Symmetries and Groups in Contemporary Physics

The following sections are included:

• PREFACE

• XXIX^th INTERNATIONAL COLLOQUIUM ON GROUP-THEORETICAL METHODS IN PHYSICS

• ICGTMP STANDING COMMITTEE

• CONTENTS

Dear Friends and Colleagues Ladies and Gentlemen I’m delighted to be here at the Chern Institute of Mathematics of the Nankai University and to address the Opening of the 29th Colloquium on Group-Theoretical Methods in Physics. This is due to two reasons…

The Wigner Medal was established in 1977/8 and was awarded for the first time at the Integrative Conference on Group Theory and Mathematical Physics (7^th International Group Theory Colloquium 1978) to Eugene P. Wigner and Valentine Bargmann. The purpose of the Wigner Medal is to recognize outstanding contributions to the understanding of physics through group theory…

The Wigner Medal was established in 1977/78 and is administered by the Group Theory and Fundamental Physics Foundation. The first Wigner Medal was awarded to Eugene P. Wigner and Valentine Bargmann at the Integrative Conference on Group Theory and Mathematical Physics /VII International Group Theory Colloquium, 1978 in Austin, Texas, USA…

The following sections are included:

• Mead’s Background and Research Interests

• The Anomalous Sign Change in Molecules

• The Gauge Theory of Molecules

• References

In accepting this award, I feel very honored, and also very humble, because I had no idea that anyone would consider me to be worthy of it. I offer my heartfelt thanks to the foundation and to the organizers of this conference for the chance to be here tonight. I would also like to thank Brian Kendrick for that very kind and flattering speech. The St. Louis Cardinals logo was also a nice touch, and much appreciated…

The Hermann Weyl Prize was created in 2000 by the Standing Committee of the International Group Theory Colloquium. The purpose of the Weyl Prize is to provide recognition for young scientists (younger than 35 years of age) who have performed original work of significant scientific quality in the area of understanding physics through symmetries. The Hermann Weyl prize consists of a certificate citing the accomplishments of the recipient, prize money and an allowance towards attendance at the bi-annual International Group Theory Colloquium at which the award is presented. Candidates are nominated by or through members of the Standing Committee…

Razvan Gurau was born in Romania in 1980, began his university education in Bucharest and continued it in Paris, first at the École Normale Supérieure and afterwards at Paris-Sud XI, where he worked for his PhD under the supervision of Vincent Rivasseau. Since 2008 he has been a post-doctoral researcher at the Perimeter Institute. His talent and exceptional potential were evident from his student days and he has subsequently written or co-authored many papers on a variety of topics in the general area of quantum field theory. In 2009 he discovered and has subsequently developed the theory of coloured random tensors, which generalises beyond two dimensions the theory of random matrices. This unexpected discovery has opened up new avenues for investigation, with potential applications in probability theory, statistical mechanics, integrable systems, quantum field theory and random discrete geometries. It has led already to significant new steps including a novel generalisation of the Virasoro algebra beyond two dimensions and the first exact calculation of exponents in three dimensional critical statistical mechanics…

Over the past few years it has been discovered that an “observable” can be set up on the lattice which obeys the discrete Cauchy-Riemann equations. The ensuing condition of discrete holomorphicity leads to a system of linear equations which can be solved to yield the Boltzmann weights of the underlying lattice model. Surprisingly, these are the well known Boltzmann weights which satisfy the star-triangle or Yang-Baxter equations at criticality. This connection has been observed for a number of exactly solved models. I briefly review these developments and discuss how this connection can be made explicit in the context of the Z_N model. I also discuss how discrete holomorphicity has been used in recent breakthroughs in the rigorous proof of some key results in the theory of planar self-avoiding walks.

There is compelling theoretical evidence that quantum physics will change the face of information science. Exciting progress has been made during the last two decades towards the building of a large scale quantum computer. A quantum group approach stands out as a promising route to this holy grail, and provides hope that we may have quantum computers in our future.

