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.     

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.     

Particle dynamics on Snyder space

We examine Hamiltonian formalism on Euclidean Snyder space. The latter corresponds to a lattice in the quantum theory. For any given dynamical system, it may not be possible to identify time with a real number parametrizing the evolution in the quantum theory. The alternative requires the introduction of a dynamical time operator. We obtain the dynamical time operator for the relativistic (nonrelativistic) particle, and use it to construct the generators of Poincaré (Galilei) group on Snyder space.     

Non-relativistic conformal symmetries in fluid mechanics

The symmetries of a free incompressible fluid span the Galilei group, augmented with independent dilations of space and time. When the fluid is compressible, the symmetry is enlarged to the expanded Schrödinger group, which also involves, in addition, Schrödinger expansions. While incompressible fluid dynamics can be derived as an appropriate non-relativistic limit of a conformally invariant relativistic theory, the recently discussed conformal Galilei group, obtained by contraction from the relativistic conformal group, is not a symmetry. This is explained by the subtleties of the non-relativistic limit.     

Discrete Lagrangian Reduction,Discrete Euler–Poincaré Equations,and Semidirect Products

A discrete version of Lagrangian reduction is developed within the context of discrete time Lagrangian systems on G × G, where G is a Lie group. We consider the case when the Lagrange function is invariant with respect to the action of an isotropy subgroup of a fixed element in the representation space of G. Within this context, the reduction of the discrete Euler–Lagrange equations is shown to lead to the so-called discrete Euler–Poincaré equations. A constrained variational principle is derived. The Legendre transformation of the discrete Euler–Poincaré equations leads to discrete Hamiltonian (Lie–Poisson) systems on a dual space to a semiproduct Lie algebra.     

The foundations of the poincaré group and the validity of general relativity

After discussions about accepted ideas concerning the nonlocalisability of the photon, the interpretation of the Minkowski space-time, the wave-corpuscle duality ideas of Niels Bohr and the concept of elementary particle by Eugene Wigner, the validity of the Poincaré group is brought into question and some other ideas are developed. Lukierski, Nowicki and Ruegg showed that the successes of the Poincaré group are preserved if we deform the group by introducing a constant κ. Such deformation replaces the Poincaré Hopf algebra by another one. We call such a deformation a mathematical deformation. The main inconvenience of this mathematical deformation is that the coproduct is not commutative. The consequence is that a two-particle state is defined in an ambiguous way because we must say which is the first particle and which is the second one. The only mathematical deformation of the Poincaré group which preserves the commutativity of the coproduct is the trivial one, that is the Poincaré Hopf algebra itself. That is why we reject the mathematical deformation of Lukierski, Nowicki and Ruegg. That is also why we propose what we call a physical deformation of the Poincaré group, which means that we reinterpret the Poincaré Hopf algebra, with the same constant κ. Our proposal has four advantages:

1.

1. The constant x has the dimensions of a mass. When this constant becomes infinite, we are left with the Poincaré group with its main successes.

2.

2. The two-particle states are unambiguously defined.

3.

3. The constant κ may be chosen in such a way that the search for a missing mass in the universe is useless.

4.

4. It consists in the disappearing of unphysical irreducible representations of the Poincaré group.

With the constant κ, we arrive at a reformulation of special relativity where the energy is no longer additive. This would imply a change in general relativity where the density of matter must be different from the density of energy. Unfortunately, we are not able to propose a substitute for the general relativity theory. Obviously, when the constant κ goes to infinity, the new general relativity would become the standard general relativity.     

Contractions of bialgebras and super-bialgebras

The contraction scheme of Z ₂ graded super-bialgebras is discused. The existence of the contraction limit and the classical r-matrix with and without the coproduct rescaling is investigated. As an example the contraction of D = 4 Poincaré N = 2 super-bialgebra to D = 2 Poincaré N = 2 super-bialgebra is given.     

Nonstandard Poincaré and Heisenberg Algebras

Nonstandard deformations of the Poincaré group Fun(P(1+1)) and its dual enveloping algebra U (p(1+1)) are obtained as a contraction of the h-deformed (Jordanian) quantum group Fun( SL _h (2)) and its dual. A nonstandard quantization of the Heisenberg algebra U(h(1)) is also investigated.     

