Complete sets of solutions of linear Lorentz covariant field equations with an infinite number of field components

We study field equations of the Gelfand-Yaglom type where transforms as a unitary representation of the inhomogeneous Lorentz group. We construct a complete set of solutions of this equation. This set includes solutions with spacelike momentum. Our method makes use of the decomposition of unitary representations of the homogeneous Lorentz group into unitary representations of the little groupsS U (2) andS U (1, 1). The covariant operators ^µ are written as differential operators on homogeneous spaces. For some classes of equations we calculate the mass spectrum explicitly.     

Electrovac fields with geodesic eigenrays

The stationary states of electrovac fields for which the geodesic eigenrays of both the Maxwell and gravitational field coincide are investigated. The fact that the Kerr-Newman solutions belong to this class lends physical interest to the fields considered here. Particular attention is devoted to fields with shearing eigenrays since principal null congruences do not coincide with eigenrays in this case, and so earlier approaches to the problem fail. By the generalisation of a theorem on the corresponding vacuum case, it is proven that no spherical solutions exist in the shearing class, with the exception of the fields with ²¦G_O¦²-¦H_O¦²=O.The metrics with O, admitted by the theorem, can either be calculated from the corresponding vacuum solutions by a relatively simple procedure, or, if not, we list them explicitly in this paper.     

Spinor Structure and Matter Spectrum

Classification of relativistic wave equations is given on the ground of interlocking representations of the Lorentz group. A system of interlocking representations is associated with a system of eigenvector subspaces of the energy operator. Such a correspondence allows one to define matter spectrum, where the each level of this spectrum presents a some state of elementary particle. An elementary particle is understood as a superposition of state vectors in nonseparable Hilbert space. Classification of indecomposable systems of relativistic wave equations is produced for bosonic and fermionic fields on an equal footing (including Dirac and Maxwell equations). All these fields are equivalent levels of matter spectrum, which differ from each other by the value of mass and spin. It is shown that a spectrum of the energy operator, corresponding to a given matter level, is non-degenerate for the fields of type (l, 0) ⊕ (0, l), where l is a spin value, whereas for arbitrary spin chains we have degenerate spectrum. Energy spectra of the stability levels (electron and proton states) of the matter spectrum are studied in detail. It is shown that these stability levels have a nature of threshold scales of the fractal structure associated with the system of interlocking representations of the Lorentz group.     

Covariant wave functions with invariant helicity for massless particles

Considering a wave function for a massless particle, transforming according to an arbitrary irreducible representation (IR) of the homogeneous Lorentz group, we determine the basic conditions for to be an eigenfunction with a specified value of the helicity inall Lorentz frames. The method used is direct and elementary, requiring no knowledge of the IR's of the Poincaré group. It is shown that there existsno invariant helicity state in unitary representations of the Lorentz group, and one such state in any non-unitary representation (with one extra in special cases).     

Construction and classification of indecomposable finite-dimensional representations of the homogeneous Galilei group

We discuss finite-dimensional representations of the homogeneous Galilei group which, when restricted to its subgroup SO(3), are decomposed to spin 0, 1/2 and 1 representations. In particular we explain how these representations were obtained in our paper (M. de Montigny et al.: J. Phys. A39 (2006) 9365) via reduction of the classification problem to a matrix one admitting exact solutions, and via contraction of the corresponding representations of the Lorentz group. Finally, for discussed representations we derive all functional invariants.     

The runaway effect in a Lorentz gas

The Lorentz gas of charged particles in a constant and uniform electric field is studied. The gas flows through the medium of immobile, randomly distributed scatterers. Particles with velocity v suffer collisions with frequency proportional to ¦v¦ ⁿ . Forn < 0 runaway of the gas is forced by the field: the mean velocity of the flow increases without bounds. By a simple physical argument an integral relation is established between the probability of collisionless motion and the velocity distribution. It is then shown that whenn < –1 a fraction of particles moves as if the scattering centers were absent. The detailed discussion of this uncollided runaway is presented. Some qualitative features of the velocity distribution are illustrated on rigorous solutions in one dimension.     

Relativistic equations and indecomposable representations of the Lorentz groupSL(2, ℂ)

Relativistic equations in which the fields cotransform under the direct sum of ordinary indecomposable representations of the Lorentz group are derived and discussed.Talk given at the Symposium on Mathematical Methods in the Theory of Elementary Particles, Liblice castle, Czechoslovakia, June 18–23, 1978.     

