A Hamiltonian structure associated with a matrix spectral problem of arbitrary-order

Starting from a specific matrix iso-spectral problem, an associated hierarchy of multi-component Hamiltonian equations is constructed, based on zero curvature equations. The key point is to choose appropriate time parts of Lax pairs which can yield evolution equations, and the existence of a Hamiltonian structure for the obtained hierarchy is established by means of the trace identity. An example with five components is computed, along with its Hamiltonian structure.     

A Groupoidification of the Fermion Algebra

In this paper, we consider the groupoidification of the fermion algebra. We construct a groupoid as the categorical analogues of the fermionic Fock space, and the creation and annihilation operators correspond to spans of groupoids. The categorical fermionic Fock states have some extra structures comparing with the normal forms. We also construct a 2-category of spans of groupoids corresponding to the fermion algebra. The relations of the morphisms in this 2-category are consistent with those in the graphical category which is represented by string diagrams. One may use these formalisms to describe the fermion systems more finely, and study some additional properties of the fermion systems.     

Fermion masses

In this paper, we show that 2m _e m /(m _e ² +m ² = (g _V/g _A) _e ² . From this expression, the Weinberg mixing parameter is shown to be 0·2254 or 0·2746. Assuming that the electron and muon neutrino masses are degenerate, we find thatm _v = (g _V/g _A) _e ² . (m _e m )/M _WL, where M_WL is the mass of the standard W^± boson. The neutrino mass turns out to be 6·5 eV. The -neutrino mass is found to be about 81 MeV. The masses of c, t, s and b quarks are found to be respectively 1·7 GeV, 21·2 GeV, 0·57 GeV and 2·18 GeV by assumingm _d=m _u= 0·3 GeV. All these masses agree with other estimates except the b quark which has about half of its expected value.     

Fermion nodes

The knowledge of the nodes of the many-fermion wave function would enable exact calculation of the properties of fermion systems by Monte Carlo methods. It is proved that fermion nodal regions have a tiling property, there is only one distinct kind of nodal region. All others are related to it by permutational symmetry. For some free particle systems, it is shown that there are only two nodal regions. An explicit form for the nodes of the many-fermion density matrix would enable exact simulations to be carried out at finite temperature. In the high-temperature limit, its nodes are related to Voronoi polyhedra. Twodimensional cross sections of nodes are depicted. General computable families of fermion wave functions and density matrices are discussed.     

Fermion Determinant with Dynamical Chiral Symmetry Breaking

One-loop fermion determinant is discussed for the case in which the dynamical chiral symmetry breakingcaused by momentum-dependent fermion self-energy ∑(p2) takes place. The obtained series generalizes the heat kernelexpansion for hard fermion mass.     

Fermion superfluid with hybridized s-and p-wave pairings

Ever since the pioneering work of Bardeen, Cooper and Schrieffer in the 1950 s, exploring novel pairing mechanisms for fermion superfluids has become one of the central tasks in modern physics. Here, we investigate a new type of fermion superfluid with hybridized s-and p-wave pairings in an ultracold spin-1/2 Fermi gas. Its occurrence is facilitated by the co-existence of comparable s-and p-wave interactions, which is realizable in a two-component 40 K Fermi gas with close-by s-and p-wave Feshbach resonances. The hybridized superfluid state is stable over a considerable parameter region on the phase diagram, and can lead to intriguing patterns of spin densities and pairing fields in momentum space. In particular, it can induce a phase-locked p-wave pairing in the fermion species that has no p-wave interactions. The hybridized nature of this novel superfluid can also be confirmed by measuring the s-and p-wave contacts, which can be extracted from the high-momentum tail of the momentum distribution of each spin component. These results enrich our knowledge of pairing superfluidity in Fermi systems, and open the avenue for achieving novel fermion superfluids with multiple partial-wave scatterings in cold atomic gases.     

Fermion Coherence Hamiltonians

We have established that the most general form of Hamiltonian that preserves fermionic coherent states stable in time, is that of the nonstationary free fermionic oscillator. This is to be compared with the earlier result of boson coherence Hamiltonian, which is of the more general form of the nonstationary forced bosonic oscillator. If however one admits Grassmann variables as Hamiltonian parameters then the coherence Hamiltonian takes again the form of (Grassmannian fermionic) forced oscillator.     

A problem relating to positive linear maps on matrix algebras

In this paper we investigate conditions under which two finite-dimensional density matrices can be transformed simultaneously into two other ones by means of a positive linear map which maps density matrices into density matrices. The result of this paper provides a complete answer in case of the matrix algebra of two-by-two matrices.     

