Thirring model with U(n) symmetry: Scale invariant only for fixed values of a coupling constant

Generalizations of the Thirring model to Fermi fields with U(n) symmetry are treated. When interactions quadratic in the SU(n) currents are introduced, scale invariance (with anomalous dimensions) is maintained only for values of the coupling g_v=0 or $\text{g}{}_{\mathrm{<mtext>v</mtext>}}\text{=}\text{4φ}\text{(n+1)}$.     

The Cauchy problem for the O(N), CP(N − 1), and GC(N,p) models

We study the Cauchy problem for the O(N), $\text{C}$P(N ? 1) and G$\text{C}$(N, p) models in n + 1 dimensional space-time. We prove the existence and uniqueness of solutions for small time intervals and for any n. In space-time dimension two, the previous solutions can be extended to all times by the method of a priori estimates. In space-time dimensions three and four, our estimates yield only partial results on the global existence problem. In all cases the solutions are required only to belong to local spaces, which means that they satisfy local regularity conditions but have no restrictions on their behaviour at infinity in space.     

Interacting boson-fermion model of collective states II. Boson-fermion symmetries connected with the U(5) limit

The boson-fermion symmetries, which are connected with the U^(B)(5) limit of the interacting boson model are discussed. These symmetries arise when the bosons have U(5) symmetry and the fermions occupy a single-particle orbit with spin $\text{j =}\text{1}\text{2}$ (Spin(3) limit), $\text{j =}\text{3}\text{2}$ (Spin(5) limit), or $\text{j =}\text{3}\text{2}\text{,}\text{5}\text{2}$ (U^(B+F)(5) ? U^(F)(2) limit). Closed expressions for energy spectra, electromagnetic transition rates, static moments, and (one and two) nucleon transfer reaction intensities are derived.     

Analysis of irreducible representations of the group U (4) ⊃ Up(2) × Un(2)

The general method of projection operators is used to construct the noncanonical nonorthogonal basis of arbitrary irreducible representation of the group U (4) in the reduction

$\text{U (4) ? U}{}_{\mathrm{p}}\text{(2) × U}{}_{\mathrm{n}}\text{(2)}$

, where

$\text{U}{}_{\mathrm{p}}\text{(2)(U}{}_{\mathrm{n}}\text{(2))}$

is the transformation group in the proton (neutron) spin space. The completeness of this basis is proved and the matrices of the U (4) group generators and of the Bargmann-Moshinsky operator Ω in this basis are obtained. The matrix 6Ω6 exhibits a nondegenerate spectrum of the eigenvalues which may be used as the missing quantum number.     

The conflict between natural flavor conservation of Higgs couplings and cabibbo mixing in SU(2)L × U(1)

The general problem of conservation of strangeness and other quark flavors by the exchange of several neutral Higgs mesons is investigated in SU(2)_L × U(1). We find that the horizontal symmetries necessary to enforce this conservation conflict with the known Cabibbo mixing. In particular, if the quarks form an irreducible representation of the horizontal symmetry, the mixing angles are all trivial (i.e., 0 or $\text{π}\text{2}$); if they form a reducible representation, it is possible to have some nontrivial mixing angles, but only if there are several unmixed generations of quarks with exactly the same relative pattern of masses and mixings.     

The contribution to the specific heat from the spin-phonon interaction for (3d)n ions

The contribution, ΔC, to the specific heat arising from the spin-phonon interaction is calculated for general spin values. For a given set of spin-phonon coupling constants and cubic symmetry the ratio ΔC_max/ C(Schottky)_max for ions having S = 2 or $\text{5}\text{2}$ is one or two orders of magnitude greater than that for ions having S = 1 or $\text{3}\text{2}$. The relative low values of ΔC in the latter cases is traced to two distinct reasons, and the differences that the inclusion of a lower symmetry crystals field makes to the ratio is discussed.     

