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In this study, we investigate the Monopole-Antimonopole Pair (MAP) solution in the SU(2)×U(1) Weinberg- Salam theory with ¢-winding number, n = 3 for bifurcation phenomena. At n = 3, the ‘t Hooft-Polyakov monopole merges with antimonopole to form a vortex ring. Other than fundamental solution, two new bifurcating full vortex ring solution branches were found when Higgs coupling constant, λ reaches a critical value λc. The two new branches possess higher energies than the fundamental solutions. These solutions behave differently from the vortex ring configuration we studied in SU(2) Georgi-Glashow theory. We investigated on the total energy E, vortex ring diameter dρ, and magnetic dipole moment µm, for 0 ≤ λ ≤49.

We present a study of the properties of Bargmann Invariants (BIs) and Null Phase Curves (NPCs) in the theory of the geometric phase for finite dimensional systems. A recent suggestion to exploit the Majorana theorem on symmetric SU(2) multispinors is combined with the Schwinger oscillator operator construction to develop efficient operator-based methods to handle these problems. The BI is described using intrinsic unitary invariant angle parameters whose algebraic properties as functions of Hilbert space dimension are analyzed using elegant group theoretic methods. The BI-geometric phase connection, extended by the use of NPCs, is explored in detail, and interesting new experiments in this subject are pointed out.

A familiar anomaly affects SU(2) gauge theory in four dimensions: a theory with an odd number of fermion multiplets in the spin 1/2 representation of the gauge group, and more generally in representations of spin 2r+1/2, is inconsistent. We describe here a more subtle anomaly that can affect SU(2) gauge theory in four dimensions under the condition that fermions transform with half-integer spin under SU(2) and bosons with integer spin. Such a theory, formulated in a way that requires no choice of spin structure, and with an odd number of fermion multiplets in representations of spin 4r+3/2, is inconsistent. The theory is consistent if one picks a spin or spin_c structure. Under Higgsing to U(1), the new SU(2) anomaly reduces to a known anomaly of "all-fermion electrodynamics." Like that theory, an SU(2) theory with an odd number of fermion multiplets in representations of spin 4r+3/2 can provide a boundary state for a five-dimensional gapped theory whose partition function on a closed five-manifold Y is $(-1)^{\int_Y w_2w_3}$. All statements have analogs with SU(2) replaced by Sp(2N). There is also an analog in five dimensions., Comment: 39 pp, added references and minor typos corrected, added comment on dynamics, updated reference in this version

We study the time evolution of quantum systems with a time-dependent non-Hermitian Hamiltonian exhibiting a SU(1,1) and SU(2) dynamical symmetry. With a time-dependent metric, the pseudo-Hermitian ...

High fidelity and robustness in population inversion is very desirable for many quantum control applications. We expand composite pulse schemes developed for two-level dynamics, and present an analytic solution for the coherent evolution of an N-level quantum system with SU(2) symmetry, for achieving high fidelity and robust population inversion, which outperforms common solutions in N-level dynamics. Our approach offers a platform for accurate steering of the population transfer in physical multi-level systems, which is crucial for fidelity in quantum computation and achieving fundamental excitations in nuclear magnetic resonances and atomic physics. We also introduce and discuss the geometrical trajectories of these dynamics on the Majorana sphere as an interpretation, allowing to gain physical insight on the dynamics of many-body or high-dimensional quantum systems., Comment: 9 pages, 6 figures

We give a proof of perturbative renormalizability of SU(2) Yang--Mills theory in four-dimensional Euclidean space which is based on the Flow Equations of the renormalization group. The main motivation is to present a proof which does not make appear mathematically undefined objects (as for example dimensionally regularized generating functionals), which permits to parametrize the theory in terms of {\it physical} renormalization conditions, and which allows to control the singularities of the correlation functions of the theory in the infrared domain. Thus a large part of the proof is dedicated to bounds on massless correlation functions., Comment: 97 pages