Behavior of effective charges in e6 grand unification theory in curved spacetime

The behavior of effective charges _i(t) corresponding to the parameters of nonminimal coupling of scalar and gravitational fields in E₆ Grand Unification Theory (GUT) in curve spacetimes is studied. It is shown that in a strong gravitational field ¦_i(t)¦ grows indefinitely.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 74–11, September, 1990.     

Bifurcation phenomena near homoclinic systems: A two-parameter analysis

The bifurcations of periodic orbits in a class of autonomous three-variable, nonlinear-differential-equation systems possessing a homoclinic orbit associated with a saddle focus with eigenvalues ( ±i,), where ¦/¦ < 1 (Sil'nikov's condition), are studied in a two-parameter space. The perturbed homoclinic systems undergo a countable set of tangent bifurcations followed by period-doubling bifurcations leading to periodic orbits which may be attractors if ¦/¦ < 1/2. The accumulation rate of the critical parameter values at the homoclinic system is exp(-2¦/¦). A global mechanism for the onset of homoclinicity in strongly contractive flows is analyzed. Cusp bifurcations with bistability and hysteresis phenomena exist locally near the onset of homoclinicity. A countable set of these cusp bifurcations with scaling properties related to the eigenvalues±i of the stationary state are shown to occur in infinitely contractive flows. In the two-parameter space, the periodic orbit attractor domain exhibits a spiral structure globally, around the set of homoclinic systems, in which all the different periodic orbits are continuously connected.     

SO(4, 2)-covariant formulation of the Bargmann-Wigner equations

It is shown that the Bargmann-Wigner equations can be written in anSO(4, 2)-covariant form. As well as the Lorentz rotations, theSO(4, 2) group contains a space-inversion and a time-reflection operator (which are different from the usual ones). It also contains the Foldy-Wouthuysen and Cini-Touschek transformations. The spin-s theory for the massive and massless cases, and also a set of Bargmann-Wigner equations corresponding to space-like four-momentum, are all given by the sameSO(4, 2)-covariant equations, and their solutions can be obtained by transforming the solutions corresponding to the special gauge in which the four-momentum vanishes.     

Representations of affine Lie algebras,ellipticr-matrix systems,and special functions

The author considers an elliptic analogue of the Knizhnik-Zamolodchikov equations—the consistent system of linear differential equations arising from the elliptic solution of the classical Yang-Baxter equation for the Lie algebra . The solutions of this system are interpreted as traces of products of intertwining operators between certain representations of the affine Lie algebra. A new differential equation for such traces characterizing their behavior under the variation of the modulus of the underlying elliptic curve is deduced. This equation is consistent with the original system.It is shown that the system extended by the new equation is modular invariant, and the corresponding monodromy representations of the modular group are defined. Some elementary examples in which the system can be solved explicitly (in terms of elliptic and modular functions) are considered. The monodromy of the system is explicitly computed with the help of the trace interpretation of solutions. Projective representations of the braid group of the torus and representations of the double affine Hecke algebra are obtained.     

A New Approach to Spinors and Some Representations of the Lorentz Group on Them

We give a geometric realization of space-time spinors and associated representations, using the Jordan triple structure associated with the Cartan factors of type 4, the so-called spin factors. We construct certain representations of the Lorentz group, which at the same time realize bosonic spin-1 and fermionic spin- wave equations of relativistic field theory, showing some unexpected relations between various low-dimensional Lorentz representations. We include a geometrically and physically motivated introduction to Jordan triples and spin factors.     

Partial diagonalization in solving electron-nuclear problems in molecules

The problem of determining the total wave functions and energies of molecular stationary states reduces to solving a Schrödinger equation with a vibrational-rotational Hamiltonian. This is achieved by a unitary transformation of the molecular Hamiltonian H with its successive diagonalization on a nondegenerate electronic state ¦e. It is shown that the molecular wave functions related to the electronic states ¦e are of the form G¦e¦g_(e), and their corresponding energy value is the sum _e + _g ^(e), where _g ^(e) and ¦g^(e) are the eigenvalues and eigenfunctions of the vibrational-rotational Hamiltonian, determined by means of the unitary operator G. It is shown that the total energy and molecular wave functions are uniquely determined, despite the arbitrariness in choosing G. As an example the vibrational-rotational operator and molecular wave functions are given for the simplest choice of the operator G.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 11–16, March, 1975.     

