Group Of Lorentz Transformations Research Articles

Differentials of the basis in Clifford Geometric Algebra

In this paper we discuss the dynamic effects of the varying frames. The differential of frame or basis vectors is always equivalent to a linear transformation of the frame, and the linear transformation is not the same in different contexts. In differential geometry, the linear transformation is the connection operator. While in quantum mechanics, the operator algebra corresponds to the differentials of matrices. Corresponding to the variation of the metric, the variation of the frame contains a unusual fourth-order tensor. We also derive the Lie differential of the frame corresponding to the Lorentz transformation group. The definition of differential of the frame is different, so the corresponding linear transformation is also different. In this paper, the unified point of view to deal with the variation of frame or basis vectors will bring great convenience to the research and application of Clifford algebras.

Understanding Lorentz Utilizing Galilei: The Emergence of a Friendly Extended Special Relativity Theory that Admits Relativistic Multi-Particle Entanglement

Special relativity theory stems from the Lorentz transformation of signature (1,3). The incorporation into special relativity of the Lorentz transformations of signature (m,n) for all m,n∈ℕ (n = 3 in physical applications) enriches the theory. The resulting enriched special relativity is a friendly extended special relativity that admits multi-particle entanglement, as demanded by relativistic quantum mechanics. The Lorentz transformation of signature (m,n) admits a novel physical interpretation induced by the intuitively clear interpretation of the Galilei transformation of signature (m,n) for all m,n > 1. In this sense we understand Lorentz utilizing Galilei in m temporal and n spatial dimensions, resulting in the emergence of multi-particle entanglement that the enriched special theory of relativity admits. Remarkably, it turns out that, for any m,n∈ℕ, the group of Lorentz transformations of signature (m,n) is the symmetry group that underlies any multi-particle system that consists of m n - dimensional entangled particles.

Matching tetrads in f(T) gravity

The procedure underlying the matching of 1-form (tetrad) fields in theories possessing absolute parallelism---$f(T)$ gravity being within this category---is addressed and exemplified. We show that the remnant symmetries of the intervening spaces play a central role in the process, because the knowledge of the remnant group of local Lorentz transformations enables one to perform rotations and/or boosts in order to ${\mathcal{C}}^{1}$-match the corresponding tetrads on the junction surface. This automatically ensures the continuity of the Weitzenb\"ock scalar there, even though this proves to be just a necessary condition in order to obtain a global parallelization of the spacetime.

A Spacetime Symmetry Approach to Relativistic Quantum Multi-Particle Entanglement

A Lorentz transformation group SO(m, n) of signature (m, n), m, n ∈ N, in m time and n space dimensions, is the group of pseudo-rotations of a pseudo-Euclidean space of signature (m, n). Accordingly, the Lorentz group SO(1, 3) is the common Lorentz transformation group from which special relativity theory stems. It is widely acknowledged that special relativity and quantum theories are at odds. In particular, it is known that entangled particles involve Lorentz symmetry violation. We, therefore, review studies that led to the discovery that the Lorentz group SO(m, n) forms the symmetry group by which a multi-particle system of m entangled n-dimensional particles can be understood in an extended sense of relativistic settings. Consequently, we enrich special relativity by incorporating the Lorentz transformation groups of signature (m, 3) for all m ≥ 2. The resulting enriched special relativity provides the common symmetry group SO(1, 3) of the (1 + 3)-dimensional spacetime of individual particles, along with the symmetry group SO(m, 3) of the (m + 3)-dimensional spacetime of multi-particle systems of m entangled 3-dimensional particles, for all m ≥ 2. A unified parametrization of the Lorentz groups SO(m, n) for all m, n ∈ N, shakes down the underlying matrix algebra into elegant and transparent results. The special case when (m, n) = (1, 3) is supported experimentally by special relativity. It is hoped that this review article will stimulate the search for experimental support when (m, n) = (m, 3) for all m ≥ 2.