A Hamiltonian, that describes the interaction between a two-level atom (su(2) algebra) and a system governed by su(1,1) Lie algebra besides two external interaction, is considered. Two canonical transformations are used, which results into removing the external terms and changing the frequencies of the interacting systems. The solution of the equations of motion of the operators is obtained and used to discuss the atomic inversion, entanglement, squeezing and correlation functions of the present system. Initially the atom is considered to be in the excited state while the other systems is in the Perelomov coherent state. Effects of the variations in the coupling parameters to the external systems are considered. They are found to be sensitive to changing entanglement, variance and entropy squeezing.

Using the idea that the symmetry generators commuting with a Landau-like Hamiltonian containing non-Abelian gauge fields will be matrix-valued differential operators, we reconsider the eigenvalue problem of the five-dimensional (5-D) Kepler problem on a SU(2) instanton background. We quickly reproduce the result of Pletyukhov and Tolkachev [J. Math. Phys. 40, 93–100 (1999)], obtained for the energy spectrum. The eigenstates can be expressed in terms of the SU(2) monopole harmonics. The relevance of the theory of induced representations for solving similar problems is emphasized.

Seven different triple sum formulas for $9j$ coefficients of the quantum algebra $u_q(2)$ are derived, using for these purposes the usual expansion of $q$-$9j$ coefficients in terms of $q$-$6j$ coefficients and recent summation formula of twisted $q$-factorial series (resembling the very well-poised basic hypergeometric $_5\phi_4$ series) as a $q$-generalization of Dougall's summation formula of the very well-poised hypergeometric $_4F_3(-1)$ series. This way for $q=1$ the new proof of the known triple sum formula is proposed, as well as six new triple sum formulas for $9j$ coefficients of the SU(2) group, in the angular momentum theory. The mutual rearrangement possibilities of the derived triple sum formulas by means of the Chu--Vandermonde summation formulas are considered and applied to derive several versions of double sum formulas for the stretched $q$-$9j$ coefficients, which give new rearrangement and summation formulas of special Kampe de Feriet functions and their $q$-generalizations. Several fourfold sum formulas [with each sum of the $_5F_4(1)$ or $_5\phi_4$ type] for the $12j$ coefficients of the second kind (without braiding) of the SU(2) and $u_q(2)$ are proposed, as well as expressions with five sums [of the $_4F_3(1)$ and $_3F_2(1)$ or $_4\phi_3$ and $_3\phi_2$ type] for the $12j$ coefficients of the first kind (with braiding) instead of the usual expansion in terms of $q$-$6j$ coefficients. Stretched and doubly stretched $q$-$12j$ coefficients [as triple, double or single sums, related to composed or separate hypergeometric $_4F_3(1)$ and $_5F_4(1)$ or $_4\phi_3$ and $_5\phi_4$ series, respectively] are considered.

New algorithms are developed for the purpose of optimizing the efficient calculation of SU(3) Clebsch–Gordan coefficients in both SU(2)- and SO(3)-coupled bases. The new algorithms make use of the fact that highest weight states in a tensor product space are easily identified by vector coherent state methods. The methods are developed for SU(3) but apply to other compact semi-simple Lie groups.

The classical characters of the highly excited bend dynamics of acetylene are analyzed in terms of two coupled SU(2)/U(1) coset spaces corresponding to the right and left circular motion of the two C–H bends. The vibrational modes show a wide variety of behaviors that are not observed in the simple SU(2)/U(1) coset case which deals with, e.g., two coupled stretches, in which case the vibrational modes can be characterized as (low-lying) local and (high-lying) normal modes with a so-called local-normal transition in between. For the two coupled SU(2)/U(1) cosets of acetylene, the general trend is that most modes are perturbed local or normal modes, with distinct characters that are not found in the SU(2) dynamics. Details of their classical characters and the dynamical action flow between the two C–H bends were deduced. When the total action number Nb is small (less than 14), normal mode motions dominate, i.e., trans bend modes at the bottom of each polyad and cis bend at the top. At higher Nb, the vibrati...

