P. Moylan

Unitary representations of the (4+1)‐de Sitter group on irreducible representation spaces of the Poincaré group

The construction of the principal continuous series of unitary representations of the simply‐connected covering group of the (4+1)‐de Sitter group on unitary irreducible representation spaces of the Poincare group is presented. A unitary irreducible representation space of this covering group of the de Sitter group is realized as the direct sum of two irreducible representation spaces of the Poincare group. Possible physical implications are indicated. In particular, an interpretation of the instantaneous velocity operator in the Dirac theory as the spin part of the de Sitter boosts is given. We obtain a simple method of computing the matrix elements of the generators of the de Sitter group in an SO(4) basis using the matrix elements of the generators of the four‐dimensional Euclidean group. Also we obtain explicit expressions for certain matrix elements between the spinor and SO(4) basis of the representation space as functions on the coset space SO(4)/SO(3).

An elementary account of the factor of 4/3 in the electromagnetic mass

It is known that elementary and nonrelativistic electromagnetic considerations lead to a value for the mass of a spherical electrical distribution that is 4/3 times larger than its value obtained from a relativistic treatment. We have obtained a very simple account of this discrepancy, which makes use only of Coulomb’s law and the Biot–Savart law together with the well‐known formulas for the energy densities of the electrostatic and magnetic fields. The resolution of this discrepancy, of course, involves the correct (relativistically covariant) definitions for the momentum and energy of an electromagnetic field. This example shows the necessity for relativistic considerations even in nonrelativistic treatments of electromagnetism.

Finite-dimensional singletons of the quantum anti de Sitter algebra

Abstract We obtain positive-energy irreducible representations of the q -deformed anti de Sitter algebra U q (so(3,2)) by deformation of the classical ones. When the deformation parameter q is an N th root of unity, all these irreducible representations become unitary and finite-dimensional. Generically, their dimensions are smaller than of the corresponding finite-dimensional non-unitary representation of so(3,2). We discuss in detail the singleton representations, i.e. the Di and Rac. When N is odd the Di has dimension (N 2 minus;1) 2 and the Rac has dimension (N 2 +1) 2 , while if N is even both the Di and Rac have dimension N 2 2 . These dimensions are classical only for N =3 when the Di and Rac are deformations of the two fundamental non-unitary representations of so(3,2).

Unitary representations of the (4+1) de Sitter group on unitary irreducible representation spaces of the Poincaré group: Equivalence with their realizations as induced representations

In a previous work we have constructed realizations of the principal continuous series of unitary irreducible representations of the simply connected covering group of the (4+1) de Sitter group on unitary irreducible representation spaces of the simply connected covering group of the Poincare group. In this work we demonstrate the equivalence of the representations constructed in the previous work with their realizations as induced representations.

On the integrability of certain symmetric representations of the Lie algebra of SO0(4,1)

A proof of the existence of an essentially self‐adjoint extension of a symmetric ∼(SO0(4,1)) Nelson operator, which is constructed out of the generators of a positive mass, arbitrary spin unitary irreducible representation of the Poincare group, is presented. Our analysis of ∼(SO0(4,1)) and its Lie algebra provides us with an example of an observation of Harish‐Chandra: There exist subspaces of the space of differentiable vectors of a representation of a noncompact group which are invariant under the Lie algebra, but the closures of the subspaces are not invariant under the group. The chief results of this paper should hold true for ∼(SO0(n,1)). In particular, we should have a realization of an arbitrary principal series irreducible unitary representation of SO0(n,1) on the direct sum of two identical unitary irreducible representation spaces of the motion group in an n‐dimensional Minkowski space, which has one timelike dimension.

Positive-energy irreps of the quantum anti de Sitter algebra

We obtain positive-energy irreducible representations of theq-deformed anti de Sitter algebraUq(so(3, 2)) by deformation of the classical ones. When the deformation parameterq isN-th root of unity, all these irreducible representations become unitary and finite-dimensional. Generically, their dimensions are smaller than those of the corresponding finite-dimensional non-unitary representations ofso(3, 2). We discuss in detail the singleton representations, i.e. the Di and Rac. WhenN is odd, the Di has dimension 1/2(N2−1) and the Rac has dimension 1/2(N2+1), while ifN is even, both the Di and Rac have dimension 1/2N2. These dimensions are classical only forN=3 when the Di and Rac are deformations of the two fundamental non-unitary representations ofso(3, 2).

Harmonic analysis on spannors

The spannor representation of the universal covering group of the conformal group of Minkowski space–time is studied herein. It is an example of an indecomposable representation. Various parallelizations for spannors are described. Normalized K‐finite basis fields for the spannors are introduced, and a computation of the actions of the scale generator and other generators of the conformal group on these basis fields is given. The irreducibility of all irreducible composition factors is established, and it is decided which ones are infinitesimally unitary. Two sets of generators for maximal Abelian subalgebras of the enveloping algebra of the conformal group are introduced. The action of one of the sets on K‐finite basis fields in the spannor representation is studied; this is important for understanding of the problem of degeneracy in the spannor representation, since K types occur with multiplicity 2. Many calculations make use of the lowest (highest) weight module structures of the unitarizable and p...

A topological criterion for group decompositions

Given a connected Lie group G and a closed connected subgroup H of G we prove a necessary and sufficient condition that G decomposes into the Cartesian product of H with G/H is that a similar decomposition holds for the maximal compact subgroups of G and H . Our criterion is applied to the three series of groups for which G/H is SO 0 ( p, q )/ SO 0 ( p, q − 1), SU(q + 1, q + 1)/ S[U(q + 1, q ) × U (1)], and SU(q + 1, q + 1)/ SL(n , ℂ) ⋊ H(n ) ( p, q ≥ 1), and we list the values of p and q for which G ≅ H × G/H in each of the three cases. We describe certain decompositions for some of the groups. We show the usefulness of our criterion in obtaining characterization of the space of differentiable vectors for a unitary induced group representation, and, finally, we show by example of SU (2, 2), how the asymptotic properties of certain function spaces for induced group representations are readily obtained using our results. Our results should be of interest to those working in de Sitter and conformal field theories.

Construction of representations of Poincaré group using Lie fields

In this paper, we give an explicit construction of the unitary irreducible representations of the Poincare groups in 2, 3, and 4 space-time dimensions on Hilbert spaces associated with the Schrodinger representation of the Weyl algebra for n = 1, 2, and 3, respectively. Our method of constructing the representations uses extension and localization of the enveloping algebras associated with these Weyl algebras and the Poincare algebras.

Localization and the Weyl algebras

Let Wn(ℝ) be the Weyl algebra of index n. It is well known that so(p, q) Lie algebras can be viewed as quadratic polynomial (Lie) algebras in Wn(ℝ) for p + q = n with the Lie algebra multiplication being given by the bracket [a, b] = ab − ba for a, b quadratic polynomials in Wn(ℝ). What does not seem to be so well known is that the converse statement is, in a certain sense, also true, namely, that, by using extension and localization, it is possible, at least in some cases, to construct homomorphisms of Wn(ℝ) onto its image in a localization of U(so(p + 2, q)), the universal enveloping algebra of so(p + 2, q), and m = p + q. Since Weyl algebras are simple, these homomorphisms must either be trivial or isomorphisms onto their images. We illustrate this remark for the so(2, q) case and construct a mappping from Wq(ℝ) onto its image in a localization of U(so(2, q)). We prove that this mapping is a homomorphism when q = 1 or q = 2. Some specific results about representations for the lowest dimensional case of W1(ℝ) and U(so(2, 1)) are given.