Lagrangian electromagnetic field

The electromagnetic field may now formally be interpreted as the gauge field which must be introduced to ensure invariance under local U( 1) gauge transformation. In the most general case the field variables are introduced in terms of the Lagrangian density of the field, which itself is gauge invariant. In the case of the electromagnetic field, as before,... 

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 total Lagrangian if I X I if2 is now invariant under the local gauge transformation because of the introduction of the 4-potential A, which couples to the current of the complex A of the pure gauge vacuum. The field A also contributes to the Lagrangian, and since if + ifj + if2 is invariant, an extra term if3 appears, which must also be gauge-invariant. This can be so only if the electromagnetic field is introduced... 

The total Lagrangian X = JS G + JS D + JS , then involves the interaction between fermions and the gauge field. The Dirac field will be generically considered to be the electron and the gauge theory will be considered to be the non-Abelian electromagnetic field. The theory then describes the interaction between electrons and photons. A gauge theory involves the conveyance of momentum form one particle (electron) to another by the virtual creation and destruction of a vector boson (photon) that couples to the two electrons. The process can be diagrammatically represented as... 

This paper presents an account of the dynamics of electric charges coupled to electromagnetic fields. The main approximation is to use non-relativistic forms for the charge and current density. A quantum theory requires either a Lagrangian or a Hamiltonian formulation of the dynamics in atomic and molecular physics the latter is almost universal so the main thrust of the paper is the development of a general Hamiltonian. It is this Hamiltonian that provides the basis for a recent demonstration that the S-matrix on the energy shell is gauge-invariant to all orders of perturbation theory. 

A Lagrangian for a collection of charged particles in an electromagnetic field can be written down directly using the polarization fields... 

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... 

We shall first find that Lagrangian for a system of charged particles in an electromagnetic field which, through the principle of least action, gives the correct equation of motion. The electric and magnetic fields of the electromagnetic field, d and B, respectively, are related to the scalar and vector potentials, and A, by the equations... 

Thus, the atomic Lagrangian and action integrals in the presence of an electromagnetic field, like their field-free counterparts, vanish as a consequence of the zero-flux surface condition (eqn (8.109)). These properties are common to the corresponding integrals for the total system and it is a consequence of this equivalence in properties that the action integrals for the total system and each of the atoms which comprise it have similar variational properties. 

The key feature of the theory of QED—whether it is cast in nonrelativis-tic or fully covariant forms is that the electromagnetic field obeys quantum mechanical laws. A frequent first step in the construction of either version of the theory is the writing of the classical Lagrangian function for the interaction of a charged particle with a radiation field. For a particle of mass m, electronic charge —e, located at position vector q, and moving with velocity d /df c in a position-dependent potential V( ) subject to electromagnetic radiation described by scalar and vector potentials cp0) and a(r), at field point... 

We now turn to the electromagnetic field. In the presence of a charge distribution it s Lagrangian density splits into two terms. The first term is the Lagrangian density of the free field... 

The electric and magnetic polarization fields P(x), M(x) are invoked to describe the electrodynamic properties of charged particles they enter the theory through a representation of the lagrangian interaction potential for a closed system of electric charges and the electromagnetic field... 

The remaining parts of the lagrangian L describe the free field and fi-ee particles, and are not needed here. The requirements of V are that under lagrangian variation of the fields (for fixed partiele variables) it should contribute the appropriate terms to the Maxwell equations, and that variation of the particle variables (for a fixed electromagnetic field) gives the Lorentz force on the particles. 

The equation of motion for a nonrelativistic charged particle with charge cj and mass m subject to time-dependent electromagnetic fields will now be derived within the Lagrangian and also within the Hamiltonian formalism. The... 

The Lagrangian density em for the electromagnetic field is therefore given as the sum of the kinetic term 3 and the interaction term... 

We will now examine one of the most central features of the interaction between the electromagnetic field and charged matter as described by the interaction Lagrangian density int given by Eq. (3.189). As has been explicitly shown above, this form for jnt is the simplest choice compatible with all... 

An assembly of nuclei and electrons could be described very accurately within QED. There would still just be a cluster of particles, our molecule, and any structure would have to arise out of the dynamics of the system. For reasons pointed out earlier, QED—if viable at all— would be a very expensive path to calculation of the electronic stracture and chemical properties of molecules. For electrons, we circumvented this problem by going to a many-particle treatment based on the Dirac equation, as discussed in chapter 5, and we could presumably do the same here for our cluster of electrons and nuclei. In doing this, we choose a Hamiltonian description of the system, but alternative approaches based on a Lagrangian formalism are also possible. In this process we draw a formal distinction between the molecule and the electromagnetic field, which leaves us with the normal Coulomb interactions between the particles in the molecule and the radiation field as an entity external to the molecule. 

The problem for us is therefore to derive the classical Hamiltonian function for an electron in the presence of electromagnetic fields, which is normally done from the classical Lagrangian. Hamilton s and Lagrange s generalizations of classical mechanics are essentially the same theory as Newton s formulation but are more elegant and often computationally easier to use. In our context, their importance lies in the fact that they serve as a springboard to quantum mechanics. 

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