Sp(2)-covariant quantization of the Freedman-Townsend model

A model antisymmetric rank two tensor field is quantized in both the Lagrangian and the Hamiltonian versions of the expanded BRST quantization. It is demonstrated that the Lagrangian quantization results in a unitary S-matrix.Study supported in part by the Russian Fund for Fundamental Research, Project No. 94-02-03234, and by the International Science Fund, Grant RI 1000.Tomsk Pedagogical Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 72–78, February, 1995.     

Dynamical and symmetry groups of the SU(2) Kepler problem

It is well known that the MIC–Kepler problem, an extension of the three-dimensional Kepler problems, admits the same dynamical and symmetry groups as the Kepler problem. This paper aims to study dynamical and symmetry groups of the SU(2) Kepler problem, where the SU(2) Kepler problem is defined to be the dynamical system reduced from the eight-dimensional conformal Kepler problem through an SU(2) symmetry and turns out to be an extension of the five-dimensional Kepler problem. It is shown that the SU(2) Kepler problem admits a dynamical group SO^*(8) and that the phase space of the SU(2) Kepler problem is symplectomorphic with a co-adjoint orbit of SO^*(8), on which the Kirillov–Kostant–Souriau form is defined. It is further shown that the subgroups, SU(4), SU^*(4), and Sp(2)×_SR⁵, of SO^*(8) provide the symmetry groups, SU(4)/Z₂SO(6), SU^*(4)/Z₂SO₀(1,5), and (Sp(2)×_SR⁵)/Z₂SO(5)×_SR⁵, of the SU(2) Kepler problem with negative, positive, and zero energies, respectively, where ×_S denotes a semi-direct product. Furthermore, constants of motion for the SU(2) Kepler problem are found together with their Poisson brackets. The symmetry Lie algebra formed by constants of motion is shown to be isomorphic with so(6)su(4), so(1,5)su^*(4), or so(5)_SR⁵sp(2)_SR⁵, depending on whether the energy is negative, positive, or zero, where _S denotes a semi-direct sum. These Lie algebras are subalgebras of so^*(8)so(2,6).     

On representations of the groups Sp(n, 1) and Sp(n)

New classes of unitary irreducible representations of Sp(n, 1) which can be useful for applications in physics are obtained. The infinitesimal operators of these representations of Sp(n, 1) and of irreducible representations of Sp(n+1) with highest weights (m, m, m₃,…,m_n+1) and (m₁, m₂, 0,…,0) are expressed in terms of the simple Clebsch–Gordancoefficients for Sp(n). For Sp(3) and Sp(2, 1) they are found in an explicit form.     

SU(2) symmetry in a Hubbard model with spin-orbit coupling

We study the underlying symmetry in a spin-orbit coupled tight-binding model with Hubbard interaction. It is shown that, in the absence of the on-site interaction, the system possesses the SU(2) symmetry arising from the time reversal symmetry. The influence of the on-site interaction on the symmetry depends on the topology of the networks: The SU(2) symmetry is shown to be the spin rotation symmetry of a simply-connected lattice even in the presence of the Hubbard interaction. On the contrary, the on-site interaction breaks the SU(2) symmetry of a multi-connected lattice. This fact indicates that a discrete spin-orbit coupled system has exclusive features from its counterpart in a continuous system. The obtained rigorous result is illustrated by a simple ring system.     

Vibrational symmetry classification and torsional tunneling splitting patterns in G6(EM), G12, and G36(EM) molecules

It is shown that the torsional splitting patterns in methanol-like molecules, with the excitation of small amplitude vibrational modes in the methyl group, are determined by mechanisms that can be formulated in an almost identical fashion to that for ethane-like molecules. This is achieved by treating ethane-like molecules by the internal axis method (IAM) and methanol-like molecules by the principal axis method (PAM) or rho-axis method (RAM). Using the extended molecular groups G₆(EM) or C_6v(M) for methanol and G₃₆(EM) for ethane, vibrations perpendicular to the internal rotation axis are conveniently described by modes of higher degeneracy (E for methanol and G_s for ethane) in the absence of coupling of top and frame. Head–tail coupling operators, except the cos-type barrier terms, lower the degeneracy, causing vibrational splittings. Coupled vibrational pairs with torsional splitting patterns that we call ‘regular’ (pure A₁, A₂ pairs for methanol and pure E_1d, E_2d pairs for ethane) or ‘inverted’ (pure B₁, B₂ pairs for methanol and pure E_1s, E_2s pairs for ethane) can be formed as limit cases. Actual splitting patterns occur between the above limits, and are basically determined by torsional Coriolis coupling, which can tune more or less to resonance pairs of uncoupled basis levels linked by specific head-tail coupling operators. The inversion of torsional splitting patterns, observed in perpendicular vibrational modes of the methyl group of methanol, can be predicted by these theoretical considerations. Similar considerations apply to molecules of G₁₂ symmetry.     