Reflections on the evolution of physical theories

Conclusion Let me come back to the successes of the Poincaré group in particle physics. This is a group with ten generators. The translation generators are responsible for the energy-momentum conservation laws, the rotation generators of the conservation of angular momentum, and the boost generators of the conservation ofinitial position. If positions are slightly different from the ones described by Minkowski space, it means that we have to change slightly the notion of boosts. If we remember that boosts were questionable in Minkowski space (see Section 9), we are not surprised. We are naturally led to a deformation of the Poincaré group which would preserve translations and rotations [such a deformation has been proposed by Lukierskiet al. (n.d.)]. By duality, small changes at short distances must correspond to small changes in large momenta. The fact that cutoffs for momenta are involved in QED is perhaps related to a noncommutative structure for our space. With such a structure, making the size of an electron go to zero is meaningless and consequently the difficulty of an electron with infinite energy also becomes meaningless. A noncommutative space is probably a way to solve the difficulties mentioned in the epigraphs to this paper.     

The Hamiltonian in relativistic systems of interacting particles

The dynamics of a system of relativistically interacting particles is determined by a set of constraints, some combination of which has been frequently identified with the Hamiltonian. These constraints differ from the generators of the Poincaré transformations, among whichp ₀ generates translations along the time axis and hence is to be considered as the energy of the system. There are thus grounds for consideringP ₀ as the appropriate Hamiltonian. In this paper we establish a close relationship between transformations generated by the constraints and those generated by the Poincaré generators. In particular we find that the true Hamiltonian is a rather complicated but well-defined function ofp ₀ and all the constraints. We show that the generators of the entire algebra of the Poincaré group can be realized in such a fashion that the Hamiltonian is correctly included among them, and such that particle world lines in Minkowski space-time generated by this Hamiltonian transform correctly under the Poincaré group.This work was partially supported by the National Science Foundation Grant No. PHY 79-0887 to Syracuse University and by Grant No. PHY 79-09405 to Yeshiva University.     

Clebsch-Gordan coefficients for the extended quantum-mechanical Poincaré group and angular correlations of decay products

This paper describes Clebsch-Gordan coefficients (CGCs) for unitary irreducible representations (UIRs) of the extended quantum-mechanical Poincaré group . ‘Extended’ refers to the extension of the 10 parameter Lie group that is the Poincaré group by the discrete symmetries C, P, and T; ‘quantum mechanical’ refers to the fact that we consider projective representations of the group. The particular set of CGCs presented here is applicable to the problem of the reduction of the direct product of two massive, unitary irreducible representations (UIRs) of with positive energy to irreducible components. Of the 16 inequivalent representations of the discrete symmetries, the two standard representations with U_CU_P = ±1 are considered. Also included in the analysis are additive internal quantum numbers specifying the superselection sector. As an example, these CGCs are applied to the decay process of the ? (4S) meson.     

Gauge semi-simple extension of the Poincaré group

An approach to the cosmological term problem is proposed, using the gauge semi-simple tensor extension of the D-dimensional Poincaré group as a basis.     

Canonical quantization of Galilean covariant field theories

The Galilean-invariant field theories are quantized by using the canonical method and the five-dimensional Lorentz-like covariant expressions of non-relativistic field equations. This method is motivated by the fact that the extended Galilei group in 3 + 1 dimensions is a subgroup of the inhomogeneous Lorentz group in 4 + 1 dimensions. First, we consider complex scalar fields, where the Schrödinger field follows from a reduction of the Klein-Gordon equation in the extended space. The underlying discrete symmetries are discussed, and we calculate the scattering cross-sections for the Coulomb interaction and for the self-interacting term λΦ⁴. Then, we turn to the Dirac equation, which, upon dimensional reduction, leads to the Lévy-Leblond equations. Like its relativistic analogue, the model allows for the existence of antiparticles. Scattering amplitudes and cross-sections are calculated for the Coulomb interaction, the electron-electron and the electron-positron scattering. These examples show that the so-called ‘non-relativistic’ approximations, obtained in low-velocity limits, must be treated with great care to be Galilei-invariant. The non-relativistic Proca field is discussed briefly.     