Two types of (4 × 4)-dimensional solutions of YBE can be mapped to 2-dimensional YBE by acting the Temperely-Lieb algebra on the topological bases |e₁〉 and |e₂〉. The fermionic expressions for |e₁〉 and |e₂〉 have been shown. The 3-d topological base have been found for Birman-Wenzl-Murakami algebra. The extreme of L₁-norm of Wigner’s D-function is introduced to explain why there are only the two types in Physics that are related to entanglement in quantum information.

The tensor track approach to quantum gravity,¹ is based on a new class of quantum field theories, hereafter called tensor group field theories (TGFTs).^2–6 We provide a brief review of recent progress and list some desirable properties of TGFTs. In order to narrow the search for interesting models, we also propose a set of guidelines for TGFT’s loosely inspired by the Osterwalder-Schrader axioms of ordinary Euclidean QFT.

Traveling backward in time along the timeline of the Universe, we underline what is known experimentally, how sound is our understanding at various moments of its evolution and, towards the origin, at which point hypotheses start to replace certainties.

In this article we discuss some aspects of the geometric phase in molecular systems, starting with the Herzberg - Longuet-Higgins sign change appearing when a Born-Oppenheimer (BO) electronic wave function is carried through a closed path surrounding a conical intersection, and the gauge potential that arises when one rephases the BO function so as to make it single-valued. The discussion is generalized to the nonabelian gauge potential arising when the electronic wave function is a Kramers doublet. We also discuss the behavior of electronic wave functions under permutations of identical nuclei.

Random matrix models encode a theory of random two dimensional surfaces with applications to string theory, conformal field theory, statistical physics in random geometry and quantum gravity in two dimensions. The key to their success lies in the 1/N expansion introduced by ’t Hooft. Random tensor models generalize random matrices to theories of random higher dimensional spaces. For a long time, no viable 1/N expansion for tensors was known and their success was limited.

A series of recent results has changed this situation and the extension of the 1/N expansion to tensors has been achieved. We review these results in this paper.

The following sections are included:

• Outline

• Modeling an integrable discontinuity

• Solitons and defects

• Generalisations

• Defects in quantum field theory

• Boundaries revisited: the sine-Gordon model

• Remarks

• Acknowledgements

• References

In this paper I start from the early history of application of group theory to nuclear physics, then I discuss the seniority scheme in nuclear shell model under strong pairing interaction. Then I go to the Interacting Boson Model (IBM) and its applications. As the microscopic foundation, I describe nucleon pair approximation and its application to complex nuclei. Finally I discuss the microscopic method to derive the IBM Hamiltonian in strongly deformed nuclei from the mean-field theory.

We attempt in this paper to present a quick overview of coherent states and coherent state quantization, emphasizing the role of group theory. Coherent states feature in many areas of quantum physics and mathematics, but here the focus is on the mathematical aspects.

We demonstrate a novel approach that allows the determination of very general classes of exactly solvable Hamiltonians via Bethe ansatz methods. This approach combines aspects of both the co-ordinate Bethe ansatz and algebraic Bethe ansatz. The eigenfunctions are formulated as factorisable operators acting on a suitable reference state. Yet, we require no prior knowledge of transfer matrices or conserved operators. By taking a variational form for the Hamiltonian and eigenstates we obtain general exact solvability conditions. The procedure is conducted in the framework of Hamiltonians describing the crossover between the low-temperature phenomena of superconductivity, in the Bardeen-Cooper-Schrieffer (BCS) theory, and Bose-Einstein condensation (BEC).

The spatiotemporal propagation of a momentum excitation traveling along the Fermi-Pasta-Ulam β (FPU-β) lattice is investigated. The lifetime of the solitary wave against the anharmonic parameter β exhibits a hierarchical multiple-peak fractal structure for the case of an open boundary condition. The energy of the solitary wave decays exponentially during the final stage of the collapse process. The decay rate against β is found to exhibit a crossover between two different scaling laws. The hierarchical multiple-peak structure corresponds to a weak stochasticity where the KAM tori are dominant. Our studies of the solitary-wave dynamics support the discussions of energy carriers in the heat conduction processes of low-dimensional lattices, where phonons play a dominant role.

