Lorentz homogeneous

It relates the space time coordinates xf of an event as labeled by an observer 0, to the space-time coordinates of the same event as labeled by an observer O . The most general homogeneous Lorentz transformation is the real linear transformation (9-8) which leaves invariant the quadratic form... 

Note that the scalar product is formally the same as in the nonrela-tivistic case it is, however, now required to be invariant under all orthochronous inhomogeneous Lorentz transformations. The requirement of invariance under orthochronous inhomogeneous Lorentz transformations stems of course from the homogeneity and isotropy of space-time, send corresponds to the assertion that all origins and orientation of the four-dimensional space time manifold are fully equivalent for the description of physical phenomena. 

We next inquire as to the transformation property of the quantity ip(z) under a homogeneous Lorentz transformation... 

Hence, under a proper homogeneous Lorentz transformation -without time inversion the quantity transforms like a scalar ... 

The right side is again determined by the fact that tjr(x) must transform like a spinor under a homogeneous Lorentz transformation. Its form is made transparent by recalling that for an infinitesimal homogeneous Louentz transformation... 

Lorentz approximation, 46 Lorentz condition, 551 Lorentz gauge, 657,664 Lorentz group homogeneous, 490... 

Lorentz invariant scalar product, 499 of two vectors, 489 Lorentz transformation homogeneous, 489,532 improper, 490 inhomogeneous, 491 transformation of matrix elements, 671... 

The basic mechanism of spin-orbit coupling is magnetic induction. It is therefore a truly relativistic effect, as will be discussed shortly. The potentials of a moving charge can be found from Maxwell s equations, as direct solutions or from Lorentz transformations of potentials of a static charge to a moving frame. Maxwell s equations can be divided into the homogeneous parr... 

The return to equilibrium of a polarized region is quite different in the Debye and Lorentz models. Suppose that a material composed of Lorentz oscillators is electrically polarized and the static electric field is suddenly removed. The charges equilibrate by executing damped harmonic motion about their equilibrium positions. This can be seen by setting the right side of (9.3) equal to zero and solving the homogeneous differential equation with the initial conditions x = x0 and x = 0 at t = 0 the result is the damped harmonic oscillator equation ... 

In conclusion, the homogeneous field equation of 0(3) electrodynamics is Lorentz-invariant, and all its classical solutions must be also Lorentz-invariant. The same result is obtained therefore in QED. 

The first ellipsometric measurement of the thickness of the adsorbed layer and the adsorbance of a polyelectrolyte and a negative adsorbance of salt onto a solid surface was reported by Takahashi et al.U4) They measured the adsorption of sodium poly(acrylate) (M = 950 x 103) onto a platinum plate as a function of the concentration of added sodium bromide. In an aqueous polyelectrolyte solution with an added simple salt, the bulk phase is a three-component system which consists of a polyelectrolyte, a simple salt, and water. The adsorbed layer on the solid surface is a three-component phase as well. The adsorbance of polyelectrolytes thus cannot easily be determined from measurements of the refractive index nf of the adsorbed phase. Hence, it was assumed that the adsorbed layer is a homogeneous layer of thickness t and further that nf is represented by the Lorenz-Lorentz equation as follows ... 

This amplitude is found from the equation of motion of a harmonic oscillator affected by an a.c. field E(t). This approach yields the Lorentz and Van Vleck-Weisskopf lines, respectively, for a homogeneous and Boltzmann distributions of the initial a.c. displacements x(l0) established after instant to of a strong collision. The susceptibility corresponding to the Van Vleck-Weisskopf line in terms of our parameters is given by [66]... 

In the relations between the macroscopic susceptibilities y , y and the microscopic or molecular properties a, ft, y, local field corrections have to be considered as explained above. The molecule experiences the external electric field E altered by the polarization of the surrounding material leading to a local electric field E[oc. In the most widely used approach to approximate the local electric field the molecule sits in a spherical cavity of a homogenous media. According to Lorentz the local electric field [9] is... 

In a Penning trap, ions are constrained spatially by a combination of electric and magnetic fields. In the presence of a homogeneous magnetic field (B) ions that have a component of velocity perpendicular to B are subjected to the Lorentz force and subsequently follow a circular trajectory [70]. The rotational frequency of this trajectory is proportional to B and to the ion charge (q) and inversely proportional to the ion mass (m). When expressed in radians, this rotational frequency is known as the cyclotron frequency, coc, where... 

In addition to the field generated by the adjacent dipoles, there is a macroscopic fieldE due to the presence of charges and of the average polarization P of the medium. In the Lorentz treatment of polarization, for a constant macroscopic field in a linear and homogeneous medium of dielectric constant e (hence satisfying P = e0(e — 1)E), the local field E n(fl at a site of a selected dipole is related to the macroscopic field E via19... 

Let us first review the basics of the Lorentz theory for polarization. If one assumes that a constant macroscopic field is applied to a homogeneous medium of dielectric constant s, the polarization through the medium will be uniform. However, the polarization of a molecule is not proportional to the macroscopic electric field (created by sources external to the dielectric), but to the local electric field, which contains also the field generated by all the other molecules of the dielectric. To account for the latter, one can separate the medium in a spherical cavity (in which the central molecule and its molecular neighbors reside, see Fig. 1 A) and the rest of the medium, which... 

The refraction index n is the experimentally measurable quantity which determines the interaction between light and a homogeneous condensed material. From the value of this index, one determines the polarizability a of the polarizable element. In the case of isotropic elements, with concentration Nj V in a homogeneous material, the relation is given by the Lorentz-Lorenz1 formula... 

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