Lorentz‐invariant potentials and the nonrelativistic limit

We discuss a relativistic particle affected simultaneously by two static potentials, a Lorentz scalar and the time component of a four‐vector. We show that the scalar potential has effects on the motion of a particle that are distinctively different from those of a four‐vector potential. With the aid of an example we show that, in both classical and quantum mechanics, there are two inequivalent but compatible criteria of nonrelativistic behavior: (mean) kinetic energy much smaller than rest energy and nonrelativistic potential energy independent of total energy. The second of these is the correct condition for use of nonrelativistic methods, but it is the first that is usually invoked to justify their use. The two criteria are equivalent only when the scalar potential has a (mean) value much less in magnitude than the particle’s rest energy. If this condition is not satisfied, nonrelativistic formalisms such as ordinary Newtonian mechanics and the Schrödinger equation may be inadequate even if v/c is small. On the other hand, it is possible that a large S may permit relativistic energy relations to be written in a nonrelativistic form even if v/c is not small. Such an influence of the size of the scalar potential on what is considered nonrelativistic behavior is unfamiliar and is rarely mentioned in textbooks and other publications. In our opinion, it should be better known.