Instanton correction of prepotential of one-dimensional SL(2) Ruijsenaars model is presented with the help of Picard-Fuchs equation of Pakuliak-Perelomov type. It is shown that the instanton induced prepotential reduces to that of the SU(2) gauge theory coupled with a massive adjoint hypermultiplet., Comment: revtex, 15 pages, to be published in Journal of Mathematical Physics

The Green’s function for the five-dimensional counterpart of the MIC–Kepler problem [Kepler potential plus SU(2) Yang–Mills instanton plus Zwanziger-like 1/R2 centrifugal term] is constructed on the basis of the Green’s function for the eight-dimensional harmonic oscillator.

The relativistic particles described by the Bhabha equation (Bhabha particles) can possess several values of masses (Mτ=n/2τ) and spins s⩽n/2. In order to find the spin-mass content of such particles corresponding to the general irrep 〈n1n2〉 of the group Sp(4)∼O(5) which is the symmetry group of the Bhabha equation, the noncanonical chain of groups Sp(4)⊃SUs(2)×Uτ(1) is considered. In this paper, we construct the basis of the irrep 〈n1n2〉 in a boson realization form using the method of elementary permissible diagrams. This is achieved by the introduction of special “symplectic” bosons. The maximum and minimum values of spin s are established for each value of τ. The multiplicity of degenerate pairs (s,t) can be easily calculated with the help of the generating function obtained in this paper.

A novel realization of the classical SU(2) algebra is introduced for the Dirac relativistic hydrogen atom defining a set of operators that, besides, allow the factorization of the problem. An extra phase is needed as a new variable in order to define the algebra. We take advantage of the operators to solve the Dirac equation using algebraic methods. To acomplish this, a similar path to the one used in the angular momentum case is employed; hence, the radial eigenfuntions calculated comprise non unitary representations of the algebra. One of the interesting properties of such non unitary representations is that they are not labeled by integer nor by half-integer numbers as happens in the usual angular momentum representation., Comment: 20 pages 1 eps figure in a single zipped file, submitted to J. Math. Phys

Differential equations for scaling relation of prepotential in N=2 supersymmetric SU(2) Yang-Mills theory coupled with massive matter hypermultiplet are proposed and are explicitly demonstrated in one flavour ($N_f =1$) theory. By applying Whitham dynamics, the first order derivative of the prepotential over the $T_0$ variable corresponding to the mass of the hypermultiplet, which has a line integral representation, is found to satisfy a differential equation. As the result, the closed form of this derivative can be obtained by solving this equation. In this way, the scaling relation of massive prepotential is established. Furthermore, as an application of another differential equation for the massive scaling relation, the massive prepotential in strong coupling region is derived., Comment: revtex

A recent paper gave an explicit construction for inducing shift tensors of a compact reductive Lie group from shift tensors of a suitably defined subgroup. The shift tensors were defined on model spaces of holomorphic vector-coherent-state wave functions. In this paper, we use these shift tensors to obtain an algorithm for computing Clebsch–Gordan coefficients. The approach reproduces the known analytical results for SU(2) and gives a simple algorithm for computing SU(3) coefficients. The algorithm is shown to yield analytical expressions for the multiplicity-free SU(3) couplings of type (λ20)⊗(λ10).

We summarize the algebraic structure of spherically symmetric Yang–Mills potentials for a general compact gauge group, and investigate the particular case of gauge groups with Lie algebra su(n) in detail. We develop techniques that lead to a complete classification of the possible spherical symmetry ansatze, including descriptions of the reduced gauge group Z, the space of magnetic potentials H, and for those ansatze that admit extensions across the symmetry axis, a description of the space of vacuum potentials H0 and its little group Z0. These results are illustrated by listing all irreducible models for su(n), n⩽6.

