A numerical study of the 2-flavour Schwinger model with dynamical overlap hypercube fermions

We present numerical results for the 2-flavour Schwinger model with dynamical chiral lattice fermions. We insert an approximately chiral hypercube Dirac operator into the overlap formula to construct the overlap hypercube operator. This is an exact solution to the Ginsparg?CWilson relation, with an excellent level of locality and scaling. Due to its similarity with the hypercubic kernel, a low polynomial in this kernel provides a numerically efficient Hybrid Monte Carlo force. We measure the microscopic Dirac spectrum and discuss the corresponding scale-invariant parameter, which takes a surprising form. This is an interesting case, since Random Matrix Theory is unexplored for this setting, where the chiral condensate ?? vanishes in the chiral limit. We also measure ?? and the ??pion?? mass, in distinct topological sectors. In this context we discuss and probe the topological summation of observables by various methods, as well as the evaluation of the topological susceptibility. The feasibility of this summation is essential for the prospects of dynamical overlap fermions in QCD.     

Topological anomalies: Explicit examples

We discuss the mathematical picture of anomalies. By solving the Dirac equation in the background of non-trivial families of gauge connections, we show explicitly the interplay between spectral flows, zero modes of the Dirac operator and projective representations of the gauge group, and the existence of both perturbative and non-perturbative anomalies. We give an explicit expression for the fermion determinant for chiral QCD in two dimensions when an anomaly is present.     

A simple derivation of the overlap Dirac operator

We derive the vector-like four-dimensional overlap Dirac operator starting from a five-dimensional Dirac action in the presence of a delta-function space–time defect. The effective operator is obtained by first integrating out all the fermionic modes in the fixed gauge background, and then identifying the contribution from the localized modes as the determinant of an operator in one dimension less. We define physically relevant degrees of freedom on the defect by introducing an auxiliary defect-bound fermion field and integrating out the original five-dimensional bulk fields.     

The soliton of the effective chiral action

The baryon-number-1 soliton solution of the effective chiral meson action of QCD is found numerically. For this purpose the underlying fermion determinant (quark loop) is calculated in a non-perturbative way by diagonalizing the corresponding Dirac hamiltonian. The vacuum energy is defined within a static version of the proper-time regularization. Topological soliton solutions are found for cut-offs below some critical value, above which only topologically trivial solutions exist.     

Large-mass expansions or one-loop effective actions and fermion currents

The large-mass expansion of the functional determinants for second-order elliptic operators and general Dirac operators is calculated for four-dimensional flat euclidean space using zeta function regularisation and heat kernel methods. The results are applicable to one-loop boson and fermion effective actions. In addition the expansions of covariantly regularised fermion currents are derived. It is also possible for the corresponding Pauli-Villars regularised forms to be then simply obtained and the modified currents then reproduce the usual Bardeen anomaly. Although covariant methods are used it is shown how to derive the expansion for the phase of the fermion determinant, which is non-covariant and produces the anomaly, in terms of a representation as a five-dimensional integral which is related to the spectral asymmetry for a suitable spinor hamiltonian. This relation is essentially exact and is demonstrated by considering the variation of the phase with respect to the Dirac operator.     

Dirac Operator on Fuzzy Sphere with U（1） Dirac Monopole Background

We treat the Dirac operator on the Fuzzy sphere with the help of the generalized coherent state. It is shown that the derivatives can be constructed by Moyal product with symbol of the operator. We obtain the eigenvalue of the free fermion Dirac operator as same as the result by [Hajime Aoki, Satoshi Iso, and Kelichi Nagao, Phys. Rev.D67 (2003) 065018]. Meanwhile, we also give the eigenvalue of Dirac operator with U(1) Dirac monopole background.     

Dirac Operator on Fuzzy Sphere with U(1) Dirac Monopole Background

We treat the Dirac operator on the Fuzzy sphere with the help of the generalized coherent state. It is shown that the derivatives can be constructed by Moyal product with symbol of the operator. We obtain the eigenvalue of the free fermion Dirac operator as same as the result by [Hajime Aoki, Satoshi Iso, and Kelichi Nagao, Phys. Rev.D67 (2003) 065018]. Meanwhile, we also give the eigenvalue of Dirac operator with U(1) Dirac monopole background.     

Renormalization of the Regularized Relativistic Electron-Positron Field

We consider the relativistic electron-positron field interacting with itself via the Coulomb potential defined with the physically motivated, positive, density-density quartic interaction. The more usual normal-ordered Hamiltonian differs from the bare Hamiltonian by a quadratic term and, by choosing the normal ordering in a suitable, self-consistent manner, the quadratic term can be seen to be equivalent to a renormalization of the Dirac operator. Formally, this amounts to a Bogolubov-Valatin transformation, but in reality it is non-perturbative, for it leads to an inequivalent, fine-structure dependent representation of the canonical anticommutation relations. This non-perturbative redefinition of the electron/positron states can be interpreted as a mass, wave-function and charge renormalization, among other possibilities, but the main point is that a non-perturbative definition of normal ordering might be a useful starting point for developing a consistent quantum electrodynamics. Received: 8 March 2000 / Accepted: 7 July 2000     

含外规范场和动力学费米子自能的狄拉克算符及费米子凝聚

将非阿贝尔规范理论中狄拉克算符行列式的计算从传统的只能含有硬费米子质量项的情况推广到可以含有动量相关的费米子自能的情况，并且行列式与费米子凝聚的计算都被推广到使之能够含有任意的外规范场. 关键词： 费米子自能 外规范场 狄拉克算符的行列式 费米子凝聚     

Thermal and thermoelectric response from Keldysh formalism with application to gapped Dirac fermions

Based on the Keldysh Green's functions theory, we present a general formula of the thermal and thermoelectric transport. In the clean limit, our formula recovers the previous results obtained from the semiclassical transport theory.In our approach, we propose an appropriate energy current operator and electric current operator, and the unphysical divergence from the direct application of the Kubo formula is eliminated. As an application, we study the thermal and the thermoelectric Hall conductivities of a gapped Dirac fermion model in the presence of impurity scattering.     

