from one or more compound states, probably in the 'P and S configurations. '~'The position of the hydrogen resonance on the energy scale is in very good agreement with theoretical predictions, which range from 9.6 to 9.8 ev.Because of the difficulty of the present experiment the author had to seek advice on many aspects of the experiment. He is indebted to A. O.
It is demonstrated that there exists the possibility of defining scale and conformal transformations in such a way that these constitute exact invariance operations of the Schradinger equation. Unlilte the relativistic case there i s only a single conformal transformation and the usual eleven-parameter extended Galilei group is consequently enlarged to a thirteen-parameter group. The generalization to the case of fields of arbitrary spin is carried out within the framework of minimal-component theories whose interactions respect scale and conformal invariance. One finds that the bare-internal-energy term can be used to break these additional invariance operations in much the same way as the mass term in special relativity. The generators and conservation laws associated with all space-time symmetries of minimalcomponent Galilean-invariant field theories are derived, it being shown that, in analogy to the relativistic case, the operators which appear in these equations can be redefined so as to allow the formulation of scale and conformal invariance entirely in terms of those operator densities relevant to the transformations of the Galilei group.
The scattering of relativistic spin-one-half particles in an Aharonov-Bohrn potential is considered. It is shown that earlier approaches to this problem have neglected a crucial delta-function contribution to the potential. By formulating the problem with a source of finite radius which is then allowed to go to zero, it is established that it is the delta function alone that causes solutions that are singular at the origin to become relevant. The changes in the amplitude which arise from the inclusion of spin are seen to modify the cross section for the case of polarized beams. Finally, the calculated Aharonov-Bohm amplitude is shown to describe the scattering of particles with arbitrary spin in the c~limit. PACS numbers: 03.65.BzThe Aharonov-Bohm effect' has long been recognized for its crucial role in demonstrating the importance of potentials in quantum mechanics. More recently, interest in this topic has been stimulated by the considerable effort currently being expended on the study of (2+ 1)-dimensional models in both superconductivity and particle theory. With the increased application of the results of Ref. 1 to other problems (e.g. , Ref.2) it has become important that certain limitations on the original Aharonov-Bohm calculation be removed. This has been accomplished in a recent series of papers.A concern which has been increasingly addressed of late has to do with the question of how the inclusion of spin modifies the results of Ref. l. In particular, Alford and Wilczek have applied their calculations for Dirac particles to the interaction of cosmic strings with matter. They assert that the scattering amplitude is unaff'ected by the spin, a result which certainly follows if one accepts their requirement that the upper component of the two-component spinor be regular at the origin. On the other hand, Gerbert and Jackiw have suggested a more general boundary condition which introduces a new parameter into the calculation. This has been applied to the question considered in Ref. 5 with quite different results being obtained. This paper approaches the same problem by attempting to infer the behavior of the wave function at the origin in terms of the underlying physics.One begins by writing the Dirac equation for a particle of mass M which, in terms of two-component spinors P', 1S Etir [MP+Py" II]y where the matrices P and Py; are conveniently defined in terms of the Pauli spin matrices as p 03 py; = (o,sa2) and s is twice the spin value (+1 for spin "up" and -1 for spin "down"). The form (1) follows most simply by using the decoupling of the usual four-component Dirac equation in the absence of a third spatial coordinate into two uncoupled two-component equations for s =+ 1 and where the potential A, is related to the magnetic field in the usual way H VxA.(2) Since one is generally interested in the situation in which H is restricted to a flux tube of zero radius, it is conventional to write eH -(a/r )b (r ),with r being the two-dimensional radius vector. In the Coulomb gauge this yields for 8; the result ...
It is shown that there is an exact equivalence between the Aharonov-Bohm effect for spin-j particles and the Aharonov-Casher effect. The demonstration of this precise relationship between the two is seen furthermore to be independent of whether relativistic or nonrelativistic kinematics are used. The only remaining substantive distinction between the two effects may well be the fact that the scattering cross section for polarized beams has a considerably greater structure in the Aharonov-Casher case despite the mathematical equivalence of the scattering amplitudes for the two effects. PACS numbers: 03.65.Bz, 05.30.-d In 1959 it was shown by Aharonov and Bohm 1 (AB) that standard Schrodinger-equation analysis of the scattering of charged particles by a thin impenetrable solenoid implies the remarkable result that such particles are deflected even when classical forces are absent.Despite many attempts to disprove and/or deny the physical reality of this phenomenon, a growing body of experimental data has led to its increased acceptance as an observable effect. More recently, however, it has been observed by Aharonov and Casher 2 (AC) that the symmetries inherent in the Maxwell equations should imply results similar to those of the AB effect when magnetic dipoles are scattered from a filament carrying a uniform charge density. However, the correspondence they obtained was not an exact one. In particular, it required that the product of the magnetic moment times the electric field be much less than the mechanical momentum of the projectile. On the other hand, the electric field is neither constant nor bounded in the AC application. Furthermore, the required exclusion of slow particles cannot be readily accepted since nonrelativistic kinematics form the basis for the analysis of Ref. 2. Thus, while it is fairly clear that some kind of analogy exists between the two processes, it has never been stated precisely what the limitations on the correspondence between them happen to be.
XEFT is a low-energy effective field theory for charm mesons and pions that provides a systematically improvable description of the X(3872) resonance. A Galilean-invariant formulation of XEFT is introduced to exploit the fact that mass is very nearly conserved in the transition D * 0 → D 0 π 0. The transitions D * 0 → D 0 π 0 and X → D 0 ¯ D 0 π 0 are described explicitly in XEFT. The effects of the decay D * 0 → D 0 γ and of short-distance decay modes of the X(3872), such as J/ψ π + π − , can be taken into account by using complex on-shell renormalization schemes for the D * 0 propagator and for the D * 0 ¯ D 0 propagator in which the positions of their complex poles are specified. Galilean-invariant XEFT is used to calculate the D * 0 ¯ D 0 scattering length to next-to-leading order. Galilean invariance ensures the cancellation of ultraviolet divergences without the need for truncating an expansion in powers of the ratio of the pion and charm meson masses.
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