The character problems of SU(2) and SU(1,1) are reexamined from the standpoint of a physicist by employing the Hilbert space method which is shown to yield a completely unified treatment for SU(2) and the discrete series of representations of SU(1,1). For both the groups the problem is reduced to the evaluation of an integral which is invariant under rotation for SU(2) and Lorentz transformation for SU(1,1). The integrals are accordingly evaluated by applying a rotation to a unit position vector in SU(2) and a Lorentz transformation to a unit SO(2,1) vector which is time-like for the elliptic elements and space-like for the hyperbolic elements in SU(1,1). The details of the procedure for the principal series of representations of SU(1,1) differ substantially from those of the discrete series., Comment: 31 pages, RevTeX, typos corrected. To be published in Journal of Mathematical Physics

We study relations between the deformed cotangent bundle (T*B)_q for the Borel subgroup B of a given simple Lie group G, the quantum Lie algebra J_q associated with the corresponding quantum group G_q and the matrices generating Clebsch-Gordan coefficients (CGC) for J_q. We reveal the connection of these objects to quantum analogue of the model space M and q-tensor operators.

We discuss the N=2 SU(2) Yang-Mills theory coupled with a massive matter in the weak coupling. In particular, we obtain the instanton expansion of its prepotential. Instanton contributions in the mass-less limit are completely reproduced. We study also the double scaling limit of this massive theory and find that the prepotential with instanton corrections in the double scaling limit coincides with that of N=2 SU(2) Yang-Mills theory without matter., Comment: 14 pages, latex, 2 figures; Revised version; The expression for the prepotential is modified

A new set of polynomial states (to be called character states) are derived for $Sp(4)$ reduced to its $SU(2) \times U(1)$ subgroup, and the relevant generator matrix elements are evaluated. The group--subgroup in question is that of the seniority model of nuclear physics.

It is shown that the complete symmetry group for the Kepler problem, as introduced by Krause, can be derived by Lie group analysis. The same result is true for any autonomous system.

SU(2) coherent states on an m‐sheeted covering of the sphere are introduced, and properties like overcompleteness and resolution of the identity are studied. Operators J+(m), J−(m), Jz(m) that obey the SU(2) algebra but shift the ‖jn〉 states by m steps are considered, and it is shown that the properties of our coherent states with respect to them are analogous to the properties of the standard SU(2) coherent states with respect to the usual angular momentum operators J+, J−, Jz. An extended SU(2) Bargmann representation on an m‐sheeted sphere is introduced in which the transformations of an m‐sheeted covering of the SU(2) group are implemented as extended Mobius conformal mappings. The formalism is applied in the study of Hamiltonians that contain the operators J+(m), J−(m), Jz(m) and describe processes where the state of a particle is shifted by m steps. The method is also used in the context of both the Holstein–Primakoff and Schwinger representation of SU(2).

The SU(2) coherent‐state path integral is used to represent the matrix element of a propagator in the SU(2) coherent‐state basis. It is argued that the continuum representation of this integral is correct provided the necessary boundary term is taken into account. In the case of the SU(2) dynamical symmetry the path integral is explicitly computed by means of a change of variables, the SU(2) motion of the underlying phase space. The correct stationary‐phase expansion for the propagator in terms of the total action including boundary term and classical trajectories is obtained.

A large class of braid group representations are proven unitary which means a Hermitian bilinear form left invariant by the representation under consideration exists. This generalizes and improves known results on the Burau and Gassner representations. The elementary methods are naturally suggested by the study of monodromy invariant Hermitian forms. Applications to conformal quantum field theory in two dimensions are mentioned.

In this article, the second of three, we discuss and develop the basis of a Weyl quantisation for compact Lie groups aiming at loop quantum gravity-type models. This Weyl quantisation may serve as the main mathematical tool to implement the program of space adiabatic perturbation theory in such models. As we already argued in our first article, space adiabatic perturbation theory offers an ideal framework to overcome the obstacles that hinder the direct implementation of the conventional Born-Oppenheimer approach in the canonical formulation of loop quantum gravity. Additionally, we conjecture the existence of a new form of the Segal-Bargmann-Hall "coherent state" transform for compact Lie groups $G$, which we prove for $G=U(1)^{n}$ and support by numerical evidence for $G=SU(2)$. The reason for conjoining this conjecture with the main topic of this article originates in the observation, that the coherent state transform can be used as a basic building block of a coherent state quantisation (Berezin quantisation) for compact Lie groups $G$. But, as Weyl and Berezin quantisation for $\mathbb{R}^{2d}$ are intimately related by heat kernel evolution, it is natural to ask, whether a similar connection exists for compact Lie groups, as well. Moreover, since the formulation of space adiabatic perturbation theory requires a (deformation) quantisation as minimal input, we analyse the question to what extent the coherent state quantisation, defined by the Segal-Bargmann-Hall transform, can serve as basis of the former.

All local extensions of the chiral algebra of observables in the SU(2)⊗r conformal current algebra models are classified for r≤4 and all levels ki. This method is based on analyzing the polynomial solutions of the corresponding Knizhnik–Zamolodchikov equations and can be applied also for explicit construction of all local extensions corresponding to conformal embeddings of SU(2)⊗r for any r.

Useful and new representations of the transfer matrices are explicitly obtained for systems with and without time reversal or spin rotation symmetries. In the symplectic, orthogonal, and unitary cases, the whole group of transfer matrices contains a compact and a noncompact subgroup, and the generalized Bargman’s polar parametrization is obtained in a straightforward way. It is shown that the transfer matrices used by Luttinger, to prove the Saxon–Hutner conjecture on the forbidden energy levels, belong to the noncompact Sp(2,C) subgroup.

The complete classification of WZNW modular invariant partition functions is known for very few affine algebras and levels, the most significant being all levels of $A_1$ and $A_2$ and level 1 of all simple algebras. Here, we address the classification problem for the nicest high rank semi-simple affine algebras: $(A_1^{(1)})^{\oplus_r}$. Among other things, we explicitly find all automorphism invariants, for all levels $k=(k_1,\ldots,k_r)$, and complete the classification for $A_1^{(1)}\oplus A_1^{(1)}$, for all levels $k_1,k_2$. We also solve the classification problem for $(A_1^{(1)})^{\oplus_r}$, for any levels $k_i$ with the property that for $i\ne j$ each $gcd(k_i+2,k_j+2)\leq 3$. In addition, we find some physical invariants which seem to be new. Together with some recent work by Stanev, the classification for all $(A^{(1)}_1)^{\oplus_r}_k$ could now be within sight.

A quantum deformation of a Poincare group was constructed using as a starting point SU(2,2) conformal group and twistorlike definition of the Minkowski space. This leads to a pure twisted quantum Poincare group. The relative simplicity of the q deformation permits one to find out the important properties of the q groups of the space–time transformations. In particular, it is shown that the geometrical meaning of the twisting is the transition to the another space–time coordinates frame, so that one can freely choose the most convenient deformation among the ones related to each other by twists. The small subgroup used in the induced representation construction is proven to be twisted with respect to initial q group. This last fact means that commutation relations for creation and destruction operators are not strongly defined by the q group of the space–time transformations.

Irreducible representations of three simplest Lie algebras are realized over Lagrangian submanifolds Λ (over closed curves, tori, and spheres) in coadjoint orbits. The intertwining operator is constructed which enables to pass from an abstract representation to a ‘‘Λ representation.’’ The integral kernel of the intertwining operator involves special ‘‘Λ‐coherent states.’’ The simple integral formulas are obtained for exact and semiclassical eigenvectors and for solutions of the Cauchy problem over Lie algebras, mentioned in the title. The Hamiltonians considered are nonanalytic in general.

