Gauge term

The same result is obtained from Eq. (443) using the same proportionality factor g = / Note carefully that without the gauge term igA i, this energy would vanish, and so the energy is due to the vacuum configuration and topology, in this case assumed to be described by the U(l) group. 

If we are working with local gauge transformations where A is flat, we can work with the pure gauge term (dg)d 1 = idX as the gauge connection. 

In Eq. (3.47)

field operator of two-component structure, a are the Pauli matrices and the electron-electron interaction reduces to the Coulomb interaction, denoted by H e- As usual, the gauge term proportional to... 

The starting point for this generalization is the Gordon decomposition, in which the total current is split into the paramagnetic (orbital) component yp, a gauge term proportional to the scalar density Ps, and the curl of the magnetization density m. 

The basic variables of a density functional scheme are derived from the structure of the coupling of the electrons to the external fields. However, there are two possible ways to express the Hamiltonian (122) in terms of density and current operators, which differ by the treatment of the gauge term. Using n and j one obtains... 

This contradiction is resolved by noticing that the order 1/c has not been treated consistently in the weakly relativistic expansion which leads to (122). In fact, consistent neglect of all terms of the order 1 /c in the Hamiltonian (128), i.e. of the gauge term, allows a proof of an existence theorem with the variables n and j, at the price of loosing gauge invariance. In other words For any weakly relativistic Hamiltonian one has to choose between consistency in 1 /c and gauge invariance. Only a fully relativistic approach combines both properties. 

To understand the difference between the spurious gauge terms (i), which are removed in the ideal case of optimal variational wavefunctions [65], and terms (ii), which account for the essential origin dependence of the property, let us first discuss die origin dependence of the quadrupole polarizability of magnetic susceptibility (22) within the conventional common-origin representation. 

In conclusion, summing the RHS of equations (49) and (50), the spurious gauge terms cancel each other out if the hypervirial conditions (33) and (54), and the sum rules for gauge invariance, equations (52), (53), (55), and (56) are fulfilled. Then the CO quadmpole polarizabilities of magnetic susceptibility change according to a relationship analogous to equation (41) for the electric polarizability,... 

The last term in is the gauge term. The addition to the Coulomb interaction in the Feynman gauge is called the Gaunt interaction, and in the Coulomb gauge it is the Breit interaction. 

This is precisely the interaction we obtained in (5.42) by substitution of ca for u in the classical interaction expanded in powers of 1 /c. In some places, the last term in square brackets is referred to as the retardation correction, but this is not correct because the whole term in square brackets can be derived from an unretarded classical interaction. Moreover, the retardation is expressed here by the finite photon frequency, which does not contribute to this interaction. It is better to describe the Coulomb gauge interaction as a sum of the Coulomb interaction, a current-current interaction, and a gauge term. 

Because 1 is a solution for the system without the gauge term, (13.15), this expression reduces to... 

Thus, the wave function must incorporate the gauge term exponentially in order for the energy to remain invariant under gauge transformations. We should add that the same factor must be included also for the nonrelativistic treatment of the magnetic fields. For a more general derivation for the nonrelativistic case, the reader should consult the book by Sakurai (1967). 

Disregarding for the moment the gauge term, we consider the contribution to the functional U from the two other terms. This may be written as... 

The modified operators for the Gaunt and Breit interactions can be derived in an analogous manner. The derivation is somewhat more involved than for the Coulomb interaction due to the presence of the a matrices. Here we derive the Gaunt terms, but the gauge term for the Breit interaction is considerably more complicated (and the derivation may be found in appendix F). 

The gauge term, which is the difference between the Gaunt interaction and the Breit interaction, produces a spin-free operator that can be interpreted as an orbit-orbit interaction. Thus, both the Gaunt interaction and the gauge term of the Breit interaction give rise to spin-free contributions to the modified Dirac operator. We will use the developments of this section in chapter 17 to derive the Breit-Pauli Hamiltonian. 

The contributions to the two-electron operator from the gauge term of the Breit interaction may be developed in the same manner, using the representation for the modified Dirac operator. The derivation is more complicated, and details may be found in appendix I. The only term that contributes to 0(c ) is a spin-free term ... 

We may now combine all the spin-free contributions from the Coulomb, Gaunt, and gauge terms. As noted before, the delta functions from the Coulomb and Gaunt... 

Note that we would have a very different operator if we did not include the contributions from the gauge term. However, this says little about the magnitude of the gauge term, which is usually small. 

Two-Electron Gauge Terms for the Modified Dirac Operator... 

The gauge term, which comprises the difference between the Gaunt and the Breit interactions (see (5.48) and (5.49)), is more complicated than the Gaunt term due to the scalar quadruple product involving the alpha mafiices ... 

Each of the operators in the gauge term consists of a scalar quadruple product, which may be rearranged as was the spin-spin term in the Gaunt interaction. 

Gauge Term Contributions from the Breit Interaction to the Breit-Pauli Hamiltonian... 

The reduction of the remaining contributions from the gauge term is extremely tedious, and in fact gives a zero contribution to the operator to 0(c ). Thus, there is no spin-dependent contribution from the gauge term for the relativistic correction to the electron-electron interaction to the Breit-Pauli Hamiltonian. 

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