On the irreducible representations of the lorentz group

All continuous irreducible representations of the SL(2, C) group (as given by Naimark) are obtained by means of methods developed by Harish-Chandra and Kihlberg. The analysis is done in the SU(2) basis and a single closed expression for the matrix elements of the noncompact generators for an arbitrary irreducible representation of SL(2, C) is given. For the unitary irreducible representations the scalar product for each irreducible Hilbert space is found explicitly. The connection between the unitary irreducible representations of SL(2, C) and those of

is discussed by means of Inönü and Wigner contraction procedure and the Gell-Mann formula. Finally, due to physical interest, the addition of a four-vector operator to SL(2, C) unitary irreducible representations in a minimal way is considered; and all group extensions of the parity and time reversal operators by SL(2, C) are explicitly obtained and some aspects of their representations are treated.     

On irreducible representations of the ultrahyperbolic BMS group

The ordinary Bondi–Metzner–Sachs (BMS) group B is the common asymptotic symmetry group of all asymptotically flat Lorentzian space–times. As such, B is the best candidate for the universal symmetry group of General Relativity. However, in studying quantum gravity, space–times with signatures other than the usual Lorentzian one, and complex space–times, are frequently considered. Generalisations of B appropriate to these other signatures have been defined earlier. Here, the generalisation B(2,2) appropriate to the ultrahyperbolic signature (+,+,−,−) is described in detail, and the irreducible unitary representations (IRs) of B(2,2) are analysed. It is proved that all induced IRs of B(2,2) arise from IRs of compact “little groups”. These little groups, which are closed subgroups of K=SO(2)×SO(2), are classified here in detail, with particular attention paid to those of infinite order.     

On the dimension of some modular irreducible representations of the symmetric group

We compute the dimension of some irreducible representations of the symmetric groups in characteristic p (Theorem 2). The representations considered here are associated with Young diagrams m: m ₁m₂...m_lsuch that m ₁–m_l(p–l). The formula is based on a variant of Verlinde's formula which computes some tensor product multiplicities of indecomposable modules for GL₁(F _p ).     

A series of irreducible unitary representations of a group of diffeomorphisms of the half-line

We present a family of unitary representations of a group of diffeomorphisms of a finite-dimensional real Euclidean space using a family of quasi-invariant measures. In the one-dimensional case, for a special kind of group diffeomorphisms of the halfline, we prove the irreducibility of the representations thus obtained.     

Fine structure, partial widths and spreading widths of isobaric analogue resonances

Formulas for the microscopic calculation of partial widths and spreading widths, derived from several theories of the isobaric analogue resonances, are compared and critically examined. It is shown that the statistical assumptions used in the derivation of these formulas are not in contradiction with the fine-structure data, in contrast to recent speculations. The accuracy with which partial and spreading widths can be calculated is discussed. The theoretical values of the partial widths, and thus the determination of the spectroscopic factors, involve poorly known phenomenological parameters (essentially the shape of the symmetry potential). Thence, only relative values of the spectroscopic factors can at present be accurately determined. The calculation of the spreading width depends critically upon our knowledge of isospin violation in the nucleus. Recent calculations of the spreading widths are re-examined in the light of these results.     

Propagation of a relativistic particle in terms of the unitary irreducible representations of the Lorentz group

In a generalized Heisenberg/Schr?dinger picture we use an invariant space-time transformation to describe the motion of a relativistic particle. We discuss the relation with the relativistic mechanics and find that the propagation of the particle may be defined as space-time transition between states with equal eigenvalues of the first and second Casimir operators of the Lorentz algebra. In addition we use a vector on the light-cone. A massive relativistic particle with spin 0 is considered. We also consider the nonrelativistic limit. Received: 20 September 2001 / Published online: 23 November 2001     

Dynamics on the Virasoro group, 2D gravity and hidden symmetries

We derive the action of 2D gravity from the two-cocycle of the Virasoro group by using a previously introduced method for constructing a dynamical system from a Lie group. A natural construction and explanation of the hidden SL(2, )-Kac-Moody symmetry, of general applicability, is provided. We also show how the proper quantization goes beyond the ordinary co-adjoint orbit method, and the possible connection with the Poisson-Lie groups.     