Non-commutative spacetime in very special relativity

The purpose of Very Special Relativity is to show that the ISIM(2) subgroup of the Poincaré group is sufficient to describe the spacetime symmetries of the so far observed physical phenomena. A deformation of such group, called DISIMb(2), was later introduced. In the present work, we present a novel non-commutative spacetime structure, underlying the DISIMb(2), that allows us to construct explicitly the generators of the group. Exploiting the Darboux map technique, we then construct a point particle Lagrangian that lives in the non-commutative phase space proposed by us.     

From the Galilei to the Lorentz group in a general light-cone frame

The most general coordinate system in which the Galilean symmetries of the ordinary and the Bell and Ruegg light-cone frames appear is found. With a modification of this general coordinate system it is possible to go from the Galilei group in two space dimensions to the Lorentz group, a process inverse to the contraction of the Lorentz group with respect to the subgroup of rotations. As an example of the applicability of this modified general light-cone frame, the two-dimensional Schrödinger-Pauli equation is obtained to all orders.     

Poincaré covariance and κ-Minkowski spacetime

A fully Poincaré covariant model is constructed as an extension of the κ-Minkowski spacetime. Covariance is implemented by a unitary representation of the Poincaré group, and thus complies with the original Wigner approach to quantum symmetries. This provides yet another example (besides the DFR model), where Poincaré covariance is realised à la Wigner in the presence of two characteristic dimensionful parameters: the light speed and the Planck length. In other words, a Doubly Special Relativity (DSR) framework may well be realised without deforming the meaning of “Poincaré covariance”.     

Symmetries of discontinuous flows and the dual Rankine-Hugoniot conditions in fluid dynamics

It has recently been shown that the maximal kinematical invariance group of polytropic fluids, for smooth subsonic flows, is the semidirect product of SL (2, R) and the static Galilei group G. This result purports to offer a theoretical explanation for an intriguing similarity, that was recently observed, between a supernova explosion and a plasma implosion. In this paper we extend this result to discuss the symmetries of discontinuous flows, which further validates the explanation by taking into account shock waves, which are the driving force behind both the explosion and implosion. This is accomplished by constructing a new set of Rankine-Hugoniot conditions, which follow from Noether’s conservation laws. The new set is dual to the standard Rankine-Hugoniot conditions and is related to them through the SL (2, R) transformations. The entropy condition, that the shock needs to satisfy for physical reasons, is also seen to remain invariant under the transformations.     

Wilson loops T-dual to short strings

We show that closed string solutions in the bulk of AdS space are related by T-duality to solutions representing an open string ending at the boundary of AdS. By combining the limit in which a closed string becomes small with a large boost, we find that the near-flat space short string in the bulk maps to a periodic open string world surface ending on a wavy line at the boundary. This open string solution was previously found by Mikhailov and corresponds to a time-like near-BPS Wilson loop differing by small fluctuations from a straight line. A simple relation is found between the shape of the Wilson loop and the shape of the closed string at the moment when it crosses the horizon of the Poincaré patch. As a result, the energy and spin of the closed string are encoded in properties of the Wilson loop. This suggests that closed string amplitudes with one of the closed strings falling into the Poincaré horizon should be dual to gauge theory correlators involving local operators and a Wilson loop of the T-dual (“momentum”) theory.     

The Poincaré-Lyapounov-Nekhoroshev Theorem

We give a detailed and mainly geometric proof of a theorem by N. N. Nekhoroshev for hamiltonian systems in n degrees of freedom with k constants of motion in involution, where 1≤k≤n. This state's persistence of k-dimensional invariant tori, and local existence of partial action-angle coordinates, under suitable nondegeneracy conditions. Thus it admits as special cases the Poincaré-Lyapounov theorem (corresponding to k=1) and the Liouville-Arnold one (corresponding to k=n) and interpolates between them. The crucial tool for the proof is a generalization of the Poincaré map, also introduced by Nekhoroshev.