Parametrization of generalized Heisenberg groups

Let M be a left module over a ring R with identity and let $$\beta $$ be a skew-symmetric R-bilinear form on M. The generalized Heisenberg group consists of the set $$M\times M\times R = \{(x, y, t):x, y\in M, t\in R\}$$ with group law $$\begin{aligned} (x_1, y_1, t_1)(x_2, y_2, t_2) = (x_1+x_2, y_1+y_2, t_1+\beta (x_1, y_2)+t_2). \end{aligned}$$ Under the assumption of 2 being a unit in R, we prove that the generalized Heisenberg group decomposes into a product of its subset and subgroup, similar to the well-known polar decomposition in linear algebra. This leads to a parametrization of the generalized Heisenberg group that resembles a parametrization of the Lorentz transformation group by relative velocities and space rotations.

A new perspective on spacetime 4D rotations and the SO(4) transformation group

In quantum field theory, every conservation law is enforced by adequate symmetries. The Lorentz group, which represents two continuous symmetries: rotations in 3D Euclidean space and Lorentz boosts, generates conservation of the angular momentum tensor. This article shows that both Lorentz boosts and 3D rotations can be replaced by spatio-temporal rotations in Minkowski spacetime. Noether’s theorem certifies that the consequences of 4D rotations are identical to the Lorentz transformation group SO(4). I demonstrate how to build a spatio-temporal rotation matrix that preserves the spacetime interval, and that the resulting matrix represents the special orthogonal group of order 4 (SO(4)).

Fundamental physics with cosmic particles

Cosmic rays (charged particles), γ‐rays and neutrinos offer unique opportunities to extend the search for dark matter, to test quantum gravity phenomenology through the study of the invariance of the group of Lorentz transformations, and to probe the existence of sterile neutrinos. We discuss here these tests of beyond the Standard Model particle physics with present and planned future missions including the Pierre Auger Observatory, the space‐borne mission Alpha Magnetic Spectrometer (AMS‐02), Fermi‐LAT, the High Altitude Water Cherenkov (HAWC), the IceCube South Pole Neutrino Observatory as presently operating experiments, and the Cherenkov Telescope Array (CTA) and the IceCube‐Gen2 as future large scale infrastructures. image

Remnant group of local Lorentz transformations inf(T)theories

It is shown that the extended teleparallel gravitational theories, known as $f(T)$ theories, inherit some on shell local Lorentz invariance associated with the tetrad field defining the spacetime structure. We discuss some enlightening examples, such as Minkowski spacetime and cosmological (Friedmann-Robertson-Walker and Bianchi type I) manifolds. In the first case, we show that the absence of gravity reveals itself as an incapability in the selection of a preferred parallelization at a local level, due to the fact that the infinitesimal local Lorentz subgroup acts as a symmetry group of the frame characterizing Minkowski spacetime. Finite transformations are also discussed in these examples and, contrary to the common lore on the subject, we conclude that the set of tetrads responsible for the parallelization of these manifolds is quite vast and that the remnant group of local Lorentz transformations includes one- and two-dimensional Abelian subgroups of the Lorentz group.

Potential for general relativity and its geometry

The unique ghost-free mass and nonlinear potential terms for general relativity are presented in a diffeomorphism and local Lorentz invariant vierbein formalism. This construction requires an additional two-index Stuckelberg field, beyond the four scalar fields used in the metric formulation, and unveils a new local SL(4) symmetry group of the mass and potential terms, not shared by the Einstein-Hilbert term. The new field is auxiliary but transforms as a vector under two different Lorentz groups, one of them the group of local Lorentz transformations, the other an additional global group. This formulation enables a geometric interpretation of the mass and potential terms for gravity in terms of certain volume forms. Furthermore, we find that the decoupling limit is much simpler to extract in this approach; in particular, we are able to derive expressions for the interactions of the vector modes. We also note that it is possible to extend the theory by promoting the two-index auxiliary field into a Nambu-Goldstone boson nonlinearly realizing a certain space-time symmetry, and show how it is "eaten up" by the antisymmetric part of the vierbein.

