Deformation and contraction of Poincaré group representations

The irreducible representations of the Poincaré group with p ≠ 0 and discrete spin are analytically deformed to representations of the Galilei group. Via decomposition, reducible representations are also deformed. Consequences for physical interpretation of taking the limit c → ∞ are discussed.     

Galilei-Newton law by symplectic realizations

We classify all the coadjoint orbits of the central extension of the one spatial dimensional Galilei group G. Taking them in support of the generic symplectic realizations of the Galilei group G, we give their possible physical interpretations and also find that mass and force have their origins in the cohomological theory of the Galilei group.This has been presented as a seminar in Trieste (ICTP) during the workshop on Lie groups and their representations (10–20 November 1986).     

Geometric quantization in the presence of an electromagnetic field

Some aspects of the formalism of geometric quantization are described emphasizing the role played by the symmetry group of the quantum system which, for the free particle, turns out to be a central extensionG(_m) of the Galilei groupG. The resulting formalism is then applied to the case of a particle interacting with the electromagnetic field, which appears as a necessary modification of the connection 1-form of the quantum bundle when its invariance group is generalized to alocal extension ofG. Finally, the quantization of the electric charge in the presence of a Dirac monopole is also briefly considered.     

On the Structure of Inhomogeneous Quantum Groups

We investigate inhomogeneous quantum groups G built from a quantum group H and translations. The corresponding commutation relations contain inhomogeneous terms. Under certain conditions (which are satisfied in our study of quantum Poincaré groups [12]) we prove that our construction has correct ‘size’, find the R-matrices and the analogues of Minkowski space for G. Received: 3 April 1995 / Accepted: 23 September 1996     

Orthosymplectic unification of the Galilei group and BRST

The inclusion of BRST generators into the Poincarè group in D dimensions is known to lead IOsp[D,2|2]. Similarly, conformal symmetry gets extended into Osp[D×1,3|2]. For the non-relativistic case we find that the Galilei symmetry gets extended, by inclusion of the BRST generators, into an orthosymplectic symmetry possessing Osp[D,1|2] as a subgroup. All such extensions express the possibility of formulating the classical theories in reparametrization invariant ways. They include besides the generators of the initial kinematical symmetry (Poincarè, or conformal, or Galilei), the generators of Parisi-Sourlas transformations. The extended symmetries follow directly through BRST quantization.     

De Sitter Relativity: a New Road to Quantum Gravity?

The Poincaré group generalizes the Galilei group for high-velocity kinematics. The de Sitter group is assumed to go one step further, generalizing Poincaré as the group governing high-energy kinematics. In other words, ordinary special relativity is here replaced by de Sitter relativity. In this theory, the cosmological constant Λ is no longer a free parameter, and can be determined in terms of other quantities. When applied to the whole universe, it is able to predict the value of Λ and to explain the cosmic coincidence. When applied to the propagation of ultra-high energy photons, it gives a good estimate of the time delay observed in extragalactic gamma-ray flares. It can, for this reason, be considered a new paradigm to approach the quantum gravity problem.     

Noncommutative geometry in string and twisted Hopf algebra of diffeomorphism

We discuss the Hopf algebra structure in string theory and present the twist quantization as a unified formulation of the world sheet quantization of the string and the symmetry of the target spacetime. Applying it to the case with a nonzero B-field background, we explain a method to decompose the twist into two successive twists. There are two different possibilities of decomposition: The first is a natural decomposition from the viewpoint of the twist quantization, leading to a new type of twisted Poincaré symmetry. The second decomposition reveals the relation of our formulation to the twisted Poincaré symmetry on the Moyal type noncommutative space.     

Another Majorana idea: Real and imaginary in the Weinberg theory

The Majorana concept of neutrality is applied to the solutions ofj=1 Weinberg equations in the (j, 0)⊕(0,j) representation of the Poincaré group.     

Cohomology,central extensions,and (dynamical) groups

We analyze in this paper the process of group contraction which allows the transition from the Einstenian quantum dynamics to the Galilean one in terms of the cohomology of the Poincaré and Galilei groups. It is shown that the cohomological constructions on both groups do not commute with the contraction process. As a result, the extension coboundaries of the Poincaré group which lead to extension cocycles of the Galilei group in the nonrelativistic limit are characterized geometrically. Finally, the above results are applied to a quantization procedure based on a group manifold.     