SU(2) andSU(1,1) algebra eigenstates: A unified analytic approach to coherent and intelligent states

We introduce the concept of algebra eigenstates which are defined for an arbitrary Lie group as eigenstates of elements of the corresponding complex Lie algebra. We show that this concept unifies different definitions of coherent states associated with a dynamical symmetry group. On the one hand, algebra eigenstates include different sets of Perelomov's generalized coherent states. On the other hand, intelligent states (which are squeezed states for a system of general symmetry) also form a subset of algebra eigenstates. We develop the general formalism and apply it to theSU(2) andSU(1,1) simple Lie groups. Complete solutions to the general eigenvalue problem are found in both cases by a method that employs analytic representations of the algebra eigenstates. This analytic method also enables us to obtain exact closed expressions for quantum statistical properties of an arbitrary algebra eigenstate. Important special cases such as standard coherent states and intelligent states are examined and relations between them are studied by using their analytic representations.     

Mass measurements in a U(1) gauge-Higgs model with global SU(2) symmetry

Mass measurements are carried out in a lattice U(1) gauge-Higgs system with the Higgs fields transforming as a doublet under a global SU(2) symmetry. The model exhibits three distinct phases: Coulomb, Higgs and confining. Evidence is found for a massless vector state in the Coulomb phase, and massless Goldstone scalars in the Higgs phase. Also the behavior is studied of the massive states near the transitions between the Coulomb and Higgs phases and between the confining and Higgs phases.     

Calculation of the glueball mass spectrum ofSU (2) andSU (3) non-abelian lattice gauge theories II:SU (3)

We calculate the glueball mass spectrum in theSU (3) lattice regularized gauge theory. We find fourlight glueballs: the 0⁺⁺, 2⁺⁺, 0^?+ and, most interestingly from the experimental point of view, the oddball 1^?+. We calculate the 0⁺⁺ and 2⁺⁺ masses over a range of β values and find thatboth states conform to continuum renormalization group behaviour to a very significant degree. The question of metastable states and temperature is addressed in detail. Finally we discuss and resolve contrary claims in the recent literature.     

Hierarchy of symmetry breakings and neutral currents in anSU(6) grand unification model

In anSU(6) grand unification model with eight quarks and eight leptons belonging to 15-plet and singlet representations, the symmetry is spontaneously broken by the sequenceSU(6)SU(3) ^c ×SU(2)×U(1)×U(1)SU(3) ^c ×U(1). Fror two cases of symmetry breakings the effective weak neutral current coupling constants are compared with experiment. For theSU(3) ^c ×SU(2)×U(1)×U(1)×SU(3) ^c ×U(1) symmetry breaking, the coupling constants reproduce the Weinberg-Salam model with a small correction term. Agreement with the experimental mean values is improved with the correction term. Parity violation in atomic physics is also discussed.     

SL(4) as a common symmetry group of electromagnetic and gravitational interactions

A theory of electromagnetic and gravitational interactions between elementary particles is proposed, based upon the symmetry groupSL(4). According to this theory, gravitational fields are formed as the result of interactions between charged particles with opposite charge.     

Callan-Symanzik and renormalization group equation in a model with spontaneous breaking of the symmetry

In a simple model with spontaneous breaking of the axialU(1)-symmetry via the Higgs mechanism we construct the Callan-Symanzik and renormalization group equation in the Goldstone mode. Aiming at questions of renormalization group improvement and the like we compare two different parametrizations the model can be described with. We show that in the presence of fermions a β-function for a physical mass or some equivalent of it enters unavoidably the Callan-Symanzik equation, which leads to significant differences to the symmetric theory starting with two loops. On the other hand in the asymptotic region the equivalence to the symmetric theory is manifest.     

Finite symmetry transformation group and localized structures of the (2+1)-dimensional coupled Burgers equation

In this paper, the finite symmetry transformation group of the (2+1)-dimensional coupled Burgers equation is studied by the modified direct method, and with the help of the truncated Painleve′ expansion approach, some special localized structures for the (2+1)-dimensional coupled Burgers equation are obtained, in particular, the dromion-like and solitoff-like structures.     

Breaking of permutation symmetry and diagonal group action: Nielsen model and the standard model as low-energy limit

We point out that, on the basis of a theorem by L. Michel, every model based on the symmetryG ... G has a generic low-energy limit the breaking to diagonal action, i.e., toG. This applies in particular to the proposal by Nielsen to consider such a model withG the standard model group.     

Non-local fields in theZ(2) Higgs model: The global gauge symmetry breaking and the confinement problem

Non-local order parameters are constructed in theZ(2) Higgs model to probe the existence or non-existence of charged states. The non-local field corresponding to the Higgs field in a complete gauge fixing is found to be an order parameter for the locally unobservable global gauge symmetry breaking. This symmetry breakdown is shown to imply perfect screening for the bare charge.