The relativistic semi-group,Lorentz group,and tachyons. I

It is claimed that the current interest of tachyons and some of their unusual properties may be the result of an insufficient mathematical formalism based on Lorentz groups. The introduction of tachyons requires not only a generalization of the principle of causality, but also an extension of the principle of relativistic invariance. A method is proposed to remove a number of problematic cases in the treatment of tachyons.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 85–88, May, 1976.The author wishes to express his appreciation to D. D. Ivanenko for consideration and constructive criticism.     

The relativistic semi-group,Lorentz group,and tachyons. II

The algebraic structure and topological properties of the hyperbolic tangent group are considered and it is proved that a group- (Lorentz-) theoretic realization will not sufficiently reflect the hyperbolic tangent group as a whole.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 89–93, May, 1976.The author wishes to express his appreciation to D. D. Ivanenko for remarks and useful criticism.     

Relativistic semigroup,the lorentz group,and tachyons. IV

The algebraic and topological properties of the relativistic semigroup are discussed. Its probability-theoretical features establish that the relativistic semigroup belongs to the type of complex Markov structures. From the functional point of view, the relativistic semigroup is a compact Lie semigroup which is contracting in partial spaces. Principles of measurability, observability, and stochasticity are formulated, and these lead to a space-time structure of complex Markov kind. Thus, a certain probability-theoretical gnosiology is also possible in the theory of relativity.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 55–58, August, 1977.I thank D. D. Ivanenko for valuable advice and numerous discussions.     

The complex Lorentz group

Russian Physics Journal -     

Propagation of a relativistic particle in terms of the unitary irreducible representations of the Lorentz group

In a generalized Heisenberg/Schr?dinger picture we use an invariant space-time transformation to describe the motion of a relativistic particle. We discuss the relation with the relativistic mechanics and find that the propagation of the particle may be defined as space-time transition between states with equal eigenvalues of the first and second Casimir operators of the Lorentz algebra. In addition we use a vector on the light-cone. A massive relativistic particle with spin 0 is considered. We also consider the nonrelativistic limit. Received: 20 September 2001 / Published online: 23 November 2001     

Time ordering and the Lorentz group

A simplified definition of point local clocks and the relationship between an inertial reference frame and a class of such clocks, at rest with respect to each other, are used for an algebraic determination of the geometry of Minkowski's space-time on the set of point events. The group of all automorphisms that preserve the time ordering induced by the set of all equivalent local clocks is shown to be generated by the inhomogeneous orthochronous Lorentz group and dilatations, consistently with a well-known result of E. C. Zeeman. The existence of a universal limit velocity and of an invariant clock cone through any event is among the implications of our proof, thus effectively exploiting a suggestion due to F. Severi.     

On the quantum Lorentz group

The quantum analog of Pauli matrices are introduced and investigated. From these matrices and an appropriate trace over spinorial indices we construct a quantum Minkowski metric. In this framework we show explicitly the correspondence between the SL(2,C) and Lorentz quantum groups. Five matrices of the quantum Lorentz group are constructed in terms of the R matrix of SL(2,C) group. These matrices satisfy Yang–Baxter equations and two of which have adequate properties tied to the quantum Minkowski space structure as the reality conditions of the coordinates and the symmetrization of the metric. It is also shown that the Minkowski metric leads to invariant and central lengths of four-vectors.     

Strings, tachyons and deconfinement

We argue that the tachyons which exist in some string models are a signal of deconfinement. We compare the deconfinement temperature obtained from Monte Carlo calculations with some strings models.     

Noncommutativity and Lorentz violation in relativistic heavy ion collisions

We show that relativistic heavy ion collisions at LHC energies could be used as an experimental probe to detect fundamental properties of spacetime long speculated about. Our results rely on the recent proposal that magnetic fields of intensity much larger than that of magnetars should be produced at the beginning of the collisions and this could have an important impact on the experimental manifestation of a noncommutative spacetime. Indeed, in the noncommutative generalization of electrodynamics the interplay between a nonzero noncommutative parameter and an external magnetic field leads us to predict the production of lepton pairs of low invariant mass by free photons (an event forbidden by Lorentz invariant electrodynamics) in relativistic heavy ion collisions at present and future available energies. This unique channel can be clearly considered as a signature of noncommutativity. On the other hand, the search for such decays is worth anyway because their absence would ameliorate of three orders of magnitude the current bound on the noncommutative parameter.     

