The inhomogeneous quantum invariance group of commuting fermions

We consider a model of d fermions where creation and annihilation operators of different fermions commute. We show that this particle algebra is invariant under an inhomogeneous quantum group.      

Dimensional reduction of fermions in the Gross-Neveu model under the conditions of violated Lorentz invariance

The Gross-Neveu model with violated Lorentz invariance is analyzed. The effective potential for the 3D and 2D cases is calculated. The gap equation is obtained and the symmetry features of the model are considered. Dimensional reduction allows the establishment of the concordance between the results for a different number of measurements.     

On the consistency of Lorentz invariance violation in QED induced by fermions in constant axial-vector background

We show for the first time that the induced parity-even Lorentz invariance violation can be unambiguously calculated in the physically justified and minimally broken, dimensional regularization scheme, suitably tailored for a spontaneous Lorentz symmetry breaking in a field theory model. The quantization of the Lorentz invariance violating quantum electrodynamics is critically examined and shown to be consistent either for a light-like cosmic anisotropy axial-vector or for a time-like one, when in the presence of a bare photon mass.     

Conformal invariance,current algebra and modular invariance

A large class of conformally invariant models in two dimensions is realised by constraining free fermion theories. The Fock spaces of the constrained theories are described, using the representation theory of affine Kac-Moody algebras. The results are extended to superconformally invariant theories. Projections of the models, producing consistent two-dimensional field theories, are discussed.     

Staggered ladder spectra

We exactly solve a Fokker-Planck equation by determining its eigenvalues and eigenfunctions: we construct nonlinear second-order differential operators which act as raising and lowering operators, generating ladder spectra for the odd- and even-parity states. The ladders are staggered: the odd-even separation differs from even-odd. The Fokker-Planck equation corresponds, in the limit of weak damping, to a generalized Ornstein-Uhlenbeck process where the random force depends upon position as well as time. The process describes damped stochastic acceleration, and exhibits anomalous diffusion at short times and a stationary non-Maxwellian momentum distribution.     

Free fermions and tau-functions

We review the formalism of free fermions used for construction of tau-functions of classical integrable hierarchies and give a detailed derivation of group-like properties of the normally ordered exponents, transformations between different normal orderings, the bilinear relations, the generalized Wick theorem and the bosonization rules. We also consider various examples of tau-functions and give their fermionic realization.     

Geometric fermions

We study the Kähler-Dirac equation which linearizes the laplacian on the space of antisymmetric tensor fields. In flat space-time it is equivalent to the Dirac equation with internal symmetry and on the lattice it reproduces Susskind fermions. The KD equation in curved space-time differs from the Dirac equation by coupling the gravitational field to the internal symmetry generators. This new way of treating fermionic degrees of freedom may lead to a solution of the generation puzzle but is in conflict with the equivalence principle and with Lorentz invariance on the Planck-mass scale.     

Symmetry and invariance

The concept of invariance relates to both the intrinsic symmetries of physical systems and the symmetry of the set of equivalent reference frames used to observe them. Standard algebraic expressions for electrostatic potentials and crystal-field effective operators display both types of invariance. The concept of a reference frame is generalized to that of an ‘observing system’, which can, for example, be the basis states of a quantum system. This idea is related to Racah’s mathematical machinery for evaluating the matrix elements of many-electron 4f open-shell states in lanthanide ions. It is argued, on the basis of computational flexibility and ease of interpretation, that all equations that represent physical processes be expressible in terms of invariants of the set of observing systems. This ‘Principle of Invariance’ is then applied to special relativity, leading to a simple geometrical interpretation of Maxwell’s electromagnetic field equations. The close relationship between Dirac’s relativistic wave equation and Maxwell’s equations is then exposed. This leads to the concept of an inner structure of space-time and the reinterpretation of particle spin. Finally, it is shown that the use of invariants in relativity theory identifies a set of observing systems with a higher symmetry than that of Minkowski space-time.     

Vortices with fermions

Vortex-like solutions of two models involving fermions are found. A method is presented for extracting the anticommuting c-number parameters in the solutions and reducing the equations to ones containing only ordinary numbers.     

