Gauge condition

The constant b will be determined later. A (noncovariant) spectral representation for the potentials satisfying the gauge condition... 

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 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. 

Now we can establish the correspondence between relativistic expressions (4.3), (4.4) and the non-relativistic ones obtained above, as well as the particular cases when specifying gauge condition K. The relationship between the values of K = 0, —y/(k + 1 )/k and the forms of the electron... 

The transformation of the relativistic expression for the operator of magnetic multipole radiation (4.8) may be done similarly to the case of electric transitions. As has already been mentioned, in this case the corresponding potential of electromagnetic field does not depend on the gauge condition, therefore, there is only the following expression for the non-relativistic operator of Mk-transitions (in a.u.) ... 

As we have seen in Chapter 4, relativistic operators of the Ek-transitions in the general case have several forms and are dependent on the gauge condition of the electromagnetic field potential. These forms are equivalent and do not depend on gauge for exact wave functions. Unfortunately, we are always dealing with the more or less approximate wave functions of many-electron systems, therefore we need general expressions for the appropriate matrix elements. 

Fig. 30.2. The dependence of the oscillator strength of the 2s2p 1P — Ip1 lS transition in OV on the gauge condition K for various theoretical approches, HF, SC [243] and NCMET [244], The curves refer to HF (1), SC without and with polarization and inner-shell effects, (2-4), NCMET (5). The experimental value / = 0.103 + 0.007 [246] is indicated with a star.

Let us also notice that slow variations of K with Z imply that the gauge condition K may be treated as a semi-empirical parameter in practical calculations to reproduce, with a chosen K, the accurate oscillator strength values for the whole isoelectronic sequence. Thus, dependence of transition quantities on K may serve as the criterion of the accuracy of wave functions used instead of the comparison of two forms of 1-transition operators. In particular, the relative quantities of the coefficients of the equation fEi = aK2 + bK +c (the smaller the a value, the more exact the result), the position of the minimum of the parabola Kf = 0 (the larger the K value for which / = 0, the more exact is the approximation used, in the ideal case / = 0 for K = +oo) may also help to estimate the accuracy of the method utilized. 

Such expressions can be easily generalized to cover the case of the electric multipole transition operator with an unspecified value of the gauge condition K of electromagnetic field potential (4.10) or (4.11). 

It should be mentioned that in the approach with nonzero electric divergence, the photon mass is also related to the space charges in vacuo. Now, in the approach with a / 0, we have j = ctE but jeff = 0. Let us now assume j = aE and j 7 0, which means fs 0. In such a case, jo is assumed to be associated with p, where p is the charge density in vacuo. So, in such an approach one can think of the existence of a kind of space charge in vacuo that is to be considered to be associated to nonzero electric field divergence. This will result in a displacement current in vacuum similar to that measured by Bartlett and Corle [43]. The assumption of the existence of space charge in vacuo makes our theory not only fully relativistic but also helps us to understand gauge condition. In the conventional framework of Maxwell s equations... 

A particular set of potentials can be specified by imposing a linear functional constraint on the vector potential. Such a constraint is usually referred to as a gauge condition a general gauge condition is provided by the equation... 

While Ho has the familiar form of the sum of the non-relativistic atomic/molecular Hamiltonian, (55), based purely on Coulombic interactions, and the Hamiltonian for free radiation (48), H has the unfamiliar feature of involving the essentially arbitrary Green s function g(x,x ) because no gauge for the vector potential is specified. In particular the form (55) does not require the Coulomb gauge condition. Of course, overall H is gauge-invariant, and observables must be as well, so we need to consider gauge-invariant calculation. 

In classical electrodynamics, the field equations for the Maxwell field A/( depend only on the antisymmetric tensor which is invariant under a gauge transformation A/l A/l + ticduxix), where x is an arbitrary scalar field in space-time. Thus the vector field A/( is not completely determined by the theory. It is customary to impose an auxiliary gauge condition, such as 9/x/Fx = 0, in order to simplify the field equations. In the presence of an externally determined electric current density 4-vector j11, the Maxwell Lagrangian density is... 

Each field mode must also satisfy the Coulomb gauge condition, V Ak(r, t) = 0, which, when substituted into Eq. (1.20), implies that... 

A certain pressure cooker is designed to operate at 20 lb/in2 gauge. It is well known that an item of food will cook faster in such a device because of the higher steam temperature at the higher pressure. Consider a certain item of food as a horizontal 4-in-diameter cylinder at a temperature of 95°F when placed in the cooker. Calculate the percentage increase in heat transfer to this cylinder for the 20-lb/in2-gauge condition compared with condensation on the cylinder at standard atmospheric pressure. 

Other convenient specifications for the divergence can be conceived (see below) the gauge defined by equation (8.41) is known as the Coulomb gauge. With use of this gauge condition, equation (8.40) can be rewritten in the form ... 

As already mentioned, the vector potential can be chosen to satisfy the Coulomb gauge condition... 

This Hamiltonian results from the standard canonical quantization of electrodynamics if it is assumed that particle speeds are negligible compared to the speed of light, and all charge-photon interactions are discarded the Coulomb gauge condition must also be imposed [8],... 

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