Gauges Coulomb

In this book we will mostly employ the Coulomb gauge of Eq. (2.133), which is also called transverse or radiation gauge. With this choice of gauge Eq. (2.128) is reduced to Poisson s equation 

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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... 

The matter field operators, although denoted by the same symbol, should not be confused with those defined in the Coulomb gauge by the equations of motion (11-47). The same is true for the in, out operators. The two are related by a transformation of the form... 

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... 

It is possible to formulate the Coulomb gauge theory in terms of radiation operators which satisfy the subsidiary condition... 

Co-representation matrices explicit forms, 733 multiplication of, 731 of the nonunitary group, 732 Corliss, L. M., 757 Corson, E. M., 498 Coulomb field Dirac equation in, 637 Coulomb gauge, 643,657,664 Counting functions, 165 Covariance matrix, 160 Covariant amplitude of one-particle system, 511 of one, two, etc. particle systems, 511... 

Maxwell distribution, 18 Maxwell equations in Coulomb gauge, 645... 

To obtain the total Hamiltonian for ird system, in the Coulomb gauge we exclude the Ao field by using the equations of motion, and neglect high-dimensional operators that arise as a result of this operation. The result for the canonical Hamiltonian of the non-relativistic ird system can be written as... 

The total Hamiltonian in the Coulomb gauge (Cf.sect.2.7) has the momentum... 

The solution of these equations can he simplified with the proper choice of gauge [20]. In Coulomb gauge, V-A —0, one obtains the Poisson equation for the scalar potential ... 

Subsequent expansion in terms of vie and transformation to Coulomb gauge gives... 

All potentials satisfy Coulomb gauge. The corresponding potentials are introduced into the relativistic Hamiltonian, and the perturbation operators are obtained as... 

The two parts of this formula are derived from the same QED Feynman diagram for interaction of two electrons in the Coulomb gauge. The first term is the Coulomb potential and the second part, the Breit interaction, represents the mutual energy of the electron currents on the assumption that the virtual photon responsible for the interaction has a wavelength long compared with system dimensions. The DCB hamiltonian reduces to the complete standard Breit-Pauli Hamiltonian [9, 21.1], including all the relativistic and spin-dependent correction terms, when the electrons move nonrelativistically. 

From relations (105)-(110) expressions are also obtained for the charge density p and the potentials A and < /, as given elsewhere [19]. These expressions in the rest frame K can be considered to correspond to the Coulomb gauge. 

Equation (93) illustrates the fact that Coulomb gauge V A = 0 is necessarily verified when the value does not depend on space. In this specific case, Beltrami fields turn into fields of the Trkalian type, which are solutions of Helmholtz equations in Coulomb s gauge ... 

Another convenient choice is the Coulomb gauge for which we have [14,17,23]... 

In the Coulomb gauge, the vector VWl vanishes, meaning that there is no correspondence between the particle and field theory for the Coulomb gauge, or the... 

The helicity is gauge-invariant. In the Coulomb gauge, it is obvious that A and C satisfy the d Alembert equation, whose solutions can be written in terms of Fourier transforms... 

The divergenceless condition of the field A in the Coulomb gauge means that the complex vector a(k) is transverse, so that k a(k) = 0. Then, for every value of k, we can choose an orthonormal trihedron with by the real vectors k/co, ei(k) and e2(k), and we can represent the field a as... 

Since the derivation is completely algebraic, the symmetrized set of equations (58)—(61) should be identical in every respect to the conventional MEs (8)-(l 1). Surprisingly, there are some slight differences as discussed in Munera and Guzman [87]. One of them is related to the Coulomb gauge, as follows. 

The process p + Pi —> P3 + Pa then involves the conservation of momentum, for there is no creation of any averaged momentum from the virtual quantum fluctuation. This process can be examined within the Coulomb gauge V A = 0. The field equation is then... 

This corresponds to a longitudinal stationary wave. It gives rise to E = B = 0. It is clear from Roscoe s analysis that the magnetic vector potential supports two kinds of waves in free space a propagating transverse wave that corresponds exactly to the Coulomb gauge solutions of the conventional formalism and a stationary longitudinal wave that has no counterpart in the conventional formalism. A general solution can be constructed as... 

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