Group Theoretical Perspective Research Articles

Special Relativity and Its Newtonian Limit from a Group Theoretical Perspective

In this pedagogical article, we explore a powerful language for describing the notion of spacetime and particle dynamics intrinsic to a given fundamental physical theory, focusing on special relativity and its Newtonian limit. The starting point of the formulation is the representations of the relativity symmetries. Moreover, that seriously furnishes—via the notion of symmetry contractions—a natural way in which one can understand how the Newtonian theory arises as an approximation to Einstein’s theory. We begin with the Poincaré symmetry underlying special relativity and the nature of Minkowski spacetime as a coset representation space of the algebra and the group. Then, we proceed to the parallel for the phase space of a spin zero particle, in relation to which we present the full scheme for its dynamics under the Hamiltonian formulation, illustrating that as essentially the symmetry feature of the phase space geometry. Lastly, the reduction of all that to the Newtonian theory as an approximation with its space-time, phase space, and dynamics under the appropriate relativity symmetry contraction is presented. While all notions involved are well established, the systematic presentation of that story as one coherent picture fills a gap in the literature on the subject matter.

CAT(0) cube complexes and inner amenability

We here consider inner amenability from a geometric and group theoretical perspective. We prove that for every non-elementary action of a group $G$ on a finite dimensional irreducible CAT(0) cube complex, there is a nonempty $G$-invariant closed convex subset such that every conjugation invariant mean on $G$ gives full measure to the stabilizer of each point of this subset. Specializing our result to trees leads to a complete characterization of inner amenability for HNN-extensions and amalgamated free products. One novelty of the proof is that it makes use of the existence of certain idempotent conjugation-invariant means on $G$. We additionally obtain a complete characterization of inner amenability for permutational wreath product groups. One of the main ingredients used for this is a general lemma which we call the location lemma, which allows us to "locate" conjugation invariant means on a group $G$ relative to a given normal subgroup $N$ of $G$. We give several further applications of the location lemma beyond the aforementioned characterization of inner amenable wreath products.

Coherent states for [formula omitted]-component fractional quantum Hall systems and their nonlinear sigma models

We revise the subject of N-component fractional quantum Hall systems and its field-theoretic description in terms of U(N)-invariant nonlinear sigma models under a group-theoretical perspective. The Berry Lagrangian, which determines the dynamics and encodes the quantum commutation relations for the order parameter, is quantized and the Hilbert space is interpreted in terms of states of N-component composite fermions (M electrons bound to f magnetic flux lines) for fractional filling factor ν=M/f, in accordance with Jain’s picture. An explicit bosonic realization of coherent states on the Grassmannian manifold U(N)/[U(M)×U(N−M)] is provided, which describes coherent excitations of these composite fermions.

On the complete integrability of a nonlinear oscillator from group theoretical perspective

In this paper, we investigate the integrability aspects of a physically important nonlinear oscillator which lacks sufficient number of Lie point symmetries but can be integrated by quadrature. We explore the hidden symmetry, construct a second integral, and derive the general solution of this oscillator by employing the recently introduced λ-symmetry approach and thereby establish the complete integrability of this nonlinear oscillator equation from a group theoretical perspective.

Erratum:CPviolation conditions inN-Higgs-doublet potentials [Phys. Rev. D74, 036003 (2006)

Conditions for CP violation in the scalar potential sector of general N-Higgs-doublet models (NHDMs) are analyzed from a group theoretical perspective. For the simplest two-Higgs-doublet model (2HDM) potential, a minimum set of conditions for explicit and spontaneous CP violation is presented. The conditions can be given a clear geometrical interpretation in terms of quantities in the adjoint representation of the basis transformation group for the two doublets. Such conditions depend on CP-odd pseudoscalar invariants. When the potential is CP invariant, the explicit procedure to reach the real CP-basis and the explicit CP transformation can also be obtained. The procedure to find the real basis and the conditions for CP violation are then extended to general NHDM potentials. The analysis becomes more involved and only a formal procedure to reach the real basis is found. Necessary conditions for CP invariance can still be formulated in terms of group invariants: the CP-odd generalized pseudoscalars. The problem can be completely solved for three Higgs-doublets.

CPviolation conditions inN-Higgs-doublet potentials

Conditions for $CP$ violation in the scalar potential sector of general $N$-Higgs-doublet models are analyzed from a group theoretical perspective. For the simplest two-Higgs-doublet model potential, a minimum set of conditions for explicit and spontaneous $CP$ violation is presented. The conditions can be given a clear geometrical interpretation in terms of quantities in the adjoint representation of the basis transformation group for the two doublets. Such conditions depend on $CP$-odd pseudoscalar invariants. When the potential is $CP$ invariant, the explicit procedure to reach the real $CP$-basis and the explicit $CP$ transformation can also be obtained. The procedure to find the real basis and the conditions for $CP$ violation are then extended to general $N$-Higgs-doublet model potentials. The analysis becomes more involved and only a formal procedure to reach the real basis is found. Necessary conditions for $CP$ invariance can still be formulated in terms of group invariants: the $CP$-odd generalized pseudoscalars. The problem can be completely solved for three Higgs-doublets.

Empirical Evidence Supporting the Conceptual Relatedness of Object Representations and Internal Working Models

The proliferation of new developmental and clinical theories broadly termed "psychoanalytic" demands a methodology for making systematic comparisons to establish the commonalities and distinctions among them. This article presents an empirical method for the comparative evaluation of such theories. The Mother-To-Child Object Representation / Internal Working Model Q-sort is a 100-item instrument constructed to assess the quality of the prototypical mother's object representation and internal working model of her child at age five. Object relations judges were asked to sort these items for a complex, differentiated, integrated object representation of a five-year-old, attachment judges for a secure, coherent, freely valuing internal working model of a five-year-old. Judges' criterion Q-sorts in each group yielded a composite criterion Q-sort of the quality of the prototypical mother's mental representation of her child according to each group's theoretical perspective. The Spearman-Brown correlation between the two composite criterion Q-sorts was r = .90, p < .001 , suggesting that when confined to a 100-item common vocabulary, judges representing each theoretical construct agreed on the conceptual relatedness of object representations and internal working models. The theoretical constructs of object representations and internal working models share common assumptions in need of further exploration.

INTEGRABILITY OF THE RICCATI EQUATION FROM A GROUP-THEORETICAL VIEWPOINT

In this paper we develop some group-theoretical methods which are shown to be very useful for a better understanding of the properties of the Riccati equation, and we discuss some of its integrability conditions from a group-theoretical perspective. The nonlinear superposition principle also arises in a simple way.

THE NONLINEAR SUPERPOSITION PRINCIPLE AND THE WEI-NORMAN METHOD

Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei–Norman method is applied to obtain the associated differential equation in the group SL(2, ℝ). The superposition principle for first order differential equation systems and Lie–Scheffers theorem are also analyzed from this group theoretical perspective. Finally, the theory is applied in the solution of second order differential equations like time independent Schrödinger equation.

Bound states in a Yukawa potential: A Sturmian group-theoretical approach

The Schr\"odinger eigenvalue problem for a Yukawa potential is reexamined from a group-theoretical perspective. By using the Fock transformation, the Schr\"odinger operator is transformed into a compact or "inverse Sturmian" operator which is a linear superposition of local representation operators of SL(2,$R$). It may be approximated by finite-rank techniques, which provide a very useful method for obtaining accurate numerical results.