Resistance minimum and heavy fermions

The phenomenon of the resistance minimum in dilute magnetic alloys is explained in terms of the s-d interaction which takes account of scattering of the conduction electron off the magnetic impurities in metals. Some of the intermetallic compounds which involve rare earth elements or uranium show a very large electronic specific heat and remain non-magnetic even though they show a Curie-like susceptibility at higher temperatures. These phenomena are also explained based on the s-d interaction model.     

Superheavy fermions and horizontal symmetries

The mechanism which was formerly used to obtain neutrino masses is generalized to all light fermions. Correspondingly, several sets of superheavy fermions are introduced. Assignments under a horizontal symmetry group are arranged such that the heaviest among the light fermions acquire their masses, not from the ordinary Higgs-Yukawa couplings, but from couplings to the heavy fermions. Masses of the other light fermions are then obtained through horizontal gauge interactions. Accordingly, the resulting light fermion masses exhibit a hierarchical generation structure. Because of the construction, light Higgs fields do not induce dangerous flavor-changing neutral-current interactions.     

Light composite fermions

In this paper we consider the possibility that QCD-like theories can lead to massless or near-massless composite fermions. The method of analysis relies on a conjectured equivalence between the confined and Higgs phases of certain non-abelian gauge theories. This “complementarity” principle allows us to analyze a theory as if the Higgs phenomenon occurred and then reinterpret the results in the language of composite gauge singlets. Those fermions which remain massless in the Higgs picture may then be interpreted as massless fermionic composites.The principle of complementarity, when applied to a class of extended technicolor models, implies that quarks and leptons are composites bound at a scale of order 1–100 TeV.     

FLIC fermions and hadron phenomenology

A pedagogical overview of the formulation of the Fat-Link-Irrelevant-Clover (FLIC) fermion action and its associated phenomenology is described. The scaling analysis indicates FLIC fermions provide a new form of nonperturbative improvement where near-continuum results are obtained at finite lattice spacing. Spin-(1/2) and spin-(3/2), even- and odd-parity baryon resonances are investigated in quenched QCD, where the nature of the Roper resonance and are of particular interest. FLIC fermions allow efficient access to the light-quark-mass regime, where evidence of chiral nonanalytic behavior in the mass is observed.Received: 30 September 2002, Published online: 22 October 2003PACS: 12.38.Gc Lattice QCD calculations - 12.38.Aw General properties of QCD (dynamics, confinement, etc.)D.B. Leinweber: Plenary talk presented by Derek Leinweber     

Light composite fermions

't Hooft's conditions for massless composite fermions are discussed. We then study the various mechanisms through which such composite fermions can acquire a small mass. Two new mass generation mechanisms are proposed.     

Quasi goldstone fermions

We discuss a mechanism by which, in theories with an explicitly broken supersymmetry, we can obtain calculable fermion masses, provided certain softly broken R symmetries are incorporated. The corresponding fermion representations are determined by the pattern of internal symmetry breakdown. This mechanism is explicitly studied in a simple U(1) model. Prospects and limitations of this idea for constructing realistic fermion spectra are discussed.     

Quasi nambu-goldstone fermions

We explore in detail the hypothesis that quarks and leptons are the approximately massless quasi Goldstone fermions of a supersymmetric preon theory. In particular, we discuss the possible patterns of states emerging from the spontaneous breakdown of global symmetries in supersymmetric theories and construct the low-energy effective lagrangians describing the interaction of these states. In contrast to what happens in the Goldstone sector, the interactions of the quasi Goldstone fermions contain arbitrary parameters which directly reflect the preon dynamics. Various models are explored, including both models in which the weak interactions are residual and models where these interactions are fundamental. A variety of issues are addressed, from the universality and approximate SU(2)_L nature of the weak interactions, for the former class of models, to the generation of states beyond the quarks and leptons and the nature of the dynamical breaking of SU(2)_L×U(1), for the latter class of models. Open questions and speculations connected with the origin of families and the nature of fermion mass generation, including supersymmetry breaking, are also discussed.