Relativistic Kepler Problem Research Articles

The dynamics of the relativistic Kepler problem

We deal with the Hamiltonian system (HS) associated to the Hamiltonian in polar coordinates H=12pr2+pϕ2r2-1r-∊2r2, where ∊ is a small parameter. This Hamiltonain comes from the correction given by the special relativity to the motion of the two-body problem, or by the first order correction to the two-body problem coming from the general relativity. This Hamiltonian system is completely integrable with the angular momentum C and the Hamiltonian H. We have two objectives.First we describe the global dynamics of the Hamiltonian system (HS) in the following sense. Let Sh and Sc are the subset of the phase space where H=h and C=c, respectively. Since C and H are first integrals, the sets Sc,Sh and Shc=Sh∩Sc are invariant by the action of the flow of the Hamiltonian system (HS). We determine the global dynamics on those sets when the values of h and c vary.Second recently Tudoran (2017) provided a criterion which detects when a non-degenerate equilibrium point of a completely integrable system is Lyapunov stable. Every equilibrium point q of the completely integrable Hamiltonian system (HS) is degenerate and has zero angular momentum, so the mentioned criterion cannot be applied to it. But we will show that this criterion is also satisfied when it is applied to the Hamiltonian system (HS) restricted to zero angular momentum.

10th Biennial Conference on Classical and Quantum Relativistic Dynamics of Particles and Fields

10th Biennial Conference on Classical and Quantum Relativistic Dynamics of Particles and Fields

Spin decoherence rate in a homogenous all-electric ring

I derive analytical expressions for the spin decoherence rate for motion in a homogenous weak focusing all-electric storage ring. Canonical dynamical variables are employed throughout. I treat on-energy betatron oscillations and off-energy dispersion orbits. I obtained exact solutions for the spin decoherence rate, for the off-energy dispersion orbits and for motion in a vertical spiral in a logarithmic potential. I also reproduce, with kind permission, Koop׳s elegant analysis of motion in a vertical spiral in a logarithmic potential. I have also recently published the exact solution for the spin precession in the relativistic Kepler problem (inverse square law), and I verify that it confirms the perturbative solution. The appendices present some comments on other published work in the field, including a discussion of conceptual and quantitative errors therein.

Exact solution for spin precession in the radiationless relativistic Kepler problem

There is interest in circulating beams of polarized particles in all-electric storage rings to search for nonzero permanent electric dipole moments of subatomic particles. To this end, it is helpful to derive exact analytical solutions of the spin precession in idealized models, both for pedagogical reasons and to serve as benchmark tests for analysis and design of experiments. This paper derives exact solutions for the spin precession in the relativistic Kepler problem. Some counterintuitive properties of the solutions are pointed out.

DYNAMICAL (SUPER)SYMMETRIES OF MONOPOLES AND VORTICES

The dynamical (super)symmetries for various monopole systems are reviewed. For a Dirac monopole, non-smooth Runge–Lenz vector can exist; there is, however, a spectrum-generating conformal o(2,1) dynamical symmetry that extends into osp(1/1) or osp(1/2) for spin 1/2 particles. Self-dual 't Hooft–Polyakov-type monopoles admit an su(2/2) dynamical supersymmetry algebra, which allows us to reduce the fluctuation equation to the spin 0 case. For large r, the system reduces to a Dirac monopole plus a suitable inverse-square potential considered before by McIntosh and Cisneros, and by Zwanziger in the spin 0 case, and to the "dyon" of D'Hoker and Vinet for spin 1/2. The asymptotic system admits a Kepler-type dynamical symmetry as well as a "helicity-supersymmetry" analogous to the one Biedenharn found in the relativistic Kepler problem. Similar results hold for the Kaluza–Klein monopole of Gross–Perry–Sorkin. For the magnetic vortex, the N = 2 supersymmetry of the Pauli Hamiltonian in a static magnetic field in the plane combines with the o(2) × o(2,1) bosonic symmetry into an o(2) × osp(1/2) dynamical superalgebra.

Semiclassical quantisation rules for the Dirac and Pauli equations

We derive explicit semiclassical quantisation conditions for the Dirac and Pauli equations. We show that the spin degree of freedom yields a contribution which is of the same order of magnitude as the Maslov correction in Einstein–Brillouin–Keller quantisation. In order to obtain this result a generalisation of the notion of integrability for a certain skew product flow of classical translational dynamics and classical spin precession has to be derived. Among the examples discussed is the relativistic Kepler problem with Thomas precession, whose treatment sheds some light on the amazing success of Sommerfeld’s theory of fine structure [Ann. Phys. (Leipzig) 51 (1916) 1].

