Bound states of three and four resonantly interacting particles

We present an exact diagrammatic approach for the problem of dimer-dimer scattering in 3D for dimers being a resonance bound state of two fermions in a spin-singlet state, with corresponding scattering length a _F . Applying this approach to the calculation of the dimmer-dimer scattering length a _B , we recover exactly the already known result a _B = 0.6 a _F . We use the developed approach to obtain new results in 2D for fermions and bosons. Namely, we calculate bound state energies for three bbb and four bbbb resonantly interacting bosons in 2D. For the case of resonance interaction between fermions and bosons, we exactly calculate bound state energies of the following complexes: two bosons plus one fermion bbf, two bosons plus two fermions bf↑bf↓, and three bosons plus one fermion bbbf.     

Composite fermions,trios, and quartets in the Fermi-Bose mixture of neutral particles

We consider the model of a Fermi-Bose mixture with strong hard-core repulsion between particles of the same sort and attraction between particles of different sorts. In this case, in addition to the standard anomalous averages of the type 〈b〉, 〈bb〉, and 〈cc〉, pairing between fermions and bosons of the type 〈bc〉 is possible. This pairing corresponds to creation of composite fermions in the system. At low temperatures and equal densities of fermions and bosons, composite fermions are further paired into quartets. At higher temperatures, trios consisting of composite fermions and elementary bosons are also present in the system. Our investigations are important in connection with the recent observation of weakly bound dimers in magnetic and optical dipole traps at ultralow temperatures and with the observation of collapse of a Fermi gas in an attractive Fermi-Bose mixture of neutral particles.     

Spatial and Angular Characteristics of the Atmospheric Showers Initiated by Photons of Energy E γ = 1018–1022 eV in the Magnetosphere of the Earth

Spatial-angular characteristics are calculated for the atmospheric electromagnetic showers generated by primary photons of ultrahigh energies (E = 10¹⁸–10²² eV) coming from outer space. Calculations carried out for vertical and oblique showers in view of the interaction of photons and electrons with the dipole magnetic field of the Earth (formation of electron-positron pairs, synchrotron radiation) and taking into account the Landau–Pomeranchuk–Migdal effect in the processes of interaction of electrons and photons with matter in the meteorological atmosphere.     

On the Bosonic Atoms

We investigate the ground state properties of atoms, in which substitute fermions—electrons by bosons, namely, π^?-mesons. We perform some calculations in the frame of modified Hartree–Fock (HF) equation. The modification takes into account symmetry, instead of antisymmetry of the pair identical bosons wavefunction. The modified HF approach thus enhances (doubles) the effect of self-action for the boson case. Therefore, we accordingly modify the HF equations by eliminating the self-action terms “by hand.” The contribution of meson–meson and meson–nucleon non-Coulomb interaction is inessential at least for atoms with low and intermediate nuclear charge, which is our main subject. We found that the binding energy of pion negative ions A_π^-, pion atoms A_π, and the number of extra bound pions ΔN_π increases with the nuclear charge Z. In particular, for Xe ΔN_π = 4. As an example of a simple process with a pion atom, we consider photoionization that differs essentially from that for electron atoms. Namely, it is not monotonic decreasing from the threshold but has instead a prominent maximum above threshold. We study also elastic scattering of pions by pion atoms.     

A one-dimensional model with phase transition

Two repellent particles are bound to occupy two among thek _n +1 adjacent sites 0=x ₀ ⁽ⁿ⁾ <x ₁ ⁽ⁿ⁾ <...<x _kn ⁽ⁿ⁾ =1, sayx _q ⁽ⁿ⁾ ,x _q+1 ⁽ⁿ⁾ . Define the Hamiltonian _q ⁽ⁿ⁾ =–ln(x _q+1 ⁽ⁿ⁾ –x _q ⁽ⁿ⁾ ) and the partition function     

Disentangling q-Exponentials: A General Approach

We revisit the q-deformed counterpart of the Zassenhaus formula, expressing the Jackson q-exponential of the sum of two non-q-commuting operators as an (in general) infinite product of q-exponential operators involving repeated q-commutators of increasing order, E_q(A+B) = E_q0(A)E_q1 (B) _i=2 E_qi. By systematically transforming the q-exponentials into exponentials of series and using the conventional Baker–Campbell–Hausdorff formula, we prove that one can make any choice for the bases ^qi, i=0, 1, 2, ..., of the q-exponentials in the infinite product. An explicit calculation of the operators C _i in the successive factors, carried out up to sixth order, also shows that the simplest q-Zassenhaus formula is obtained for ₀ = ₁ =1, and ₂ = 2, and ₃ = 3. This confirms and reinforces a result of Sridhar and Jagannathan, on the basis of fourth-order calculations.     