Nuclear structure on a Grassmann manifold

Products of particlelike representations of the homogeneous Lorentz group are used to construct the degrees of spin angular momentum of a composite system of protons and neutrons. If a canonical labeling system is adopted for each state, a shell structure emerges. Furthermore the use of the Dirac ring ensures that the spin is characterized by half-angles in accord with the neutron-rotation experiment. It is possible to construct a Clebsch-Gordan decomposition to reduce a state of complex angular momentum into simpler states which can be identified with and particles, multipole operators, etc. Finally, ground-state energy levels are calculated for all the even-even nuclei by using a differentiable manifold that is spin-graded and gauge-invariant by construction. It is shown that this manifold is Grassmann.     

Gravitational collapse with charge and small asymmetries I. Scalar perturbations

A recent analysis by Richard Price of spherical collapse with small nonspherical perturbations is here generalized to the case of an electrically charged collapsing star (0¦Q¦-M). The perturbations are confined to a scalar field generated by a nonspherical distribution of scalar charge in the star. As in the electrically neutral case, the scalar perturbations are probably a prototype for all others — electromagnetic, gravitational, and higherspin. The collapse is shown to produce a Reissner-Nordström black hole, and the scalar-field perturbations are shown to radiate completely away; but they die out more slowly the larger is the star's electric charge. For charge ¦Q¦M, the -pole part of the perturbation at fixedr and late times is dominated by a tail that dies out ast ^{–(2+ 2)}. But for ¦Q¦=M, the primary outgoing waves emitted from the star's surface are everywhere larger than the tail. At fixedr and late times they die as t^–(+2). Also, a calculation of the redshift shows that a collapsing star becomes black more slowly the larger is the star's electric charge.Supported in part by the National Science Foundation (GP-27304, GP-28027, GP-19887).     

Finite dimensional representations of the quantum Lorentz group

All finite dimensional irreducible representations of the quantum Lorentz group SL _q (2,) are described explicitly and it is proved all finite dimensional representations of SL _q (2,) are completely reducible. The conjecture of Podle and Woronowicz will be answered affirmatively.     

Description of the gamov-teller β+-decays and (n,p) transitions in spherical nuclei

The results of calculations of the total log (t)-quantities for decays of neutron-deficit nuclei and strength functions of (n, p) transitions are presented. It is shown that in spherical nuclei the ¦g _A/g _V¦1·0 conditions hold for the constants of a weak interaction.Dedicated to the memory of M. Gmitro.     

Zeros of the partition function and Gaussian inequalities for the plane rotator model

The partition function for ferromagnetic plane rotators in a complex external field , with ¦Im ¦ ¦Re ¦, is bounded below in modulus by its value at =0. The proof is based on complex combinations of duplicated variables which are positive definite on a subgroup of the configuration group. In the isotropic situation (and =0), the associated Gaussian inequalities imply that all truncated correlation functions decay at least as the two-point function.     

Generalized classical theory of magnetism

We consider an Ising model with Kac potential ^dK(¦x¦) which may have arbitrary sign, and show, following Gates and Penrose, that the free energy in the classical limit0+ can be obtained from a variational principle. When the Fourier transform of the potential has its maximum atp=0 one recovers the usual mean-field theory of magnetism. When the maximum occurs forp ₀0, however, one obtains an oscillatory or helicoidal phase in which the magnetization near the critical point oscillates with period 2/¦p ₀¦. An example with a potential possessing parameter-dependent oscillations is shown to exhibit crossover phenomena and a multicritical Lifshitz point in the classical limit.     

Contractions of representations of de Sitter groups

In order to construct the quantum field theory in a curved space with no old infinities as the curvature tends to zero, the problem of contraction of representations of the corresponding group of motions is studied. The definitions of contraction of a local group and of its representations are given in a coordinate-free manner. The contraction of the principal continuous series of the de Sitter groupsSO ₀(n, 1) to positive mass representations of both the Euclidean and Poincaré groups is carried out in detail. It is shown that all positive mass continuous unitary irreducible representations of the resulting groups can be obtained by this method. For the Poincaré groups the contraction procedure yields reducible representations which decompose into two non-equivalent irreducible representations.On leave of absence from the Institute of Physics of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia.