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First off one sees that M 2 --since it is determined by the op discrete eigenvalues of the intrinsic spin--is necessarily Poincar~ invariant. The explicit result is not hard to obtain using the aligned bosons of Chapter 2, §4. 17), and the function e is the Regge trajectory function. In this form it is obvious that M 2 is indeed Poincar6 invariant. 11) 52 where P = (P, (~2+ M2op ) ½), and A denotes any Lorentz transformation. Summary. We have d e m o n s t r a t e d in this section that there exists a t r a n s f o r m a t i o n b e t w e e n the states of a composite system w h o s e Regge t r a j e c t o r y is defined on the set of all W i g n e r irreps {M(s),s} and the same composite system r e a l i z e d in M i n k o w s k i m o m e n t u m space.

6') to the fact that instead of x elP-XelP0t .... 1') 40 In other words, these representations, corresponding to different choices f(p0), are all unitarily equivalent. (3) For the case of spin s the only change is that the Pauli matrices of the spin-½ case are replaced by the g e n e r a t o r s of the (2s + i) dimensional unitary r e p r e s e n t a t i o n of the quantal angular momentum group SU(2). nates for the general (4) Hence we have o b t a i n e d q u a s i - N e w t o n i a n coordi(M,s) r e p r e s e n t a t i o n uniformly.

Q U A S I - N E W T O N I A N C O O R D I N A T E S A N D T H E GENERATORS The W i g n e r irreps IN THOMAS F O R M {M,s} and the W i g n e r form of the generators are realized in m o m e n t u m space, and it is an i n t e r e s t i n g question as to how to obtain c o n f i g u r a t i o n - s p a c e realizations. This p r o b l e m was actually considered prior to the W i g n e r c o n s t r u c t i o n Schr6dinger, in his studies on the Dirac equation. (1939) by As we shall show here, a comprehensive view can best be o b t a i n e d d i r e c t l y from the W i a n e r c o n s t r u c t i o n of the set {M,s}; the existence of the Dirac eauation from this noint of view (for sDin ½) is irrelevant, and the historical a c c i d e n t that this eauation came first has areatlv c o n f u s e d the initial, and subseauent, discussions of the problem, based as they are on the p a r t i c u l a r i t i e s of the Dirac eauation itself.

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An Introduction to Random Vibration, Spectral and Wavelet Analysis by Newland

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