Short Description of the Nonrelativistic Limit

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. 

First of all, we note that the relativistic energy contains the rest mass mc. This expression should not appear in the nonrelativistic limit, therefore we consider the limit c oo of the operator 

In the following calculations, it is assumed, for simplicity, that the potentials VV and V- are bounded operators. The treatment of the more general case, e.g.. Coulomb potentials, poses some technical complications (use of bounded resolvents instead of unbounded operators). See [1] and the references therein for details. 

The eigenvalue E like the eigenfunction xp will, of course, depend on the parameter c. In view of the 2 x 2 block-matrix structure of (96) we write 

This is precisely the nonrelativistic Hamiltonian for a particle with spin-1/2 in a magnetic field B = V x A. This shows that the part of the energy in the magnetic field that is caused by the spin is 

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