the Hamiltonian operator H, which differs from the classical Hamiltonian function by terms of order h/2, vanishing in the classical limit. These terms, which can be explicitly computed, are zero for relatively simple Hamiltonians, such as (1/2M)[P−eA (Q)]2+V (Q), but appear when the coupling of the position and momentum operators is stronger, such as for a relativistic spinless particle in an electromagnetic field, or when configuration space is curved. They are always zero if one opts for Weyl’s rule for forming the quantum operator corresponding to a given classical Hamiltonian. The transition amplitude between two position states is found to be expressible as a path integral in configuration space alone only in very special cases, such as when the Hamiltonian is quadratic in the momenta.
116 citations
IBM1
TL;DR: In this article, the authors propose to replace the ordinary kinetic energy function for a classical system of point masses by a more general quadratic form (12 Σij=1N qiMijqj), where Mij is an arbitrary positive-definite symmetric "mass tensor", and obtain a system having different dynamics but the same equilibrium properties as the original system.
106 citations
TL;DR: In this paper, the authors investigated the anomalous properties of partial wave amplitudes of the conformal groupOD,2, (D not necessarily integer) in configuration space, and the presence of Euclidean singularities in the Wilson expansion in conformai invariant field theories.
Abstract: Analyticity properties of partial-wave amplitudes of the conformal groupOD,2, (D not necessarily integer) in configuration space are investigated. The presence of Euclidean singularities in the Wilson expansion in conformai invariant field theories is discussed, especially in connection with the program of formulating dynamical bootstrap conditions coming from the requirement of causality. The exceptional case ofD = 2 is discussed in detail.
95 citations
TL;DR: In this article, the existence and description of regions in physical space and configuration space where motion cannot occur in a three-body configuration space is discussed. But the description of these regions turns out to be most similar to the Hill regions of the circular restricted problem of three bodies.
Abstract: For the general three-body problem we show the existence and describe regions in physical space and configuration space where motion cannot occur. The description of these regions turns out to be most similar to the Hill regions of the circular restricted problem of three bodies. These regions depend upon the constants of energy and angular momentum.
89 citations
TL;DR: In this paper, the curvatures of a simple mechanical system relative to a total energy value h were studied in the setting of Riemannian geometry, where the Jacobi metric is defined relative to the energy value.
67 citations
TL;DR: In this article, a connection with the stress energy tensor of the Schrodinger field in configuration space is made with the moments of a differential force law, and the associated hypervirial theorems are written in the form of moments of momenta.
Abstract: Hermitian operators linear in momenta generate coordinate transformations. The associated hypervirial theorems are written in the form of moments of a differential force law, and a connection is made with the stress‐energy tensor of the Schrodinger field in configuration space.
TL;DR: In this article, a number of many-body calculations are carried out for transitions in BI, CII, OI, OIV, and SiX, and good agreement between all three expressions is obtained for the first time.
TL;DR: In this paper, the authors consider stationary, cylindrically-symmetric gravitational fields in the framework of harmonic mappings of Riemannian manifolds and obtain the geodesics and construct the resulting space-time geometries.
Abstract: We consider stationary, cylindrically‐symmetric gravitational fields in the framework of harmonic mappings of Riemannian manifolds. In this approach the emphasis is on a correspondence between the solution of the Einstein field equations and the geodesics in an appropriate Riemannian configuration space. Using Hamilton–Jacobi techniques, we obtain the geodesics and construct the resulting space–time geometries. We find that the light cone structure of the configuration space delineates the distinct exterior fields of Lewis and van Stockum which together form the most general solution with whole cylinder symmetry.
TL;DR: In this paper, the authors discuss methods that enable one to trace the origin of symmetry and conservation laws in mechanics to geometrical symmetries of space-time, which reveal their intimate relation to conservation laws.
TL;DR: In this paper, the authors present a consistent calculation of the low-energy spectrum and the giant multipole resonances in the model of separable residual interaction (MSI) and a simple parametrization of the effective interaction has been used which exhibits directly the influence of the restricted configuration space on the force.
Abstract: During the last three years the giant multipole resonances (GMR) with multipolarities other than E1 have been discovered in 2sspb (1.3). Some theoretical efforts havo been made to describe all those resonances in the framework of the shell model (4.7). In this paper we present a consistent calculation of the low-energy spectrum and the GMR in the model of separable residual interaction (MSI) (s.,). In this model a very simple parametrization of the effective interaction has been used which exhibits directly the influence of the restricted configuration space on the force. The idea of the model is very simple (s) : Starting from a systematic analysis of the two-body transition density in momentum space and taking into account the effect of momentum conservation in a two-body collision, we construct an interaction which contains only those Fourier components of the effective interaction which--for a given Hilbert space--are relevant for the evaluation of the matr ix elements. As a result i t turns out that the force depends primari ly on the mass number _4 of the nucleus and on the angular momentum J of the state considered. This force then can adequately be parametrized in momentum space by
TL;DR: In this paper, it is shown that soliton formation and the resulting plasma heating are nothing more than the description in configuration space of well-known parametric processes and quasi−linear theory.
Abstract: It is shown that soliton formation and the resulting plasma heating are nothing more than the description in configuration space of well‐known parametric processes and quasi−linear theory
TL;DR: In this paper, a generalized Lagrangian-Hamiltonian formalism for systems of many particles with holonomic constraints is presented, which is useful for treating problems in which the elimination of redundant co-ordinates is not convenient or the use of Lagrange multipliers is not suitable for the problem at hand.
