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International Journal of Theoretical Physics

General Dynamical Equations for Free Particles and Their Galilean Invariance

Z. E. Musielak, John L. Fry

Published Nov 10, 2008• 15 citations
Publisher
DOI
10.1007/s10773-008-9893-9

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Textbook treatments of the Galilean covariance of the time-dependent Schrödinger equation for a spinless particle seem invariably to cover the case of a free particle or one in the presence of a scalar potential. The principal objective of this paper is to examine the situation in the case of arbitrary forces, including the velocity-dependent variety resulting from a vector potential. To this end, we revisit the 1964 theorem of Jauch which purports to determine the most general form of the Hamiltonian consistent with “Galilean-invariance,” and argue that the proof is less than compelling. We then show systematically that the Schrödinger equation in the case of a Jauch-type Hamiltonian is Galilean covariant, so long as the vector and scalar potentials transform in a certain way. These transformations, which to our knowledge have appeared very rarely in the literature on quantum mechanics, correspond in the case of electrodynamical forces to the “magnetic” nonrelativistic limit of Maxwell’s equations in the sense of Le Bellac and Lévy-Leblond (1973). Finally, this Galilean covariant theory sheds light on Feynman’s “proof” of Maxwell’s equations, as reported by Dyson in 1990.
New fundamental (Galilean invariant) dynamical equations for four- component spinor wavefunctions are derived by using a different phase function than the Schrodinger phase function. The derived equations are called the generalized Levy-Leblond and generalized Schrodinger equa- tions because these equations reduce to the standard Levy-Leblond and Schrodinger equations when the Schrodinger phase function is used. The generalized and standard equations describe the same free elementary particles with spin 1/2, however, masses of the particles described by these equations are not the same as the generalized equations automati- cally account for particles with larger masses. This has important phys- ical implications as it shows that elementary particles with the same physical properties but different masses can result from using phase functions different than the Schrodinger phase function. Therefore, the main conclusion of this paper is that the existence of the three currently known families of elementary particles in the Standard Model of particle physics can be theoretically accounted for by choosing different phase functions.
The definition of the total momentum, position, and spin for Galilean- or Lorentz-invariant two-free-particle systems is discussed by using the requirement that the generators of the respective invariance groups should have the same form expressed in terms of them as for a single particle. Internal c.m. dynamical variables are introduced by applying the singular transformation due to Gartenhaus and Schwartz on the basic single-particle dynamical variables, the transformation mapping the whole Hilbert space onto the c.m. subspace spanned by states of zero total momentum. The form of the internal c.m. position operator is given for particles with spin, for what appears to be the first time. Using these dynamical variables, it is shown how an interaction can be introduced while maintaining Galilean or Lorentz invariance and satisfying the asymptotic condition of freely propagating particles for large separations.