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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Nonrelativistic limit of relativistic quantum mechanics

Lawrence Paul Horwitz, F. C. Rotbart

Published Oct 15, 1981• 42 citations
Publisher

Abstract

The nonrelativistic limit of a relativistic quantum mechanics of Stueckelberg type is studied. A new degree of freedom, which is a purely quantum effect, emerges as a residue of the relativistic theory; its role is to impose Galilean simultaneity. It is shown that the usual nonrelativistic Schr\"odinger equation is obtained as the asymptotic form of the relativistic evolution equation, and some new insight is provided into its interpretation from the point of view of the relativistic theory.

DOI
10.1103/physrevd.24.2127

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The nonrelativistic limit of a relativistic quantum mechanics of Stueckelberg type is studied. A new degree of freedom, which is a purely quantum effect, emerges as a residue of the relativistic theory; its role is to impose Galilean simultaneity. It is shown that the usual nonrelativistic Schr"odinger equation is obtained as the asymptotic form of the relativistic evolution equation, and some new insight is provided into its interpretation from the point of view of the relativistic theory.
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.
The covariance of the Schr\"odinger equation under Galilei boosts and the compatibility of nonrelativistic quantum mechanics with Einstein's equivalence principle have been constrained for so long to the existence of a superselection rule which would prevent a quantum particle from being found in superposition states of different masses. In an effort to avoid this expedient, and thus allow nonrelativistic quantum mechanics to account for unstable particles, recent works have suggested that the usual Galilean transformations are inconsistent with the nonrelativistic limit implied by the Lorentz transformation. Here we approach the issue in a fundamentally different way. Using a formalism of unitary transformations and employing quantum reference frames rather than immaterial coordinate systems, we show that the Schr\"odinger equation, although form variant, is fully compatible with the aforementioned principles of relativity.
In this paper, some concepts of nonrelativistic many-particle quantum mechanics (e.g., product states, density matrix) are generalized to the relativistic domain using a framework called relativistic Schr\"odinger theory (RST). By using a general ansatz, the RST framework is simplified considerably and some of its field equations are solved directly. The RST approach is then compared with nonrelativistic quantum mechanics (QM) for the case of a product state (conventional QM) and its RST analog. It is shown that relativistic wave equations can be derived from the RST formalism, so that they coincide in the nonrelativistic limit with the well-known Hartree equations.
The correspondence of the previously given relativistic quantum mechanics of spinless or spin 1/2 particle with the quantum nonrelativistic and the classical relativistic mechanics is proved. In the Galilean quantum case the appearance of the familiar transformation phase factor is explained and the origin of the mass superselection rule indicated. In the relativistic classical case the equation for trajectories and the covariant equation of motion for spin are obtained. The conditions are found under which the usual theory of relativistic wave equations is of relevance (in some particular reference frame).