Nonrelativistic limit velocity operator

A physical system is close to the nonrelativistic limit, if all velocities of the system are small compared to the velocity of light. Hence the nonrelativistic limit of a relativistic theory is obtained if we let c, the velocity of light, tend to infinity. In the nonrelativistic theory, there is no limit to the propagation speed of signals. For the Dirac equation, the nonrelativistic limit turns out to be rather singular. If we simply set c = oo, we would just obtain infinity in all matrix elements of the Dirac operator H. We must therefore look for cancellations. 

This step is mathematically rigorous only for a bounded potential V and for c large enough, such that the operator 1 — V-—E)/ 2m( ) is invertible). The equation (97) contains the velocity of light only in the form of the small parameter 1/c. In the nonrelativistic limit we may set 1/c = 0 and the eigenvalue problem becomes... 

In later chapters we will consider the nonrelativistic limit of other relativistic operators. Here we will only look at one, the velocity operator, to show that it has the correct nonrelativistic limit. Extracting 2mc from the square bracket in (4.82) we get... 

What is shown in Fig. 1 is the oscillator strength ig the "length gauge. As has been described in detail by Grant, the relativistic transition operators which correspond in the nonrelativistic limit to the length and velocity operators are related to each other by a gauge transformation of the photon operator. The MCHF proceedure is not gauge invariant, basically because a truncation has been carried out in the sum over creation and annihilation operators in the definition of the many-electron... 

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