Feshbach Villars Research Articles

Solution of Duffin–Kemmer–Petiau Equation for the Step Potential

The Duffin–Kemmer–Petiau (DKP) equation for spin 0 and 1 in the pressence of the step potential is solved. The problem is reduced to the solution of an equation of Feshbach–Villars type.The reflection and transmission coefficients are correctly deduced. The Klein paradox persists and is discussed using the charge interpretation.

Coherent state for a relativistic spinless particle

The Klein–Gordon equation with scalar potential is considered. In the Feshbach–Villars representation the annihilation operator for a linear potential is defined and its eigenstates are obtained. Although the energy levels in this case are not equally-spaced, depending on the eigenvalues of the annihilation operator, the states are nearly coherent and squeezed. The relativistic Poschl–Teller potential is introduced. It is shown that its energy levels are equally-spaced. The coherence of time evolution of the eigenstates of the annihilation operator for this potential is evaluated.

Solution of the Feshbach–Villars equation for the step potential

We solve the one-dimensional Feshbach–Villars equation for spin-1/2 particle subjected to a scalar smooth potential. The eight component wave function is given in terms of the hypergeometric functions and via a limiting procedure, the wave functions of the step potential are deduced. These wave functions are used to test the validity of the boundary conditions deduced from the Feshbach–Villars transformation. The creation of pairs is predicted from the boundary condition of the charge density.

A CANONICAL DECOMPOSITION IN COLLECTIVE AND RELATIVE VARIABLES OF A KLEIN–GORDON FIELD IN THE REST-FRAME WIGNER-COVARIANT INSTANT FORM

The canonical decomposition of a real Klein–Gordon field in collective and relative variables proposed by Longhi and Materassi is reformulated on spacelike hypersurfaces. This allows us to obtain the complete canonical reduction of the system on Wigner hyperplanes, namely in the rest-frame Wigner-covariant instant form of dynamics. From the study of Dixon's multipoles for the energy–momentum tensor on the Wigner hyperplanes we derive the definition of the canonical center-of-mass variable for a Klein–Gordon field configuration: it turns out that the Longhi–Materassi global variable should be interpreted as a center of phase of the field configuration. A detailed study of the kinematical "external" and "internal" properties of the field configuration on the Wigner hyperplanes is done. The construction is then extended to charged Klein–Gordon fields: the centers of phase of the two real components can be combined to define a global center of phase and a collective relative variable describing the action–reaction between the two Feshbach–Villars components of the field with definite sign of energy and charge. The Dixon multipoles for both the energy–momentum and the electromagnetic current are given. Also the coupling of the Klein–Gordon field to scalar relativistic particles is studied and it is shown that in the reduced phase space, besides the particle and field relative variables, there is also a collective relative variable describing the relative motion of the particle subsystem with respect to the field one.

Boundary conditions for one-dimensional Feshbach–Villars equation

We solve the one-dimensional Feshbach–Villars equation for spinless particle subjected to a scalar smooth potential. The wave function is given in terms of the hypergeometric function and via a limiting procedure, the wave functions of the step potential are deduced. Then, the appropriate boundary conditions for the step potential are deduced using the two-component form.

THE LIENARD–WIECHERT POTENTIAL OF CHARGED SCALAR PARTICLES AND THEIR RELATION TO SCALAR ELECTRODYNAMICS IN THE REST-FRAME INSTANT FORM

After a summary of a recently proposed new type of instant form of dynamics (the Wigner-covariant rest-frame instant form), the reduced Hamilton equations in the covariant rest-frame Coulomb gauge for the isolated system of N scalar particles with pseudoclassical Grassmann-valued electric charges plus the electromagnetic field are studied. The Lienard–Wiechert potentials of the particles are evaluated and it is shown how the causality problems of the Abraham–Lorentz–Dirac equation are solved at the pseudoclassical level. Then, the covariant rest-frame description of scalar electrodynamics is given. Applying to it the Feshbach–Villars formalism, the connection with the particle plus electromagnetic field system is found.

