Infinite-component Wave Equations Research Articles

Anyon wave equations and the noncommutative plane

The “Jackiw–Nair” nonrelativistic limit of the relativistic anyon equations provides us with infinite-component wave equations of the Dirac–Majorana–Lévy-Leblond type for the “exotic” particle, associated with the two-fold central extension of the planar Galilei group. An infinite dimensional representation of the Galilei group is found. The velocity operator is studied, and the observable coordinates describing a noncommutative plane are identified.

Fields on the Poincaré Group: Arbitrary Spin Description and Relativistic Wave Equations

In this paper, starting from a pure group-theoretic point of view, we develop an ap- proach to describing particles with different spins in the framework of a theory of scalar fields on the Poincare group. Such fields can be considered as generating functions for conventional spin-tensor fields. The case of two, three, and four dimensions are elaborated in detail. Discrete transformations C; P; T are defined for the scalar fields as automorphisms of the Poincare group. We classify the scalar functions, and ob- tain relativistic wave equations for particles with definite spin and mass. There exist two different types of scalar functions (which describe the same mass and spin), one related to a finite-dimensional nonunitary representation and the other to an infinite- dimensional unitary representation of the Lorentz subgroup. This allows us to derive both usual finite-component wave equations for spin-tensor fields and positive-energy, infinite-component wave equations.

Spinor fields and the GL(4,R) gauge structure in the nonsymmetric theory of gravitation

The spinor structure associated with the local gauge group GL(4,R) of the nonsymmetric gravitation theory (NGT) is based on a spinor wave equation constructed from a vierbein, a GL(4,R) spin connection, and the infinite-dimensional irreducible representations of the universal covering group 𝒮ℒ (4,R) of the noncompact group SL(4,R). The multiplicity-free irreducible representations of 𝒮ℒ (4,R) correspond to bivalued spinorial representations of SL(4,R) that contain an infinite number of half-odd integer spin particles. By adjoining the translations T4, the extended group 𝒜=T4×GL(4,R) replaces the Poincaré group 𝒫. The properties of the mass spectrum are obtained from an infinite-component wave equation and the physical spinor field consists of an infinite sum of finite, nonunitary representations of the Lorentz group.

Derivation of Nonperturbative Relativistic Two-Body Equations from the Action Principle in Quantumelectrodynamics

For two fermion fields Ψ1(x) and Ψ2(x) interacting with the electromagnetic field via the minimal and Pauli-couplings we introduce a composite field ϕ(x1, x2). The total action can be rewritten in terms of ϕ and its time derivatives if we use the physical retarded Green's functions Dret (corresponding to an effective to (1/2)(Dret + Dadv) = D) between two particles. The variation of the action with respect to ϕ gives then exact covariant two-body, one-time, equations with relativistic potentials. The center of motion can be covariantly separated leading to an infinite component wave equation for the composite system, which reduces to the well-known covariant wave equation for the moving H-atom in the limit m2 ∞.

Electromagnetic interactions and the relativistic infinite-component wave equation for hydrogen

We examine the problem of incorporating external electromagnetic interactions into the theory of the relativistic infinite-component SO(4,2) wave equation for the hydrogen atom proposed by Barut. We introduce the simplest set of covariant interaction terms modeled after the nonrelativistic SO(4,2) theory as an alternative to the complicated array of terms obtained from the formal replacement ${P}_{\ensuremath{\mu}}\ensuremath{\rightarrow}{P}_{\ensuremath{\mu}}\ensuremath{-}e{A}_{\ensuremath{\mu}}$. Using a covariant perturbation theory, we calculate the electric and magnetic polarizabilities of the ground state of the hydrogen atom in uniform fields and show that they have the correct nonrelativistic reductions.