We give a summary of the recent progress in the theory of positive representations of split real quantum groups. We outline the motivation of the research program and briefly describe the main ingredients of the construction, the appearance of Langlands duality, and future aspects of the theory.

Let A_q(g) be the quantized algebra of functions associated with simple Lie algebra g defined by generators obeying the so called RTT relations. We describe the embedding A_q(B₂) ↪ A_q(C₂) explicitly. As an application, a new solution of the Isaev-Kulish 3D reflection equation is constructed by combining the embedding with the previous solution for A_q(C₂) by the authors.

We briefly summarize recent constructions of the disuccessor and duplicator of a quadratic operad and relate them to the Manin products, and Rota-Baxter and average operators.

Two novel tensor network-based algorithms, say, the linearized tensor renormalization group and the optimized decimation of tensor network with superorthogonalization methods, as well as their applications to quantum spin lattice models, are discussed. Both methods can be applied to explore not only the thermodynamic but also the ground state properties of low-dimensional quantum lattice systems.

We construct the explicit free field representations of the current (super) algebras at a generic level k. The corresponding energy-momentum tensors from the Sugawara construction is also given in terms of free fields.

An overview of maximally superintegrable classical Hamitonians on spherically symmetric spaces is presented. It turns out that each of these systems can be considered either as an oscillator or as a Kepler–Coulomb Hamiltonian. We show that two possible quantization prescriptions for all these curved systems arise if we impose that superintegrability is preserved after quantization, and we prove that both possibilities are gauge equivalent.

A superintegrable finite model of the oscillator in two-dimensions is presented. It is defined on a uniform lattice of triangular shape. The wavefunctions are expressed in terms of bivariate Krawtchouk polynomials. The constants of motion form an SU(2) symmetry algebra.

A quantum superintegrable system is an integrable n-dimensional Hamiltonian system with potential: H = Δ_n + V that admits 2n − 1 algebraically independent partial differential operators commuting with the Hamiltonian, the maximum number possible. The system is of order ℓ. if the maximum order of the symmetry operators, other than H, is ℓ. Typically, the algebra generated by the symmetry operators and their commutators has been proven to close polynomially. However the degenerate 3-parameter potential for the 3D extended Kepler-Coulomb system (2nd order superintegrable) appeared to be an exception as Kalnins et al. (2007) showed that it didn’t close polynomially. The 3D 4-parameter potential for the extended Kepler-Coulomb system is not even 2nd order superintegrable. However, Verrier and Evans (2008) showed it was 4th order superintegrable, Tanoudis and Daskaloyannis (2011) showed it closed polynomially. We consider an infinite class of quantum extended Kepler-Coulomb systems that we show to be superintegrable of arbitrarily high order, compute the structure algebras and demonstrate that algebraic closure is the norm, whereas polynomial closure requires extra symmetry. This is a report on joint work with E. G. Kalnins (Waikato) and J. M. Kress (New South Wales).

The mechanism allowing a protein to search for a target sequence on DNA is currently described as an intermittent process made of 3D diffusion in bulk and 1D diffusion along the DNA molecule. Due to the relevant charge of protein and DNA, electrostatic interaction should play a crucial role during this search. In this paper, we explicitly derive the mean field theory allowing for a description of the protein-DNA electrostatics in solution. This approach leads to a unified model of the search process, where 1D and 3D diffusion appear as a natural consequence of the diffusion on an extended interaction energy profile.

The existence of the phenomenon of Horizontal Gene Transfer (HGT) is one of the important processes by means of which the progress of the evolution is explained. The purpose of this article was to propose a comprehensive and combined mathematical-statistical approach to the identification of HGT events, which is not found in the biological literature. In this article the database consisting of CLCA sequences and 13 whole bacterial genomes was used to test known methods of HGT events’ identification. Moreover some new proposals were presented. One of them is the application of new measures, i.e. the angle and the area between the vectors identified with the codon genes. The statistical ranking of all used methods has been developed on the basis of the obtained results.

