By D. J. Simms

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In section 3 we have defined a quantum mechanical system to be an elementary relativistic free particle if it is associated with anirreducible representation of the covering group M ~ of the restricted inhomogeneous Lorentz group. A choice of relativistic observer M ~ ~IR 4 induces an isomorphism : M ® Z o-- m4®sL(2,c) so that, in order to classify the possible elementary relativistic free particles, we must determine the irreducible representations of ]R4~SL(2,C) . To do this we will apply the theorem of Mackey from the previous section.

G) (p) V [gl ,v] ~ v I commutes. g ) is unitary our assertion follows. Hilbert bundles. The considerations above suggest the following definition. A triple spaces, ~ = (X,~,Y) ~ is a Hilbert bundle if a continuous surjection of a Hilbert space structure for each X X on p ~ Y . and Y Y , and X are topological -1(p) is given is then called the total space of the bundle ~ , ~ the projection, and Y the base space. The -I Hilbert space ~ (p) is the fibre over p . A section @ of the bundle is a map @: Y ~X on the base space each such that Y ~@ = ~ with its value .

0 •~"¢'! 0 II o ,r- T 0 <~ II o ~- I! <~ << I! T 0 I II tD II il ,e- II O ! T- "7 O ,,--.. ~ @ tl O V 1'4 ,el O 0 0 0 O II O O t~l c~ ,~t to ,D V 0 t!

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