Particles and events in classical off-shell electrodynamics

Despite the many successes of the relativistic quantum theory developed by Horwitz et al., certain difficulties persist in the associated covariant classical mechanics. In this paper, we explore these difficulties through an examination of the classical. Coulomb problem in the framework of off-shell electrodynamics. As the local gauge theory of a covariant quantum mechanics with evolution paratmeter τ, off-shell electrodynamics constitutes a dynamical theory of ppacetime events, interacting through five τ-dependent pre-Maxwell potentials. We present a straightforward solution of the classical equations of motion, for a test event traversing the field induced by a “fixed” event (an event moving uniformly along the time axis at a fixed point in space). This solution is seen to be unsatisfactory, and reveals the essential difficulties in the formalism at the classical levels. We then offer a new model of the particle current—as a certain distribution of the event currents on the worldline—which eliminates these difficulties and permits comparison of classisical off-shell electrodynamics with the standard Maxwell theory. In this model, the “fixed” event induces a Yukawa-type potential, permitting a semiclassical identification of the pre-Maxwell time scale λ with the inverse mass of the intervening photon. Numerical solutions to the equations of motion are compared with the standard Maxwell solutions, and are seen to coincide when λ≳10⁻⁶ seconds, providing an initial estimate of this parameter. It is also demonstrated that the proposed model provides a natural interpretation for the photon mass cut-off required for the renormalizability of the off-shell quantum electrodynamics.     

Off-shell states in dual resonance theory

We discuss ooff-shell states guided by an analogue model approach. This leads us to a more complete understanding of a model proposed recently by Schwarz with critical dimension 16. We are led, by algebraic considerations, to off-shell states in the Neveu- Schwarz-Ramond model, which obey the gauge conditions in the same critical dimension as the on-shell theory, the amplitudes factorizing on the usual positive definite states in 10 dimensions. Brief calculations reveal that some of the divergences present in the orbital model disappear in the fermion theory.     

Dimensional reduction by legendre transformation generates off-shell supersymmetric Yang-Mills theories

A new technique of dimensional reduction by Legendre transformation to derive off-shell formulations of extended supersymmetric theories is presented. The method starts from an on-shell theory in a higher dimension and retains a dependence of the fields upon one of the higher dimensions, thus allowing for a central charge in the four-dimensional supersymmetry algebra. The “hamiltonian density” of the theory plays the role of the lagrangian in one less dimension, while the equations of motion serve as constraints for the lower dimensional theory. These constraints guarantee supersymmetric and central charge invariances of the action. The technique is applied to derive off-shell formulations of the supersymmetric Yang-Mills theories.     

Lagrangian with Off-Shell Vertices and Field Redefinitions

. Meson-exchange diagrams following from a Lagrangian with off-shell meson-nucleon couplings are compared with those generated from conventional dynamics. The off-shell interactions can be transformed away with the help of a nucleon field redefinition. Contributions to the nucleon-nucleon and three-nucleon potentials and nonminimal contact electromagnetic meson-exchange currents are discussed, mostly for the important case of scalar meson exchange. Received August 1, 1997; accepted in final form February 3, 1998     

Nucleon-nucleon on-shell and off-shell contributions to the triton binding energy in an energy-dependent potential model formulation

An energy-dependent nucleon-nucleon (NN) potential model is presented which allows an unambiguous separation of NN on-shell and off-shell contributions to trinucleon observables. A potential of the form $\text{V(E; p, p′) = λ (E)}\text{V}\text{(p, p′)}$ is constructed at negative energies by comparison with various hermitian potentials. The latter comparison suggests a symmetric choice of λ(E) with respect to E = 0. It is shown that the force model constructed according to the present study is more reasonable and less ambiguous than the hermitian potential model, for which the negative-energy dependence of the corresponding t-matrix is generated in an arbitrary way by the accidental choice of the off-shell behavior of the potential. The pertinent shortcomings of the conventional potential model are elucidated by their inconsistent predictions for two- and three-nucleon systems. In the case of the proposed energy-dependent potential model the on-shell information is decoupled from the off-shell behavior at positive and negative energies. This allows one to eliminate the two-body off-shell contributions within the three-body equations in a continuous process by taking the range of forces to zero while keeping at each step the two-body on-shell information unaltered.     

Off-shell πN amplitudes via a multichannel separable potential

The off-shell pion-nucleon transition matrix is a basic ingredient in theories of pion-nuclear interactions which, in the absence of fundamental theory of πN dynamics, must be obtained by a phenomenological extrapolation from the available on-shell data. As one means of performing such an extrapolation, we explore a multichannel separable potential model with the property that the off-shell elastic scattering amplitude is generated directly from the measured elastic-channel phase shifts. The off-shell πN partial-wave transition amplitudes determined by this procedure are compared with those calculated by Landau and Tabakin using a one-channel absorptive separable potential. We find that the absorptive separable potential approach provides a physically unreasonable off-shell extrapolation at energies where the on-shell amplitude is highly inelastic, and show that the difficulty is a direct consequence of the one-channel nature of that method. The multichannel extrapolation is free of these difficulties.     

