Electromagnetic field vector potential

It is to be expected that the equations relating electromagnetic fields and potentials to the charge current, should bear some resemblance to the Lorentz transformation. Stating that the equations for A and (j> are Lorentz invariant, means that they should have the same form for any observer, irrespective of relative velocity, as long as it s constant. This will be the case if the quantity (Ax, Ay, Az, i/c) = V is a Minkowski four-vector. Easiest would be to show that the dot product of V with another four-vector, e.g. the four-gradient, is Lorentz invariant, i.e. to show that... 

One can see that the full Hamiltonian consists of three terms, two which describe separately the parts for the atom and the field, and one which represents a coupling between the field (vector potential A) and terms from the atom (operator V,-). Obviously, it is this mixed term which is responsible for the photon-atom interaction. Provided perturbation theory can be applied, this term then acts as a transition operator between undisturbed initial and final states of the atom. Following this approach, one has to verify whether the disturbance caused by the electromagnetic field in the atom is small enough such that perturbation theory is applicable. Hence, one has to compare the terms which contain the vector potential A with an energy ch that is characteristic for the atomic Hamiltonian ... 

Maxwell s equation are the basis for the calculation of electromagnetic fields. An exact solution of these equations can be given only in special cases, so that numerical approximations are used. If the problem is two-dimensional, a considerable reduction of the computation expenditure can be obtained by the introduction of the magnetic vector potential A =VxB. With the assumption that all field variables are sinusoidal, the time dependence... 

The identical transformation, equation (6), of the electromagnetic vector potentials was found before to leave the fields unaffected or gauge invariant. The fields Atl are not gauge invariant, but the fields described by the tensor, equation (33)... 

This may well appear not to produce anything new until the electron is examined in an external electromagnetic field, represented by a scalar potential V, and a vector potential A. The appropriate operators then become... 

It is important to note once again that <, and A in equation (75) are the scalar and vector potential resulting from the external electromagnetic radiation field. Also note that here, , = V(, - and S, is the spin of the ith particle. We can identify each term in equation (75) as corresponding to a certain type of physical interaction of moving charged particles. The list of physical interpretations of terms follows in the same order as the terms in equation (75). 

The interaction Hamiltonian contains the operator A, corresponding to the vector potential A of the electromagnetic field.2 Excluding magnetic scattering, the interaction Hamiltonian is given by... 

The accurate quantum mechanical first-principles description of all interactions within a transition-metal cluster represented as a collection of electrons and atomic nuclei is a prerequisite for understanding and predicting such properties. The standard semi-classical theory of the quantum mechanics of electrons and atomic nuclei interacting via electromagnetic waves, i.e., described by Maxwell electrodynamics, turns out to be the theory sufficient to describe all such interactions (21). In semi-classical theory, the motion of the elementary particles of chemistry, i.e., of electrons and nuclei, is described quantum mechanically, while their electromagnetic interactions are described by classical electric and magnetic fields, E and B, often represented in terms of the non-redundant four components of the 4-potential, namely the scalar potential and the vector potential A. 

As a next step we also need to specify the magnetic and retardation interactions experienced by an electron i and generated by all other electrons. In a first approximation retardation is neglected and we assume that electron i experiences the electromagnetic field immediately. For the scalar potential j,unret, and the vector potential A/ Unret created by electron j and felt by electron i the classical expression reads ... 

So far, we considered only the unretarded electromagnetic field. However, for the correct expression, we have to include the retardation of the vector potential due to the finite speed of light. We may obtain from Darwin s classical electromagnetic interaction energy expression (21) (correct up to 0(c 2)),... 

If there is no electromagnetic field present, the quantized vector potential fluctuates according to... 

The quaternion-valued vector potential and the 4-current J both depend directly on the curvature tensor. The electromagnetic field tensor in the Sachs theory has the form... 

As is well known in classical electromagnetics, the fields described by the Maxwell equations can be derived from a vector potential and a scalar potential. However, there are various forms that are possible, all giving the same fields. This is referred to as gauge invariance. In making measurements at some point... 

So now we have the question poased in an interesting form. There are two quite different kinds of antennas, both of which produce electric dipole fields, but different Lorenz potentials, one emphasizing the vector potential and the other, the scalar potential. In a classical electromagnetic sense, one cannot distinguish these two cases by measurements of the fields (the measurable quantities) at distances away from the source region. The gauge invariance of QED implies the same in quantum sense. 

There is also another way to think about how the vector potential, specifically its curl part, operates in QED. One can envision, on one hand, a static condition where the phase change of v[/ around a closed path, with no electromagnetic fields on the path, can be related to A. However, to establish this static condition, there is required a net time integral of E along the path (from previous times), that is, the electric impulse, to establish these new static conditions. An alter-... 

Further, in 1904 Whittaker [56] (see also Section V.C.2) showed that any electromagnetic field, wave, etc. can be replaced by two scalar potential functions, thus initiating that branch of electrodynamics called superpotential theory [58]. Whittaker s two scalar potentials were then extended by electrodynamicists such as Bromwich [59], Debye [60], Nisbet [61], and McCrea [62] and shown to be part of vector superpotentials [58], and hence connected with A. 

Here d>,A denotes the components of the four-dimensional vector-potential of the electromagnetic field corresponding to a definite state... 

Here,, 4 ( is the vector 4-potential introduced in the vacuum as part of the covariant derivative, and therefore introduced by spacetime curvature. The electromagnetic field and the topological charge g are the results of the invariance of the Lagrangian (868) under local U(l) gauge transformation, in other words, the results of spacetime curvature. 

The quantity A appears in these equations and is the vector potential of electromagnetic theory. In a very elementary discussion of the static electric field we are introduced to the theory of Coulomb. It is demonstrated that the electric field can be written as the gradient of a scalar potential E = —Vc)>, constant term to this potential leaves the electric field invariant. Where you choose to set the potential to zero is purely arbitrary. In order to describe a time-varying electric field a time dependent vector potential must be introduced A. If one takes any scalar function % and uses it in the substitutions... 

The second and third terms are the interaction terms that couple the atom, here modeled as a two-state system with Pauli matrices, to the electromagnetic field. We consider the momentum to be the operator p - - V and consider this operator as not only operating on the vector potential but on the wavefunction. Hence we find that... 

The laws of electromagnetism are based on the theory of gauge fields. The electromagnetic vector potential defines components of a gauge connection 1-form. This gauge connection defines a field strength 2-form ... 

It is then advanced that electromagnetism is expanded into a theory with three vector potentials and the conjugate product that determines an additional magnetic field,... 

Non-Abelian electrodynamics has been presented in considerable detail in a nonrelativistic setting. However, all gauge fields exist in spacetime and thus exhibits Poincare transformation. In flat spacetime these transformations are global symmetries that act to transform the electric and magnetic components of a gauge field into each other. The same is the case for non-Abelian electrodynamics. Further, the electromagnetic vector potential is written according to absorption and emission operators that act on element of a Fock space of states. It is then reasonable to require that the theory be treated in a manifestly Lorentz covariant manner. 

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