On the Dirac–Kepler problem: The Johnson–Lippmann operator, supersymmetry, and normal‐mode representations

AbstractThe relativistic Kepler problem is discussed, with emphasis on the exact supersymmetry of the problem. It is shown that the supersymmetry is generated by the Johnson–Lippmann operator. Two related operators are found to generate new supersymmetries in an extended function space. Each of these supersymmetries may be disguised as radial supersymmetries. The radial supersymmetries are discussed and it is shown that each of them defines a normal‐mode representation of the hydrogen‐atom radial functions. Thus, one obtains two different, but equivalent, analytical expressions for these functions. The expressions are well known, but are rederived here in the light of the new understanding. Finally, the nonrelativistic image of the relativistic supersymmetry is constructed and its generators shown to be identical with those recently presented in the literature. © 1995 John Wiley & Sons, Inc.

A numerical solution of the relativistic Kepler problem

A numerical solution of the relativistic Kepler problem

Classical limit of real Dirac theory: Quantization of relativistic central field orbits

The classical limit of real Dirac theory is derived as the lowest-order contribution in $$\mathchar'26\mkern-10mu\lambda = \hslash /mc$$ of a new, exact polar decomposition. The resulting classical spinor equation is completely integrated for stationary solutions to arbitrary central fields. Imposing single-valuedness on the covering space of a bivector-valued extension to these classical solutions, orbital angular momentum, energy, and spin directions are quantized. The quantization of energy turns out to yield the WKB formula of Bessey, Uhlenbeck, and Good. It is demonstrated that the success of Sommerfeld's old quantization is due to a complete mutual cancellation between wave mechanical half-integers and spin in the particular case of the relativistic Kepler problem.

Dynamical group of the relativistic Kepler problem

A manifestly covariant relativistic generation of the Lenz vector for the relativistic Kepler problem is constructed. The corresponding dynamical group for the bound state problem is SO(4,1).

Separating the dyon system.

The relation between the "dyon" system studied by D'Hoker and Vinet and its spin-0 counterpart described earlier by McIntosh and Cisneros and by Zwanziger can be made manifest by using suitable variables, introduced before by Biedenharn and by McIntosh and Berrondo in the relativistic Kepler problem. This leads to a simple determination of the spectra.

Applications of the rotating Laplace-Runge-Lenz Vector to quantum eigenvalue problems

As an extension of the familiar conserved vector of the nonrelativistic Kepler problem, the rotating Laplace-Runge-Lenz vector is introduced to describe the precessing elliptical orbits of the Kepler problem modified by an inverse cube force and the relativistic Kepler problem. This vector, which has a constant length, is always oriented to the moving perihelion points of the orbit. Using a transformation defined by an exponential operator with the vector product, this rotating vector is derived from the conserved vector. The transformations yield the dynamical systems without precessions, so that the energy eigenvalues of the two problems are derived without solving Schrodinger or Klein-Gordon equations in quantum mechanics.

Rotating Laplace-Runge-Lenz vector leading to two relativistic Kepler’s equations

For the nonrelativistic Kepler problem it is well known that there exists a transcendental equation (the so-called Kepler's equation) which gives a position in the orbit at a given time. The precessing orbit of the relativistic Kepler problem is reduced to the nonprecessing one by using the transformation related to a rotating Laplace-Runge-Lenz vector. This vector is an extension of the familiar conserved vector and is always oriented to the moving perihelion point of the precessing orbit. From the considerations on Kepler's equation and the rotating Laplace-Runge-Lenz vector, a relativistic Kepler's equation is newly defined to find a position in the precessing orbit at a given time. This equation has two expressions corresponding to the relativistic coordinate time and proper time. By using the equation, Lambert's theorem that gives the required time on the orbit is extended to the relativistic problem.

Relativistic Coulomb Sturmian matrix elements and the Coulomb Green’s function of the second-order Dirac equation

Matrix elements with respect to a discrete ‘‘Sturmian’’ basis set of the operator r1/2gr1/2, where g is the Coulomb Green’s function of the ‘‘second-order’’ Dirac equation, are investigated. Closed expressions, involving Gaussian hypergeometric functions, can be obtained for the matrix elements. This calculation illustrates contour integration techniques capable of yielding closed expressions for a number of overlap integrals associated with the relativistic Kepler problem.

A simple solution of the two-dimensional relativistic Kepler problem

A two-dimensional electron is described by an appropriate spin 1/2 bi-dimensional Dirac equation. The special case of a hydrogenic atom in two dimensions is solved in order to obtain its energy spectrum and the total spread of its fine structure levels.