Poisson Structures on the Poincaré Group

An introduction to inhomogeneous Poisson groups is given. Poisson inhomogeneous O(p,q) are shown to be coboundary, the generalized classical Yang-Baxter equation having only a one-dimensional right-hand side. Normal forms of the classical r-matrices for the Poincaré group (inhomogeneous O(1,3)) are calculated. Received: 19 February 1996 / Accepted: 10 September 1996     

Spatial contraction of the Poincaré group and Maxwell’s equations in the electric limit

The contraction of the Poincaré group with respect to the space translations subgroup gives rise to a group that bears a certain duality relation to the Galilei group, that is, the contraction limit of the Poincaré group with respect to the time translations subgroup. In view of this duality, we call the former the dual Galilei group. A rather remarkable feature of the dual Galilei group is that the time translations constitute a central subgroup. Therewith, in unitary irreducible representations (UIRs) of the group, the Hamiltonian appears as a Casimir operator proportional to the identity H = EI, with E (and a spin value s) uniquely characterizing the representation. Hence, a physical system characterized by a UIR of the dual Galilei group displays no non-trivial time evolution. Moreover, the combined U(1) gauge group and the dual Galilei group underlie a non-relativistic limit of Maxwell’s equations known as the electric limit. The analysis presented here shows that only electrostatics is possible for the electric limit, wholly in harmony with the trivial nature of time evolution governed by the dual Galilei group.     

Induced representations of quantum kinematical algebras

We construct the induced representations of the null-plane quantum Poincaré and quantum kappa Galilei algebras in (1+1) dimensions. The induction procedure makes use of the concept of module and is based on the existence of a pair of Hopf algebras with a nondegenerate pairing and dual bases.     

Automorphic Functions,Poincaré Series,and Conformal Fields on Riemann Surfaces

Poincaré series and automorphic functions for SU(1, 1) and a discrete subgroup Γ are studied with harmonic analysis. We consider automorphic functions on the open unit circle with general “spin label” m and their decomposition into irreducible automorphic functions by means of the Plancherel formula. These automorphic functions are bijectively mapped onto automorphic distributions on the boundary of the unit circle by meam of the Poisson kernel. The exponent of convergence of Poincaré series is expressed in representation theory language. The results are applied to two-point functions of conformal fields.     

Stability of instanton-induced compactification in 8 dimensions

We consider the 8-dimensional Einstein-SU(2) Yang-Mills system when the Yang-Mills system assumes a 1-instanton configuration on an internal S₄ with the vacuum solution possessing the geometry of M₄ × S₄ and the invariance group P₄(Poincaré) × SO(5). We demonstrate the classical stability of the solution by computing the spectrum of the physical states and showing the absence of ghosts and tachyons amongst them. We also discuss the possibility of obtaining SO(5) multiplets of massless fermions.     

Relativity groups in the presence of matter

We show that the action of the boosts on an infinite system can be described continuously by bundle maps of Hilbert bundles based on the manifoldsG/G ₀, whereG is the full relativity group andG ₀ its closed subgroup which can be unitarily implemented on the fibre, which is a Hilbert space. We then develop a general theory of group representations on product bundlesM × ?, whereM is a manifold and ? a Hilbert space. We construct certain bundle representations of the Galilei and the Poincaré group and find that they correspond to various classes of elementary excitations. In particular, we define nonrelativistic zero-mass systems and obtain an analogue of the Faraday effect for the passage of hot electrons through matter. Our construction remains valid for the case whenG ₀ is the product of a lattice translation group and the time translations. We conclude that many qualitative features of the physics of condensed matter can be directly understood in terms of relativity group action on a bundle space as state space, which also suggests some avenues for further work.     

Poincaré Duality and <Emphasis Type="Italic">G</Emphasis><Superscript>+++</Superscript> Algebras

Theories with General Relativity as a sub-sector exhibit enhanced symmetries upon dimensional reduction, which is suggestive of “exotic dualities”. Upon inclusion of time-like directions in the reductions one can dualize to theories in different space-time signatures. We clarify the nature of these dualities and show that they are well captured by the properties of infinite-dimensional symmetry algebras (G ⁺⁺⁺- algebras), but only after taking into account that the realization of Poincaré duality leads to restrictions on the denominator subalgebra appearing in the non-linear realization. The correct realization of Poincaré duality can be encoded in a simple algebraic constraint, that is invariant under the Weyl-group of the G ⁺⁺⁺-algebra, and therefore independent of the detailed realization of the theory under consideration. We also construct other Weyl-invariant quantities that can be used to extract information from the G ⁺⁺⁺-algebra without fixing a level decomposition. Post-doctoraal onderzoeker van het Fonds voor Wetenschappelijk Onderzoek, Vlaanderen.     

137-01137-01137-01-deformed extended Galilei algebra

We apply the pseudoextension mechanism, which in the undeformed case gives the centrally extended Galilei group {ie137-02} as a contraction of a trivial extensionP×U(1) of the Poincaré group, to the case of the-Poincaré algebra. As a result, the four-dimensional {ie137-03}-deformed extended Galilei algebra is obtained.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.