Quaternion Octonion Reformulation of Quantum Chromodynamics

We have made an attempt to develop the quaternionic formulation of Yang–Mill’s field equations and octonion reformulation of quantum chromo dynamics (QCD). Starting with the Lagrangian density, we have discussed the field equations of SU(2) and SU(3) gauge fields for both cases of global and local gauge symmetries. It has been shown that the three quaternion units explain the structure of Yang–Mill’s field while the seven octonion units provide the consistent structure of SU(3) _C gauge symmetry of quantum chromo dynamics.     

Polarization invariance and biphoton coherent states of light

A new classification of polarization states of quantum light fields is given using the concept of the polarization (P) spin due to the polarization gauge SU_p(2) invariance of free light fields [I]. Generalized coherent states (GCS) asNociated with P-scalar biphotons are discussed. We also point out some applications of the results in the optical communication theory.Presented at the International Workshop on Squeezed and Correlated States in Quantum Optics, Moscow, December 3–7, 1990.     

Discrete symmetry and GUT breaking

We study the supersymmetric GUT models in which the supersymmetry and GUT gauge symmetry can be broken by a discrete symmetry. First, with the ansatz that there exist discrete symmetries in the branes' neighborhoods, we discuss the general reflection symmetries and GUT breaking on and . In those models, the extra dimensions can be large and the KK states can be set arbitrarily heavy. Second, considering that the extra space manifold is the annulus or the disc , we can define any symmetry and break any 6-dimensional N=2 supersymmetric SU(M) models down to the 4-dimensional N=1 supersymmetric models for the zero modes. In particular, there might exist the interesting scenario on where just a few KK states are light, while the others are relatively heavy. Third, we discuss the complete global discrete symmetries on and study the GUT breaking. Received: 12 February 2002 / Published online: 14 June 2002     

Skein theory and topological quantum registers: Braiding matrices and topological entanglement entropy of non-Abelian quantum Hall states

We study topological properties of quasi-particle states in the non-Abelian quantum Hall states. We apply a skein-theoretic method to the Read-Rezayi state whose effective theory is the SU(2)_K Chern-Simons theory. As a generalization of the Pfaffian (K = 2) and the Fibonacci (K = 3) anyon states, we compute the braiding matrices of quasi-particle states with arbitrary spins. Furthermore we propose a method to compute the entanglement entropy skein-theoretically. We find that the entanglement entropy has a nontrivial contribution called the topological entanglement entropy which depends on the quantum dimension of non-Abelian quasi-particle intertwining two subsystems.     

Symmetry as a source of hidden coherent structures in quantum physics: General outlook and examples

A general algebraic approach incorporating both invariance groups and dynamical symmetry algebras is developed to reveal hidden coherent structures (closed complexes and configurations) in quantum manybody physics models due to symmetries of their Hamiltonians H. Its general ideas are manifested on some recent new examples: (1) G-invariant biphotons and a related SU(2)-invariant treatment of unpolarized light; (2) quasispin clusters in nonlinear models of quantum optics; and (3) construction of composite particles and (para)fields from G-invariant clusters due to internal symmetries.     

Quaternion-Octonion Unitary Symmetries and Analogous Casimir Operators

An attempt has been made to investigate the global SU(2) and SU(3) unitary flavor symmetries systematically in terms of quaternion and octonion respectively. It is shown that these symmetries are suitably handled with quaternions and octonions in order to obtain their generators, commutation rules and symmetry properties. Accordingly, Casimir operators for SU(2) and SU(3) flavor symmetries are also constructed for the proper testing of these symmetries in terms of quaternions and octonions.     

Geometric measure of pairwise quantum discord for superpositions of multipartite generalized coherent states

We give the explicit expressions of the pairwise quantum correlations present in superpositions of multipartite coherent states. A special attention is devoted to the evaluation of the geometric quantum discord. The dynamics of quantum correlations under a dephasing channel is analyzed. A comparison of geometric measure of quantum discord with that of concurrence shows that quantum discord in multipartite coherent states is more resilient to dissipative environments than is quantum entanglement. To illustrate our results, we consider some special superpositions of Weyl–Heisenberg, SU(2)$\mathit{SU}(2)$ and SU(1,1)$\mathit{SU}(1,1)$ coherent states which interpolate between Werner and Greenberger–Horne–Zeilinger states.     

Duality and Hidden Symmetries in Interacting Particle Systems

In the context of Markov processes, both in discrete and continuous setting, we show a general relation between duality functions and symmetries of the generator. If the generator can be written in the form of a Hamiltonian of a quantum spin system, then the “hidden” symmetries are easily derived. We illustrate our approach in processes of symmetric exclusion type, in which the symmetry is of SU(2) type, as well as for the Kipnis-Marchioro-Presutti (KMP) model for which we unveil its SU(1,1) symmetry. The KMP model is in turn an instantaneous thermalization limit of the energy process associated to a large family of models of interacting diffusions, which we call Brownian energy process (BEP) and which all possess the SU(1,1) symmetry. We treat in details the case where the system is in contact with reservoirs and the dual process becomes absorbing.     