The possible angular momenta of a system of fermions or bosons with spinj

Simple recurrence relations for the multiplicities of angular momenta are presented for a system of fermions or bosons with spinj. Tables are given for fermions withj=13/2 up to 19/2 and for bosons withj=2, 3, 4, 5 and 20 particles at most.     

Ground states and flux configurations of the two-dimensional Falicov-Kimball model

The Falicov-Kimball model is a lattice model of itinerant spinless fermions (electrons) interacting by an on-site potential with classical particles (ions). We continue the investigations of the crystalline ground states that appear for various filling of electrons and ions for large coupling. We investigate the model for square as well as triangular lattices. New ground states are found and the effects of a magnetic flux on the structure of the phase diagram are studied. The flux phase problem where one has to find the optimal flux configurations and the nuclei configurations is also solved in some cases. Finally we consider a model where the fermions are replaced by hard-core bosons. This model also has crystalline ground states. Therefore their existence does not require the Pauli principle, but only the on-site hard-core constraint for the itinerant particles.     

Unparticle searches through gamma–gamma scattering

We investigate the effects of unparticles on γγ→γγ scattering for the photon collider mode of the future multi-TeV e⁺e^- linear collider. We show the effects of unparticles on the differential, and total scattering cross sections for different polarization configurations. Considering 1-loop standard model background contributions from the charged fermions and W^± bosons to the cross section, we calculate the upper limits on the unparticle couplings λ₀ to the photons for various values of the scaling dimension d (1<d<2) at = 0.5–5 TeV.     

Microcanonical and canonical distributions and finite exchangeable random processes

A pure probabilistic approach to Gibbs' distributions is given, starting from the notion of finite exchangeable random process. The differences between bosons, fermions, and classical particles are ascribed to different values of correlation. The relationship between exchangeability and constraints on the energy distribution is investigated.     

Low-energy elementary surface excitations of symmetric films of liquid4He atT=0 K

It is known that low-energy elementary excitations of symmetric films of liquid⁴He atT=0 K are characterized by a momentum q parallel to the surface and may be described by bound states. We have evaluated wave functions and energies of these states for both best short-ranged and optimal long-ranged correlations. Quantities of physical interest may be expressed in terms of these eigenstates and, in particular, for very small momenta (q<0.2 Å^–1) they are mainly determined by the contribution due to the lowest-lying one. We propose analytic expressions for the lowest-lying excitations and fluctuations in the long-wavelength limit. It is proved that in this limiting case, the excitation energy _LW(q) and the averaged static structure functionS _LW(q) should go linearly to zero asq0, whereas the averaged direct correlationX _LW ^Dg (q) should diverge at the origin as 1/q. It is shown that numerical solutions exhibit the expected long-wavelength behavior provided that optimal correlations are used. All these results are displayed in a series of figures and are discussed in detail.     

Electroproduction of π0-mesons on protons at very low values of |q 2|

The electroproduction of π⁰-mesons on protons has been measured with electrons of 1.7 and 2.1 GeV energy in the region of the nucleon resonances at \(W = \sqrt {(q + p)^2 } = 1,400 - 2,000MeV\) at rather low values of |q ²|=|(e?e′)²|=0.01–0.1 GeV² but at high values of the nucleon recoil |t|=|(p?p′)²|≈0.6–2.2 GeV² corresponding to \(\Theta _\pi ^{cm} \approx 145^ \circ - 180^ \circ\) . Up toW≈1,650 MeV the cross section for π⁰-production with virtual photons of |q ²|≈0.03 GeV² is only slightly smaller than that with real photons. ForW?1,700 MeV the cross section at |q ²|≈0.03 GeV² is on average a factor of 2 smaller than that atq ²=0 but only slightly higher than that at |q ²|≈0.3–1 GeV².     

Absence of discrete spectrum in highly negative ions

LetH _N be the Hamiltonian for the Coulomb system consisting ofN particles of like charge in the field of a fixed point chargeZ. We show that if the particles are bosons, thenH _N has no discrete spectrum whenNN ₀=cZ ² for some constantc. If the particles are fermions, thenH _N is bounded below uniformly inN. These results can be extended to molecules and to other power law potentials.Research supported by the National Science Foundation, MCS78 -20455 USAOn leave from Department of Mathematics, University of Lowell, Lowell, MA O1854 USA     

On current superalgebras and super-Schwinger terms

We present a general construction of current superalgebras within the framework of quasi-free second quantization of bosons and fermions. Mathematically speaking, we given projective representations of certain Lie superalgebras realized as bounded operators on Z₂-graded Hilbert spaces and, more generally, on Grassmann algebra-modules. The super-Schwinger terms occurring correspond to Z₂-graded two-cocycles.Supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Project No. P5588.     

