Classical Relativistic Field Research Articles

COLLECTIVE AND RELATIVE VARIABLES FOR MASSLESS FIELDS

In this paper, a set of canonical collective and relative variables for a classical relativistic massless field are defined and discussed. The discussion is based on a harmonic analysis on the light-cone in momentum space. It is shown how to avoid a set of consistency constraints which are necessary in the massive case. As a result a canonical separation of the angular momentum in collective and relative parts and a canonical realization of an extended BMS algebra are obtained.

Solutions of 1+1 relativistic field models

A class of classical relativistic nonlinear field models has been recently introduced by two of us [A. Degasperis and F. Tinebra, J. Math. Phys. 34, 2950 (1993)]; they are solvable by an explicit mapping to free field equations. Here we show the behavior of particular solutions of these models for both scalar and spinor fields. In the real spectrum we find brachion and tachion localized solutions, and we show the large-time behavior of wave packets, while in the complex spectrum we prove that the asymptotic states may contain solitons, antisolitons, and point-like charged particles (“pointons”). We report on several processes such as soliton–antisoliton pair creation and annihilation, and pointon decay in two solitons.

CLASSICAL YANG-MILLS THEORY WITH FERMIONS II: DIRAC’S OBSERVABLES

For pure Yang-Mills theory on Minkowski space-time, formulated in functional spaces where the covariant divergence is an elliptic operator without zero modes, and for a trivial principal bundle over the fixed time Euclidean space with a compact, semisimple, connected and simply connected structure Lie group, a Green function for the covariant divergence has been found. It allows one to solve the first class constraints associated with Gauss’ laws and to identify a connection-dependent coordinatization of the trivial principal bundle. In a neighborhood of the global identity section, by using canonical coordinates of the first kind on the fibers, one has a symplectic implementation of the Lie algebra of the small gauge transformations generated by Gauss’ laws and one can make a generalized Hodge decomposition of the gauge potential one-forms based on the BRST operator. This decomposition singles out a pure gauge background connection (the BRST ghost as Maurer-Cartan one-form on the group of gauge transformations) and a transverse gauge-covariant magnetic gauge potential. After an analogous decomposition of the electric field strength into the transverse and the longitudinal part, Dirac’s observables associated with the transverse electric and magnetic components are identified as their restriction to the global identity section of the trivial principal bundle. The longitudinal part of the electric field can be re-expressed in terms of these electric and magnetic transverse parts and of the constraints without Gribov ambiguity. The physical Lagrangian, Hamiltonian, non-Abelian and topological charges have been obtained in terms of transverse Dirac’s observables, also in the presence of fermion fields, after a symplectic decoupling of the gauge degrees of freedom; one has an explicit realization of the abstract “Riemannian metric” on the orbit space. Both the Lagrangian and the Hamiltonian are nonlocal and nonpolynomial; like in the Coulomb gauge they are not Lorentz-invariant, but the invariance can be enforced on them if one introduces Wigner covariance of the observables by analyzing the various kinds of Poincare orbits of the system and by reformulating the theory on suitable spacelike hypersurfaces, following Dirac. By extending to classical relativistic field theory the problems associated with the Lorentz noncovariance of the canonical (presymplectic) center of mass for extended relativistic systems, in the sector of the field theory with P2>0 and W2≠0 one identifies a classical invariant intrinsic unit of length, determined by the Poincare Casimirs, whose quantum counterpart is the ultraviolet cutoff looked for by Dirac and Yukawa: it is the Compton wavelength of the field configuration (in an irreducible Poincare representation) multiplied by the value of its spin.

One solution of the problem of intrinsic energy of a point charge, as obtained within classical relativistic field theory

New equations are proposed for classical relativistic field theory, which insure a finite value for the intrinsic energy of a point charge.

Structure of quantum-mechanical relativistic two-body interactions for spinning particles

Relativistic constraint mechanics yields consistent systems of coupled Dirac equations for pairs of spinning particles. We explicitly connect these equations to the Bethe-Salpeter equation of quantum field theory and to the interactions of classical Fokker-Tetrode dynamics (and hence to classical relativistic field theory) to obtain versions of these equations governed by systems of (possibly noncoulombic) relativistic potentials whose detailed structures contain important relativistic effects like correct Darwin interactions. We recast the defining pair of Dirac equations in a number of equivalent but important forms—“external potential,” Sazdjian, hyperbolic, and Breit— and examine their interconnection. Since the potentials in these equations are no more singular than — 1/4r2 we are able to solve appropriate versions of them nonperturbatively for the q¯q system to obtain a very good fit to the entire meson spectrum and for the e + e − system to calculate the positronium spectrum of QED correct through order α 4 .