多模双参数变形相干态

借助一个满足量子Heisenberg-Weyl代数(H-W_q,s代数)的多模算符,给出了量子代数SU(2)_q,s和SU(1,1)_q,s的k(k≥2)模实现,并构造了相应的相干态.     

Physics at the planck scale: Strings and symmetries

The emergence of symmetries in the earlier proposed model of the Universe (3D network of strings in a thermal bath) is studied. The number is introduced which estimates the relative probability of changing the Gibbs distribution (or its nonequilibrium analogue) under action of stochastic forces. The principle of maximal stability (PMS) stating that only the most stable distributions are realized in the Universe is formulated. The nature of gauge symmetries and supersymmetries is discussed. According to PMS the groups SU(5) and SU(3) are advantageous. It is shown that in this model the Kaluza—Klein-Mandel-Fock unification of gravity and the Yang-Mills fields appears in the natural way. The list of thirty consequences of the model is given. They form the basis of modern physics (classical Hamiltonian mechanics, quantum mechanics, gauge symmetries, internal symmetries and so on).     

Hidden symmetries in two dimensional field theory

The bosonization process elegantly shows the equivalence of massless scalar and fermion fields in two space-time dimensions. However, with multiple fermions the technique often obscures global symmetries. Witten’s non-Abelian bosonization makes these symmetries explicit, but at the expense of a somewhat complicated bosonic action. Frenkel and Kac have presented an intricate mathematical formalism relating the various approaches. Here, I reduce these arguments to the simplest case of a single massless scalar field. In particular, using only elementary quantum field theory concepts, I expose a hidden SU (2) × SU (2) chiral symmetry in this trivial theory. I then discuss in what sense this field should be interpreted as a Goldstone boson.     

Critique of generalised purity as an entanglement measure: Explicit failure in simple separable Bose–Hubbard models

The SU(2) and SU(3) Lie algebras lend themselves naturally to studies of two- and three-well Bose–Einstein condensates, with the group operators being expressed in terms of bosonic annihilation and creation operators at each site. The success of these representations has led to the purities associated with these algebras to be promoted as a measure of entanglement in these systems. In this work, we show that these purities do not provide an unambiguous measure of entanglement between wells, but instead give results which depend on the quantum statistical states of the atomic ensembles in each well. Using the example of totally uncoupled wells where the atoms in one have never interacted with the atoms in the other, we quantify these purities for different states and show that completely separable states can give values which have been claimed to indicate the presence of entanglement. We also consider claims that the generalised purities measure particle rather than mode entanglement, with emphasis on the case of indistinguishable bosons, as found in these systems.     

Gauge symmetries of electroweak interactions

By considering the symmetries associated with baryon number and lepton number conservation as gauge symmetries, the underlying gauge symmetry of weak electromagnetic interactions is shown to beSU(2) _L ×U(1)×U(1)Baryon×U(1)_Lepton. If right-handed currents exist on a par with the observed left-handed ones, then the full symmetry of electroweak interactions that emerges isSU(2)_L×SU(2)_R×U(1)_Baryon×U(1)_Lepton. These symmetries offer a rich spectrum of massive neutral gauge bosons, one of which is the massive neutral boson of the standardSU(2) _L ×U(1) _Y model.     

A symmetry principle for topological quantum order

We present a unifying framework to study physical systems which exhibit topological quantum order (TQO). The major guiding principle behind our approach is that of symmetries and entanglement. These symmetries may be actual symmetries of the Hamiltonian characterizing the system, or emergent symmetries. To this end, we introduce the concept of low-dimensional Gauge-like symmetries (GLSs), and the physical conservation laws (including topological terms, fractionalization, and the absence of quasi-particle excitations) which emerge from them. We prove then sufficient conditions for TQO at both zero and finite temperatures. The physical engine for TQO are topological defects associated with the restoration of GLSs. These defects propagate freely through the system and enforce TQO. Our results are strongest for gapped systems with continuous GLSs. At zero temperature, selection rules associated with the GLSs enable us to systematically construct general states with TQO; these selection rules do not rely on the existence of a finite gap between the ground states to all other excited states. Indices associated with these symmetries correspond to different topological sectors. All currently known examples of TQO display GLSs. Other systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin-exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. The symmetry based framework discussed herein allows us to go beyond standard topological field theories and systematically engineer new physical models with finite temperature TQO (both Abelian and non-Abelian). Furthermore, we analyze the insufficiency of entanglement entropy (we introduce SU(N) Klein models on small world networks to make the argument even sharper), spectral structures, maximal string correlators, and fractionalization in establishing TQO. We show that Kitaev’s Toric code model and Wen’s plaquette model are equivalent and reduce, by a duality mapping, to an Ising chain, demonstrating that despite the spectral gap in these systems the toric operator expectation values may vanish once thermal fluctuations are present. This illustrates the fact that the quantum states themselves in a particular (operator language) representation encode TQO and that the duality mappings, being non-local in the original representation, disentangle the order. We present a general algorithm for the construction of long-range string and brane orders in general systems with entangled ground states; this algorithm relies on general ground states selection rules and becomes of the broadest applicability in gapped systems in arbitrary dimensions. We exactly recast some known non-local string correlators in terms of local correlation functions. We discuss relations to problems in graph theory.     

