Vector parameterization of the Lorentz group transformations and polar decomposition of Mueller matrices

It is shown that the representation of the coherence matrix (the polarization density matrix) of beams of electromagnetic waves as a biquaternion corresponding to the four-vector of a pseudo-Euclidean space whose components are the intensity and the Stokes parameters provides a possibility of introducing the group transformations of these quantities isomorphic to SO(3.1) group. These transformations are a subset of the set of Mueller polarization matrices which, generally speaking, form a semigroup. The reduction of the semigroup of Mueller matrices to the group of transformations opens the possibility to use the vector parameterization of SO(3.1) group for interpretation of the polar decomposition of Mueller matrices. In particular, in this approach, the elements of the Mueller matrices corresponding to phase elements and polarizers turn out to be most simply and naturally related to their eigenpolarizations.     

A q-deformed Lorentz algebra

We derive a q-deformed version of the Lorentz algebra by deforming the algebraSL(2,C). The method is based on linear representations of the algebra on the complex quantum spinor space. We find that the generators usually identified withSL _q(2,C) generateSU _q (2) only. Four additional generators are added which generate Lorentz boosts. The full algebra of all seven generators and their coproduct is presented. We show that in the limitq→1 the generators are those of the classical Lorentz algebra plus an additionalU(1). Thus we have a deformation ofSL(2,C)×U(1).     

The introduction of Superluminal Lorentz transformations: A revisitation

We revisit the introduction of the Superluminal Lorentz transformations which carry from bradyonic inertial frames to tachyonic inertial frames, i.e., which transform time-like objects into space-like objects, andvice versa. It has long been known that special relativity can be extended to Superluminal observers only by increasing the number of dimensions of the space-time or—which is in a sense equivalent—by releasing the reality condition (i.e., introducing also imaginary quantities). In the past we always adopted the latter procedure. Here we show the connection between that procedure and the former one. In other words, in order to clarify the physical meaning of the imaginary units entering the classical theory of tachyons, we have temporarily to call into play anauxiliary six-dimensional space-time M(3, 3); however, we are eventually able to go back to the four-dimensional Minkowski space-time. We revisit the introduction of the Superluminal Lorentz transformations also under another aspect. In fact, the generalized Lorentz transformations had been previously written down in a form suited only for the simple case of collinear boosts (e.g., they formed a group just for collinear boosts). We express now the Superluminal Lorentz transformations in a more general form, so that they constitute a group together with the ordinary—orthochronousand antichronous—Lorentz transformations, and reduce to the previous form in the case of collinear boosts. Our approach introduces either real or imaginary quantities, with exclusion of (generic) complex quantities. In the present context, a procedure—in two steps—for interpreting the imaginary quantities is put forth and discussed. In the case of a chain of generalized Lorentz transformations, such a procedure (when necessary) is to be applied only at the end of the chain. Finally, we justify why we call transformations also the Superluminal ones.     

洛伦兹变换的一种新推导

对于洛伦兹变换中两个惯性系S和S′系的约定提出了新思路,使x轴与x′轴反向,从而使S和S′系完全对称,简化了推导过程;并根据约定,从狭义相对论的两条基本假设出发,严格地推导出了洛伦兹变换式.     

Electrodynamics: A consequence of nonlinear realizations of the Lorentz group

Extensions from the representations of the Lorentz group to include local nonlinear diagonal transformations is sufficient to generate, via the covariant derivative, the interaction of minimal coupling. These diagonal realizations are characterized by six functions φ which must satisfy a system of transformation equations. Inequivalent categories of solutions for the φ give rise to different electromagnetic fields. The Dirac monopole and Coulomb potentials follow directly from two different categories of these nonlinear realizations. Within this theory, charge becomes simply the nonlinear counterpart of intrinsic spin for aparticular nonlinear realization of the Lorentz group. Charge is thus placed on equal footing with intrinsic spin in the sense that both phenomena can be described as consequences of our space-time symmetry. Other solutions for the six φ exist, including a spinor. We briefly discuss the possibility that with these other solutions, these realizations could represent some other basic properties of elementary particles.     

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.     

Lorentz lattice gases: Basic theory

We present several ballistic models of the Lorentz gas in two-dimensional lattices with deterministic and stochastic deflection rules, and their corresponding Liouville equations. Boltzmann-level-equation results are obtained for the diffusion coefficient and velocity autocorrelation function for models with stochastic deflection rules. The long-time behavior of the mean square displacement is briefly discussed and the possibility of abnormal diffusion indicated. Even if the diffusion coefficient exists, its low-density limit may not be given correctly by the Boltzmann equation.     