Quantum theory of infinite component fields

The quantum theory of the infinite component SO(4,2) fields is formulated as a model for relativistic composite objects. We discuss three classes (timelike, lightlike, and spacelike) of physical solutions to a general class of infinite component wave equations. These solutions provide a definite physical interpretation to infinite component wave equations and are obtained by reducing SO(4,2) with respect to its orthogonal, pseudo-orthogonal, and Euclidean subgroups. The analytic continuations among these solutions are established. In the nonrelativistic limit the timelike physical states exactly reduce to the Schrödinger solution for the hydrogen atom—the simplest composite object. The wave equations for the three classes are studied in two different realizations. In one case the equations describe a three-dimensional internal Kepler motion with a discrete and a continuous energy spectrum and in the other case the equations describe a four-dimensional internal oscillatory motion with attractive as well as repulsive potentials. It is found that the Kustaanheimo–Steifel transformation of classical mechanics exactly relates these two internal motions also in the quantum case. Thus a completely relativistic theory of composite systems is established for which the internal dynamics is the generalization of the nonrelativistic two-body dynamics.

Четырех-скорости в квантовой механике иK-пространство

From the unitary representations of the Lorentz transformations two four-vector operators,u μ andU μ, are constructed that satisfy the equationsu μ u μ = −1 andU μ U μ=+1 as operator relations, thus behaving like a timelike and a spacelike velocity, respectively. With the help of one of the Casimir operators (K) of the Lorentz group they are represented by difference operators. Eigenvalues and eigenfunctions for the componentsu 4 andU 4 are derived for spin 0 and 1. In contrast to the case of spin 1/2, in which they could be worked out only in approximation, the solutions for the integral spin values can be given in closed form. At the end, a possible use in particle physics of theK-representation is exhibited by the discussion of a linearized infinite-component wave equation.

Interpretation of space-like solutions of infinite-component wave equations and Grodsky-Streater 'No-Go' theorem

The existence of space-like solutions of infinite-component wave equations is not a 'disease' but a virtue. By comparing quantum electrodynamics with the infinite-component wave equations for bound electrons the authors show that the space-like solutions correspond to relativistic negative-energy solutions of the constituents of the composite system. Hence they have physical consequences in second-order processes such as the Compton effect and electromagnetic polarizabilities. Thus the assumption of the 'No-Go' theorem admitting only time-like solutions is too restrictive for a field theory with infinite-component equations.

Perturbation theory for infinite-component wave equations

A new covariant perturbation theory for infinite-component wave equations is constructed which is analogous to the Epstein-Waller and Lewis-Dalgarno (1955) method of atomic physics. Instead of infinite sums over discrete and continuous states the matrix elements involve Lie algebra or group elements and can be evaluated in closed form.

Deep bound states in the Kadyshevsky equation

We study bound-state solutions to the Kadyshevsky equation in relativistic space. We first show that, for the Coulomb interaction in the unequal-mass case, bound states of arbitrarily small energy may exist, provided the coupling constant is large enough. In the static limit, we show that infinitely deep bound states may occur for very general forms of the interaction as the coupling becomes infinite. In both cases, our results for the Coulomb interaction are similar to those obtained from different approaches (infinite-component wave equation or modified Dirac-Coulomb Hamiltonian).

Internal structure of composite systems described by infinite-component wave equations

It is shown that the boson representation of infinite-component SO(4,2) wave equations represents a system of four oscillators as well as three-dimensional Kepler dynamics. These two pictures are complementary to each other; that is, the Kepler dynamics is adequate for small distances, while the oscillator dynamics is more relevant for infinitely large distances, and both pictures are equivalently possible for finite distances. This is interpreted as a sort of constituent confinement.

Soliton solutions of infinite component wave equations

Three dimensional moving solitons, including relativistic Fitzgerald contraction of lengths, are explicitly constructed as stationary solutions of infinite component wave equations.

Electromagnetic polarizabilities of nucleons

A general theory of electromagnetic polarizabilities (and higher-order effects) of composite systems described by infinite-component wave equations is developed. Such wave equations were previously used to predict form factors, mass spectrum, and structure functions of the proton. As a special case the polarizabilities of the relativistic H atom, proton, and neutron are evaluated explicitly and compared with experiments.