Electric-magnetic duality in non-abelian gauge theories

The duality transformation of the vacuum expectation value of the operator which creates magnetic vortices (the 't Hooft loop operator in the Higgs phase), is performed in the radial gauge (x_uA_u^a(x) = 0). It is found that in the weak coupling region (small g) of a pure Yang-Mills theory the dual operator creates electric vortices whose strength is $\text{1}\text{g}$. The theory is self-dual in this region, and the effective coupling of the dual Lagrangian is $\text{1}\text{g}$. (It is self-dual also in the extreme strong coupling region.) Thus the above duality transformation reduces to electric-magnetic duality where the electric field in the 't Hooft loop operators transforms into a magnetic field in the dual operator. In a spontaneously broken gauge theory these results are valid only within the region where the vortices (or the monopoles) are concentrated, or in directions of the algebra space of unbroken symmetry, as self-duality holds only for this subset of fields. Noting that the 't Hooft loop operator project into the subspace of these field configurations we find that it is an electric-magnetic duality for the spontaneously broken theory as well. In the strong coupling region a strong coupling expansion in powers $\text{1}\text{g}$ is suggested.     

Quantization of linear maps on a torus-fresnel diffraction by a periodic grating

Quantization on a phase space q, p in the form of a torus (or periodized plane) with dimensions Δq, Δp requires the Planck's constant take one of the values h = ΔqΔp/N, where N is an integer. Corresponding to a linear classical map T of points q, p is a unitary operator U mapping quantum states that are periodic in q and p; the construction of U involves techniques from number theory. U has eigenvalues exp(iα). The ‘eigenangles’ α must be multiples of $\text{2π}\text{n(N)}$, where n(N) is the lowest common multiple of the lengths of the classical ‘cycles’ mapped under T by those rational points in q, p which are multiples of $\text{Δq}\text{N}$ and $\text{Δp}\text{N}$ (i.e. n(N) is the ‘period of T mod N′), at least for odd N. If T is hyperbolic, n is a very erratic function of N, and the classical limit N → ∞ is very different from the ‘Bohr-Sommerfeld’ behaviour for parabolic maps. The degeneracy structure of the eigenangle spectrum is related to the distribution of cycle lengths. Computation of the quantal Wigner function shows that eigenstates of U do not correspond to individual cycles.     

The baryon ladder of the U(6,6)-symmetry

An unified classification of the Baryons is made with the help of the so-called Baryon ladder of U(6,6). For this reason a complete set of commuting operators (introduced in a previous paper), including the squares of the spin and the isotopic spin, their third projections, the hypercharge, the Baryon number and so on, is used. Multiplets of a nontraditional type are obtained, which are in good agreement with the experimental situation. A Baryon mass operator is introduced; it is diagonal in the particles. With its help series of mass formulas are obtained for Baryons with spin-parity $\text{(}\text{1}\text{2}\text{)}{}^{\mathrm{+}}$, $\text{(}\text{3}\text{2}\text{)}{}^{\mathrm{+}}$, $\text{(}\text{1}\text{2}\text{)}{}^{\mathrm{?}}$, $\text{(}\text{3}\text{2}\text{)}{}^{\mathrm{?}}$, $\text{(}\text{5}\text{2}\text{)}{}^{\mathrm{?}}$ and they are in a very good agreement with the experiment.     

Dynamical group quantization

The projective unitary irreducible representations U(G) of the space-time symmetry group G provides a unique quantization scheme for elementary particles. By extension a direct method of quantization for more general systems by the projective unitary representations U^$\text{G}$(G) induced from a dynamical group $\text{G}$ is outlined. Reducible relativistic composite systems are defined and the geometry of $\text{G}$ is discussed.     

Reducible scale invariance at short distances

It is proven that, using reducible scale invariance at short distances, conformal symmetry implies canonical (Bjorken) scaling, provided diagonal dimensions of dilatation multiplets occuring in the operator product expansion of two electromagnetic currents have the canonical value l_n = 2 + n. If the electromagnetic current itself belongs to such multiplets then the hadron production cross section in e⁺e^? annihilation falls off faster than $\text{1}\text{s}$ at asymptotic energy.     

