Galilei group

In principle, the basis Mi can be chosen arbitrarily. However, if we want the order parameters x, (r, r ) to have well-defined transformation properties under the Galilei group we are led to the Balian-Werthamer matrices (Balian and Werthamer 1963) ... 

J. -M. L vy-Leblond, Galilei group and galilean invariance, in E. M. Loebl (Ed.), Group Theory and Its Applications, volume II, Academic Press, New York, 1971, p. 221. 

Levy-Leblond [16] has realized that not only the Lorentz group (or rather the homomorphic group SL(2) [32, 7]), but also the Galilei group has spinor-field representations. While the simplest possible spinor field with s = I and m 0 in the Lorentz framework is described by the Dirac equation, the corresponding field in a Galilei-invariant theory satisfies the Levy-Leblond equation (LLE)... 

It took some time until it was realized that the Dirac theory describes the spin correctly because it is a spinor-field theory, and not because it is relativistic [16]. In fact, if one takes the nonrelativistic limit of the Dirac equation, spin survives, and this is consistent with the observation that the Galilei group has spinor representations as well. So, without any doubt, spin is not a relativistic effect. 

The previous discussion shows that neither the spin nor the operator K are related to relativistic effects (as is often claimed), but rather they are compatible with nonrelativistic motion (Galilei group relativity) as well as relativistic motion (Poincare group relativity). This point was also made in several of the papers by Levy-Leblond [13]. 

The maximal invariance (symmetry) group for a free p>oint particle in nonrelativistic mechanics is shown to be a 12-parameter group instead of the 10-parameter Galilei group. This elementary but by no means trivial discussion may be of interest for the advanced reader but goes beyond the scope of this book. 

W. I. Fushchych, L. F. Barannik, and A. F. Barannik, Subgroup Analysis of the Galilei and Poincare Groups and Reduction of Nonlinear Equations, Naukova Dumka, Kiev, 1991 (in Russian). 

Thermodynamics, Enrico Fermi. (60361-X) 7.95 Introduction to Modern Optics, Gr2mt R. Fowles. (65957-7) 13.95 Dialogues Concerning Two New Sciences, Galileo Galilei. (60099-8) 9.95 Group Theory and Its Application to Physical Problems, Morton Hamermesh. (661814) 14.95... 

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