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Annals of Physics

Physical theories in Galilean space-time and the origin of Schrödinger-like equations

Z. E. Musielak, John L. Fry

Published Jun 28, 2008• 20 citations
Publisher
DOI
10.1016/j.aop.2008.06.006

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The Galilean symmetry of a free particle in one-dimensional space is examined under the scope of non-Abelian quantum kinematics. Within the Hilbert space that carries the regular ray representation of the Galilei group the Schrödinger operator appears as one of the three fundamental invariants of the extended kinematic algebra. By means of a superselection rule the physical Hilbert subspaces of the system are identified, in which a complementary ray representation of the Galilean transformation produces the time-dependent Schrödinger equation, and the Feynman space-time propagator. The quantization approach used in this paper is purely group theoretic and relativistic.
A search for fundamental (Galilean invariant) dynamical equations for two and four-component spinor wave functions is conducted in Galilean space-time. A dynamical equation is considered as fundamental if it is invariant under the symmetry operators of the group of the Galilei metric and if its state functions transform like the irreducible representations of the group of the metric. It is shown that there are no Galilean invariant equations for two-component spinor wave functions. A method to derive the Lévy-Leblond equation for a four-component spinor wave function is presented. It is formally proved that the Lévy-Leblond and Schrödinger equations are the only Galilean invariant four-component spinor equations that can be obtained with the Schrödinger phase factor. Physical implications of the obtained results and their relationships to the Pauli-Schrödinger equation are discussed.