Lifetime of elementary particles from analytic continuation in momentum and infinite-component wave equations

We derive the spin dependence of mass and of decay width of elementary particles described by infinite-component fields transforming irreducibly under $\mathrm{SL}(2,C)(\mathrm{Majorana})\ensuremath{\bigotimes}\mathrm{SL}(2,C)(\mathrm{Dirac})\ifmmode\times\else\texttimes\fi{}{T}_{4}$ representations. The fields describing the unstable particles are certain operator-valued distributions whose Fourier transforms are defined in the complex-momentum space via analytic continuation, and classification under the nonunitary Poincar\'e group is used to characterize the decay widths.

Relativistic transition amplitudes in a class of O(4,2) infinite multiplets. I. Continuum-continuum transitions

A class of unitary irreducible O(4, 2) representations characterized by the lowest spin $\ensuremath{\mu}$, $\ensuremath{\mu}=0, \ifmmode\pm\else\textpm\fi{}\frac{1}{2}, \ifmmode\pm\else\textpm\fi{}1, \dots{}$, and the corresponding infinite-component wave equations are considered. The continuum-continuum transition amplitudes are evaluated by the method of analytic continuation generalizing the relativistic Coulomb case ($\ensuremath{\mu}=0$). A general exact Regge-type amplitude is derived, which contains the relativistic scattering amplitudes of O(4)-symmetric Coulomb interactions of two dyons in the symmetric model and amplitudes in certain hadron models as special cases.

Proton structure functions in an infinite resonance O(4, 2) model

The proton electromagnetic inelastic structure functions have been explicitly evaluated in a noncompact O(4, 2) model for the spectrum of hadronic states, using the Barut-Corrigan-Kleinert infinite component wave equation. The resulting structure functions exhibit nontrivial scaling behavior in the Bjorken limit and give qualitative agreement with observation. They also satisfy exactly the Callan-Gross relation.

Some Unusual Applications of Lie Algebra Representations in Quantum Theory

Several new applications of Lie algebra representations in quantum theory are discussed. These include a method of self adjoint extension of the Dirac Hamiltonian, description of continuous spectra, dynamical groups and infinite-component wave equations, induced and indecomposable representations and infinite parameter Lie algebras.

Completeness of the solutions of a wave equation with a linear mass spectrum

Some of the conditions under which the solutions of an infinite-component wave equation form a complete set are investigated. The investigation is carried out by the resolvent method, and this allows us to find a connection between the NO-GO theorem and the existence or not of the operator exp [πK3]. As a concrete example the completeness relation is explicitly worked out for the solutions of an equation with a linear mass spectrum; as a consequence of the imposed conditions it is found that the intercept of the trajectory must be −1/2. The two Majorana representations are considered, and it is found that completeness holds for the bosonic case only. In this last case it is shown that to the completeness contribute a discrete set of timelike states and a continuum of states with complex energy.

Introduction of internal coordinates into the infinite-component Majorana equation

Given an infinite-component wave equation describing the global quantum numbers of a system one can introduce various internal dynamical coordinates such that ‘constituents’ will appear to move in an oscillator or in a Kepler potential, or, in principle, in other potentials. This is explicitly shown for the Majorana equation. The space-like solutions of the Majorana equations correspond to the scattering state-solutions in terms of the constituent ‘particles’. Light-like solutions and a generalized second-order Majorana equation are also treated in a similar way. Relation to Dirac’s new wave equation without negative energy solutions is discussed.

An infinite-component particle equation with a widely spaced mass spectrum

Infinite-component wave equations have been studied by several authors for the generation of mass eigenvalues, that might explain the masses of the elementary particles. So far no spectra have been found with any resemblance to the existing masses, the intervals rising far too slowly or even descending. We employ here a wave equation based on a unitary representation of the Lorentz transformation with nonvanishing Casimir operators (I andK), one of which (K) is continuous. With its help mass operators are easily formulated and analysed. We evaluate a particularly simple one constructed earlier in an attempt to quantize the motion of electrically charged particles under the influence of their own field. There results a well-defined discrete and increasing, but too widely spaced mass spectrum. The largeness of the intervals is caused by the smallness of the electromagnetic coupling constant(e2/ℏc) that enters by its reciprocal value. Hence a field with a larger coupling effect, perhaps Heisenberg's matter field, should modify the distribution in the right sense.