Point-form dynamics of quasistable states

We present a field theoretical model of point-form dynamics which exhibits resonance scattering. In particular, we construct point-form Poincaré generators explicitly from field operators and show that in the vector spaces for the in-states and out-states (endowed with certain analyticity and topological properties suggested by the structure of the S-matrix) these operators integrate to furnish differentiable representations of the causal Poincaré semigroup, the semidirect product of the semigroup of spacetime translations into the forward lightcone and the group of Lorentz transformations. We also show that there exists a class of irreducible representations of the Poincaré semigroup defined by a complex mass and a half-integer spin. The complex mass characterizing the representation naturally appears in the construction as the square root of the pole position of the propagator. These representations provide a description of resonances in the same vein as Wigner's unitary irreducible representations of the Poincaré group provide a description of stable particles.

New parametrization of lorentz transformations and tachyonic motion in special theory of relativity

Assuming the existence of an invariant velocity a slightly generalized form of Lorentz transformations is derived. The group of these transformations has a simpler composition law than the group of standard Lorentz transformations has. It is shown that this new form allows the description of both subluminal and superluminal motions. It also allows to find all velocity-dependent tensors. In particular, the tachyonic momentum as a function of superluminal velocity is derived.

THE RELATIVISTIC PROPER-VELOCITY TRANSFORMATION GROUP

The Lorentz transformation group of the special theory of relativity is commonly represented in terms of observer’s, or coordinate, time and coordinate relative velocities. The aim of this article is to uncover the representation of the Lorentz transformation group in terms of traveler’s, or proper, time and proper relative velocities. Following a recent demonstration by M. Idemen, according to which the Lorentz transformation group is inherent in Maxwell equations, our proper velocity Lorentz transformation group may pave the way to uncover the proper time Maxwell equations.

The Language of Einstein Spoken by Optical Instruments

Einstein had to learn the mathematics of Lorentz transformations in order to complete his covariant formulation of Maxwell's equations. The mathematics of Lorentz transformations, called the Lorentz group, continues playing its important role in optical sciences. It is the basic mathematical language for coherent and squeezed states. It is noted that the six-parameter Lorentz group can be represented by two-by-two matrices. Since the beam transfer matrices in ray optics is largely based on two-by-two matrices or $ABCD$ matrices, the Lorentz group is bound to be the basic language for ray optics, including polarization optics, interferometers, lens optics, multilayer optics, and the Poincar\'e sphere. Because the group of Lorentz transformations and ray optics are based on the same two-by-two matrix formalism, ray optics can perform mathematical operations which correspond to transformations in special relativity. It is shown, in particular, that one-lens optics provides a mathematical basis for unifying the internal space-time symmetries of massive and massless particles in the Lorentz-covariant world.

Hyperbolic phase transformation group and its applicaton to four-dimensional relativistic spacetime

The hyperbolic complex space RH defined by the Clifford algebra and the hyperbolic phase transformation group U4(H) acting on RH are endowed with definite physical meaning in this paper. The hyperbolic complex space RH is isomorphic to the 4-dimensional(4D) Minkowski spacetime, and the hyperbolic phase transformation group U4(H) in RH is just Lorentz transformation group on 4D relativistic spacetime. Furthermore, the general expressions of Lorentz transformation and the velocity transformation on 4D Minkowski spacetime are naturally derived using the composite transformations of the group U4(H). Hence, the well-known special Lorentz transformation in the special relativity(SR) is contained as a special case in our discussions.

Lorentz group in classical ray optics

It has been almost 100 years since Einstein formulated his special theory of relativity in 1905. He showed that the basic space–time symmetry is dictated by the Lorentz group. It is shown that this group of Lorentz transformations is not only applicable to special relativity, but also constitutes the scientific language for optical sciences. It is noted that coherent and squeezed states of light are representations of the Lorentz group. The Lorentz group is also the basic underlying language for classical ray optics, including polarization optics, interferometers, the Poincar´ es phere, one-lens optics, multi-lens optics, laser cavities, as well multilayer optics.

