Gauges Lorentz

In the Lorentz gauge, Maxwell s equations when expressed in terms of the potentials assume the following form... 

The generalization of the previous formalism to encompass systems of n (noninteracting) photons is straightforward. In the Lorentz gauge, an n photon amplitude is a tensor , ) (fcf =... 

This vector potential si should not be confused with the vector potential for the radiation field introduced in Section 9.8 of Chapter 9. The vector potential si of the present section obeys the equation Qsi = ji. We have denoted it by script cap si to indicate that it satisfies the transversality condition div si as 0 in contradistinction to the Lorentz gauge potentials A to be introduced later, which satisfy d A x) as 0 and QAp =... 

Precisely the result (11-91) would have been obtained if, instead of working in the Coulomb gauge, we had adopted the Lorentz gauge. The theory is then described by Heisenberg operators satisfying the following equations of motion2... 

Note that in the Lorentz gauge we have to adopt the Gupta-Bleuler quantization scheme, with its indefinite metric in a vector space that contains, in addition to the physically realizable states, unphysical... 

The Coulomb gauge theory and the Lorentz gauge theory thus both describe the same physical phenomena, but they handle one aspect of the physical situation, namely, the Coulomb interaction, in fundamentally different ways. In the Coulomb gauge the interaction is... 

Incorporated into the electron field, while in the Lorentz gauge it appears as being caused by the emission and absorption of longitudinal quanta. It is because the Coulomb interaction does not involve observable quanta that this freedom of choosing the gauge exists. 

Asymptotic Condition.—In Section 11.1, we exhibited the equivalence of the formulation of quantum electrodynamics in the Coulomb and Lorentz gauges in so far as observable quantities were concerned (t.e., scattering amplitudes). We also noted that both of these formulations, when based on a hamiltonian not containing mass renormalization counter terms, suffered from the difficulty that the... 

In the present section we shall make this difficulfy apparent in a somewhat different way by showing that it is not possible to satisfy the asymptotic condition when the theory is formulated in terms of an unsubtracted hamiltonian of the form jltAll(x) — JS0JV. We shall work in the Lorentz gauge, where the relativistic invariance of the theory is more obvious. 

Lorentz approximation, 46 Lorentz condition, 551 Lorentz gauge, 657,664 Lorentz group homogeneous, 490... 

Expressions for the charge density p and the potentials A and are readily obtained from relations (92)-(97) as shown in detail elsewhere [19]. These relations are thus given in the laboratory frame, and they can be considered to correspond to the Lorentz gauge, which is discussed further in Appendix A. 

We observe that the gauge transform is unique and cannot allow us to eliminate the vector potential outside the solenoid. In addition, the vector potential A derives from a multiform scalar potential F. This result contradicts the solenoidal characteristic of the A = — VA r In (r)Bo/2] vector potential. Henceforth, the gauge transform given above represents nonobservable stationary waves in vacuum since the Lorentz gauge ... 

Since A is a 4-vector field and is a scalar operator, it follows that 4p is a (polar)vector field. Let us now choose the (axial) pseudovector field Bp that accompanies 4p so that 1) it satisfies the same Lorentz gauge as 4 M, that is, oMBp = 0, and it solves the field equation (that accompanies (6) for 4p ... 

Two equivalent forms of Maxwell s field equations in terms of the standard vector formalism are Eq. (5), or (6) with the Lorentz gauge 9 4 = 0. The former is in terms of the antisymmetric second-rank tensor solution F, which is a combination of the electric and magnetic field variables. The latter is in terms of the vector potential, A, shown in Eq. (6) [as well as Eq. (7) in terms of the pseudo vector potential Bassuming that the parameter E, is nonzero]. (Experimental results to this point in time indicate that indeed this parameter is zero to within experimental accuracy [15]—even though the symmetry of relativity theory has no reason to exclude it. Henceforth, we will assume that this parameter is zero.)... 

For further applications we have to write down the electromagnetic field equations in 4-dimensional form. Then we have to use the Lorentz gauge. The Eqs(34) for the potentials become ... 

We can specify them by imposing either the Lorentz gauge (91) or the... 

The electrodynamic potentials satisfy the wave equations. in Lorentz gauge... 

The scalar potential satisfies in the nonrelativistic limit the Poisson equation (95) with no difference between the Coulomb and the Lorentz gauge. 

Incidentally, the Lorenz gauge was proposed by the Danish physicist Ludvig Lorenz, It is often erroneously designated Lorentz gauge, after the more famous Dutch physicist Hendrik Lorentz. In fact, the condition does fulfill the property known as Lorentz invariance.)... 

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