The epidemiology of person-to-person communicable disease in large, homogeneous populations is modeled by three ordinary differential equations. These represent the susceptible, infectious and recovered individuals. For smaller populations an agent-based model can capture the inhomogeneous nature of human interactions by modeling each individual as a node in a network, or vertex in a graph. In this model we look at a set of large homogeneous populations, each modeled as a classic SIR epidemic (the patches), but which have immigration between them (the graph). We use this approach to simulate the spread of influenza during an outbreak and compare results of the simulation with available data.

It is a mathematically interesting and epidemiologically useful question to ask which aspects of disease transmission are controlled by the local properties of each SIR model, and which are controlled by the global connectivity of the graph. We prove that, for a general class of patch SIR models, the stability of the disease free equilibrium is a local property.

For a set of quantum states generated by the action of a group, we consider the graph obtained by considering two group elements adjacent whenever the corresponding states are non-orthogonal. We analyze the structure of the connected components of the graph and show two applications to the optimal estimation of an unknown group action and to the search for decoherence free subspaces of quantum channels with symmetry.

This article proposes a unified method to estimation of group action by using Fourier analysis.

We present a generalization of the binomial distribution associated with a sequence of positive numbers. It involves asymmetric and symmetric expressions of probabilities for win-loss sequences of trials. Our approach is based on generating functions and produces, in the symmetric case, polynomials of the binomial type. Poisson-like limits, Leibniz triangle rules and related entropy(ies) are considered. Our generalizations are illustrated by various analytical and numerical examples.

We analyze the dynamics of a finite level quantum system using local in time master equation. Markovian dynamics is characterized in terms of divisible dynamical map. Moreover we provide a family of criteria which can distinguish Markovian and non-Markovian quantum evolution. They are based on a simple observation that Markovian dynamics implies monotonic behavior of several well known quantities like distinguishability of states, fidelity and relative entropy. Violation of any of these criteria implies that the corresponding dynamics is non-Markovian.

This is a follow up of our recent paper¹, which reorganizes the exposition and adds some comments made from a slightly different point of view. More precisely this is a purely reproducing kernel approach which allows to eliminate any use of measure in the standard constructions.

Pauli’s principle is generalized for the case of the Z₃ grading replacing the usual Z₂ grading, leading to ternary commutation relations and ternary algebras. Invariant cubic forms on such algebras are introduced, with the SL(2,C) arising naturally as their invariance group in the case of lowest dimension, with two generators only. The wave equation generalizing the Dirac operator for the Z₃-graded case is introduced, whose diagonalization leads to a third-order equation. The solutions of this equation cannot propagate because their exponents contain a non-oscillating real damping factor. We show how certain cubic products can propagate nevertheless. The model suggests the origin of the color SU(3) symmetry.

This report constitutes an introduction to three papers published by the authors in J. Phys. A [43 (2010) 115303 and 45 (2012) 244036] and J. Math. Phys. [52 (2011) 082101]. See these three papers for the relevant references.

We investigate multiple qubit Pauli groups and the quantum states/rays arising from their maximal bases. Remarkably, the real rays are carried by a Barnes-Wall lattice BW_n (n = 2^m). We focus on the smallest subsets of rays allowing a state proof of the Bell-Kochen-Specker theorem (BKS). BKS theorem rules out realistic non-contextual theories by resorting to impossible assignments of rays among a selected set of maximal orthogonal bases. We investigate the geometrical structure of small BKS-proofs ν – l involving ν rays and l 2n-dimensional bases of n-qubits. Specifically, we look at the classes of parity proofs 18 – 9 with two qubits (A. Cabello, 1996), 36 – 11 with three qubits (M. Kernaghan & A. Peres, 1995) and related classes. One finds characteristic signatures of the distances among the bases, that carry various symmetries in their graphs.

We investigate new models for a finite quantum oscillator based upon the Lie superalgebra , where the position and momentum operators are represented as odd elements of the algebra. We discuss properties of the spectrum of the position operator, and of the (discrete) position wavefunctions given by (alternating) Krawtchouk polynomials.