Dark viscous fluid coupled with dark matter and future singularity

Motivated by Kerner and Man’s fermions tunneling method of dimension 4 black holes, in this paper, we further improve the analysis to investigate Hawking radiation of charged Dirac particles with spin 1/2 from general non-extremal rotating charged black holes with two parameters and a non-zero cosmological constant in minimal five-dimensional gauged supergravity. For space-times with different horizon topology and different dimensions, constructing a set of appropriate γ ^μ matrices for general covariant Dirac equation is an important technique for the fermion tunneling method. By introducing a set of appropriate matrices γ ^μ and employing the ansatz for the spin-up spinor field, we successfully recovered the tunneling probability of charged Dirac particles and the expected Hawking temperature of the black hole, which is exactly consistent with that obtained by other methods. Moreover, the fermion tunneling method can be directly applied to the other five-dimensional charged black holes, which strengthens the validity and power of the fermion tunneling method.     

On fermion masses in a dimensional reduction scheme

A candidate model for Grand Unification, arising from a Coset Space Dimensional Reduction scheme based on anE(7) gauge theory, is found to have a promising set of fermionic quantum numbers. Unfortunately, these fermions all develop large (geometric) masses. We derive formulae for the square of the Dirac operator and for fermion masses for a large class of CSDR schemes, revealing this as a general feature.     

Explicit SO(10) supersymmetric grand unified model for the higgs and yukawa sectors

A complete set of fermion and Higgs superfields is introduced with well-defined SO(10) properties and U(1)xZ2xZ2 family charges from which the Higgs and Yukawa superpotentials are constructed. The structures derived for the four Dirac fermion and right-handed Majorana neutrino mass matrices coincide with those previously obtained from an effective operator approach. Ten mass matrix input parameters accurately yield the twenty masses and mixings of the quarks and leptons with the bimaximal atmospheric and solar neutrino vacuum solutions favored in this simplest version.     

Evaluating fermion determinants through the chiral anomaly

A class of fermion operators whose determinants can be calculated exactly has recently been noted. We observe that typically such operators can be chirally rotated into the free Dirac operator; hence, their determinants are given by the chiral anomaly. Four-dimensional fermion determinants of this type are computed; the appearance of the Wess-Zumino anomaly term is noted.     

Axial anomalies of Lifshitz fermions

We compute the axial anomaly of a Lifshitz fermion theory with anisotropic scaling z = 3 which is minimally coupled to geometry in 3+1 space‐time dimensions. We find that the result is identical to the relativistic case using path integral methods. An independent verification is provided by showing with spectral methods that the η‐invariant of the Dirac and Lifshitz fermion operators in three dimensions are equal. Thus, by the integrated form of the anomaly, the index of the Dirac operator still accounts for the possible breakdown of chiral symmetry in non‐relativistic theories of gravity. We apply this framework to the recently constructed gravitational instanton backgrounds of Hořava–Lifshitz theory and find that the index is non‐zero provided that the space‐time foliation admits leaves with harmonic spinors. Using Hitchin's construction of harmonic spinors on Berger spheres, we obtain explicit results for the index of the fermion operator on all such gravitational instanton backgrounds with SU(2) × U(1) isometry. In contrast to the instantons of Einstein gravity, chiral symmetry breaking becomes possible in the unimodular phase of Hořava–Lifshitz theory arising at λ = 1/3 provided that the volume of space is bounded from below by the ratio of the Ricci to Cotton tensor couplings raised to the third power. Some other aspects of the anomalies in non‐relativistic quantum field theories are also discussed.     

Motion of neutral fermions with anomalous magnetic and electric moments in external fields

A covariant spin operator is found for fermions with anomalous magnetic and electric dipole moments in constant external fields. The spin behavior of a neutral fermion in constant magnetic and electric fields is investigated using exact solutions obtained for the Dirac equation.     

Quantum holonomy theory

We present quantum holonomy theory, which is a non‐perturbative theory of quantum gravity coupled to fermionic degrees of freedom. The theory is based on a ‐algebra that involves holonomy‐diffeo‐morphisms on a 3‐dimensional manifold and which encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Employing a Dirac type operator on the configuration space of Ashtekar connections we obtain a semi‐classical state and a kinematical Hilbert space via its GNS construction. We use the Dirac type operator, which provides a metric structure over the space of Ashtekar connections, to define a scalar curvature operator, from which we obtain a candidate for a Hamilton operator. We show that the classical Hamilton constraint of general relativity emerges from this in a semi‐classical limit and we then compute the operator constraint algebra. Also, we find states in the kinematical Hilbert space on which the expectation value of the Dirac type operator gives the Dirac Hamiltonian in a semi‐classical limit and thus provides a connection to fermionic quantum field theory. Finally, an almost‐commutative algebra emerges from the holonomy‐diffeomorphism algebra in the same limit.     

Systematic Renormalization Scheme in Non-Perturbative Calculations Within Covariant Light-Front Dynamics

We advocate the use of the recently developed Taylor-Lagrange renormalization scheme for non-perturbative calculations of the properties of bound state systems. As an example, we show how this scheme can be applied to the calculation of non-perturbative corrections to the mass of a fermion within the covariant formulation of light-front dynamics.     

A mathematical analysis of the axial anomaly

Letters in Mathematical Physics - As is well known to physicists, the axial anomaly of the massless free fermion in Euclidean signature is given by the index of the corresponding Dirac operator. We...