On extracting the infinite-dimensional unitary representations ofSU 2,2 in a free-particle basis. — I

Abstract

The infinite-dimensional unitary irreducible representations ofSU2,2, which includes the Poincare group properly, are sought in a basis in which the free-particle quantum numbers momentum, spin, and helicity are sharp. None of the standard representation formalisms apply, such as the Wigner-Mackey semidirect product theory or the Weyl-Cartan theory for semisimple groups, so a general method based on Schur’s lemma is used. This paper discusses new insights into the problem of synthesizing the Poincare group and internal symmetries gained by working out this specific problem. These are, very briefly: 1) the explanation of continuous mass spectra within supermultiplets; 2) new sources and possibilities for internal quantum numbers; 3) the existence of massiveSU2,2-invariant fields in spite of the general belief to the contrary (SU2,2=C ≡ conformal group of space-time); 4) the convenience of space-time-dependent similarity transformations to change to familiar bases of irreducible states.