Formula omitted] a new symmetry possibly realizing in hadron spectroscopy

Starting from the multi-local Klein–Gordon equations with Lorentz-scalar squared-mass operator we give a covariant quark representation of the general composite mesons and baryons with definite Lorentz transformation property. The phenomenologically observed hadron mass spectra is pointed out to satisfy possibly the approximate symmetry under the U(12) transformation group of light constituent quarks. Here U(12)⊃ U(4) DS ⊗SU(3) F , and U(4) DS is a homogeneous Lorentz transformation group for Dirac spinors, while SU(3) F is the flavor unitary group. This symmetry predicts the existence of new type of chiral mesons and baryons out of the conventional level classification scheme: typical ones are for light q q ̄ system the scalar σ- and axial-vector a 1-nonets to be assigned as relativistic S-wave states, while for light-quark–baryon-system the Roper resonance N(1440) (J P= 1 2 +) and the Λ(1405) (J P= 1 2 −) with comparatively light mass to be assigned as the ground states.

Iwasawa effects in multilayer optics.

There are many two-by-two matrices in layer optics. It is shown that they can be formulated in terms of a three-parameter group whose algebraic property is the same as the group of Lorentz transformations in a space with two spacelike and one timelike dimensions, or the Sp(2) group which is a standard theoretical tool in optics. Among the interesting mathematical properties of this group, the Iwasawa decomposition drastically simplifies the matrix algebra under certain conditions, and leads to a concise expression for the S matrix for transmitted and reflected waves. It is shown that the Iwasawa effect can be observed in multilayer optics, and a sample calculation of the S matrix is given.

Topology and a lorentz-invariant pseudo-riemannian metric of the manifold of directions in the physical space

In the mathematical model of the special relativity theory, a two-dimensional Minkowski subspace is treated as a one-dimensional direction in the physical space. The manifold of such planes is naturally endowed with the structure of a pseudo-Riemannian manifold on which the group of isochronous Lorentz transformations acts transitively by isometries. In this paper, the topology and the metric geometry of this manifold are studied. Bibliography: 4 titles.

Reflections, spinors, and projections on a Minkowski space underlie Dirac's equation

Reflections and spinors on a Minkowski space are obtained by the methods of Cartan. The group of all such reflection transformations has as its subgroup the Poincaré group of Lorentz transformations. Two types of spinors are shown to exist for a Minkowski space. Spinor reflections and Lorentz transformations exist for both types of spinors. Associated with these spinors are two Hermitian, orthogonal projection operators which together spectrally resolve the identity operator of a two-dimensional complex vector space. These spinors and their associated projection operators are applied to find the structure of a 4 × 4 matrix G which is equivalent to the relativistic conservation law of the energy and momentum of a single moving particle. These spinor-calculus procedures demonstrate that G is singular of rank 2, and as a consequence the solutions of G θ = 0 consists of all bispinor elements of the null space of G. This equation and its solutions are equivalent to those of Dirac's quantum-mechanical equation of an electron in the Fourier domain of frequency and wavenumber. Finally some properties of 2-spinors and 4-spinors found herein are shown to extend naturally to n dimensions.

Two-component spinor fields in time-nonorientable spacetimes

This paper examines spinor structures and two-component spinor fields in in a class of spacetimes that are space-orientable but not time-orientable. The space-oriented frames form a principal bundle acted on by the group of proper nonorthochronous Lorentz transformations. This group has two double coverings, Sin and Sin, but only Sin acts on the usual two-component spinors associated with Weyl neutrinos in Minkowski space. Consideration is initially restricted to Lorentzian universes-from-nothing, geometries, like antipodally identified deSitter space, that have a single spacelike boundary and a smooth metric with Lorentzian signature. Every such spacetime has a Sin structure, but only a subclass has a Sin structure. Inequivalent Sin- and Sin-spinor structures correspond to members of two classes of homomorphisms from to , where is the orientable double covering of the spacetime manifold M. For general time-nonorientable spacetimes, a similar classification is obtained of Sin structures in terms of homomorphisms from to where E is the bundle of space-oriented frames of M.