Proper Lorentz Group Research Articles

Equivalence of Möbius Transformation and Lorentz Transformation

The equivalence between the Möbius group acting on and the proper orthochronous Lorentz group on world vectors is reviewed. This relationship will be used to characterize how celestial spheres change with Lorentz transformations.

A Solvable Algebra for Massless Fermions

We derive the stabiliser group of the four-vector, also known as Wigner’s little group, in case of massless particle states, as the maximal solvable subgroup of the proper orthochronous Lorentz group of dimension four, known as the Borel subgroup. In the absence of mass, particle states are disentangled into left- and right-handed chiral states, governed by the maximal solvable subgroups sol2± of order two. Induced Lorentz transformations are constructed and applied to general representations of particle states. Finally, in our conclusions, it is argued how the spin-flip contribution might be closely related to the occurrence of nonphysical spin operators.

Covariant orbital-spin scheme for any spin based on irreducible tensor

In hadron spectrum physics, the partial wave analysis is a primary method used to extract properties of hadronic resonances. The covariant orbital-spin coupling scheme holds unique advantages over other partial wave methods due to its Lorentz covariant form and determined orbital-spin quantum numbers. This paper presents a general form of the covariant orbital-spin coupling scheme based on the irreducible tensor of the homogeneous proper Lorentz group and its little groups. A systematic procedure for constructing partial wave amplitude in a Lorentz covariant way is provided, which can be applied to both massive and massless particles. Specific examples are also included.

Some Formulas for Ordinary and Hyper Bessel–Clifford Functions Related to the Proper Lorentz Group

In this paper, we show that the matrix elements of some Lorentz group representation operators and bases transform operators acting in the representation space may be expressed in terms of the modified Bessel–Clifford functions and their multi-index analogs introduced by the authors via Delerue hyper Bessel functions.

A Parametrization of the General Lorentz Group

We obtain the two new variants of an explicit parametrization for the general Lorentz group. Formulas are given for the direct and inverse four-dimensional Lorentz transformations. These formulas use the orthogonal three- or four-dimensional matrices. We find the infinitesimal operators of the proper Lorentz group and the multiplication formulas (commutators) of the infinitesimal operators. The orthogonal three- and four-dimensional matrices are parameterized by lower triangular matrices containing three or six independent parameters.

Some integrals involving Coulomb functions associated with the three-dimensional proper Lorentz group

For two continual bases in the representation space, we obtain the matrix elements of the linear operator transforming the first basis into the second. These elements are expressed in terms of Coulomb wave functions. Computing the matrix elements of subrepresentations to some subgroups or their separate elements and using the connection between above bases, we evaluate some integrals involving Coulomb wave functions.

Fradkin Equation for a Spin-3/2 Particle in the Presence of External Electromagnetic and Gravitational Fields

Fradkin’s model for a spin-3/2 particle in the presence of external fields is investigated. Applying the general Gel’fand–Yaglom formalism, we develop this model on the base of a set of six irreducible representations of the proper Lorentz group, making up a 20-component wave function. Applying the standard requirements such as the relativistic invariance, single nonzero mass, spin S =3/2, P-symmetry, and existence of a Lagrangian for the model, we derive a set of spinor equations, firstly in the absence of external fields. The 20-component wave function consists of a bispinor and a vector-bispinor. In the absence of external fields, the Fradkin model reduces to the minimal Pauli–Fierz (or Rarita–Schwinger) theory. Details of this equivalence are given. Then we take the presence of external electromagnetic fields into account. It turns out that the Fradkin equation in the minimal form contains an additional interaction term governed by electromagnetic tensor Fab. In addition, we consider the external curved space-time background. In the generally covariant case, the Fradkin equation contains the additional gravitational interaction term governed by the Ricci tensor Rab. If the electric charge of a particle is zero, the Fradkin model remains correct and describes a neutral Majorana-type spin-3/2 particle interacting additionally with the geometric background through the Ricci tensor.

