A rigorous construction of the chiral anomaly on a spin manifold

The quantum field for Dirac fermion is rigorously formulated on a compact spin manifold by using the functional integral defined as the continuum limit of a lattice approximation with a new action. Within this framework, the chiral anomaly for a fermion interacting with gauge as well as gravitational fields is calculated with mathematical rigor.     

Equations of motion for a classical color particle in external non-Abelian fermionic and bosonic fields

On the basis of principles of gauge and reparametrization invariance of a real-valued Lagrangian, construction of the action describing the dynamics of a classical color-charged particle interacting with background non-Abelian gauge and fermion fields is considered. The cases of the linear and quadratic dependence of a Lagrangian on the background fermion field are discussed. It is shown that, in both cases, there exist an infinite number of interaction terms. From an iteration scheme, examples of construction of the first few currents and sources induced by a moving particle with non-Abelian charge are given. It is shown that these quantities by a suitable choice of parameters exactly reproduce additional currents and sources obtained previously in [1] on the basis of heuristic considerations.     

Negative tunnel magnetoresistance in a quantum dot induced by interplay of a Majorana fermion and thermal-driven ferromagnetic leads

We investigate the spin-related currents and tunnel magnetoresistance through a quantum dot, which is side-coupled with a Majorana fermion zero mode and two thermal-driven ferromagnetic electrodes. It is found that the interplay of Majorana fermion and electrodes' spin polarization can induce a nonlinear thermal-bias spin current. This interplay also decreases the total magnitude of spin or charge current, in either parallel or antiparallel configuration. In addition, a thermal-driven negative tunnel magnetoresistance is found, which is an unique feature to characterize Majorana fermion.With large temperature difference, a step phenomenon is observed in gate tuned spin-up current. When the coupling between quantum dot and topological superconductor is strong enough, this step will evolve into a linear relation, revealing Majorana fermion's robustness.     

Classical versus quantum gravity

Is Einstein's metric theory of gravitation to be quantized to yield a complete and logically consistent picture of the geometry of the real world in the presence of quantized material sources? To answer this question, we give arguments that there is a consistent way to extend general relativity to small distances by incorporating further geometric quantities at the level of the connection into the theory and introducing corresponding field equations for their determination, allowing thereby the metric and the Levi-Civita connection to remain classical quantities. The dualism between matter and geometry is extended to quantized fields with the help of a Hibert bundle ? raised over a Riemann-Cartan spacetime. Quantized subnuclear matter fields (generalized quantum mechanical wave functions) are sections on ? which determine generalized bilinear currents acting as sourc currents for the bundle geometry at small distances. The established dualism between matter and the underlying bundle geometry contains general relativity as a classical part.     

Maximal symmetry and mass generation of Dirac fermions and gravitational gauge field theory in six-dimensional spacetime

The relativistic Dirac equation in four-dimensional spacetime reveals a coherent relation between the dimensions of spacetime and the degrees of freedom of fermionic spinors. A massless Dirac fermion generates new symmetries corresponding to chirality spin and charge spin as well as conformal scaling transformations. With the introduction of intrinsic W-parity, a massless Dirac fermion can be treated as a Majorana-type or Weyl-type spinor in a six-dimensional spacetime that reflects the intrinsic quantum numbers of chirality spin. A generalized Dirac equation is obtained in the six-dimensional spacetime with a maximal symmetry. Based on the framework of gravitational quantum field theory proposed in Ref. [1] with the postulate of gauge invariance and coordinate independence, we arrive at a maximally symmetric gravitational gauge field theory for the massless Dirac fermion in six-dimensional spacetime. Such a theory is governed by the local spin gauge symmetry SP(1,5) and the global Poincar′e symmetry P(1,5)= SO(1,5) P~(1,5) as well as the charge spin gauge symmetry SU(2). The theory leads to the prediction of doubly electrically charged bosons. A scalar field and conformal scaling gauge field are introduced to maintain both global and local conformal scaling symmetries. A generalized gravitational Dirac equation for the massless Dirac fermion is derived in the six-dimensional spacetime. The equations of motion for gauge fields are obtained with conserved currents in the presence of gravitational effects. The dynamics of the gauge-type gravifield as a Goldstone-like boson is shown to be governed by a conserved energy-momentum tensor, and its symmetric part provides a generalized Einstein equation of gravity. An alternative geometrical symmetry breaking mechanism for the mass generation of Dirac fermions is demonstrated.     

