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1. M. Gel'fand, R. A. Minlos and Z. Ya. , New York, 1963). 12. S. Strom, Arkiv for Fysik 29, 467 (1965); ibid 33, 465 (1966). 3 THE HOMOGENEOUS LORENTZ GROUP, II. 1 INTRODUCTION In the preceding chapter the basic spinors ~ and 'Y) (Eq. ) respectively. The existence of the space-time continuum enters only when one makes the appropriate identification of the SL(2, C) parameters with the Lorentz transformations and regards ~ and 'Y) as spinor fields. The foregoing representations and their generalizations (Eq.
Streater and A. S. Wightman, "PCT, Spin and Statistics, and All That" (W. A. , New York, 1964). : ( ,'1'1111/1, I . 1'11' ItI('(ls. lliIl S 19 1. H. II · 'I1WI111 . " I ,i ' (: lO ull' 1,)1' I'il y~ ' ' I ~ I ," ( W . /\ . Ih:llj,II'lin , In '. ) . 1\ . I ~. M . ('orso n , " Inlrotiucl io n 10 T 'lbo rs, Spin ors ;1I1d Rebl iy isl ic Wayc I-,qualion," ( Illnck ic Gnd Son, London , 1953). 5. II. , Amstcrda m, 1956). 6. P. , Amsterdam , J960). 7. W. L. Bade and H. Jehle, Rev. Mod. Phys. 25, 714 (1953).
In order to relate the Casimir operators F and G to the reduced matrix clements of the boost operators K, we expand them in terms of Land K as defined following Eq. 27). 82) G=L·K It will be noted that this breakup is a non-covariant breakup; the operators Land K do not have tensorial properties under Lorentz transformations. If we rewrite K2 and L . K in terms of spherical basis vectors, as in Eq. 84) It will soon become evident that the Casimir operators F and G are not completely independent.