Off-shell sensitivity,repulsive correlations and the pion-nucleus optical potential

Repulsive nucleon-nucleon correlations tend to reduce the dependence of pion-nucleus elastic scattering upon the off-shell pion-nucleon dynamics. However, optical potential calculations can in practice be quite sensitive to the particular choice of off-shell model parameters. It is argued that this sensitivity results from the nature of the optical potential as a one-body operator which introduces extra off-shell dependence not found in the physical many-body process itself. Thus, one must be very careful in any attempt to extract correlation or off-shell information, or to predict pion-nucleus phase shifts, by means of an optical potential theory. Results of model calculations are presented for purposes of illustration.     

Off-shell extension of the π-nucleon transition matrix and separable π-nucleon potentials

The off-shell extension of the P-wave π-nucleon transition matrix given by the Chew-Low model is compared with the one obtained from a separable potential. It is shown that the two off-shell extensions are rather different; π-deuteron scattering results are compared for the two cases.     

Off-shell effects in quasielastic electron nucleus scattering

Using the Ward-Takahashi identity, on-shell condition, bound Dirac equation and off-shell expansion, a reduced version of half off-shell virtual photon nucleon vertex has been suggested. The vertex are decomposed into several different order terms: the on-shell terms, first and second off-shell terms. The off-shell behaviour of the form factors is discussed in the one meson loop model. Using the reduced vertex and parametrized off-shell form factors the quasielastic response functions are calculated for several nuclei at ¦q¦–k_f and for⁵⁶Fe at large ¦q¦ up to 1.14 GeV/c and at ?q²=0.9 (GeV/c)². The Coulomb sums are evaluated and a comparison of the theoretical prediction with data is given. The off-shell electron nucleon cross section is calculated and compared with the “cc1” off-shell extrapolation.     

Phase-function method for velocity-dependent potentials

We have adapted the phase-function method for studying on- and off-shell properties of velocity-dependent potentials. The main result presented in this paper is an ansatz for the interpolatingT-matrix function (on- or off- the energy-shell as the case may). Based on this ansatz we have presented an efficient method for computing the off-shell extension function which plays a role in the theories of three particle system. We have demonstrated this by means of a model calculation.     

One-dimensional integral equations for a system of three identical particles in the boundary condition model and the possibility of changing the off-shell behaviour of the two-particle T-matrix

It is shown that in the framework of the boundary condition model (BCM) for the two-particle interaction the Schrödinger equation for the system of three identical bosons can be reduced to the one-dimensional integral equation in an exact way. The method used for obtaining such an equation is based on a special consideration of the two-particle off-shell wave functions. The binding energy of the simple three-particle system is calculated. It is indicated that by means of the equation obtained it is possible to change the off-shell behaviour of the two-particle (-matrix and therefore to simulate three-particle effects.     

Off-shell corrections to the glauber theory of nuclear multiple scattering

We study the first-order off-shell corrections to Glauber's multiple scattering model. These consist of a direct and a rescattering term, and are obtained from the eikonal corrections studied in a previous paper by a simple substitution. For an independent particle nuclear model the direct term becomes identical to an expression previously derived by Moniz. We show that the rescattering term is in general not small compared to the direct term. For high incident energy a comparison is made with other corrections to the Glauber model. An optical potential is proposed which consists of the Glauber potential and its off-shell, eikonal, Fresnel and spatial correlation corrections.     

Off-shell extension function from a generalised phase-function method

It is shown that the Combined-variable-phase-off-shell scattering theory or the generalised phase-function method provides efficient algorithms to calculate the off-shell extension function which plays a role in the theories of three-particle scattering. In order to assert this we have chosen to work with a model potential for theα-α interaction. The functions which occur in the context of our approach exhibit certain interesting features with regard to effect of the potential in producing on-and off-shell quantities like the phase shift and quasi-phase. Interpolating off-shell extension function is seen to exhibit a discontinuity where the phase function goes through a zero.     

Exact form factors in integrable quantum field theories: the sine-Gordon model (II)

A general model independent approach using the ‘off-shell Bethe Ansatz’ is presented to obtain an integral representation of generalized form factors. The general techniques are applied to the quantum sine-Gordon model alias the massive Thirring model. Exact expressions of all matrix elements are obtained for several local operators. In particular soliton form factors of charge-less operators as for example all higher currents are investigated. It turns out that the various local operators correspond to specific scalar functions called p-functions. The identification of the local operators is performed. In particular the exact results are checked with Feynman graph expansion and full agreement is found. Furthermore all eigenvalues of the infinitely many conserved charges are calculated and the results agree with what is expected from the classical case. Within the frame work of integrable quantum field theories a general model independent ‘crossing’ formula is derived. Furthermore the ‘bound state intertwiners’ are introduced and the bound state form factors are investigated. The general results are again applied to the sine-Gordon model. The integrations are performed and in particular for the lowest breathers a simple formula for generalized form factors is obtained.     