Following constructions in the case of canonical and affine coherent states, a universal propagator for a general Hamiltonian appropriate to a single spin degree of freedom is introduced. The universal propagator is a single function, independent of any particular choice of fiducial vector, which, nonetheless, propagates all coherent state Hilbert space representatives correctly. Furthermore, we explicitly construct the universal propagator for several examples.

For positive q, the scalar product defined by the integral on Funq(SU(2)) is used to complete Funq(SU(2)) to a Hilbert space Hq. The dual map to the coproduct on Funq(SU(2)) makes Hq into a Banach algebra without unit. Multiplication from the left defines a *‐representation of Funq(SU(2)) on Hq and thus Funq(SU(2)) can be completed to a C*‐algebra Fq. This C*‐algebra is shown to coincide with the C*‐algebra SUν(2) considered by Woronowicz [Publ. RIMS, Kyoto Univ. 23, 117 (1987)]. General representations of Fq are discussed and the spectrum of Fq and its Jacobson topology are calculated.

The sum rules among 6‐j symbols are derived on the basis of the multifold group integral for the SU(2) characters. The formulas for simple multifold group integrals are derived by means of the graphic method.

Continual integrable Heisenberg models are constructed on real subalgebras of the superalgebra spl(2/1). Two Heisenberg models are shown to exist on the compact subalgebra uspl(2/1)≊su(2/1). One of these, SU(2/1)/S(U(2)×U(1)), is gauge equivalent to SU(2) nonlinear vector Schrodinger equation (NLSE) expressed in odd Grassman variables, the other, SU(2/1)/S(L(1/1)×U(1)), to ‘‘super’’ NLSE which is invariant under global supersymmetry transformations of SL(1/1). Also constructed are a Heisenberg model on the noncompact subalgebra ospu(1,1/1), with higher nonlinearities, and its gauge equivalent analog. Hamiltonian structure and classical solutions are studied and the possible connection of the given models with a version of the Hubbard one is discussed.

The representation of the two‐dimensional harmonic oscillator by the unitary group SU (2) simple Coriolis‐adapted vibrational basis states for the treatment of vibration–rotation interaction in polyatomic molecules. The vibrational part of the zeroth‐order vibration–rotation Hamiltonian is expressed in terms of the generators (Sx,Sy,Sz) of the group SU(2), leading to a coupled angular momentum representation of the vibration–rotation Hamiltonian. In the prolate limit, this leads to an effective k‐dependent zeroth‐order vibrational Hamiltonian that is linear in the group generators. The problem can be solved exactly in this limit by a simple axis transformation in the vibrational ‘‘spin’’ space. Because of the underlying SU(2) structure, the transformation matrix elements and overlaps of basis states of different effective Hamiltonians corresponding to different values of k are given by simple expressions involving Wigner d matrices.

The three degrees of freedom Smorodinsky–Winternitz system is a degenerate or super‐integrable Hamiltonian that possesses five functionally independent globally defined and single‐valued integrals of the motion in both classical and quantum mechanics. This is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the Hamiltonian under a group of symmetry transformations isomorphic to the three‐dimensional unitary unimodular group, SU(3). In turn, this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other, the so‐called dynamical group. All the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group. So, in common with the hydrogen atom and the harmonic oscillator, the quantum mechanics of the Smorodinsky–Winternitz system may be completely solved within the framework of group theory alone.

New sum rules are derived for the SU(2) Racah coefficients by means of the group integral. The general graphic rules of calculating the multifold group integral for the product of the SU(2) characters in terms of the Racah coefficients are presented.

Generalized vector coherent state constructions of totally symmetric U(3) tensors are used to gain new expressions for the SU(3)⊇SU(2)×U(1) Wigner coefficients for the coupling (λ1μ1)×(λ20)→(λ3μ3). These expressions show how the extremely simple formulas of Le Blanc and Biedenharn, involving a single 9‐j coefficient, arise as special cases of a general result that involves 12‐j coefficients. A simpler general result involving only 9‐j coefficients and K‐normalization factors is derived in a way that can, in principle, be generalized to the generic coupling with multiplicity.