A classification of the unitary irreducible representations ofSU(N, 1)

All inequivalent continuous unitary irreducible representations ofS U(N, 1) (N2) have been determined and classified. The matrix elements of the infinitesimal generators realized on a certain Hilbert space have been derived. Representations of the groups ,S U(N, 1)/Z _N+1, andU(N, 1) are classified in a similar manner.     

The matrix elements of irreducible representations of the group SO(n,1) in integral form

Using the method of induced representations, the matrix elements of unitary irreducible representations of the group SO(n,1) are found in integral form.     

A classification of the unitary irreducible representations ofSO 0(N, 1)

All inequivalent continuous unitary irreducible representations of the groupSO ₀(N, 1),N3, and its universal covering group are classified.     

Induced representations,reproducing kernels and the conformal group

The properties of induced (or multiplier) representations of groups which act in Hilbert spaces with a reproducing kernel are investigated. A resumé of earlier work is followed by a discussion of criteria for the irreducibility of such representations. The notions of reproducing kernel and positive definite spherical function are found to overlap. As a result, functional equations (analogous to those of Godement for spherical functions) are found for the reproducing kernel. The abstract theory is illustrated by certain discrete series representations of the conformal group and by their limit points. In particular the so-called ladder representations (which give rise to the conformal symmetry of zero mass particles) are analysed from this viewpoint.     

Non-unitary representations and fourier transform on the poincaré group

Two families of representations of the Poincaré group $\text{P}$ associated to complex orbits (with complex linear momenta) are introduced. The conditions for equivalence and irreducibility are stated. Fourier transforms related to them are defined and inversion formulas obtained for functions infinitely differentiable and of compact support on $\text{P}$.     

A study of the distribution of partial radiative widths and amplitudes for 149Sm(n, γ) 150Sm

A new approach to the study of width distributions is developed. The approach makes use of level-level interference effects in (n,γ) reactions to produce a set of radiative width amplitudes. This set is compared to the width distribution and to the hypothesis of Porter and Thomas for the case of ¹⁴⁹Sm(n, γ)¹⁵⁰Sm. Both width and amplitude distributions show small but significant deviations from the hypothesis for transitions to low-lying 3⁺ and 4⁺ states in ¹⁵⁰Sm. No significant width correlations are observed, nor is there apparent any violation of the assumption of random phases for the amplitudes.     

The spectrum of the algebra of skew differential operators and the irreducible representations of the quantum Heisenberg algebra

The left spectrum of a wide class of the algebras of skew differential operators is described. As a sequence, we determine and classify all the algebraically irreducible representations of the quantum Heisenberg algebra over an arbitrary field.     

The relativistic semi-group,Lorentz group,and tachyons. I

It is claimed that the current interest of tachyons and some of their unusual properties may be the result of an insufficient mathematical formalism based on Lorentz groups. The introduction of tachyons requires not only a generalization of the principle of causality, but also an extension of the principle of relativistic invariance. A method is proposed to remove a number of problematic cases in the treatment of tachyons.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 85–88, May, 1976.The author wishes to express his appreciation to D. D. Ivanenko for consideration and constructive criticism.     

Distribution of partial neutron widths for nuclei close to a maximum of the neutron strength function

For nuclei near a maximum of the neutron strength function, the secular dependence on energy E of s-wave partial neutron widths differs from the canonical form √E. We derive the universal form of that dependence and show that it is expected to significantly influence the analysis of neutron resonance data.     

Unitary and non-unitary representations of the complex inhomogeneous Lorentz group

We construct unitary and non-unitary representations of the complex inhomogeneous Lorentz group, including all its unitary, irreducible representations. We discuss the decomposition of these representations when they are restricted to the real inhomogeneous Lorentz group. We also discuss the representations of the Poincaré group for which the translation subgroup transforms under a not necessarily unitary representation. We summarize briefly the physical motivation for this study.Supported by the National Science Foundation (NSF-GP-58) while at Brandeis University and by the National Science Foundation, Grant NSF-GP-3465, while at the Courant Institute of Mathematical Sciences.