A Review About Invariance Induced Gravity: Gravity and Spin from Local Conformal-Affine Symmetry

In this review paper, we discuss how gravity and spin can be obtained as the realization of the local Conformal-Affine group of symmetry transformations. In particular, we show how gravitation is a gauge theory which can be obtained starting from some local invariance as the Poincaré local symmetry. We review previous results where the inhomogeneous connection coefficients, transforming under the Lorentz group, give rise to gravitational gauge potentials which can be used to define covariant derivatives accommodating minimal couplings of matter, gauge fields (and then spin connections). After we show, in a self-contained approach, how the tetrads and the Lorentz group can be used to induce the spacetime metric and then the Invariance Induced Gravity can be directly obtained both in holonomic and anholonomic pictures. Besides, we show how tensor valued connection forms act as auxiliary dynamical fields associated with the dilation, special conformal and deformation (shear) degrees of freedom, inherent to the bundle manifold. As a result, this allows to determine the bundle curvature of the theory and then to construct boundary topological invariants which give rise to a prototype (source free) gravitational Lagrangian. Finally, the Bianchi identities, the covariant field equations and the gauge currents are obtained determining completely the dynamics.     

Spherically symmetric solution in higher-dimensional teleparallel equivalent of general relativity

A theory of(N+1)-dimensional gravity is developed on the basis of the teleparallel equivalent of general relativity(TEGR).The fundamental gravitational field variables are the(N+1)-dimensional vector fields,defined globally on a manifold M,and the gravitational field is attributed to the torsion.The form of Lagrangian density is quadratic in torsion tensor.We then give an exact five-dimensional spherically symmetric solution(Schwarzschild(4+1)-dimensions).Finally,we calculate energy and spatial momentum using gravitational energy-momentum tensor and superpotential 2-form.     

Hestenes' Tetrad and Spin Connections

Defining a spin connection is necessary for formulating Dirac's bispinor equation in a curved space-time. Hestenes has shown that a bispinor field is equivalent to an orthonormal tetrad of vector fields together with a complex scalar field. In this paper, we show that using Hestenes' tetrad for the spin connection in a Riemannian space-time leads to a Yang-Mills formulation of the Dirac Lagrangian in which the bispinor field Ψ is mapped to a set of SL(2,R)× U(1) gauge potentials F_α^K and a complex scalar field ρ. This result was previously proved for a Minkowski space-time using Fierz identities. As an application we derive several different non-Riemannian spin connections found in the literature directly from an arbitrary linear connection acting on the tensor fields (F_α^K, ρ). We also derive spin connections for which Dirac's bispinor equation is form invariant. Previous work has not considered form invariance of the Dirac equation as a criterion for defining a general spin connection.     

On the role of a torsion‐like field in a scenario for the Spin Hall Effect

Starting from a field theory action that describes a Dirac fermion, we propose and analyze a model based on a low‐relativistic Pauli equation coupled to a torsion‐like term to study Spin Hall Effect (SHE). We point out a very particular connection between the modified Pauli equation and the (SHE), where what we refer to torsion as field playing an important role in the spin‐orbit (SO) coupling process. In this scenario, we present a proposal of a spin‐type current, considering the tiny contributions of torsion in connection with intrinsic anisotropy of the crystal electric field.     

THE CORRESPONDENCE PRINCIPLE IN (1+1)-DIMENSIONAL FIELD THEORY AND THE COLEMAN THEOREM

The correspondence relations between a fermion field and a boson field in (1+1)-dimensional quantum field theory is discussed in general. Emphases have been laid on the renorinalization with respect.to an arbitrary mass parameter in boson version as well as the nonlocal property of currents in fermion version. After establishing the equivalence between the continuous chiral transformation in fermion version and the translational transformation in boson version, we are able to prove the Coleman theorem correspondingly.     

Poincaré gauge theory from self-coupling

Poincaré gauge theory is derived from a linear theory by the method suggested by Gupta for deriving Einstein’s general relativity from the linear theory of a spin-2 field. Non-linearity is introduced by requiring that a set of tensor fields be coupled to the Noether currents of the Poincaré group (energy-momentum and spin).     

On defining the spin moment in the tetrad theory of gravitation

A general definition of the spin moment is presented in the tetrad formulation of the relativistic theory of gravitation; it is based on the conditions for the invariance of the corresponding action integral relative to infinitesimal tetrad transformations (the so-called tetrad spin moment) and infinitesimal coordinate transformations (the so-called coordinate spin moment). It is shown that the tetrad formulation of the general theory of relativity (TFGTR) and the tetrad theory of gravitation (TTG) in a space of absolute parallelism lead to fundamentally different definitions of spin, since in the Riemannian geometry of the TFGTR only the coordinate spin moment is physically meaningful, whereas in the space of absolute parallelism of the TTG only the tetrad spin moment has essential significance. It is also indicated that the Pellegrini-Plebanski theory (PPT) leads to an unsatisfactory hybrid definition of spin in the form of the coordinate spin moment of the gravitational and boson fields and the tetrad spin moment of the gravitational and fermion fields, the gravitational field entering into these spin moments of the PPT with opposite signs.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 68–71, May, 1976.     

