From the Galilei to the Lorentz group in a general light-cone frame

The most general coordinate system in which the Galilean symmetries of the ordinary and the Bell and Ruegg light-cone frames appear is found. With a modification of this general coordinate system it is possible to go from the Galilei group in two space dimensions to the Lorentz group, a process inverse to the contraction of the Lorentz group with respect to the subgroup of rotations. As an example of the applicability of this modified general light-cone frame, the two-dimensional Schrödinger-Pauli equation is obtained to all orders.     

On a new condition distinguishing Weyl and Lorentz space-times

The paper is concerned with the derivability of a Lorentz instead of only a Weyl manifold as space-time structure from postulates about free fall and light propagation. For this purpose it identifies a property distinguishing both kinds of space-times. The property is one of a particular metric of the conformal class of the Weyl manifold. viz. that in suitably chosen locally geodesic coordinates theg _i4 components,i=1, 2, 3 vanish along the time axis. Although seemingly somewhat hidden, one is led to this property in looking for a metric which can play a distinguished role. We demonstrate that for a Lorentzian manifold such a condition is always given; thus it is a necessary one. It is sufficient since for a Weyl space it has the consequence that the metric connection of the selectedg is projectively equivalent to the Weyl connection. Thus, if a Weyl space-time complies with it, it is a reducible one. The results of this paper lay the ground for deriving in a second step this condition from a simple, empirically testable postulate about free-fall worldlines and “radar” measurements by light signals.     

On dynamical realizations of l-conformal Galilei groups

We consider the dynamics invariant under the action of l-conformal Galilei group using the method of nonlinear realizations. We find that by an appropriate choice of the coset space parametrization one can achieve the complete decoupling of the equations of motion. The Lagrangian and Hamiltonian are constructed. The results are compared with those obtained by Galajinsky and Masterov (2013) [14].     

Hierarchical order of Galilei and Lorentz invariance in the structure of matter

The structure of matter shows a hierarchical order: (1) from Lorentz invariance in high-energy physics; (2) to Galilei invariance in the low-energy nonrelativistic limit of high-energy physics; and (3) again to Lorentz invariance in condensed matter physics (where the velocity of sound takes the place of the velocity of light). The hierarchical order can be continued downward further to: (4) non-relativistic (velocity small compared to the velocity of sound) condensed matter excitons, obeying Galilei invariance; and (5) to excitonic matter obeying Lorentz invariance with an excitonic matter sound velocity. It was previously conjectured that Lorentz invariance of high-energy physics is preceded by Galilei invariance at the Planck scale. Still further, the conjectured Galilei invariance at the Planck scale may be the result of an underlying five-dimensional non-Euclidean conform invariant metric structure, with three spatial and two time dimensions, compactified onto three spatial and one time dimension.     

Discontinuous automorphisms of the proper Galilei and Euclidean groups

It is shown that discontinuous automorphisms of the proper Galilei and Euclidean groups (i.e., without space inversions) exist in three dimensions, but not in higher dimensions.     

On the representations of deformed Galilei group

The projective representations of k-Galilei group G _k are found by contracting the relevant representations of –Poincare group. The projective multiplier is found. It is shown that it is not possible to replace the projective representations of G _k by vector representations of some its extension.     

A Laplace transform on the Lorentz groups

We extend the results of a previous paper to arbitrary non-integrable but polynomially bounded functions defined over any connected semi-simple Lie group of real-rank one. Our approach is based on the method of bilateral horospheres and is a direct generalisation of that used earlier. All the features of the more restricted transform are retained in this more general formalism.     

A Laplace transform on the Lorentz groups

As a first step in the generalisation of the Laplace transform to a non abelian group, we examine the representations of the groupsSO(n, 1) by means of transformations of (not necessarily integrable) functions defined over the hyperboloidsO(n, 1)/O(n). We define a regularised version of the Gel'fand-Graev transformation from then-dimensional hyperboloid to its associated cone, which is valid (under certain restrictions) for polynomially bounded functions. Upon the cone we then carry out a pair of classical Laplace transforms parallel to a generator. We give inversion formulas for both these procedures, and express the Laplace transform/inversion pair directly in terms of the function on the hyperboloid.For integrable functions our results reduce to those already known; in the nonintegrable case they are new. New features include the divergence of the transform for certain discrete asymptotic behaviours; the existence of a finite dimensional kernel subspace which is annihilated; good asymptotic behaviour of both Laplace projection and inversion formulas; and the existence of discrete terms contributing to the inversion formula for even dimension. Our results are valid for all dimensions and are completely independent of the usual Laplace transforms involving projection by means of second-kind representation functions; in a final section of the paper we examine briefly the significance of that approach in the light of our own.     

Octonions and the Lorentz and conformal groups of ten-dimensional space-time

Replacing complex numbers by octonions enables spinors and twistors to be defined for ten-dimensional space-time in close analogy to the four-dimensional case.     

On the possibility of a three-temporal Lorentz transformation

It is suggested that, strictly speaking, light speed invariance would not seem to be fully consistent with ordinary space geometry unless proper time is allowed to undergo one independent dilatation for each degree of space freedom.     

On the Classification of Subalgebras of the Galilei Algebras

Abstract

We investigate the structure of certain types of subalgebras of Galilei algebras and the relationship between the conjugacies of these subalgebras under different groups of automorphisms.     

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.     

Three-dimensional Gross-Neveu model under the condition of Lorentz invariance violation

A 3-D Gross-Neveu model with Lorentz-violating term b _μ is studied. The effective potential V _eff with a real or imaginary Lorentz-violating term b _μ is calculated. The gap equation is obtained and the influence of the presence of an additional term b _μ on the symmetry of the theory is examined.     

On a Lorentz Non-Invariant Generalization of the Dirac Equation

A generalized Dirac equation is presented as a model theory of disturbed Lorentz invariance. The physical properties of this model and experimental consequences are discussed. A program is described how such Lorentz noninvariant equations may be produced by cosmological induction of local causal structure.     

On a possible subtraction for the Lorentz non-invariant model

We define a subtraction of infinities appearing in the Lorentz non-invariant (LNI) renormalization of the electron kinetic term. This makes our previous model of LNI more precise. A relic of LNI appearing in the low-energy electron propagation is calculated, which modifies the Dirac equation as suggested in our previous paper.     

On Lorentz Invariance and the Minimum Length

JETP Letters - It was shown by Kirzhnits and Chechen, following an earlier paper by Mead, that the minimum length scale l is constrained by the Mössbauer effect, which leads to the...     

On the Meaning of Lorentz Covariance

In classical mechanics, Galilean covariance and the principle of relativity are completely equivalent and hold for all possible dynamical processes. In contrast, in relativistic physics the situation is much more complex. It will be shown that Lorentz covariance and the principle of relativity are not completely equivalent. The reason is that the principle of relativity actually only holds for the equilibrium quantities that characterize the equilibrium state of dissipative systems. In the light of this fact it will be argued that Lorentz covariance should not be regarded as a fundamental symmetry of the laws of physics.     

On Lorentz invariant distributions

n-point Lorentz invariant tempered distributions with the supports for one-point only in are described.     

On the generator of Lorentz boost

Traditionally, the theory related to the spatial angular momentum has been studied completely, while the investigation in the generator of Lorentz boost is inadequate. This paper shows that the generator of Lorentz boost has a nontrivial physical significance: it endows a charged system with an electric moment, and has an important significance for the electrical manipulations of electron spin in spintronics. An alternative treatment and interpretation for the traditional Darwin term and spin--orbit coupling are given.