Irreducibility of some unitary representations of the Poincaré group with respect to the Poincaré subsemigroup, III

Abstract

The aim of this paper is to prove that irreducible unitary representations (E7'' f ,φ'' ) of the Poincare group P = i? 4 X S SL(2, C) are reducible as the representations of the Poincare subsemigroup P + = V + X S SL(2, C) with V + = {xl -x\ -x\ > 0, x 0 > 0}. The representations mentioned above are those associated with the one-sheeted hyperboloid V tM = {yl -yl -yl -y\ --M 2 } (M > 0) and the irreducible unitary representations ττ (^ε) of SU(1,1) not belonging to the discrete series (see the end of this introduction for the definition of the discrete series). To attain our purpose we shall determine all P + -invariant, closed proper subspaces for the representations (C/ Λε , £> Λε ) (Theorems 1.1 and 4.1). Other irreducible unitary representations of P are known to be irreducible even when they are restricted to P+ [6].In [6], [7] and this paper we are concerned with the question whether (Q) there exists a P + -invariant, closed proper subspace for an irreducible unitary representation of P.A physical aspect of this problem is as follows. From E. Wigner's view point of relativistic quantum mechanics an irreducible unitary representation (U, §) describes the dynamics of an elementary particle. In particular the one-parameter unitary group U(t, 0, 0, 0, e) (t e R) on φ stands for the dynamical transformation group. On the other hand some elementary particles (a neutral pion, for example) are known to decay spontaneously. If one tries to explain the phenomena from Wigner's point of view, one naturally expects that there exists a proper closed subspace 3f of φ such that 3f is invariant under U(t, 0, 0, 0, e) (t > 0) and [7(0, 0, 0, 0, g) (g e SL(2, C)), equivalently such that Of is P + -invariant. We are very likely to suspect the existence of an irreducible unitary representation of P with (see (1,13), (1,14)). Of course these operators are connected with the Laplacians Δ and Δ' of SL(2, C) respectively.In § 1, after the definition of the representation (C7 /ίβ , φ'Ό we shall show that, if the statement (Q) above is valid for this representation, there exists a non-trivial sequence {D k } kez++ε of closed subspaces in L\R) 2k+ί or L 2 (R) such that it satisfies certain conditions (Q.I) and (Q.2) in Lemma 1.4. Conversely, once such sequences are given (Proposition 1.5, Theorems 2.2 and 3.1), we can construct P + -invariant subspaces 2** of φ' (Theorems 1.1 and 4.1) mainly due to Proposition 1.6. To determine all nontrivial sequences {D k } kQZ++ε satisfying the conditions (Q.I) and (Q.2) is, therefore, the core of our argument. The simplest case, in which τr (Λβ) is the unit representation of SU(1,1), namely (£, ε) = (0, 0), is discussed in § 1, while the other cases are investigated in § § 2 and 3. In the final section, §4, we shall describe all the P + -invariant, closed proper subspaces of £> Λe for (£, ε) Φ (0, 0). Notation and terminology.Z is the set of integers and Z + = {n e Z; n > 0}.R is the set of real numbers, i? + == {λ e R; λ > 0} and Jί* = i?\{0}.C is the set of complex numbers and C* = C\{0}. ...