Method of action-angle variables and the classical dynamics of a nonlinear lagrangian

The method of action-angle variables is used to obtain the complete periodic solutions of a nonlinear chiral Lagrangian system with the Lagrangian of the formL=1/2 {q² [(q · q)²/(1 - q²)] - [k ₀q²/(1 - q²)]q=(q ₁,q ₂,q ₃) by making suitable canonical transformations. Usual semiclassical quantization procedure may then be applied to obtain the energy levels, which is shown to be in good agreement with exact results.     

Realizability of a model in infinite statistics

Following Greenberg and others, we study a space with a collection of operatorsa(k) satisfying the q-mutator relationsa(l)a (k)a(l)= _k,l (corresponding forq=±1 to classical Bose and Fermi statistics). We show that then!×n! matrixA _n (q) representing the scalar products ofn-particle states is positive definite for alln ifq lies between –1 and +1, so that the commutator relations have a Hilbert space representation in this case (this has also been proved by Fivel and by Bozejko and Speicher). We also give an explicit factorization ofA _n (q) as a product of matrices of the form(1–q ^jT)^±1 with 1jn andT a permutation matrix. In particular,A _n (q) is singular if and only ifq ^M=1 for some integerM of the formk ²–k, 2kn.     

Canonical versus grand canonical occupation numbers for simple systems

An ideal gas ofN indistinguishable particles is described by a canonical ensemble (c.e.) and also by a grand canonical ensemble (g.c.e.) which hasN as themean total number of particles, the temperature and volume being the same in both cases. Exact mean occupation numbersn _j(N) are found if the system has only two states 1 and 2 of energiesE ₂E ₁. This should apply to quantum wells and similar simple systems. For systems which have captured one particle, the theory gives the simplest answers, and one find a maximum discrepancy of 17% between the two ensembles for the fermion case. It occurs whenE ₂–E ₁53 meV at room temperature. ForN=1 the mean occupation number for the c.e. is identical for fermions and for bosons, being in both cases given byn ₂(1)={exp[(E ₂-E ₁)/kT]+1}^-1,n ₁(1)=1-n ₂(1) For largeN one reverts to the usual situation and the discrepancy between the ensembles becomes small.     

The Influence of the Symmetry of Identical Particles on Flight Times

In this work, our purpose is to show how the symmetry of identical particles can influence the time evolution of free particles in the nonrelativistic and relativistic domains as well as in the scattering by a potential δ-barrier. For this goal, we consider a system of either two distinguishable or indistinguishable (bosons and fermions) particles. Two sets of initial conditions have been studied: different initial locations with the same momenta, and the same locations with different momenta. The flight time distribution of particles arriving at a ‘screen’ is calculated in each case from the density and flux. Fermions display broader distributions as compared with either distinguishable particles or bosons, leading to earlier and later arrivals for all the cases analyzed here. The symmetry of the wave function seems to speed up or slow down the propagation of particles. Due to the cross terms, certain initial conditions lead to bimodality in the fermionic case. Within the nonrelativistic domain, and when the short-time survival probability is analyzed, if the cross term becomes important, one finds that the decay of the overlap of fermions is faster than for distinguishable particles which in turn is faster than for bosons. These results are of interest in the short time limit since they imply that the well-known quantum Zeno effect would be stronger for bosons than for fermions. Fermions also arrive earlier and later than bosons when they are scattered by a δ-barrier. Although the particle symmetry does affect the mean tunneling flight time, in the limit of narrow in momentum initial Gaussian wave functions, the mean times are not affected by symmetry but tend to the phase time for distinguishable particles.     

The sl2 Loop Algebra Symmetry of the Six-Vertex Model at Roots of Unity

We demonstrate that the six vertex model (XXZ spin chain) with =(q+q^-1)/2 and q^2N=1 has an invariance under the loop algebra of sl₂ which produces a special set of degenerate eigenvalues. For =0 we compute the multiplicity of the degeneracies using Jordan–Wigner techniques.     

Grand unified models including extraZ bosons

The grand unified theories (GUT) of the simple Lie groups including extraZ bosons are discussed. There are onlySU _5+m,SO _6+4n, andE ₆ under our hypothesis. First we give a general discussion forSU _5+m, then forSU ₆ andSU ₇ for illustration. We use15 +6 ^* +6 ^* fermion representations inSU ₆ but not with the fermion content, Yukawa coupling, and the hierarchy of other authors. We suggest that there is a series of clans of particles. These clans consist of the extraZ bosons and the corresponding fermions of the scale.