Three-Body Dynamics in One Dimension: A Test Model for the Three-Nucleon System with Irreducible Pionic Diagrams

 We formulate the three-body problem in one dimension in terms of the (Faddeev-type) integral equation approach. As an application, we develop a spinless, one-dimensional (1-D) model that mimics three-nucleon dynamics in one dimension. Using simple two-body potentials that reproduce the deuteron binding, we obtain that the three-body system binds at about 7.5 MeV. We then consider two types of residual pionic corrections in the dynamical equation; one related to the 2π-exchange three-body diagram, the other to the 1π-exchange three-body diagram. We find that the first contribution can produce an additional binding effect of about 0.9 MeV. The second term produces smaller binding effects, which are, however, dependent on the uncertainty in the off-shell extrapolation of the two-body t-matrix. This presents interesting analogies with what occurs in three dimensions. The paper also discusses the general three-particle quantum scattering problem, for motion restricted to the full line. Received March 5, 2002; accepted July 19, 2002     

Pion-induced knock-out reactions

A strong absorption model for pion-induced knock-out reactions is proposed. The distortion of the incoming and outgoing pions has been included by (i) computing the pion wave number in a nuclear medium (dispersive effect) and (ii) excluding the central region of the nucleus where the real pion absorption is dominant (absorption effect). In order to study the dependence of the (π⁺, π⁺p) reaction on the off-shell pion-nucleon t-matrix, different off-shell extrapolations are used. The magnitude of the cross sections seems to be sensitive to the type of off-shell extrapolation; their shapes, however, are similar. The theoretical results are compared with experimental data. The agreement between the theoretical results for separable off-shell extrapolation and the data is good.     

Improved theoretical pion-nucleus optical potentials

An approach which makes the first order pion-nucleus optical potential theoretically sound is presented. This study should permit higher order improvements to the potential to be more meaningful and the nuclear structure information extracted from pi-nucleus scattering to be more reliable. Based on multiple scattering theory, three optical potentials are constructed and studied in momentum space. These models are the popular Kisslinger potential, the local “Laplacian” potential, and an “improved off-shell potential;” the latter one is derived from absorptive separable pion-nucleon potentials which exactly reproduce on-shell πN scattering. By working in momentum space and explicitly including πN resonances and off-shell effects in the definition of the optical potential, the approach described here is capable of handling any number of pi-nucleon partial waves, is applicable over a very wide energy region, is based on a physical model for off-shell behavior, and is extended easily to include higher order effects. The optical potentials are inserted into two different relativistic wave equations to determine the total cross section and elastic differential cross section for pi-nucleus scattering. It is found that the various models for off-shell πN scattering determine significantly different πC¹² scattering, with the improved off-shell model preferred on theoretical grounds. Also discussed is the importance of properly transforming πN scattering to the pi-nucleus c.m. system, the origin of the shift in the peak position of the π^?C total cross section, and the reason for the increased diffractive nature of the differential cross section at 180 MeV.     

Locality and currents for the dual string

We wish to describe the coupling of a local current to the dual string as the amplitude for the string to collapse to a point. At the quantum level we note that states which are localized in the transverse directions are forced by the gauge constraints to have longitudinal extension. The best we can do in the conventional model is, therefore, to construct off-shell states which have zero helicity (but no unique spin). We calculate the propagator for such states and compare it with the model of Schwarz. In their respective critical dimensions (26 for the string and 16 for the off-shell model of Schwarz) the two results become identical. We also calculate the Green functions for the coupling of one of our off-shell states to any number of on-shell particles by means of the functional formalism. Our results have a close connection with the previous models of Schwarz and Wu and of Corrigan and Fairlie.     

An integral transform of Green’s function, off-shell Jost solution and T-matrix for Coulomb-Yamaguchi potential in coordinate representation

By exploiting the theory of ordinary differential equations together with certain properties of higher transcendental functions, a useful analytical expression for the integral transform of the Green’s function for motion in Coulomb-Yamaguchi potential is derived via the r-space approach. This integral transform is applied to construct an analytical expression for off-shell Jost solution in its ‘maximal reduced form’ involving confluent and Gaussian hypergeometric functions. Corresponding Jost functions automatically follow from this solution. Finally, as another application of the off-shell Jost solution, the off-shell T-matrix is calculated by using a modified relation between off-shell physical wave function and T-matrix which does not involve the potential explicitly, thereby avoiding certain difficult integrals, and expressed it in terms of rational functions and simple hypergeometric functions which is in exact agreement with the results given previously by other authors.      

The supercurrent in N = 1 supersymmetric Yang-Mills theories: (I). The classical case

We construct, in the tree approximation, the complete off-shell supercurrent, including Faddeev-Popov fields and gauge contributions for a general supersymmetric Yang-Mills theory maintaining parity. Due to the conformally non-covariant character of the Faddeev-Popov procedure this supercurrent cannot be improved. The other currents of the superconformal group can therefore no longer be obtained by simply forming moments.