Matrix logarithms and range of the exponential maps for the symmetry groups , and the Lorentz group

Physicists know that covering the continuously connected component of the Lorentz group can be achieved through two Lie algebra exponentials, whereas one exponential is sufficient for compact symmetry groups like SU(N) or SO(N). On the other hand, both the general Baker-Campbell-Hausdorff formula for the combination of matrix exponentials in a series of higher order commutators, and the possibility to define the logarithm of a general matrix through the Jordan normal form, seem to naively suggest that even for non-compact groups a single exponential should be sufficient. We provide explicit constructions of for all matrices in the fundamental representations of the non-compact groups , and SO(1, 2). The construction for also yields logarithms for SO(1, 3) through the spinor representations. However, it is well known that single Lie algebra exponentials are not sufficient to cover the Lie groups and . Therefore we revisit the maximal neighbourhoods and which can be covered through single exponentials with or , respectively, to clarify why or outside of the corresponding domains or . On the other hand, for the Lorentz groups SO(1, 2) and SO(1, 3), we confirm through construction of the logarithm that every transformation in the connectivity component of the identity element can be represented in the form with or , respectively. We also examine why the proper orthochronous Lorentz group can be covered by single Lie algebra exponentials, whereas this property does not hold for its covering group : The logarithms in correspond to logarithms on the first sheet of the covering map , which is contained in . The special linear groups and the Lorentz group therefore provide instructive examples for different global behaviour of non-compact Lie groups under the exponential map.

A Study of Gaugeon Formalism for QED in Lorentz Violating Background

At the energy regimes close to Planck scales, the usual structure of Lorentz symmetry fails to address certain fundamental issues and eventually breaks down, thus paving the way for an alternative road map. It is thus argued that some subgroup of proper Lorentz group could stand consistent and might possibly help us to circumvent this problem. It is this subgroup that goes by the name of Very Special Relativity (VSR). Apart from violating rotational symmetry, VSR is believed to preserve the very tenets of special relativity. The gaugeon formalism due to type-I Yokoyama and type-II Izawa are found to be invariant under BRST symmetry. In this paper, we analyze the scope of this invariance in the scheme of VSR. Furthermore, we will obtain VSR modified Lagrangian density using path integral derivation. We will explore the consistency of VSR with regard to these theories.

Mass, Zero Mass and ...Nophysics

In this paper we demonstrate that massless particles cannot be considered as limiting case of massive particles. Instead, the usual symmetry structure based on semisimple groups like $U(1)$, $SU(2)$ and $SU(3)$ has to be replaced by less usual solvable groups like the minimal nonabelian group ${\rm sol}_2$. Starting from the proper orthochronous Lorentz group ${\rm Lor}_{1,3}$ we extend Wigner's little group by an additional generator, obtaining the maximal solvable or Borel subgroup ${\rm Bor}_{1,3}$ which is equivalent to the Kronecker sum of two copies of ${\rm sol}_2$, telling something about the helicity of particle and antiparticle states.

Weyl, Majorana and Dirac Fields from a Unified Perspective

A self-contained derivation of the formalism describing Weyl, Majorana and Dirac fields from a unified perspective is given based on a concise description of the representation theory of the proper orthochronous Lorentz group. Lagrangian methods play no role in the present exposition, which covers several fundamental aspects of relativistic field theory, which are commonly not included in introductory courses when treating fermionic fields via the Dirac equation in the first place.

The deformation of Poincaré subgroups concerning very special relativity

We investigate here various kinds of semi-product subgroups of Poincare group in the scheme of Cohen-Glashows very special relativity along the deformation approach by Gibbons- Gomis-Pope. For each proper Poincare subgroup which is a semi-product of proper lorentz group with the spacetime translation group T (4), we investigate all possible deformations and obtain all the possible natural representations inherited from the 5- d representation of Poincare group. We find from the obtained natural representation that rotation operation may have additional accompanied scale transformation when the original Lorentz subgroup is deformed and the boost operation gets the additional accompanied scale transformation in all the deformation cases. The additional accompanied scale transformation has a strong constrain on the possible invariant metric function of the corresponding geometry and the field theories in the spacetime with the corresponding geometry.

Cyclic Structures of Cliffordian Supergroups and Particle Representations of Spin+(1, 3)

Supergroups are defined in the framework of \({\mathbb{Z}_2}\) 2-graded Clifford algebras over the fields of real and complex numbers, respectively. It is shown that cyclic structures of complex and real supergroups are defined by Brauer-Wall groups related with the modulo 2 and modulo 8 periodicities of the complex and real Clifford algebras. Particle (fermionic and bosonic) representations of a universal covering (spinor group Spin +(1, 3)) of the proper orthochronous Lorentz group are constructed via the Clifford algebra formalism. Complex and real supergroups are defined on the representation system of Spin +(1, 3). It is shown that a cyclic (modulo 2) structure of the complex supergroup is equivalent to a supersymmetric action, that is, it converts fermionic representations into bosonic representations and vice versa. The cyclic action of the real supergroup leads to a much more high-graded symmetry related with the modulo 8 periodicity of the real Clifford algebras. This symmetry acts on the system of real representations of Spin +(1, 3).

