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Journal of Physics A Mathematical and General

Galilean relativistic wave equations

José F. Cariñena, Mariano Santander

Published Feb 1, 1982• 9 citations
Publisher

Abstract

The locally operating realisations of the Galilei group are studied. Levy-Leblond's equation for spin-one-half Galilei relativistic massive systems is rederived by making use of group theoretical methods and a generalisation to spin one is proposed. Finally, the relation to the corresponding Poincare relativistic equations is analysed.

DOI
10.1088/0305-4470/15/2/008

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The locally operating realisations of the Galilei group are studied. Levy-Leblond's equation for spin-one-half Galilei relativistic massive systems is rederived by making use of group theoretical methods and a generalisation to spin one is proposed. Finally, the relation to the corresponding Poincare relativistic equations is analysed.
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A search for fundamental (Galilean invariant) dynamical equations for two- and four-component spinor wavefunctions is conducted in Galilean spacetime. 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 wavefunctions. A method to derive the Levy-Leblond equation for a four-component spinor wavefunction is presented. It is formally proved that the Levy-Leblond and Schrodinger equations are the only Galilean invariant four-component spinor equations that can be obtained with the Schrodinger phase factor. Physical implications of the obtained results and their relationships to the Pauli–Schrodinger equation are discussed.
An elementary derivation of the system of phase factors defining the ray representations (those faithful to within a phase factor) of the Galilei group is presented. The proof employs the group elements themselves. In particular, the operation of conjugation (which corresponds to coordinate transformation) is used extensively to effect the desired result, i.e., in the notation of Levy-Leblond, U(b′,a′,v′,R′)U(b,a,v,R)=±exp [i(12m)(a′·R′v−v′·R′a+bv′·R′v)]×U(b′+b,a′+R′a+bv′,v′+R′v,R′R).
Nonrelativistic Galilean quantum mechanics and the standard transition to relativistic Poincar\'e quantum mechanics is analyzed in terms of group theory. Special emphasis is given to the discussion of the relation between dynamics and geometry. Certain unsatisfactory features are pointed out and a new relativistic group G5{\mathcal{G}}_{5}G5​ is suggested as the symmetry group of dynamics. G5{\mathcal{G}}_{5}G5​ contains both the nonrelativistic Galilei group and the Poincar\'e group as subgroups, and it is a group extension of the restricted Lorentz group. For use in relativistic quantum mechanics, the central extension of G5{\mathcal{G}}_{5}G5​ by a phase group must be employed. The Lie algebra of this relativistic quantum-mechanical Galilei group ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ contains an acceptable covariant space-time position operator and a nontrivial relativistic mass operator. The latter also serves to describe dynamical development. The irreducible unitary projective representations of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ correspond to infinite towers of states with increasing spin.