Some exact solutions of reduced scalar Yukawa theory

The scalar Yukawa model, in which a complex scalar field, ϕ, interacts via a real scalar field, χ, is reduced by using covariant Green functions. It is shown that exact few-particle eigenstates of the truncated QFT Hamiltonian can be obtained in the Feshbach–Villars formulation if an unorthodox "empty" vacuum state is used. Analytic solutions for the two-body case are obtained for massless chion exchange in 3+1 dimensions and for massive chion exchange in 1+1 dimensions. Comparison is made to ladder Bethe–Salpeter, Feynman–Schwinger, and quasipotential results for massive chion exchange in 3+1. Equations for the three-body case are also obtained. PACS Nos.: 11.10.Ef, 11.10.Qr, and 03.70.+k

The N- and 1-Time Classical Descriptions of N-Body Relativistic Kinematics and the Electromagnetic Interaction

Given N relativistic scalar free particles described by N mass-shell first class constraints in their 8N-dimensional phase space, their N-time description is obtained by means of a series of canonical transformations to a quasi-Shanmugadhasan basis adapted to the constraints. Then the same system is reformulated on spacelike hypersurfaces: the restriction to the family of hyperplanes orthogonal to the total timelike momentum gives rise to a covariant intrinsic 1-time formulation called the "rest-frame instant form" of dynamics. The relation between the N- and 1-time descriptions, the mass spectrum of the system and the way to introduce mutual interactions among the particles are studied. Then the 1-time description of the isolated system of N charged scalar particles plus the electromagnetic field is obtained. The use of Grassmann variables to describe the charges together with the determination of the field and particle Dirac observables leads to a formulation without infinite self-energies and with mutual Coulomb interactions extracted from classical electromagnetic field theory. A comparison with the Feshbach–Villars Hamiltonian formulation of the Klein–Gordon equation is made. Finally a 1-time covariant formulation of relativistic statistical mechanics is found.

Analysis of the Klein-Gordon Coulomb problem in the Feshbach-Villars representation.

We consider the problem of a massive charged scalar particle bound in a given external Coulomb field of charge Z with varying strength parameter \ensuremath{\alpha}=Z\ensuremath{\alpha}' (\ensuremath{\alpha}'\ensuremath{\approxeq}${137}^{\mathrm{\ensuremath{-}}1}$) in the Feshbach-Villars (FV) representation. It is shown how the expansion in \ensuremath{\alpha} of the momentum space integral equations can be handled to reproduce the exactly known spectrum up to some given order in \ensuremath{\alpha}. In particular, it is shown in detail how the O(${\mathrm{\ensuremath{\alpha}}}^{5}$) contributions to the ground-state energy that arise in the truncated FV equations from the kinetic and potential energies are canceled by virtual pair contributions. The findings are of relevance to quasipotential approaches to the relativistic few-body problem.

The Association of the Sakata-Taketani (Feshbach-Villars) Field with the Kemmer Field, under Symmetry Breaking

We discuss both the original and two new derivations of the covariant Sakata-Taketani system of and of the identical relativistic, but non-covariant Sakata-Taketani (S-T) and Feshbach-Villars (F-V) equations. (These last two equations were originally derived from the Duffin-Kemmer-Petiau and Klein-Gordon (K-G) equations, respectively.) By considering the scaling properties of the Kemmer, K-G and S-T fields, we conclude that in the presence of SU(3) symmetry breaking the S-T field can no longer be associated with both the Kemmer and K-G fields, but only with the Kemmer field. This has important conceptual implications concerning the particle-anti-particle nature of mesons.

The high energy behavior of scattering amplitudes in perturbation theory: P. G. Lederbush and M. T. Grisaru, Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts

We solve the one-dimensional Feshbach–Villars equation for spinless particle subjected to a scalar smooth potential. The wave function is given in terms of the hypergeometric function and via a limiting procedure, the wave functions of the step potential are deduced. Then, the appropriate boundary conditions for the step potential are deduced using the two-component form.