Gravitational Fermion interactions

The scattering of unpolarized Fermions and scalar mesons by single graviton exchange is considered by means of a Feynman graph type perturbation theory scheme. In the limit of scattering of the Fermions by very heavy mesons, one obtains the cross section for Fermions scattering in a Schwarzschild metric. The result obtained conflicts with an earlier result of Mitskevich. In the limit of scattering of massless Fermions (neutrinos) with massless scalar mesons it is seen, using Weinberg's treatment of soft graviton Bremsstrahlung, that the cosmological red shift of light cannot be explained by interactions of the light with intergalactic neutrinos.This works was supported by the Fonds zur Förderung der wissenschaftlichen Forschung.     

Propagator of the Lattice Domain Wall Fermion and the Staggered Fermion

We calculate the propagator of the domain wall fermion (DWF) of the RBC/UKQCD collaboration with 2 + 1 dynamical flavors of 16³ × 32 × 16 lattice in Coulomb gauge, by applying the conjugate gradient method. We find that the fluctuation of the propagator is small when the momenta are taken along the diagonal of the 4-dimensional lattice. Restricting momenta in this momentum region, which is called the cylinder cut, we compare the mass function and the running coupling of the quark-gluon coupling α _s,g1(q) with those of the staggered fermion of the MILC collaboration in Landau gauge. In the case of DWF, the ambiguity of the phase of the wave function is adjusted such that the overlap of the solution of the conjugate gradient method and the plane wave at the source becomes real. The quark-gluon coupling α _s,g1(q) of the DWF in the region q > 1.3 GeV agrees with ghost-gluon coupling α _s (q) that we measured by using the configuration of the MILC collaboration, i.e., enhancement by a factor (1 + c/q ²) with c ≃ 2.8 GeV² on the pQCD result. In the case of staggered fermion, in contrast to the ghost-gluon coupling α _s (q) in Landau gauge which showed infrared suppression, the quark-gluon coupling α _s,g1(q) in the infrared region increases monotonically as q→ 0. Above 2 GeV, the quark-gluon coupling α _s,g1(q) of staggered fermion calculated by naive crossing becomes smaller than that of DWF, probably due to the complex phase of the propagator which is not connected with the low energy physics of the fermion taste. An erratum to this article can be found at     

A Comparative Study on q-Deformed Fermion Oscillators

In this paper, the algebras, representations, and thermostatistics of four types of fermionic q-oscillator models, called fermionic Newton (FN), Chaichian-Kulish-Ng (CKN), Parthasarathy-Viswanathan-Chaichian (PVC), Viswanathan-Parthasarathy-Jagannathan-Chaichian (VPJC), are discussed. Similarities and differences among the properties of these models are revealed. Particular emphasis is given to the VPJC-oscillators model so that its Fock space representation is analyzed in detail. Possible physical applications of these models are concisely pointed out.     

Fermion to boson mapping

The enveloping algebra, D_n, of fermions is extended on the lattice to include the discrete space invariance. This extended algebra, denoted X, has the space symmetry as a factor: X/D_n = space group.     

Fermion spectra from superstrings

Some compactifications of the ten-dimensional anomaly-free E₈ × E₈ and SO(32) theories that correspond to superstrings are studied. Compactification is achieved by setting the classical gauge field equal to the spin connection. The resulting chiral fermion spectra are obtained for any six-dimensional manifold, under the condition Tr F² = 30 Tr R², plus a quantization condition for U(1) charges. For E₈ × E₈ these conditions lead to potentially realistic models for any irreducible six-dimensional manifold and any embedding of the holonomy group. Apart from a few more exotic examples, the four-dimensional models we obtain are more or less standard SU(5), SO(10), SU(4) × SU(2) × SU(2) or E₆ models.     

Fermion number creation via electromagnetism

Let an external current, whose support is confined to the space-like slab |x ⁰| < T in two-dimensional spacetime, build up a localized charge density which vanishes for times |x ⁰| > T. We show that the zero mass Dirac quantum field reacts to this current by a c-number shift of the fermion number, i.e. Q _out=Q _in+Q, with , where q(x ⁰) denotes the total external charge. For the shift of the axial charge we obtain an extension of existing results.     

Light Front Fermion Model Propagation

In this work we consider the propagation of two fermion fields interacting with each other by the exchange of intermediate scalar bosons in the light front. We obtain the corrections up to fourth order in the coupling constant using hierarchical equations in order to obtain the bound state equation (Bethe-alpeter equation).     

Generic phase diagram of Fermion superfluids with population imbalance

It is shown by microscopic calculations for trapped imbalanced Fermi superfluids that the gap function always has sign changes, i.e., the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like state, up to a critical imbalance P(c), beyond which normal state becomes stable, at temperature T=0. A temperature-versus-pressure phase diagram is constructed, where the BCS state without sign change is stable only at T not equal to 0. We reproduce the observed bimodality in the density profile to identify its origin and evaluate P(c) as functions of T and the coupling strength. These dependencies match with the recent experiments.     

Density matrix approach to the Hartree-Fock problem

The problem of determining the best one-body density matrix which minimizes the total energy of a many fermion system is studied, with the idea of providing an alternative approach to the method of Roothaan for solving the Hartree-Fock equation.