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

Abstract

Since E. Wigner set up a framework of the relativistically covariant quantum mechanics, several aspects of unitary representations of the Poincaré group have been investigated (see [8], [16]). In this paper it will be shown that some unitary representations of the Poincaré group are irreducible, even if they are restricted to the Poincaré semigroup (Theorem 1, 2 and 3).

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

“…To be more precise, we shall determine all P + (3)-invariant, closed proper subspaces for the irreducible unitary representations (t/* >e , $ v ' ε )(η e R, ε = 0,1/2), which are associated with the one-sheeted hyperboloid V iM (S) -{y\ -y\ -y\ = -M 2 } (M > 0). As for the other irreducible unitary representations of P (3) it is easy to show that they are irreducible even when they are restricted to P + (3) (see [5], Theorem 5). Recall that all the irreducible unitary representations of the 2-dimensional space-time Poincare group are irreducible even when they are restricted to the Poincare subsemigroup ( [5], Theorem 1).…”

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

“…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.…”

“…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.…”

“…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.…”