Finite discrete Hamiltonian systems are mothered by the Lie algebra su(2) and have wavefunctions (signals) of unit-spaced positions . These wavefunctions are N-vectors (N = 2j + 1) that can be subject to the group U(2) of unitary linear transformations and overall phases. But moreover, they can also be subject to the much larger group of all unitary N × N matrices in U(N); those outside U(2) are ‘nonlinear’ and correspond to aberrations in geometric optics. Here we examine in particular the contraction j → ∞ of su(2) to iso(2), where the latter mothers infinite discrete free Hamiltonian systems. We find that the aberrations can be applied only to signals that are of appropriate decrease or bounded.

The inversion operators on a lattice in finite phase plane are used for building a complete set of mutually orthogonal Hermitian operators. The lattice is given by tc in the x direction and by in the p-direction; c is an arbitrary length constant and M is the dimension of the space; s and t assume the values from 0 to M – 1. For M odd the M² inversion operators on the lattice form a complete set of mutually orthogonal operators. For M even we assign a sum of 4 inversion operators (a quartet) to each site of the lattice (t, s). We prove that these quartets for t, s = 0, 1, … ,M – 1 form a mutually orthogonal set of M² Hermitian operators.

Bipartite graphs, especially drawn on Riemann surfaces, have of late assumed an active rôle in theoretical physics, ranging from MHV scattering amplitudes to brane tilings, from dimer models and topological strings to toric AdS/CFT, from matrix models to dessins d’enfants in gauge theory. Here, we take a brief and casual promenade in the realm of brane tilings, quiver SUSY gauge theories and dessins, serving as a rapid introduction to the reader.

This article provides a mini-introduction to three-dimensional field theories with rigid (p, q) anti-de Sitter supersymmetry.

We have studied the fermi gas system associated to the ABJM matrix model, aiming at understanding some hidden structures of M-theory. In trying to reduce the number of integrations in computing the partition functions Z_k(N), we have found an identity relating two density matrices of the opposite parities. Using this relation, we have computed exact values of the partition functions for k = 1 up to N = 9.

We discuss an interpretation of a simple supersymmetric matrix model with a double-well potential as two-dimensional type IIA superstrings on a nontrivial Ramond-Ramond background. We find direct correspondence between single-trace operators in the matrix model and integrated vertex operators in type IIA theory.

We present a new perspective on early cosmology based on Loop Quantum Gravity. We use projected spinnetworks, coherent states and spinfoam tecniques, to implement a quantum reduction of the full Kinematical Hilbert space of LQG, suitable to descrive inhomogeneous cosmological models. Some preliminar results on the solutions of the Scalar constraint of the reduced theory are also presented.

The topic of this proceeding is some themes of spinfoam formalism, covariant formalism of loop quantum gravity. The focus is on the imposing of simplicity constraint and the computation of the Lorentzian correlation functions. The former is to find the way to connect different aspects related to the Engle-Pereira-Rovelli-Livine spinfoam model, and the latter is to test the resulting model and try to extract physics from that.

In this article we present a method for finding all foams with given boundary, one internal vertex and no edges connecting this vertex with itself. We apply the method to the Dipole Cosmology model and find all spinfoams contributing to the transition amplitude in first order of vertex expansion.

Recently, a rank four tensor group field theory has been proved renormalizable. We provide here the key points on the renormalizability of this model and its UV asymptotic freedom.

In this talk, we elaborate on the operation of graph contraction introduced by Gurau in his study of the Schwinger-Dyson equations. After a brief review of colored tensor models, we identify the Lie algebra appearing in the Schwinger-Dyson equations as a Lie algebra associated to a Hopf algebra of the Connes-Kreimer type. Then, we show how this operation also leads to an analogue of the Wilsonian flow for the effective action. Finally, we sketch how this formalism may be adapted to group field theories.

We remark the importance of adding suitable pre-geometric content to tensor models, obtaining what has recently been called tensorial group field theories, to have a formalism that could describe the structure and dynamics of quantum spacetime. We also review briefly some recent results concerning the definition of such pre-geometric content, and of models incorporating it.

