Lorentz field Matrices

The previous results become somewhat more transparent when consideration is given to the manner in which matrix elements transform under Lorentz transformations. The matrix elements are c numbers and express the results of measurements. Since relativistic invariance is a statement concerning the observable consequences of the theory, it is perhaps more natural to state the requirements of invariance as a requirement that matrix elements transform properly. If Au(x) is a vector field, call... 

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 crudest approximation to the density matrix for the system is obtained by assuming that there are no statistical correlations between the elementary excitations (perfect fluid), so that can be written as a simple product of molecular density matrices A. A better approximation is obtained if one does a quantum field theory calculation of the local field effects in the system which in a certain approximation gives the Lorentz-Lorenz correction L(TT) in terms of the refractive index n53). One then writes,... 

The dipole strength of an induced electric dipole transition is proportional to the square of the matrix element in the dipole operator and therefore also to the square of the electric field at the lanthanide site. However, in intensity studies, the lanthanide ions are not in a vacuum, but embedded in a dielectric medium. The lanthanide ion in a dielectric medium not only feels the radiation field of the incident light, but also the field from the dipoles in the medium outside a spherical surface. The total field consisting of the electric field E of the incident light (electric field in the vacuum), plus the electric field of the dipoles is called the effective field eff> i e. the field effective in inducing the electric dipole transition. The square of the matrix element in the electric dipole operator has to be multiplied by a factor E fflEf. In a first approximation, ( efr/ = ( + 2) /9. The factor (n + 2fl9 is the Lorentz local field correction and accounts for dipole-dipole corrections. 

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