Restrictions imposed on relativistic two-body interactions by classical relativistic field theory.

We show that various relativistic potential models (all sharing exact relativistic two-body kinematics and a common nonrelativistic limit) can be distinguished by agreement or disagreement with relativistic corrections produced by classical field theory. We find that the only one of these models whose relativisic corrections duplicate those of classical field theory is the minimal Todorov equation. Conversely, we derive the Todorov equation from the semirelativistic dynamics of classical field theory, thus exposing the classical field-theoretic origins of its characteristic minimal potential structures and dependences on effective one-body variables.

A note on Lorentz transformation properties of conserved charges

It is shown that in classical relativistic field theories there may exist global conserved charges which are not Lorentz-covariant quantities.

Hilbert space sectors for solutions of non-linear relativistic field equations

In the set of Cauchy data corresponding to the solutions of non-linear classical relativistic field equations having locally finite kinetic energy a structure of Hilbert space sectors is introduced. Each sector is invariant under time evolution and a total energy and linear momentum functionals can be defined as global quantities. Within this framework the existence of conserved dynamical charges is proved and the mechanism by which a symmetry can be spontaneously broken is explained.

Energy - Momentum Tensors for Dispersive Electromagnetic Waves

Classical relativistic field theory is used as a basis for a general discussion of the problem of splitting up the total energy–momentum tensor of a system into contributions from its component subsystems. Both the Minkowski and Abraham forms (including electrostriction) arise naturally in alternative split-up procedures applied to a non dispersive dielectric fluid. The case of an electromagnetic wave in a (spatially and temporally) dispersive medium in arbitrary but slowly varying motion is then treated. In the dispersive case the results cannot be found by replacing the dielectric constant ε with ε(κ, ω) but include derivatives with respect to the wave vector κ and the frequency ω. Ponderomotive force expressions are obtained and the perturbation in the total energy–momentum tensor due to a one-dimensional wavepacket is found. A nonlinear Schrödinger equation is obtained for the evolution of a three-dimensional wavepacket. Both hot and cold plasmas are treated.

Second quantization of classical nonlinear relativistic field theory

The construction of a relativistic interacting local quantum field is given in two steps: first the classical nonlinear relativistic field theory is written down in terms of Poisson brackets, with initial conditions as canonical variables: next a representation of Poisson bracket Lie algebra by means of linear operators in the topological vector space is given and an explicit form of a local interacting relativistic quantum field $$\hat \Phi $$ is obtained. The construction of asymptotic local relativistic fields $$\hat \Phi _{in} $$ and $$\hat \Phi _{out} $$ associated with $$\hat \Phi $$ is also given.

Integrable Hamiltonian systems and interactions through quadratic constraints

O n -invariant classical relativistic field theories in one time and one space dimension with interactions that are entirely due to quadratic constraints are shown to be closely related to integrable Hamiltonian systems.

Resonances within nonperturbative methods in field theories

We study the stability of solitonlike solutions of a classical relativistic field theory with logarithmic nonlinearities. For a given charge (not greater than a certain value ${q}_{max}$) the model exhibits a stable state and an unstable one. The latter state, instead of being nonphysical, can be interpreted as a resonance. Problems related to the semiclassical quantization of this theory are also discussed. All calculations are done in any number of spatial dimensions.

Second quantization of classical nonlinear relativistic field theory. I. Canonical formalism

The canonical formalism for nonlinear relativistic classical field theory is presented. It is shown that the solutions Φ (x) of the nonlinear equation (⧠+m2) Φ (x) =λΦ3(x) as well as the asymptotic fields Φin(x) and Φout(x) are local relativistic fields with respect to Poisson brackets, with initial data as canonical variables. A convenient form for the generators of the Poincaré group is derived, and the properties of the scattering operator are discussed.