A canonical formalism of the rank-three tensor model with the notion of local time is proposed. The consistency of the local time evolution is guaranteed by imposing that local Hamiltonians and the so(N) kinematical symmetry of the tensor model should form a first class constraint algebra. By imposing some physically reasonable assumptions, it is shown that there exist only two such local Hamiltonians with a slight difference in index contraction. The first class constraint algebra is shown to approach the DeWitt constraint algebra of the general relativity in a certain locality limit. Quantization of the system is briefly discussed.

In this talk we prove that the perturbation series of the 2-dimensional Grosse-Wulkenhaar model is Borel summable. The interested reader could look at¹ for more details.

We describe a class of examples of braided monoidal categories which are built from Hopf algebras in symmetric categories. The construction is motivated by a calculation in two-dimensional conformal field theory and is tailored to contain the braided monoidal categories occurring in the study of the Ising model, their generalisation to Tamabara-Yamagami categories, and categories occurring for symplectic fermions.

Higher genus partition functions of two-dimensional conformal field theories have to be invariants under linear actions of mapping class groups. We illustrate recent results ^1,3 on the construction of such invariants by concrete expressions obtained for the case of Drinfeld doubles of finite groups. The results for doubles are independent of the characteristic of the underlying field, and the general results do not require any assumptions of semisimplicity.

An interpretation of the gauge anomaly of the two-dimensional multi-phase σ-model is presented in terms of an obstruction to the existence of a topological defect network implementing a local trivialisation of the gauged σ-model.

The simple finite Lie superalgebras D(2, 1; α), G(3), D(4, 1), D(2, 2), A(3, 1) and F(4) admit D-module representations, given by a set of differential operators of a single variable t ∈ ℝ, at a critical value of the scaling dimension λ. These superalgebras are one-dimensional N-extended superconformal algebras with N = 4 (D(2, 1; α)), N = 7 (G(3)) and N = 8 (the remaining ones). The critical D-module reps induce invariant actions in the Lagrangian framework for superconformal mechanics in D target dimensions. The N = 8 critical scalings are linked to the D-dimensional harmonic functions with D = 1, 2, 3, 5, 6, 7, 8. This talk is based on J. Math. Phys. 53 (2012) 043513 (arXiv:1112.0995), J. Math. Phys. 53 (2012) 103518 (arXiv:1208.3612) and some extra material.

We introduce the extended Griess algebra for a vertex operator superalgebra and derive its Matsuo-Norton trace formula based on conformal design structure. This article is an exposition of the paper [Y12].

The Galilean conformal algebra (GCA) is a class of non-semisimple Lie algebra specified by spin ℓ and dimension of space d. We investigate N = 2 supersymmetric extensions of GCA and introduce a novel super-GCA for d = 2 but arbitrary ℓ We also give a realization of the novel superalgebra in terms of bosons and fermions.

Singletons are those unitary irreducible modules of the Poincaré or (anti) de Sitter group that can be lifted to unitary modules of the conformal group. Higher-spin algebras are the corresponding realizations of the universal enveloping algebra of the conformal algebra on these modules. The ambient formulation of the maximal symmetry algebra of tensorial singletons is reviewed.

We present the class of deformations of simple Euclidean superalgebra, which describe the supersymmetrization of some Lie algebraic noncommutativity of D = 4 Euclidean space-time. The presented deformations are generated by the supertwists. We provide new explicit formulae for a chosen twisted D = 4 Euclidean Hopf superalgebra and describe the corresponding quantum covariant deformation of chiral Euclidean superspace.

We review the features of the background field method for three-dimensional gauge theories in N = 2 superspace. As an application, we compute one-loop low-energy effective actions in N = 2, 4, 8, d = 3 SYM theories.

Using our previous results on the systematic construction of invariant differential operators for non-compact semisimple Lie groups we classify the conservation laws in the case of SO(p,q).

We present elements of the superfield approach to constructing N=4 supersymmetric quantum mechanics developed in arXiv:0812.4276, arXiv:0905.4951, arXiv:0912.3508, arXiv:1112.1947 and arXiv:1204.4474.

We briefly sketch our formulation¹ of finite twisted Poincaré transformations and of the corresponding covariant (scalar) Quantum Field Theory on Grönewold-Moyal-Weyl noncommutative spacetime.