Biphotons,squeezing, and new states of light in polarization quantum optics

The concept of squeezing is discussed for multimode quantum light beams with consideration of polarization using the polarization gaugeSU (2) invariance of free electromagnetic fields. We separate the polarization and biphoton degrees of freedom from other ones, and consider uncertainty relations characterizing polarization and biphoton observables. As a consequence, we obtain a new classification of polarization states of light within quantum optics. We also discuss briefly some interrelations of our analysis with experiments related to some fundamental problems of physics.     

Random matrix ensembles with random interactions: Results for EGUE(2)-SU(4)

We introduce in this paper embedded Gaussian unitary ensemble of random matrices, for m fermions in Ω number of single particle orbits, generated by random twobody interactions that are SU(4) scalar, called EGUE(2)-SU(4). Here the SU(4) algebra corresponds to Wigner’s supermultiplet SU(4) symmetry in nuclei. Formulation based on Wigner-Racah algebra of the embedding algebra U(4Ω) ? U(Ω) ? SU(4) allows for analytical treatment of this ensemble and using this analytical formulas are derived for the covariances in energy centroids and spectral variances. It is found that these covariances increase in magnitude as we go from EGUE(2) to EGUE(2)-s to EGUE(2)-SU(4) implying that symmetries may be responsible for chaos in finite interacting quantum systems.     

Classification of qubit entanglement: SL(2,ℂ) versus SU(2) invariance

The role of SU(2) invariants for the classification of multiparty entanglement is discussed and exemplified for the Kempe invariant I ₅ of pure three-qubit states. It is found being an independent invariant only in presence of both W-type entanglement and three-tangle. In this case, constant I ₅ allows for a wide range of both three-tangle and concurrences. This means that I ₅ provides no information on the entanglement in the system in addition to that contained already in the tangles (concurrences and three-tangle) themselves. Furthermore, norm-preserving SL ^⊗3 orbits of states with equal tangles but continuously varying I ₅ are shown to exist. As a consequence, I ₅ can be increased (and decreased) by general local operations. Nevertheless, numerical analysis of random SLOCC’s has not shown any violation of the monotone property of I ₅. In case I ₅ finally turned out to being an entanglement monotone, this would imply that both SU(2) invariance and the stronger monotone property are too weak requirements for the characterization and quantification of entanglement for systems of three qubits, and that SL(2,ℂ) invariance is required. This conclusion can be extended to general multipartite systems (including higher local dimension) because the entanglement classes of three-qubit systems appear as subclasses.     

No role for muons in the DAMA annual modulation results

We construct the non-linear realized Lagrangian for the Goldstone bosons associated to the breaking pattern of SU(4) to SO(4). This pattern is expected to occur in any Technicolor extension of the standard model featuring two Dirac fermions transforming according to real representations of the underlying gauge group. We concentrate on the Minimal Walking Technicolor quantum number assignments with respect to the standard model symmetries. We demonstrate that for, any choice of the quantum numbers, consistent with gauge and Witten anomalies the spectrum of the pseudo Goldstone Bosons contains electrically doubly charged states which can be discovered at the Large Hadron Collider.     

Lie—Bäcklund Symmetries of Two-Dimensional SU(2) Yang—Mills System and Nonhomogeneous Lax Pair

We obtain the Lie—Bäcklund-type symmetries of the two-dimensional SU(2)Yang—Mills equation with the help of a generalized formal series method. Both(x, t)-dependent and independent symmetries are obtained and it is shown thatthey form a closed algebra. Finally a nonhomogeneous Lax equation is derivedusing these symmetries.     

HXXZ模型与量子SUq(2)群的表示

本文利用Bethe Ansatz方法讨论具有特定的边界条件的H_xxz模型与量子SU_q(2)群表示,证明对任意q值,BA态是量子SU_q(2)的最高权态,由此生成量子群的不可约表示,对q=e^ir为单位根,证明Bethe Ansatz方程存在新的解组,利用极限方法,导出|b′>,构造了量子SU_q(2)群的不完全可约表示(Ⅰ型)和不可约表示(Ⅱ型)。 关键词：