Maxwell field

Notice that the results obtained so far (Eqs. (19) and (23) for the Maxwell field and Eqs. (20) and (24) for the Dirac field) are particular cases of the above expressions, corresponding to a = (/3,0,0,0) and a = (0,0,0,i2L) respectively. Another important aspect is that T 11) ) is traceless in both cases, as it should be. Now, we will apply these general results to some specific examples. 

Hence, it follows that C( 1, 3) x -invariant ansatzes for the Maxwell fields, which reduce (99) to systems of ordinary differential equations, can be represented in the form (22), namely,... 

After some algebra, we obtain the following form of the conformally invariant ansatz for the Maxwell fields ... 

Given any Maxwell field in vacuum, we define the current density of electromagnetic helicity Jf x as one-half times the sum of the current densities of electric and magnetic helicities (41) and (42) ... 

By construction, and taking into account Eqs. (43), the density of electromagnetic helicity is a conserved current for any Maxwell field in vacuum (with the above indicated behavior at infinity) ... 

With the connection of PDEs, and especially soliton forms, to group symmetries established, one can conclude that if the Maxwell equation of motion that includes electric and magnetic conductivity is in soliton (SGE) form, the group symmetry of the Maxwell field is SU(2). Furthermore, because solitons define Hamiltonian flows, their energy conservation is due to their symplectic structure. 

The general dispersion formula obtained for the coupling of the vibrational equations with the Maxwell field can be brought into the form of Fresnel s wellknown equation for the wave normal from crystal optics. It is usually written in the form... 

The solutions to the Maxwell field equations that are most often used in the applications discussed in this book are referred to as plane waves of monochromatic light. These are derived from the Maxwell curl equations. In a system free of charges and currents, these are... 

This book has been written for the practitioner, as well as researchers seeking to either predict the optical response of complex liquids or to interpret optical data in terms of microstructural attributes. For these purposes, the book is meant to be self contained, beginning with sections on the fundamental Maxwell field equations describing the interaction of electromagnetic waves with anisotropic media. These interactions include... 

In order to formulate a theory for the evaluation of vibrational intensities within the framework of continuum solvation models, it is necessary to consider that formally the radiation electric field (static, Eloc and optical E[jc) acting on the molecule in the cavity differ from the corresponding Maxwell fields in the medium, E and Em. However, the response of the molecule to the external perturbation depends on the field locally acting on it. This problem, usually referred to as the local field effect, is normally solved by resorting to the Onsager-Lorentz theory of dielectric polarization [21,44], In such an approach the macroscopic quantities are related to the microscopic electric response of... 

The key differences between the PCM and the Onsager s model are that the PCM makes use of molecular-shaped cavities (instead of spherical cavities) and that in the PCM the solvent-solute interaction is not simply reduced to the dipole term. In addition, the PCM is a quantum mechanical approach, i.e. the solute is described by means of its electronic wavefunction. Similarly to classical approaches, the basis of the PCM approach to the local field relies on the assumption that the effective field experienced by the molecule in the cavity can be seen as the sum of a reaction field term and a cavity field term. The reaction field is connected to the response (polarization) of the dielectric to the solute charge distribution, whereas the cavity field depends on the polarization of the dielectric induced by the applied field once the cavity has been created. In the PCM, cavity field effects are accounted for by introducing the concept of effective molecular response properties, which directly describe the response of the molecular solutes to the Maxwell field in the liquid, both static E and dynamic E, [8,47,48] (see also the contribution by Cammi and Mennucci). 

In parallel the the field-dependent part of the free energy of the molecule in the presence of the Maxwell field is ... 

In the presence of a Maxwell field E(r) the electronic Hamiltonian of the solute can... 

The OWB model describes the solute as a classical polarizable point dipole located in a spherical or ellipsoidal cavity in an isotropic and homogeneous dielectric medium representing the solvent. In the presence of a macroscopic Maxwell field E, the solute experiences an internal (or local) field E given by a superposition of a cavity field Ec and a reaction field ER. In terms of Fourier components E -n, Ec,n, ER,n of the fields we have... 

The cavity and the reaction fields are related to the Maxwell field in the medium and to the total (permanent+induced) dipole moment /xa of the molecule at the frequency O by... 

The OWB equations obtained in this semiclassical scheme analyse the effective polarizabilities in term of solvent effects on the polarizabilities of the isolated molecules. Three main effects arise due to (a) a contribution from the static reaction field which results in a solute polarizability, different from that of the isolated molecules, (b) a coupling between the induced dipole moments and the dielectric medium, represented by the reaction field factors FR n, (c) the boundary of the cavity which modifies the cavity field with respect the macroscopic field in the medium (the Maxwell field) and this effect is represented by the cavity field factors /c,n. 

All these effects are considered in a more consistent and general way in the PCM framework, where the coupling between the induced electronic charge distribution (not limited to the dipolar component but described by the QM wavefunction) and the external medium is represented by the reaction potential produced by the apparent charges, while the boundary effect on the Maxwell field is represented by the matrices m . 

From the point of view of theory, the formulae of Table 2.6 are equally applicable to both gas and condensed phase samples, as they include the local field factors, which account for local modifications to the Maxwell fields due to bulk interactions within the Onsager-Lorentz model. 

The quantity on the left is the Fourier component of the dipole moment induced by the optical field Max(w). These equations can be generalized to mixed frequency-dependent electric dipole, electric quadrupole, magnetic dipole properties, and similar equations can be written for the Fourier components of the permanent electric quadrupole, aj8(magnetic dipole, ma(co). For static Maxwell fields similar expansions yield effective (starred) properties, defined as derivatives of the electrostatic free energies. 

The Dirac and Maxwell fields are coupled through the covariant derivative... 

In classical electrodynamics, the field equations for the Maxwell field A/( depend only on the antisymmetric tensor which is invariant under a gauge transformation A/l A/l + ticduxix), where x is an arbitrary scalar field in space-time. Thus the vector field A/( is not completely determined by the theory. It is customary to impose an auxiliary gauge condition, such as 9/x/Fx = 0, in order to simplify the field equations. In the presence of an externally determined electric current density 4-vector j11, the Maxwell Lagrangian density is... 

For the Dirac field in an externally determined Maxwell field, the Lagrangian density including a renormalized mass term is... 

For the Maxwell field, the energy-momentum tensor Tfi(A) derived from Noether s theorem is unsymmetric, and not gauge invariant, in contrast to the symmetric stress tensor derived directly from Maxwell s equations [318], Consider the symmetric tensor 0 = T + AT, where... 

In terms of these vector fields, the Maxwell field Lagrangian density is (1 /8 n) 2 — B2), and the field energy and momentum are... 

For interacting fields, the Maxwell field energy is not separately conserved. A gauge-covariant derivation follows from the inhomogeneous field equations (Maxwell equations in vacuo),... 

Since local energy and momentum density are well-defined for the classical Maxwell field, respectively... 

The local conservation law for the interacting gauge field can be derived from the covariant field equations, as was done above for the Maxwell field. Using the SU(2) field equations and expanding (3VW0>-) WV/l as... 

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