Spinor Wave Equation of Electromagnetic Field

In this paper, we give two spinor wave equations of free electromagnetic field, corresponding to the reducibility and irreducibility representations D (10)+D (01) and D (10) of the proper Lorentz group, which are the differential equations of space-time one order. The spinor equations are covariant and are equivalent to Maxwell equations.

CPT Groups of Higher Spin Fields

$CPT$ groups of higher spin fields are defined in the framework of automorphism groups of Clifford algebras associated with the complex representations of the proper orthochronous Lorentz group. Higher spin fields are understood as the fields on the Poincar\'{e} group which describe orientable (extended) objects. A general method for construction of $CPT$ groups of the fields of any spin is given. $CPT$ groups of the fields of spin-1/2, spin-1 and spin-3/2 are considered in detail. $CPT$ groups of the fields of tensor type are discussed. It is shown that tensor fields correspond to particles of the same spin with different masses.

Dirac, Majorana, and Weyl fermions

We discuss the Dirac, Majorana, and Weyl fermion fields. The definitions and motivations for introducing each kind of field is discussed, along with the connections between them. It is pointed out that these definitions have to do with the proper Lorentz group and not with any discrete symmetry. The action of discrete symmetries, such as charge conjugation and CP on various types of fermion fields, which are particularly important for Majorana fermions, is also clarified.

The Poincaré and related groups are algebraically determined Polish groups

The purpose of this paper is to prove a new topological fact about the Poincare and related groups. IfG is a group, say that G is an algebraically determined Polish (i.e., complete separable metric topological) group if, wheneverH is a Polish group and ϕ:H → G is an algebraic isomorphism, then ϕ is a topological isomorphism. The proper Lorentz group, the proper orthochronous Lorentz group and the Heisenberg group are examples of Polish groups that are not algebraically determined. On the other hand it will be shown that the Lorentz group, the orthochronous Lorentz group and the Poincare group and the other closely associated semi-direct products are algebraically determined Polish groups.

Electromagnetic properties of non-Dirac particles with rest spin 1/2

We resolve a number of questions related to an analytic description of electromagnetic form factors of non-Dirac particles with the rest spin 1/2. We find the general structure of a matrix antisymmetric tensor operator. We obtain two recurrence relations for matrix elements of finite transformations of the proper Lorentz group and explicit formulas for a certain set of such elements. Within the theory of fields with double symmetry, we discuss writing the components of wave vectors of particles in the form of infinite continued fractions. We show that for $Q^{2} \leq 0.5$ (GeV/c)$^{2}$, where $Q^{2}$ is the transferred momentum squared, electromagnetic form factors that decrease as $Q^{2}$ increases and are close to those experimentally observed in the proton can be obtained without explicitly introducing an internal particle structure.

A MATHEMATICAL REVISION OF THE LANDAU-LIFSHITZ EQUATION

For the Landau-Lifshitz equation a Lie type linear system representation in the Minkowski space M(superscript 3+1) has been derived previously [25]. The internal symmetry group is a proper orthochronous Lorentz group SO0(3,1), and the numerical method based on the internal symmetry was developed in [29]. This paper derives another four new representations of the Landau-Lifshitz equation. We prove that this equation admits two generators: one conservative and one dissipative, as well as two brackets: Poisson bracket and dissipative bracket. Upon embedding the Landau-Lifshitz equation into a skew-symmetric matrix space, we can develop a double-bracket flow representation. The conserved magnetization' magnitude is just the result of the isospectrality for an iso-spectral flow equation. Finally, on the cotangent bundle of an invariant manifold of the constant magnetization magnitude, we introduce the Lie-Poisson bracket to construct an evolutional differential equations system. The magnetization trajectory traces a coadjoint orbit in the Poisson manifold under a coadjoint action of the rotation group SO(3). The six different representations including the one by Bloch et al. [3] are compared.

A general theory of matrix transformation and its application to quantum mechanical problems

A general formalism is given to construct a transformation matrix TAB which connects two matrices A and B of order n × n satisfying any given polynomial equations of degree r(≤n). The matrix TAB is given explicitly by a polynomial of degree (r − 1) in A and B. A special case where B is a diagonal matrix equivalent to A leads to a general theory of matrix diagonalization. The formalism contains the projection operator method as a special case. Illustrative examples are given on the Dirac theory of the electron and the basic 2 × 2 spinor representation of the proper orthochronous Lorentz group; the latter is completely prameterized by the spatial rotation and the velocity of the parallel translation.