Radiation by relativistic dipoles. III

The classical radiation of a point magnetic moment (a magneton) moving at a constant velocity in an arbitrary direction with respect to the field lines of a uniform magnetic field is analyzed. All characteristics of the radiation agree with the Ternov-Bagrov-Khapaev relativistic quantum thoery of the radiation by a neutron. It is thus demonstrated that the classical model of radiation with spin flip is valid. The correspondence principle in the theory of radiation with spin flip will be discussed in more detail in subsequent papers.V. V. Kuibyshev Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 53–59, January 1994.     

On the Lorentz group SO(3,1), geometrical supersymmetric action for particles and square-root operators II. Squeezed states and relativistic wave equations

The geometrical relativistic superparticle action is analyzed from the theoretical group point of view. To this end an alternative technique of quantization, outlined by the authors in a previous work and based on the correct interpretation of the square-root Hamiltonian, is used. We show that the obtained spectrum of physical states and the Fock construction in this previous work consist of squeezed states with the even and odd representations with the lowest weights λ = 1/4 and λ= 3/4 corresponding to four possible (nontrivial) fractional representations for the group decomposition of the spin structure. The conserved currents are computed, and a new relativistic wave equation is proposed and explicitly solved for the time-dependent case. The relation between the relativistic Schrödinger equation and the time-dependent harmonic oscillator is analyzed and discussed.     

Poisson structures on the Lorentz group

Poisson-Lie structures on the Lorentz group are completely classified. A method applicable to an arbitrary semisimple complex Lie group (treated as real) is developed.     

One representation of the proper Lorentz group,and quaternionic analyticity

A representation of the proper Lorentz group G is considered, with operators realized as transformations of complex-valued functions of real variables x, y, z, and t. These transformations leave invariant the set of solutions of the Moisil-Fueter equation, defining quaternionic analytic functions. A subset of quaternionic analytic functions is found, which realizes irreducible representations of the group G. A slightly different representation is also considered, in the form of a transformation of functions which leaves invariant the set of solutions of the Dirac equation for a massless particle in the quaternionic form.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 73–79, October, 1982.     

The laws of relativistic thermodynamics: III. The entropy law

The covariant entropy law is derived for an arbitrary choice of the hydrodynamic velocity field. The effects of the choices of Eckart and of Landau and Lifshits are shown.     

A discussion of causality and the Lorentz group

Zeeman (1964) has shown that the group of automorphisms for the relation of causality on Minkowski space is that generated by the orthochronous Poincaré (Halpern, 1968) group and dilatations. Here we prove that the group of automorphisms that preserve the time-like vectors of Minkowski space normwise is the complete Poincaré group. We prove that the timelike structure within the null cone of a single event does define the whole structure of Minkowski space. Further, it is shown that only inertial observers can use Minkowski space to describe space-time.     

Radiation of a relativistic magneton. III

An expression is given for the angular distribution of the change in the four-dimensional momentum of the radiation of a charged magneton in unit proper time. The expression is given for both the tensor and the vector form of expression of the intrinsic magnetic moment. A method of covariant integration of the angular distribution of the radiation in the laboratory coordinate system is developed. The results of the integration in the tensor and vector variants go over into each other exactly.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 33–38, October, 1980.     

Consequences of the Noncompactness of the Lorentz group

The following four statements for indefinitemetrics of Lorentz signature do not possess an analogousstatement in the definite Euclidean signature case: (1)All curvature invariants of a gravitational wave vanish, in spite of the fact that itrepresents a nonflat spacetime. (2) The eigennullframecomponents of the curvature tensor (the Cartanscalars) do not represent curvaturescalars. (3) The Euclidean topology in the Minkowskispacetime doesnot possessa basis composedofLorentzinvariant neighborhoods. (4) There are pointsin the de Sitter spacetime which cannot be joined toeach other by any geodesic. We show that these fourstatements all follow from the noncompactness of theLorentz group. We conclude that the popular (and oftenuseful) imaginary-coordinate rotation from Euclidean to Lorentzian signature (called Wick rotation) isnot an isomorphism.