Lorenz gauge

If gauge freedom is lost, however, the Lorenz condition is no longer valid, and a far more comprehensive view of the electromagnetic entity would be obtained by solving the 0(3) equations numerically. On the 0(3) level, there is no gauge freedom, and no Lorenz condition. 

The same conclusion regarding the Lorenz gauge is reached by Jackson [5], who shows that ... 

The electromagnetic field defined above is a solution of Maxwell s equations, provided the new potentials . C are solutions of wave equations and satisfy the Lorenz gauge. 

The most common form taken uses the Lorenz potentials that satisfy the Lorenz gauge ... 

In formulating QED a least-action principle involving a Lagrangian is often used [9,18,20]. This involves the potentials in various forms. Not only is relativistic invariance (Lorenz potentials) desired, but also gauge invariance. At least in the current state of QED, gauge invariance is included as a fundamental part [21,22]. 

Then, for both Lorenz and electric gauges (and others such as Coulomb as well), the change from initial (1) to final (2) conditions is characterized by (35). Referring to (4), we can see that this change is characterized by the electric impulse... 

So both Lorenz potentials are zero for late time (as well as initially), but are, of course, nonzero for intermediate times. The electric gauge vector potential is, however, nonzero at late times and is the negative of the electric impulse. 

Now consider the case of no scalar potential (Lorenz gauge) for all time. This implies that outside Vs... 

Then we have the same electric impulse in both cases. This gives the same electric gauge vector potential, 4,2. However, the Lorenz gauge potentials are quite different. For the electric dipole in Section VI, both 42 and 2 are zero. For the toroidal antenna equivalent electric dipole in Section VII, while 2 is zero, 42 is non zero. How then are these two cases different Within the gauge condition... 

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. 

So our choices of the two antennas is not unique for separately emphasizing the Lorenz vector and scalar potentials. All that is required is for the two to have the same exterior fields (say, electric dipole fields, or more general multipole fields) with different potentials (related by the gauge condition). In a classical electromagnetic sense, these antennas cannot be distinguished by exterior measurements. This is a classical nonuniqueness of sources. In a QED sense, the same is the case due to gauge invariance in its currently accepted form. 

The name of this gauge refers to the Dutch physicist Hendrik Antoon Lorentz (1853-1928), whereas the gauge was really introduced by the Danish physicist Ludvig Valentin Lorenz who not only introduced this gauge as well as their retarded solutions (91), but also independently developed Maxwell s equations [39]. 

In Lorenz gauge, the function x(r, t) in Eq. (11.114) is chosen such that (= 0. The scalar and vector potentials are then related hy the Lorenz condition... 

Maxwell s equations in Lorenz gauge can be expressed by a single Minkowski-space equation... 

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.)... 

In Lorenz gauge, given in Eq. (2.132), the inhomogeneous Maxwell equations (2.128) and (2.129) will be decoupled and read... 

In Lorenz gauge, however, both scalar and vector potentials explicitly incorporate a retardation term, because in this gauge the Maxwell Eqs. (2.136) and (2.137) have the most general solutions... 

So far we have just defined another four-component quantity Af, but by now it is not clear whether it properly transforms under Lorentz transformations in order to justify the phrase 4-vector. In order to prove the transformation property of the gauge field, we re-express the inhomogeneous Maxwell equations in Lorenz gauge as given by Eq. (2.138) in explicitly covariant form by employment of the charge-current density and the gauge field A, ... 

Depending on gauge choices, Quiney et al. write two expressions became popular in atomic physics and quantum chemistry [213] the interaction operator in Lorenz (Feynman) gauge in configuration-space representation as... 

R. Nevels, C.-S. Shin. Lorenz, Lorenlz, and the Gauge. IEEE Antennas Propagation Mag., 43 (2001) 70-72. 

Finally, in the Lorenz gauge the gauge function x( i> ) is chosen in such a way that... 

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