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     

The unitary irreducible representations of the quantum heisenberg algebra

The goal of this work is to describe the irreducible representations of the quantum Heisenberg algebra and the unitary irreducible representation of one of its real forms. The solution of this problem is obtained through the investigation of theleft spectrum of the quantum Heisenberg algebra using the result about spectra of generic algebras of skew differential operators (cf. [R]).     

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.     

u q(2,1) noncompact quantum algebra: Intermediate discrete series of unitary irreducible representations

The unitary irreducible representations of the u _q(2,1) quantum algebra that belong to the intermediate discrete series are considered. The q analog of the Mickelsson-Zhelobenko algebra is developed. Use is made of the U basis corresponding to the reduction u _q(2,1) ? u _q(2). Explicit formulas for the matrix elements of the generators are obtained in this basis. The projection operator that projects an arbitrary vector onto the extremal vector of the intermediate-series representation is found.     

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.     

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.     

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.     

Spectra of generators in irreducible unitary representations of non-compact semi-simple groups

Several theorems concerning the spectra of elements of the complexified Lie algebra in unitary representations of non-compact semi-simple groups are proved. The principal theorem gives purely Lie algebraic sufficient conditions for the type of spectrum (point or continuous) of any element of the real Lie algebra. For elements of special self adjoint Cartan subalgebras these conditions are rephrased in terms of the basis-dependent information most readily available to the physicist, namely their hermiticity properties and the values of the structure constants: roots, etc.International Atomic Energy Agency International Center for Theoretical PhysicsOn leave of absence from New Mexico State University, NM, USA.     

Irreducible unitary representations of quantum Lorentz group

A complete classification of irreducible unitary representations of a one parameter deformationS _q L(2,C) (0<q<1) ofSL(2,C) is given. It shows that in spite of a popular belief the representation theory forS _q L(2,C) is not a smooth deformation of the one forSL(2,C).     

All linear unitary irreducible representations of De Sitter supersymmetry with positive energy

All linear unitary irreducible representations of the graded Lie algebra osp (4.1) with positive energy are calculated by acting with the odd elements on a vacuum state. With the exception of the Dirac supermultiplet, the result corresponds to the representations of Poincaré supersymmetry.     

Asymptotic behaviour of the unitary representations of the Poincaré group

The results of R. Jost and K. Hepp [2] and D. Maison [3] concerning the asymptotic behaviour of the translation operators are generalized.     

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 ).     

The unitary irreducible representations of\overline {SO} _0 (4,2)

An exhaustive classification of all irreducible Harish-Chandra \(\mathfrak{s}\mathfrak{o}\) (4,2)-modules, integrable to unitarizable projective representations of the conformal group, is established by infinitesimal methods: the classification is based

1. on the reduction upon the maximal compact subalgebra, associated with a lattice of points in ?³, and
2. on a set of additional parameters upon which the eigenvalues of central elements of the enveloping algebra depend polynomially.

     

Degenerate representations of non-compact unitary groups. II. Continuous series

Three degenerate principal series of irreducible unitary representations of an arbitrary non-compact unitary groupU(p, q) are derived. There series are determined by the eigenvalues of the first and second-order invariant operators, the former having a discrete spectrum and the latter a continuous one. The explicit form of the corresponding harmonic functions is derived and the properties of the continuous representations are discussed.On leave of absence from Institute of Physics of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia.On leave of absence from Institute of Nuclear Research, Warsaw, Poland.     

Perturbations of unitary representations of the Galilei group: Mass operators with continuous spectra

We present a systematic procedure for constructing mass operators with continuous spectra for a system of particles in a manner consistent with Galilean relativity. These mass operators can be used to construct what may be called point-form Galilean dynamics. As in the relativistic case introduced by Dirac, the point-form dynamics for the Galilean case is characterized by both the Hamiltonian and momenta being altered by interactions. An interesting property of such perturbative terms to the Hamiltonian and momentum operators is that, while having well-defined transformation properties under the Galilei group, they also satisfy Maxwell’s equations. This result is an alternative to the well-known Feynman-Dyson derivation of Maxwell’s equations from non-relativistic quantum physics.     

On a method of constructing irreducible multiplier representations

A systematic method is presented for constructing the irreducible multiplier representations (ray representations) of a class of finite groups corresponding to a given factor system. The method consists in first identifying a normal subgroup of prime index, classifying its irreducible multiplier representations into orbits and then inducing the required representations from these orbits. The proposed technique finds useful application in solid state physics where irreducible multiplier representations of the point group underlying the group of the wave vector are frequently required.     

Infinite-dimensional representations of the Lsorentz group: The complete series

The method used by Carmeli to obtain a new form for the principal series of representations of the groupSL(2, C) is further generalized to all completely irreducible (finite and infinite-dimensional) representations of that group. This is done, following Naimark, by extending the meaning of one of the parameters appearing in the formula for the operators of the principal series of representations. As a result a new form for the complete series of representations of the groupSL(2, C) is obtained which describes the transformation law of an infinite set of quantities under the group translation in a way which is very similar, but as a generalization, to the way spinors appear in the finite-dimensional case. The finite-dimensional representation is then discussed in details and the relation between the new set of quantities (which becomes finite in this case) and 2-component spinors is found explicitly.     

All unitary ray representations of the conformal group SU(2,2) with positive energy

We find all those unitary irreducible representations of the -sheeted covering group of the conformal group SU(2,2)/₄ which have positive energyP ⁰0. They are all finite component field representations and are labelled by dimensiond and a finite dimensional irreducible representation (j ₁,j ₂) of the Lorentz group SL(2). They all decompose into a finite number of unitary irreducible representations of the Poincaré subgroup with dilations.     

On the geometry of some unitary Riemann surface braid group representations and Laughlin-type wave functions

In this note we construct the simplest unitary Riemann surface braid group representations geometrically by means of stable holomorphic vector bundles over complex tori and the prime form on Riemann surfaces. Generalised Laughlin wave functions are then introduced. The genus one case is discussed in some detail also with the help of noncommutative geometric tools, and an application of Fourier–Mukai–Nahm techniques is also given, explaining the emergence of an intriguing Riemann surface braid group duality.