The Cayley–Hamilton–Newton theorem for half-quantum matrices is proven.

A series of integrable spin chains related to special tensor product representations of Temperley - Lieb algebra is constructed. The generator of the TL-algebra is a rank one projector acting on ℂⁿ ⊗ ℂⁿ. The spectra of these spin chains with appropriate boundary conditions are the same and equivalent to the spin 1/2 XXZ-model while the multiplicities of the eigenvalues are different and depend on n. The generators of the TL-algebra with rankX_j bigger than 1 are also mentioned.

We discuss general conditions which are imposed on the form of actions for duality-symmetric theories by the requirement of manifest duality symmetry and space-time invariance.

Cubic extensions of the Poincaré algebra can be constructed in consistency with physical assumptions and provide new insights for the description of phenomena. In this paper we review some recent results associated to these structures and give the salient steps to construct their associated groups.

In this paper, we study the structure theory of a class of generalized map Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of generalized map centerless Virasoro algebras are determined.

In this paper we present a study of the nucleon-pair approximation with the isospin symmetry. In this model the building blocks are collective nucleon-pairs with given spin and isospin. We exemplify our work by using the ⁹⁶Cd nucleus.

We show that pseudospin symmetry is a symmetry of the Dirac Hamiltonian for which the sum of the scalar and vector potentials are a constant. In this paper we discuss some of the implications of this relativistic symmetry and the experimental data that support these predictions. We show that pseudo-U(3) symmetry is a symmetry of the Dirac Hamiltonian for which the sum of harmonic oscillator vector and scalar potentials are equal to a constant, and we give the generators of pseudo-U(3) symmetry. We also show that pseudospin for nuclei implies spin symmetry for anti-nucleons moving in a nuclear environment.

The Heine-Stieltjes correspondence is extended and applied to solve Bethe ansatz equations of the SU(1,1) and SU(2) Gaudin models, from which the extended Heine-Stieltjes polynomial approach to these models is proposed. As examples for the application, exact solutions of the standard two-site Bose-Hubbard model and the standard pairing model are formulated from the corresponding polynomials.

A group theoretic method for the systematic study of multi-quark states is developed. The calculation of matrix elements of many-body Hamiltonians is simplified by transforming the physical bases (quark cluster bases) to symmetry bases (group chain classified bases), where the fractional parentage expansion method can be used. The technique is applied to 5- and 6-quark systems.

In this contribution I present an analytic proof for the partial conservation of the seniority symmetry in j = 9/2 shells. It is found that all non-diagonal (and the relevant diagonal) matrix elements can be re-expressed in closed forms and are proportional to certain one-particle cfp’s. This remarkable occurrence of partial dynamic symmetry is the consequence of the peculiar property of the j = 9/2 shell, where all ν = 3 and 5 states are uniquely defined.

We developed a group representation theory and calculation method based on quantum representation theory. All the calculations, such as Character, Irreducible bases and Matrices, Projection operators, Clebsch-Gordan series and coefficients are reduced to solve a set of eigen-equations of a complete set of commuting operators consisting of the class operators of the group. This approach had been successfully applied to all of the finite groups and compact Lie groups and various useful coefficients for physical application of group representation theory had been calculated and tabulated.

The E(2) dynamical symmetry is realized with both the differential operators and boson operators. It is shown that the discrete spectrum of the E(2) dynamical symmetry just correspond to the solutions of the infinite square well Hamiltonian in two dimensions, of which the eigenfunction can be related with the Bessel equation of integral order.

In this contribution I would like to discuss the SO(3) symmetry in the nuclear shell model and its applications, based on our recent works. We shall focus on the enumeration of number of states with given spin, J-th pairing interaction, and sum rules of angular momentum coupling coefficients such as six-j and nine-j symbols.

The following sections are included:

• Introduction

• Partial inner product spaces
  • Scales of Hilbert or Banach spaces
  • Sequence spaces
  • Spaces of locally integrable functions
  • The spaces L^p(ℝ, dx), 1 < p < ∞

• Operators on pip-spaces

• Special classes of operators on PIP-spaces
  • Homomorphisms
  • Symmetric operators
  • Orthogonal projections
  • Representations of Lie groups and Lie algebras

• References

Enhanced quantization is an improved program for overcoming difficulties which may arise during an ordinary canonical quantization procedure. We review here how this program applies for a particle on circle.