Canonical quantization of classical nonlinear relativistic field theory

The construction of a relativistic interacting local quantum field is given in two steps: first the classical nonlinear relativistic field theory is written down in terms of Poisson brackets, with initial conditions as canonical variables: next a representation of Poisson bracket Lie algebra by means of linear operators in the topological vector space is given and an explicit form of a local interacting relativistic quantum field\(\hat \Phi \) is obtained. The construction of asymptotic local relativistic fields\(\hat \Phi _{in} \) and\(\hat \Phi _{out} \) associated with\(\hat \Phi \) is also given.

The relationship between causality, in a classical relativistic field theory, and Lorentz invariance in the corresponding quantum theory, for a massive spin-one field in external potentials

A massive spin-one field, interacting with external potentials, is described in terms of the Stuckelberg formalism (1938). Some points, concerning necessary and sufficient conditions for the classical theory to be causal, in the sense of Velo and Zwanziger (1969 and 1971) are clarified. It is then shown that with a simple restriction on the type of interaction considered, the classical theory is causal, in the above sense, if and only if the corresponding quantum theory is Lorentz invariant, in that the S operator in the interaction picture is normal independent. Incidental in this demonstration is the result that the characteristic and Lee-Yang determinants share the same form. It is indicated how this result generalized to interactions between Stuckelberg, Klein-Gordon and electromagnetic fields and the validity of the result, in the more usual vector formalism is established.

Some Exact Results for One-Dimensional Models of Solids

Exact results are presented for one-dimensional models of a band problem, an elastic solid, and a classical relativistic field theory with broken symmetry.

The Nuclear Repulsive-Core Effect as a Possible Consequence of Matter in Interaction with a Classical Relativistic Scalar Field: A New Approach to the Question

The N-body problem is analyzed within the framework of a new formalism for relativistic point masses interacting via a scalar field, in which the problems of infinite self-energies are absent. A Lagrangian formalism is exhibited which yields the particle equations of motion in the form of a parameterized class of equations. The parameter determines the choice of boundary conditions which is chosen on the scalar-field equations. The existence or nonexistence of the relativistic nuclear hard-core effect, associated with the scalar-field interactions, is shown to depend critically on the particular set of boundary conditions which are imposed on the scalar-field equations. In particular, time-symmetric boundary conditions yield no hard-core repulsion, while retarded boundary conditions are shown to yield a hard-core repulsion at very short range.

Non-L2 Solutions of Exactly Soluble Relativistic Wave Equations and the Lee Model

In nonrelativistic potential theory there can exist singularities of the S matrix which are associated with wavefunctions belonging to a non-L2 class. In this paper the non-L2 character of these singularities is shown to persist in several relativistic models: in (1) two schemes, including that of the Klein-Gordon equation and the coupling of a classical relativistic field to an external exponential source, in (2) the pair theory, classical and quantized, with separable interactions, and in (3) the Lee model. For (2) and (3), poles of the ``proper'' Jost function f̄_(k, 0), i.e., the Fredholm determinant for the outgoing scattering state, correspond to non-L2 class solutions of the associated dynamical field equation in coordinate space. These non-L2 solutions will be called ``shadow'' fields. They are of special importance and bear the same relation as the ``shadow'' states in potential theory because they also are of dynamical origin and do not appear in the completeness or unitarity relations by virtue of their non-L2 status.

On classical scalar field theories and the relativistic kepler problem

Two versions of classical relativistic field theory corresponding to massless scalar exchange are compared. It is found that they lead to identical field equations but different equations of motion. The motion of one particle bound by the field of another, infinitely massive particle, i.e., the Kepler problem, is examined. In one case the path is an ellipse precessing through an angle equal to minus one-sixth the value predicted by general relativity. In the other case the path is an ellipse which does not precess. The relativistic Kepler problem in the latter case is identical to that formulated by Fronsdal, who started from an approximation to the Bethe-Salpeter equation.

Classical, Stable, Nonlinear Spinor Field with Positive Rest Energy

The influence of nonlinear (fourth-order) terms in the Lagrangians of some classical relativistic field theories is investigated, independently of the mass and coupling-constant parameters. In the spinor case, stable localized configurations with a lowest energy state are shown to exist always for positive values of the coupling constant.