On the difference between pp and pp topological cross sections up to 100 GeV/c

New 100 GeV/c$\text{p}$p data are used to find moments of the difference between the $\text{p}$p and pp topological cross sections. The mean multiplicity for annihilations at 100 GeV/c is estimated to be 9.06 ± 0.56, and the value of the quantity 〈n〉/D to be 2.75 ± 0.33. It is shown that $\text{R}{}_{\mathrm{n}}\text{= {σ}{}_{\mathrm{n}}\text{(}\text{p}\text{p) ? σ}{}_{\mathrm{n}}\text{(pp)}/σ}{}_{\mathrm{n}}\text{(pp)}$ appears at 100 GeV/c to have acquired an asymptotic form, R_n = s^?αβⁿ, with α and β constant.     

Hydrodynamics of an n-component phonon system

The dynamic properties of an n-component phonon system in d dimensions, which serves as a model for a structural phase transition of second order, are investigated. The symmetry group of the hamiltonian is the group of orthogonal transformations $\text{O}$(n). For n ≥ 2 a continuous symmetry is broken for T<T_c, where T_c is the transition temperature. We derive the hydrodynamic equations for the generators of this group, the $\text{1}\text{2}\text{n (n ? 1)}$ angular-momentum variables. Besides the usual hydrodynamics of a phonon system, there are $\text{1}\text{2}\text{n (n ? 1)}$ additional independent diffusive modes for T > T_c. In the ordered phase we find 2 (n ? 1) propagating modes with linear dispersion and quadratic damping. Formally the hydrodynamics is similar in the isotropic Heisenberg ferromagnet (n = 2) or the isotropic antiferromagnet (n ≥ 3). The relaxing modes for T < T_c require special care. We study the critical dynamics by means of the dynamical scaling hypothesis and by a mode-coupling calculation, both of which give the critical dynamical exponent $\text{z =}\text{1}\text{2}\text{d}$. The results are compared with the 1/n expansion. It is shown that for large n there is a non-asymptotic region characterized by an effective exponent $\text{z}\text{?}\text{= φ/2ν}$, where φ is the crossover exponent for a uniaxial perturbation, and ν the critical exponent of the correlation length.     

Quark-lepton symmetry and B − L as the U(1) generator of the electroweak symmetry group

We seek an interpretation of the U(1) part of the electroweak symmetry group in terms of the quantum number B ? L. We show that the electroweak symmetry group, for which U(1) can be interpreted as a local B ? L symmetry, is the left-right symmetry group SU(2)_L × SU(2)_R × U(1)_L+R. The equating of U_L+R(1) to U_B?L(1) should lead to physical consequences which are not shared by standard gauge theory. B ? L may also help to explain the inversion of quark and lepton mass spectra.     

Quantum maps

We quantize area-preserving maps M of the phase plane q, p by devising a unitary operator $\text{U}$ transforming states | φ_n〉 into | φ_n+1〉. The result is a system with one degree of freedom q on which to study the quantum implications of generic classical motion, including stochasticity. We derive exact expressions for the equation iterating wavefunctions ψ_n(q), the propagator for Wigner functions W_n(q,p), the eigenstates of the discrete analog of the quantum harmonic oscillator, and general complex Gaussian wave packets iterated by a $\text{U}$ derived from a linear M. For | ψ_n〉 associated with curves $\text{L}$_n in q, p, we derive a semiclassical theory for evolving states and stationary states, analogous to the familiar WKB method. This theory breaks down when $\text{L}$_n gets so complicated as to develop convolutions of area ? or smaller. Such complication is generic; its principal morphotologies are“whorls” and “tendrils,” associated respectively with elliptic and hyperbolic fixed points of M. Under $\text{U}$, ψ_n(q) eventually transforms into a new sort of wave that no longer perceives the details of $\text{L}$_n. For all regimes, however, the smoothed | ψ_n(q)|² appears semiclassically appears to be given accurately by the smoothed projection of $\text{L}$_n onto the q axis, both smoothings being over a de Broglie wavelength. The classical, quantum, and semiclassical theory is illustrated by computations on the discrete quartic oscillator map. We display for the first time stochastic wavefunctions, dominated by dense clusters of caustics and characterized by multiple scales of oscillation.     

