The Gaunt Operator for Unretarded Interactions

The next problem is to choose for one electron the scalar and vector potentials generated by the other electron. We approach this issue step by step and first study the unretarded classical scalar potential created by electron 2, 

This expression is obviously in accord with the standard Coulomb law, but could also be derived from the most general Eq. (3.211) in the limit of infinite speed of light, i.e., after expansion and truncation of the absolute value of the 

Similar expressions for the fields produced by electron 1 are obtained by exchanging the subscripts 1 and 2. In order to find the quantum mechanical operator expressions, we invoke the only guiding principle that we are familiar with in quantum mechanics, namely the correspondence principle. The velocities ri and f2 are then to be substituted by the velocity operators caj and CCC2 of Dirac s theory derived in section 5.3.3. Note that we switch from an explicit trajectory picture, which has been the basis of all derivations in section 3.5, to the operator formulation in quantum mechanics and solely trust in the correspondence principle that this is the correct way to obtain a trajectory-free quantum picture of the interaction of charged moving particles. 

Moreover, one might also think about introducing momentum operators instead of velocity operators. Obviously, that would yield different expressions since no a matrices are then introduced at all. However, Eq. (8.7) gives us a hint whether we should use the velocity operator or the momentum operator since it contains by virtue of the minimal-coupling expression terms such as which will only be symmetric in the two coordinates i and j if we introduce the velocity operator and not the momentum operator. Recall that we have encountered this line of reasoning already in section 5.4.2. 

Consequently, we choose the unretarded electromagnetic potential operators to become 

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