Lorentz tensor

Because W/tv is an antisymmetric Lorentz tensor, the total iso vector current density must satisfy dvJvw = 0. 

Consequently, the transformation property of a covariant Lorentz tensor under Lorentz transformations is therefore given as the one of an n-fold product of covariant vectors. 

Although the metric g has been defined as constant by Eq. (3.8), we have just shown that it perfectly fits into the definition of a tensor and features the correct transformation property under Lorentz transformations. Another very useful fourth-rank Lorentz tensor is the totally antisymmetric Levi-Civit (pseudo-)tensor whose contravariant components are defined by... 

As for every contravariant Lorentz tensor its indices can be lowered by application of the metric g,... 

The generalization of the previous formalism to encompass systems of n (noninteracting) photons is straightforward. In the Lorentz gauge, an n photon amplitude is a tensor , ) (fcf =... 

I returned to the University of Toronto in the summer of 1940, having completed a Master s degree at Princeton, to enroll in a Ph.D. program under Leopold Infeld for which I wrote a thesis entitled A Study in Relativistic Quantum Mechanics Based on Sir A.S. Eddington s Relativity Theory of Protons and Electrons. This book summarized his thought about the constants of Nature to which he had been led by his shock that Dirac s equation demonstrated that a theory which was invariant under Lorentz transformation need not be expressed in terms of tensors. 

Therefore the fact that 9 is arbitrary in U(l) theory compels that theory to assert that photon mass is zero. This is an unphysical result based on the Lorentz group. When we come to consider the Poincare group, as in section XIII, we find that the Wigner little group for a particle with identically zero mass is E(2), and this is unphysical. Since 9 in the U(l) gauge transform is entirely arbitrary, it is also unphysical. On the U(l) level, the Euler-Lagrange equation (825) seems to contain four unknowns, the four components of , and the field tensor H v seems to contain six unknowns. This situation is simply the result of the term 7/MV in the initial Lagrangian (824) from which Eq. (826) is obtained. However, the fundamental field tensor is defined by the 4-curl ... 

In this second technical appendix, it is shown that the Maxwell-Heaviside equations can be written in terms of a field 4-vector = (0, cB + iE) rather than as a tensor. Under Lorentz transformation, GM transforms as a 4-vector. This shows that the field in electromagnetic theory is not uniquely defined as a... 

The only common factor is that the charge-current 4-tensor transforms in the same way. The vector representation develops a time-like component under Lorentz transformation, while the tensor representation does not. However, the underlying equations in both cases are the Maxwell-Heaviside equations, which transform covariantly in both cases and obviously in the same way for both vector and tensor representations. 

The Lorentz-Lorenz equation can be used to express the components of the refractive index tensor in terms of the polarizability tensor. Recognizing that the birefringence normalized by the mean refractive index is normally very small, ( A/i / 1), it is assumed that Aa /a 1, where the mean polarizability is a = (al + 2oc2)/3 and the polarizability anisotropy is Aa = a1-a2. It is expected that the macroscopic refractive... 

Magnetic field acts on moving electrons by the Lorentz force directed perpendicularly to both the electron velocity and magnetic field. This leads to the appearance of non-diagonal components of the conductivity tensor. Namely, these components cause an electric field perpendicular to both the current flowing through a sample and to the external magnetic field ... 

Noether s theorem will be proved here for a classical relativistic theory defined by a generic field , which may have spinor or tensor indices. The Lagrangian density (, 9/x) is assumed to be Lorentz invariant and to depend only on scalar forms defined by spinor or tensor fields. It is assumed that coordinate displacements are described by Jacobi s theorem S(d4x) = d4x 9/xgeneral variation of the action integral, evaluated over a closed space-time region 2, is... 

In the case of a scalar field, the irreducible matrix D is a unit matrix, and drops out of. I1. For rotation through an angle S9t about the Cartesian axis ek, the rotational submatrix of the Lorentz matrix is given by Xkx = ()Hkekl]x], where el]k is the totally antisymmetric Levi-Civita tensor. For the one-electron Schrodinger field f, Noether s theorem defines three conserved components of a spatial axial vector,... 

The Ewald series for the three-dimensional crystal can also be differentiated. The first derivative yields expressions for the Madelung electric field Fm (due to local charges). The second derivative yields the Madelung field gradient, or, equivalently, the internal or dipolar or Lorentz field FD (due to local dipoles) [68-71],This second derivative can also generates the dimensionless 3x3 Lorentz factor tensor L with its nine components Lv/t ... 

Tensor forms cfClausius93-Mossotti94 and Lorentz-Lorenz95 equations. If there are no permanent local dipoles, but only induced dipoles because atoms or molecules have a tensor polarizability a, then... 

The mass forces may be the gravitational force, the force due to the rotational motion of a system, and the Lorentz force that is proportional to the vector product of the molecular velocity of component i and the magnetic field strength. The normal stress tensor a produces a surface force. No shear stresses occur (t = 0) in a fluid, which is in mechanical equilibrium. 

Many complex fluids contain orientable molecules, particles, and microstmctures that rotate underflow, and under electric and magnetic fields. If these molecules or microstructures have anisotropic polarizabilities, then the index of refraction of the sample will be orientation-dependent, and thus the sample will be birefringent. In general, the anisotropic part of the index of refraction is a tensor n that is related to the polarizability a of the sample. The polarizability is the tendency of the sample to become polarized when an electric field is applied thus P = a E, where P is the polarization and E is the imposed electric field. When the anisotropic part of the index of refraction is much smaller than the isotropic part (the usual case), the index-of-refraction tensor n can be related to a by the Lorentz-Lorenz formula ... 

We now can use the Lorentz lemma to derive the reciprocity relations for the Green s electromagnetic tensor. Let us assume that the electric dipoles with moments a and b are located at points with the radius-vectors r and r",... 

The EM Green s tensors exhibit symmetry and can be shown, using the Lorentz lemma, to satisfy the following reciprocity relations (Felsen and Marcuvitz, 1994) ... 

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