According to a story told to me by Larry Schulman Wigner was going around Princeton in the 1960s asking people: Tell me, what is the 2 P^1/2 (decaying) state of the H-atom? Wigner knew from his work on the Weisskopf-Wigner approximation (1930) and on the Breit-Wigner lineshape (1936) that there could not be an exponentially decaying state (Gamow 1926) vector or a Breit-Wigner (Lorentzian) resonance state within the mathematical framework of quantum mechanics at his time. We shall discuss how new mathematical tools, Hardy spaces predicting semigroup time evolution, can overcome this problem and lead to a unified description of resonance and decay phenomena.

A unified method of calculating structure functions from commutation relations of deformed single-mode oscillator algebras is presented. A natural approach to building coherent states associated to deformed algebras is then deduced.

We revisit the theorem of Wigner, Araki and Yanase (WAY) describing limitations to repeatable quantum measurements that arise from the presence of conservation laws. We will review a strengthening of this theorem by exhibiting and discussing a condition that has hitherto not been identified as a relevant factor. We will also show that an extension of the theorem to continuous variables such as position and momentum can be obtained if the degree of repeatability is suitably quantified.

In this note, we report on ongoing research concerning geometric realisations of the simplest unitary Riemann surface braid group representations. We discuss in particular the genus one case by means of noncommutative geometric techniques.

For any skew-Hermitian irreducible infinite dimensional representation η of iso(3), we find a sequence of (finite dimensional) irreducible representations ρ_n of so(4) which contract to η.

We summarize the recent development of realizing arbitrary Jack symmetric functions by a sequence of vertex operators associated with rectangular Young tableaux.

The Bannai–Ito (BI) polynomials are shown to arise as Racah coefficients of the Hopf algebra sl₋₁(2).

Due to some ambiguity in the definition of mutual Tsallis entropy in classical probability theory, its generalization to quantum theory is discussed. We define a q-discord derived from the q-expectation value of mutual Tsallis entropy. We provide analytical expressions for q-discord for two-qubit Werner states as well as for isotropic states and we show that both are positive, at least for states under investigation, for all q > 0.

The Cayley number, which is closely related to the exceptional Lie groups, is somewhat difficult to deal with, due to the lack of its associativity, where the associativity can be represented by the commutativity between the left and right multiplications L_x and R_y, respectively. We obtain a relation between L_x and R_y by generalizing the commutator [L_x, R_y]. However, the resultant relation depends upon the geometrical configuration between x and y.

In this paper the rotational invariants constructed by the products of three spherical harmonic polynomials are expressed generally as homogeneous polynomials with respect to three coordinate vectors, where the coefficients are calculated explicitly in compact forms. The details can be found in arXiv 1203.6702.

Understanding the nature of the roots of the Bethe ansatz equations is central to understanding the mathematical physics underpinning quantum integrable and exactly solvable models. Here we analyse an exactly solvable, nonhermitian BCS pairing Hamiltonian dependent on two real-valued coupling parameters. For particular constraints on these prameters, we show that a class of solutions for the Bethe ansatz equations can be obtained analytically. A discussion is given on the physical implications of these solutions.

The “one particle” and the “two particles” interferometers can be (quantum mechanically) described in terms of Hilbert spaces of states and scattering operators. In this case the scattering operators realize a unitary representation of SU(2) and the Euler angles of the SU(2) group are related to the interferometers parameters (transmission coefficients, phase shifts).

In this brief report we connect the notion of quantum (state) t-designs, motivated by the theory of measurement in quantum information theory, to generalised coherent states, specifically involving finite group actions on finite dimensional Hilbert space. We use this to show that the Clifford group, which plays an important role in many areas, can give only 3-designs in the case of one qubit.

This article introduces briefly some recent developments of reduction theory of controlled Hamiltonian systems with symmetry.