Mass eigenfunction expansions for the relativistic Kepler problem and arbitrary static magnetic field in relativistic quantum theory

We investigate the existence of orthogonality and completeness relations for the eigenvalue problem associated with the differential operator Λ=−ΠμΠμ −ieσ ⋅ (E+iB), Πμ=−i ∂μ−eAμ. The operator Λ acts on 2×1 Pauli-type spinor fields defined over all Minkowski space, and may be interpreted as the square of the mass of a charged Dirac particle moving in an external c-number electromagnetic field. We show that Λ is self-adjoint with respect to the not positive-definite inner product (φb; φa)=∫ d4x φ̄bφa, where φ̄b is defined as φ̄b=φ†b(−iΠ↙4−σ⋅Π↙). A proof is provided for the Coulomb case that the mass eigenfunctions form a complete set in spite of the indefinite metric in Hilbert space. The mass eigenfunction expansion of the propagator is worked out explicitly for the Kepler case. This mass eigenfunction expansion is expected to be quite useful for bound state calculations in quantum electrodynamics, since it involves the covariant denominators (m′)2−(m)2.

Relativistische Verallgemeinerung des Lenzschen Vektors (I) / Relativistic generalisation of the Lenz vector (I)

The non-relativistic motion of a particle in a central field with 1/r potential, e.g. the motion of an electron in the Coulomb field of a charged nucleus at rest, is described by the equation of motion (non-relativistic Kepler problem) m x″ = α · x /r3 with α = ez e (product of the charges of the central body ez and the electron e). From this equation of motion, three statements of conservation can be derived: in respect of the energy E, of the angular momentum L and of the Lenz vector Λ = m {x′× L + α ·x/r}. The geometric meaning of Λ is that of a vector pointing in the direction of the perihelion of the particle orbits (conic sections). It will be demonstrated that also at the relativistic Kepler problem, which is based on the equation of motion an analogous Lenz vector exists. It represents a quantity of conservation - in the same way as the relativistic energy and the relativistic angular momentum. For the transitional case → ∞, where the relativistic problem turns into the non-relativistic problem, the relativistic Lenz vector also turns into the non-relativistic Lenz vector. The generalised (relativistic) Lenz vector has also a geometric meaning. Its direction coincides with the oriented axis of symmetry of the orbits (rosettes, spirals, hyperbola-type curves etc.). The quantity of conservation Λ occupies a special position in respect of the quantities of conservation energy and angular momentum. Whereas the energy and the angular momentum correspond with a symmetry of time and space, the Lenz quantity of conservation corresponds with a symmetry of the orbits. The fact that the Lenz vector can relativistically be generalised touches thereby on principal aspects.

Relativistische Verallgemeinerung des Lenzschen Vektors (II) / Relativistic generalisation of the Lenz vector (II)

The Lenz vector Λ= m {x′ × L + α · x/r} is a special integral of motion of the non-relativistic Kepler problem. Accordingly, there also exists a special integral of motion Λ (x, x') of the relativistic Kepler problem as pointed out in Part I of this paper. In the following Part II, the system of equations is compiled to the extent required for the relativistic generalisation of the Lenz vector and its interpretation. Starting point are the statements of conservation for the energy and the angular momentum. From these two quantities, it is possible to derive the complete system of solutions x(t) in accordance with standardised methods. Some details and some modifications of the approach are of special importance for the purposes of this paper. In the centre of the consideration are the construction and interpretation of the standardised polar angle θ as a function of r, r'. The explicit time dependance of the functions r(t), θ(t) is not required for constructing the relativistic Lenz vector in the form Λ (x, x'). In the literature, one does Find only few, not sufficient for our purposes, data on the relativistic orbits (rosettes, spirals, hyperbola-type curves etc.). It is, therefore, necessary to discuss these orbits in detail. For this purpose, a series of relationships are derived which apparantly have not been published before.

Complex-energy solutions of the relativistic Kepler problem at high energies

In the relativistic Kepler problem the analytic continuation of the principal quantum number $n$ into $i\ensuremath{\nu}$ describes the scattering states. When ${\ensuremath{\nu}}^{2}$ ranges from $+\ensuremath{\infty}$ to ${\ensuremath{\alpha}}^{2}$ the energy ranges from (${m}_{1}+{m}_{2}$) to infinity. In the remaining gap $0<{\ensuremath{\nu}}^{2}<{\ensuremath{\alpha}}^{2}$, the energy becomes complex corresponding to a continuum of resonances or perhaps pair production. This phenomenon also exists in the classical relativistic problem, and we suggest that one should look experimentally for the manifestation of this gap, for example when $Z>~137$.

Relativistic Kepler problem. I. Behavior in the distant past of orbits with gravitational radiation damping

The method of perturbation of orbital elements is used to estimate the effect of the emission of gravitational radiation on the behavior, in the distant past, of the orbit of two weakly gravitationally interacting bodies. The perturbing acceleration due to radiation reaction is computed from time derivatives of the quadrupole moment. All such systems which were bound at finite times were necessarily unbound in the past. This result, while independent of the precise details of the radiation-reaction law, does not follow solely from energy loss due to radiation emission. A consequence of possible astrophysical significance is the possibility of the mutual capture of two bodies from scattering orbits by emission of gravitational radiation.