Lorentz manifolds modelled on a Lorentz symmetric space

We give examples of Lorentz manifolds modelled on an indecomposable Lorentz symmetric space which are geodesically complete and not locally homogeneous.     

Lorentz cobordism

A Lorentz cobordism between two (in general nondiffeomorphic) 3-manifoldsM ₀,M ₁ is a pair (M,v), whereM is a differentiable 4-manifold andv is a differentiable vector field onM, such that 1) the boundary ofM is the disjoint union ofM ₀ andM ₁, 2)v is everywhere nonzero, 3)v is interior normal onM ₀ and exterior normal onM ₁. Such a manifoldM admits a Lorentz tensor with respect to whichM ₀ andM ₁ are spacelike hypersurfaces; thus a Lorentz cobordism is a model of a portion of a spacetime in which the topology of spacelike hypersurfaces is changing. We discuss the form that these changes can take, and give two methods for constructing a Lorentz cobordism between two nondiffeomorphic 3-manifolds. We comment on the possible relevance of Lorentz cobordism to the problem of gravitational collapse.     

Realisation of a Lorentz algebra in Lorentz violating theory

A Lorentz non-invariant higher derivative effective action in flat spacetime, characterised by a constant vector, can be made invariant under infinitesimal Lorentz transformations by restricting the allowed field configurations. These restricted fields are defined as functions of the background vector in such a way that background dependence of the dynamics of the physical system is no longer manifest. We show here that they also provide a field basis for the realisation of a Lorentz algebra and allow the construction of a Poincaré invariant symplectic two-form on the covariant phase space of the theory.     

A dynamical partition function for the Lorentz gas

In this paper we introduce a dynamically defined partition function for the Lorentz gas and investigate its connection with the classical ensembles and the phase-space probability measure derived from periodic orbit expansions. Numerical evidence is presented to support the equivalence of these measures and to link them to the thermodynamic quantities for the Lorentz gas. This also suggests a new dynamical basis for the assumption of equala priori probabilities in the microcanonical ensemble.     

自制洛兹力投影演示仪

电磁现象的演示,对提高教学效果有重要意义.我们自制了洛伦兹力投影演示仪,并在教学实践中不断改进、应用,收到很好效果.现介绍如下.     

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.     

洛伦兹速度变换的一种直接推导方法

用光速不变原理直接导出洛伦兹速度变换式，并证明相对论性速度构成双曲几何的Beltrami-Klein模型。     

Lorentz Transformations from Reflections: Some Applications

We point out, by exhibiting two examples and mentioning a third one, that it is sometimes useful to consider Lorentz transformations as generated from hyperplane or line reflections. One example concerns the construction of boosts linking two given 4-vectors, the other one concerns the Minkowski geometric understanding of V. Moretti's polar decomposition of orthochronous Lorentz matrices.     

Lorentz electron microscopy

This paper reviews the theory and practice of Lorentz electron microscopy in the investigations of magnetic domain structure. The transmission electron microscope, the necessary operational procedures and imaging theory required for Lorentz microscopy are described. Magnetization devices and other facilities that have been developed over the last two years to study domain magnetization in situ in the electron microscope are discussed. Results so far obtained on such materials as thin single crystals and polycrystals of ferromagnetic metals, magnetic oxides, evaporated and electrodeposited single crystal and polycrystal films, and whisker and single domain particles are reviewed. The methods available for preparing specimens in a suitable form for transmission electron microscopy are described.     

Lorentz force velocimetry

We describe a noncontact technique for velocity measurement in electrically conducting fluids. The technique, which we term Lorentz force velocimetry (LFV), is based on exposing the fluid to a magnetic field and measuring the drag force acting upon the magnetic field lines. Two series of measurements are reported, one in which the force is determined through the angular velocity of a rotary magnet system and one in which the force on a fixed magnet system is measured directly. Both experiments confirm that the measured signal is a linear function of the flow velocity. We then derive the scaling law that relates the force on a localized distribution of magnetized material to the velocity of an electrically conducting fluid. This law shows that LFV, if properly designed, has a wide range of potential applications in metallurgy, semiconductor crystal growth, and glass manufacturing.     

Lorentz effect imaging

This paper presents a method that can detect minute electrical activity in a strong magnetic field. It uses displacement encoding to detect small spatial displacement induced by Lorentz force on the conducting materials, hence the term Lorentz effect imaging (LEI). With increased sensitivity from improved hardware capabilities or signal averaging, this technique may be used to detect spatial displacements induced by small currents comparable to neuronal electrical current. The initial results using the LEI technique may provide insight in assessing the feasibility of using MRI to non-invasively detect the neuronal electrical activities.