Contractions of Lie algebras and of their representations are generalized to define new quantum groups. An explicit and complete exposition is made for the one‐dimensional Heisenberg H(1)q and the two‐dimensional Euclidean quantum group E(2)q obtained by contracting SU(2)q.

A general way of writing the gauge potentials Aaμ for which the SU(2) Yang–Mills equations of motion can be simplified and become solvable is shown. A number of exact solutions can be obtained from these simplified equations of motion.

Lie group symmetries and invariants of the generalized Henon–Heiles equations are found. The coupled second‐order equations are invariant under translations in time, in general, and the stretching group (dilation) if the linear terms in the ‘‘force’’ are absent. The equivalent set of four coupled first‐order equations is found to be invariant under a one‐parameter group for three cases and the group generators are given. Three different approaches are reported: the ‘‘classical method’’ for determining Lie group symmetries, a modified method for finding Lie group symmetries with vector fields and the direct method for calculating the invariants. For the Henon–Heiles equations the direct method is the most efficient.

This paper analyzes the irreducible unitary representations of SU(2,1) in a basis labeled as ‖p, q; jmy〉, where p,q correspond to quantum numbers associated with the quadratic and cubic Casimir operators, j,m label states of the SU(2) subgroup, and y labels the quantum number with respect to the U(1) subgroup. All the irreducible representations are found and the allowed range of these quantum numbers for each representation are given. The results are expressed in the form of diagrams that show the allowed values in a (j,y) plot for fixed values of p,q. A (p,q) plot is also provided that indicates the allowed values of these quantum numbers.

The discrete family of global solutions of the static spherically symmetric SU(2) Einstein–Yang–Mills equations that were recently numerically obtained by Bartnik and McKinnon [Phys. Rev. Lett. 6 1, 141 (1988)] is studied in greater detail, both numerically and analytically. A similar discrete sequence of numerical solutions outside a regular event horizon is shown to exist for every radius of the horizon.

Based on the six-dimensional real special orthogonal Lie algebra SO(4), we construct two different isospectral problems. And the two hierarchies of soliton equations from zero curvature equations can be generated. At last, we use the property of the isomorphism SU (2) ⊗ SU (2) ≅ SO(4), and obtain the relations between the hierarchy of SU (2) ⊗ SU (2) and the hierarchy of SO(4).

The eigenvalue problem for radial potentials is considered in a space whose spatial coordinates satisfy the SU(2) Lie algebra. As the consequence, the space has a lattice nature and the maximum value of momentum is bounded from above. The model shows interesting features due to the bound, namely, a repulsive potential can develop bound-states, or an attractive region may be forbidden for particles to propagate with higher energies. The exact radial eigen-functions in momentum space are given by means of the associated Chebyshev functions. For the radial stepwise potentials the exact energy condition and the eigen-functions are presented. For a general radial potential it is shown that the discrete energy spectrum can be obtained in desired accuracy by means of given forms of continued fractions., Comment: 1+20 pages, 2 figs, LaTeX

It is remarked that, if the gauge group is not compact, finite−time blowing−up Yang–Mills fields over the Minkowski spaces may exist. The example taken is G=SU(2,C). It is shown that blow−up can occur in an arbitrarily small time interval when the initial data of the fields are suitably chosen.

We introduce a fully coherent spin network amplitude whose expansion generates all SU(2) spin networks associated with a given graph. We then give an explicit evaluation of this amplitude for an arbitrary graph. We show how this coherent amplitude can be obtained from the specialization of a generating functional obtained by the contraction of parametrized intertwiners a la Schwinger. We finally give the explicit evaluation of this generating functional for arbitrary graphs.

We give explicit expressions for the three-quark exchange operators, crossing matrices and Fierz transforms for the SU(2) and SU(3) groups. We identify the invariant terms in these operators and express them in terms of Casimir operators.