Constraining conformal field theory with higher spin symmetry in four dimensions

We analyze the constraints on the general form and the singularity structure of the correlation functions of the symmetric, traceless and conserved stress-energy tensor implied by conformal invariance and higher spin symmetry in four dimensions. In particular, we show that all these correlation functions will have at most double pole singularities. We then compute the 4-, 5- and 6-point functions of the stress-energy tensor and find that they are linear combinations of the three free field expressions (scalar, fermion and Maxwell field). This is a strong indication that all such theories are essentially free.     

Multi-Trace Superpotentials vs. Matrix Models

We consider ᵊ9=1 supersymmetric U(N) field theories in four dimensions with adjoint chiral matter and a multi-trace tree-level superpotential. We show that the computation of the effective action as a function of the glueball superfield localizes to computing matrix integrals. Unlike the single-trace case, holomorphy and symmetries do not forbid non-planar contributions. Nevertheless, only a special subset of the planar diagrams contributes to the exact result. In addition, the computation of the superpotential localizes to doing matrix integrals. In view of the results of Dijkgraaf and Vafa for single-trace theories, one might have naively expected that these matrix integrals are related to the free energy of a multi-trace matrix model. We explain why this naive identification does not work. Rather, an auxiliary single-trace matrix model with additional singlet fields can be used to exactly compute the field theory superpotential. Along the way we also describe a general technique for computing the large-N limits of multi-trace Matrix models and raise the challenge of finding the field theories whose effective actions they may compute. Since our models can be treated as ᵊ9=1 deformations of pure ᵊ9=2 gauge theory, we show that the effective superpotential that we compute also follows from the ᵊ9=2 Seiberg-Witten solution. Finally, we observe an interesting connection between multi-trace local theories and non-local field theory.     

A remark on the contact structure on Weyl manifold and Fedosov connection

Using the contact structure on Weyl manifold, we introduce degree operator fields. The degree operator field gives a fiberwise decomposition of Weyl manifold with respect to eigenvalues. We remark that this decomposition canonically gives a symplectic connection and also a Fedosov connection on the Weyl manifold.     

Localization of matter and fermion resonances on double walls

We investigate the possibility of localizing various matter fields on the double walls. For spin 0 scalar field, massless zero mode can be normalized on the double walls. However, for spin 1 vector field, the zero mode is not localized on the double walls. In the paper [C.A.S. Almeida, M.M. Ferreira Jr., A.R. Gomes, R. Casana, arXiv:0901.3543 [hep-th]], the authors investigated fermion localization on a Bloch brane, especially, they found fermion resonances on the Bloch brane for both chiralities and related their appearance to branes with internal structure. Inspired by their work, for spin 1/2 spinor field, we focus our attention mainly on the fermion resonances, and also found fermion resonances for both left-handed fermions and right-handed ones on the double walls, which further supports the arguments presented in the paper [C.A.S. Almeida, M.M. Ferreira Jr., A.R. Gomes, R. Casana, arXiv:0901.3543 [hep-th]].     

Wilson loops,geometric operators and fermions in 3d group field theory

Group field theories whose Feynman diagrams describe 3d gravity with a varying configuration of Wilson loop observables and 3d gravity with volume observables at each vertex are defined. The volume observables are created by the usual spin network grasping operators which require the introduction of vector fields on the group. We then use this to define group field theories that give a previously defined spin foam model for fermion fields coupled to gravity, and the simpler “quenched” approximation, by using tensor fields on the group. The group field theory naturally includes the sum over fermionic loops at each order of the perturbation theory.     

First-order invariant Einstein-Cartan variational structures

A first-order invariant Einstein-Cartan structure is a Lagrangian structure on a differential manifold defined by a generally invariant Lagrangian depending on a metric field, a connection field, and the first derivatives of these fields. Moreover, it is assumed that the metric and connection fields satisfy the so-called compatibility condition. In this paper the problem of finding all such invariant Einstein-Cartan structures is discussed. It is shown that each Lagrangian of these structures depends only on certain tensors constructed from the metric and the connection fields, which means that all the Lagrangians can be described within the framework of the classical theory of invariants. The maximal number of functionally independent Lagrangians is determined as a function of the dimension of the underlying manifold.     

Entropy of Gibbons—Maeda Dilaton Black Hole Due to Arbitrary Spin Fields

Using the membrane model which is based on brick-wall model, we calculated the free energy and entropy of Gibbons—Maeda dilation black hole due to arbitrary spin fields. The result shows that the entropy of a scalar field and the entropy of a fermion field have similar formulas. There is only a coefficient difference between them. Furthermore, both entropies depend on the degeneracy of the field.