Abstract: Using configuration-dependent projectors we construct a generalized Lagrangian-Hamiltonian formalism for systems of many particles with holonomic constraints. The formalism in useful for treating problems in which the elimination of redundant co-ordinates is not convenient or the use of Lagrange multipliers is not suitable for the problem at hand. The method developed is a useful mathematical tool for treating collective processes in the many-body problem and can be the starting point for a quantization process for the many-body system restricted by holonomic constraints in configuration space.
TL;DR: In this paper, a statistical theory, previously developed for electrostatic waves, was extended to the case of electromagnetic waves in the presence of a constant magnetic field and the effects of mode-mode coupling.
Abstract: A statistical theory, previously developed for electrostatic waves is extended to the case of electromagnetic waves in the presence of a constant magnetic field. The theory is nonlinear and contains the effects of mode-mode coupling. The extension is not straightforward as assumed previously by many authors and involves calculation of perturbed trajectories in both velocity and configuration space. A diffusion equation is derived for the average particle distribution function, the associated diffusion tensor is calculated and a nonlinear wave dispersion relation is found. All these results contain the usual quasilinear theory as the lowest order approximation.
TL;DR: In this article, the invariance properties of second-order variational problems with local Lie groups of transformations are investigated, and a new set of fundamental invariance identities are derived for single and multiple integral problems, and new proofs of Zermelo conditions and Noether's theorem are presented.
Abstract: This paper investigates the invariance properties of second‐order variational problems when the configuration space is subjected to an r‐parameter local Lie group of transformations. In particular, the recent results of Hanno Rund on first‐order problems are extended to the higher order case: A new set of fundamental invariance identities are derived for single and multiple integral problems, and new proofs of the Zermelo conditions and Noether’s theorem are presented. The results are applied to a variational problem whose second‐order Lagrangian depends upon a scalar field in Minkowski space, and some conformal identities are obtained.
TL;DR: The generalized phase-space descriptions of a quantum system are constructed as special linear representations of the space of the linear operators, acting on the state vector space of a system.
Abstract: The generalized phase-space descriptions of a quantum system are constructed as special linear representations of the space of the linear operators, acting on the state vector space of the system. The relationship between quantum mechanics and classical mechanics is studied in terms of the phase-space descriptions.
TL;DR: In this article, the authors proved that the quantum mechanical observable Q(X), which corresponds to the function C(X) identical to Xi(q)pi on phase space in classical mechanics, is equal to the anticommutator 1/2(Q(Xi),Q(pi)).
Abstract: Starting with Segal's postulates (1960) for quantum mechanics, augmented by the postulate that the commutators of the free Hamiltonian with position observables are canonical, the authors prove that the quantum mechanical observable Q(X), which corresponds to the function C(X) identical to Xi(q)pi on phase space in classical mechanics, is equal to the anticommutator 1/2(Q(Xi),Q(pi)). In coordinate free language, this means the Q( phi X)=1/2(Q( phi ),Q(X)) for any scalar field phi and any vector field X on the configuration space.
TL;DR: In this article, the transition matrix elements between the 16 O-16 O system and 32 S compound states are estimated in a simple oscillator model and the configuration space for the sulfur states includes the 3ω shell.
01 Jan 1975
TL;DR: The Loge Theory as mentioned in this paper is a procedure to extract from an electronic wave function describing a certain system a certain kind of information about the localizability of the electrons of that system.
Abstract: The loge theory is a procedure to extract from an electronic wave function describing a certain system a certain kind of information about the localizability of the electrons of that system.
01 Jan 1975
TL;DR: In this paper, it was shown that the conserved magnetic charge discovered by ’t Hooft in non−Abelian gauge theories with spontaneous symmetry breaking is not associated with the invariance of the action under a symmetry group.
Abstract: It is shown that the conserved magnetic charge discovered by ’t Hooft in non−Abelian gauge theories with spontaneous symmetry breaking is not associated with the invariance of the action under a symmetry group. Rather, it is a topological characteristic of an isotriplet of Higgs fields in a three−dimensional space: the Brouwer degree of the mapping between a large sphere in configuration space and the unit sphere in field space provided by the normalized Higgs field ? a = φ a (φ b φ b )−1/2. The use of topological methods in determining magnetic charge configurations is outlined. A peculiar interplay between Dirac strings and zeros of the Higgs field under gauge transformations is pointed out. The monopole−antimonopole system is studied.
TL;DR: In this article, the Lie algebra of Schouten concomitants of symmetric contravariant tensor fields on the configuration space of the system is used to formulate Hamilton's principle of stationary action.
Abstract: Hamilton's principle of stationary action is formulated in classical mechanics using the Lie algebra of Schouten concomitants of symmetric contravariant tensor fields on the configuration space of the system Such a formulation is global and coordinate free It is shown that a directly parallel formulation holds in quantum mechanics so long as all the Poisson brackets involved can be replaced in the quantum version by commutators in a canonical way An example (where the Hamiltonian possesses a velocity dependent potential) in which this cannot be done is discussed and concluded that in this case the action is stationary only for a subclass of variations, namely those corresponding to Killing vector fields on the configuration manifold
TL;DR: In this paper, conditions of applicability of the eikonal approximation for the nonrelativistic one-particle propagator are investigated, and it is shown that the direction of Eikonalization cannot be chosen arbitrarily.
TL;DR: In this article, a new class of special orbits is introduced which are recurrent in the configuration space only, and an algorithm is suggested for their numerical determination in the case of a non-integrable dynamical system.
Abstract: A new class of special orbits is introduced which are recurrent in the configuration space only. An algorithm is suggested for their numerical determination in the case of a non-integrable dynamical system.