Potentials four-vector

Consider a system that can be decomposed into n-electrons and m-nuclei. The total hamiltonian H includes the field mediating the interactions between the charged constituents. Charged particles interact via the four vector potential (0, A), where 0 is the Coulomb potential and A is the transverse electromagnetic potential. This Hamiltonian is obtained as a non-relativistic limit of Dirac s Hamiltonian [10] ... 

In this form, we see that if we multiply the top equation by ijc and add the two equations, we recover the four-vector current density j from (3.6) on the right-hand side. If we define a four-vector potential A accordingly, we get... 

Because and j are invariant, it follows that the four-vector potential A is also Lorentz invariant. Note that this is really implied in the Lorentz condition (3.23), which is a product of the Lorentz-invariant four-gradient and the four-vector A and can be written... 

We know that the four-vector momentum and the four-vector potential are conserved quantities under a Lorentz transformation, so we expect that a is also conserved, as may be shown with the conditions below. 

Now the Lagrangean associated with the nuclear motion is not invariant under a local gauge transformation. Eor this to be the case, the Lagrangean needs to include also an interaction field. This field can be represented either as a vector field (actually a four-vector, familiar from electromagnetism), or as a tensorial, YM type field. Whatever the form of the field, there are always two parts to it. First, the field induced by the nuclear motion itself and second, an externally induced field, actually produced by some other particles E, R, which are not part of the original formalism. (At our convenience, we could include these and then these would be part of the extended coordinates r, R. The procedure would then result in the appearance of a potential interaction, but not having the field. ) At a first glance, the field (whether induced internally... 

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... 

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. 

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

The most general form of the vector potential can be obtained by writing the first four terms of Eq. (B.4) as... 

In this chapter we discuss the close relationship between the Born-Oppenheimer treatment of molecular systems and field theory as applied to elementary particles. The theory is based on the Born-Oppenheimer non-adiabatic coupling terms which are known to behave as vector potentials in electromagnetic dynamics. Treating the time-dependent Schrodinger equation for the electrons and the nuclei we show that enforcing diabatization produces for non-Abelian time-dependent systems the four-component Curl equation as obtained by Yang and Mills (Phys. Rev. 95, 631 (1954)). 

The diabatization within the time-dependent framework produced the expected potential matrix W presented in equation (56) but enforced the four vector curl equation which is given in equation (54). This set of equations contains not only derivatives with respect to the spatial coordinates but also with respect to time. In fact this non-Abelian curl equation is completely identical to YM curl equation which has its origin in field theory. 

An electromagnetic field is described in relativistic theory by a four-vector A, where the three space components Aij2,3 = Aare called the vector potential A and the fourth (time) component A4 is equal to i where

scalar potential. The Lagrangian for a particle in an electromagnetic field is now given by... 

In summary, we have transformed a problem requiring the solution for six unknown variables (the six scalar components of the vectors E and H) to a problem requiring solution for four unknown variables (the three scalar components of the vector potential A and the single variable, the scalar potential U). As we will see below, this reduction in the number of variables allows us to simplify markedly the solution of electromagnetic forward and inverse problems in many cases. 

The standard representation of the four-component relativistic Dirac equa-tion for a single particle in an electrostatic potential V(r) and a vector potential (r) reads... 

The idea is as follows. The electromagnetic field intensity solution of Maxwell s equations, expressed in terms of the four-dimensional curl of the vector potential, is... 

The bracket in the second equation in (5) denotes a cyclic sum, and we use units (henceforth in this article) with c = 1. Combining the definition of /p as the four-dimensional curl of a 4 vector, as in Eq. (4), Maxwell s equations in terms of the vector potential are ... 

Here p = —iV is the electron momentum operator, d, P are the standard Dirac matrices, A is the vector potential and V is the scalar potential of the external field. The wave function (r,t) is the four-component spinor. For the stationary state ... 

Even within the no-pair approximation the RKS equations (62) are more involved than the nonrelativistic KS equations due to the four vector structure of the RKS potential. Thus the question arises whether one can find simplified forms in which the RKS potential reduces to one or two components. Fortunately, in most applications the external magnetic field... 

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