Decomposable operators application to K.M.S. weights in a decomposable von Neumann algebra

Let $\text{U}\text{=}\text{?}\text{⊕}\text{U}\text{(ξ) dμ(ξ)}$be a decomposable von Neumann algebra, ? a faithful strictly semi-finite weight on $\text{U}$, σ^?_! the modular automorphism group associated with ?; if $\text{U}$ admits a cyclic and separating vector (or if π_?($\text{U}$)′ is countably decomposable, (π_?, H_?) being the representation of $\text{U}$, given by ?), then we get $\text{?=}\text{?}\text{⊕}\text{?ξ dμ(ξ)}$ (the ?ξ being faithful strictly semi-finite weights) and $\text{σ}{}^{\mathrm{?}}{}_{\mathrm{!}}\text{=}\text{?}\text{?}\text{σ}{}^{\mathrm{?}}{}_{\mathrm{!}}\text{dμ(ξ)}$ (σ^?_! is the modular automorphism group in $\text{U}$(ξ), associated with ?).These results are obtained by developing the functional calculus for decomposable operators. More precisely, let $\text{Δ=}\text{?}\text{⊕}\text{Δ(ξ)dμ(ξ)}$ be a Hermitian (bounded) decomposable operators, and f a complex essentially bounded function on R; then $\text{f(Δ) =}\text{?}\text{?}\text{f(Δ(ξ))dμ(ξ)}$.Extensions of these results are given for nonessentially bounded measurable fields of bounded operators and for unbounded self-adjoint operators commuting with diagonalizable operators.     

Integral equations for the scattering of N identical particles

Integral equations are obtained for the scattering of N identical particles using a form of the N-particle scattering equations derived previously. The equations couple together only transition operators between physical two cluster channels, the breakup amplitudes being expressed in terms of quadratures over two-cluster amplitudes. The kernel of the equations becomes connected after a single iteration. The number of coupled equations for identical particles is $\text{1}\text{2}\text{N}\text{or}\text{1}\text{2}\text{(N?1)}$ when N is even or odd respectively.     

Global SO(3) × SO(3) × U(1) symmetry of the Hubbard model on bipartite lattices

In this paper the global symmetry of the Hubbard model on a bipartite lattice is found to be larger than SO(4). The model is one of the most studied many-particle quantum problems, yet except in one dimension it has no exact solution, so that there remain many open questions about its properties. Symmetry plays an important role in physics and often can be used to extract useful information on unsolved non-perturbative quantum problems. Specifically, here it is found that for on-site interaction U ≠ 0 the local SU(2) × SU(2) × U(1) gauge symmetry of the Hubbard model on a bipartite lattice with N_a^D sites and vanishing transfer integral t = 0 can be lifted to a global [SU(2) × SU(2) × U(1)]/Z₂² = SO(3) × SO(3) × U(1) symmetry in the presence of the kinetic-energy hopping term of the Hamiltonian with t > 0. (Examples of a bipartite lattice are the D-dimensional cubic lattices of lattice constant a and edge length L = N_aa for which D = 1, 2, 3,... in the number N_a^D of sites.) The generator of the new found hidden independent charge global U(1) symmetry, which is not related to the ordinary U(1) gauge subgroup of electromagnetism, is one half the rotated-electron number of singly occupied sites operator. Although addition of chemical-potential and magnetic-field operator terms to the model Hamiltonian lowers its symmetry, such terms commute with it. Therefore, its 4^N_aD energy eigenstates refer to representations of the new found global [SU(2) × SU(2) × U(1)]/Z₂² = SO(3) × SO(3) × U(1) symmetry. Consistently, we find that for the Hubbard model on a bipartite lattice the number of independent representations of the group SO(3) × SO(3) × U(1) equals the Hilbert-space dimension 4^N_aD. It is confirmed elsewhere that the new found symmetry has important physical consequences.