Gravity, non-commutative geometry and the Wodzicki residue

We derive an action for gravity in the framework of non-commutative geometry by using the Wodzicki residue. We prove that for a Dirac operator D on an n dimensional compact Riemannian manifold with n ≥ 4, n even, the Wodzicki residue Res(D⁻ⁿ⁺²) is the integral of the second coefficient of the heat kernel expansion of D². We use this result to derive a gravity action for commutative geometry which is the usual Einstein-Hilbert action and we also apply our results to a non-commutative extension which is given by the tensor product of the algebra of smooth functions on a manifold and a finite dimensional matrix algebra. In this case we obtain gravity with a cosmological constant.     

Eigenvalue estimates for the Dirac operator depending on the Weyl tensor

We prove new lower bounds for the first eigenvalue of the Dirac operator on compact manifolds whose Weyl tensor or curvature tensor, respectively, is divergence-free. In the special case of Einstein manifolds, we obtain estimates depending on the Weyl tensor.     

The first eigenvalue of the Dirac operator on locally reducible Riemannian manifolds

We prove a lower estimate for the first eigenvalue of the Dirac operator on a compact locally reducible Riemannian spin manifold with positive scalar curvature. We determine also the universal covers of the manifolds on which the smallest possible eigenvalue is attained.     

Some remarks on the Hijazi inequality and generalizations of the Killing equation for spinors

We generalize the well-known lower estimates for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold proved by Friedrich [Math. Nachr. 97 (1980) 117–146] and Hijazi [Math. Phys. 104 (1986) 151–162; J. Geom. Phys. 16 (1995) 27–38]. The special solutions of the Einstein–Dirac equation constructed recently by Friedrich/Kim are examples for the limiting case of these inequalities. The discussion of the limiting case of these estimates yields two new field equations generalizing the Killing equation as well as the weak Killing equation for spinor fields. Finally, we discuss the two-and three-dimensional case in more detail.     

The Spectral Action for Dirac Operators with Skew-Symmetric Torsion

We derive a formula for the gravitational part of the spectral action for Dirac operators on 4-dimensional manifolds with totally anti-symmetric torsion. We find that the torsion becomes dynamical and couples to the traceless part of the Riemann curvature tensor. Finally we deduce the Lagrangian for the Standard Model of particle physics in the presence of torsion from the Chamseddine-Connes Dirac operator.     

Lower bounds for the eigenvalues of the Dirac operator

On a compact Riemannian spin manifold we give new lower bounds for the eigenvalues of the Dirac operator in terms of the curvature and of the norm of an appropriate endomorphism of the tangent bundle. As a corollary, one gets all known results in this direction. The limiting-case is then studied.     

The first eigenvalue of the transversal Dirac operator

On a foliated Riemannian manifold with a transverse spin structure, we give a lower bound for the square of the eigenvalues of the transversal Dirac operator. We prove, in the limiting case, that the foliation is a minimal, transversally Einsteinian with constant transversal scalar curvature.     

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.     

A conformal lower bound for the smallest eigenvalue of the Dirac operator and killing spinors

On a Riemannian spin manifold, we give a lower bound for the square of the eigenvalues of the Dirac operator by the smallest eigenvalue of the conformal Laplacian (the Yamabe operator). We prove, in the limiting case, that the eigenspinor field is a killing spinor, i.e., parallel with respect to a natural connection. In particular, if the scalar curvature is positive, the eigenspinor field is annihilated by harmonic forms and the metric is Einstein.     

The Dirac Operator on Untrapped Surfaces

We establish a sharp extrinsic lower bound for the first eigenvalue of the Dirac operator of an untrapped surface in initial data sets without apparent horizon in terms of the norm of its mean curvature vector. The equality case leads to rigidity results for the constraint equations with spherical boundary as well as uniqueness results for surfaces with constant mean curvature vector field in Minkowski space.     

1-forms over the moduli space of irreducible connections defined by the spectrum of Dirac operators

Let (P) be the moduli space of irreducible connections of a G-principal bundle P over a closed Riemannian spin manifold M. Let D_A be the Dirac operator of M coupled to a connection A of P and f a smooth function on M. We consider a smooth variation A(u) of A with tangent vector ω and denote T_ω:= (D_A(u)−f) (u=0. The coefficients of the asymptotic expansion of trace (T_ω · e^-t(D_A−f)2) near t=0 define 1-forms a^(k)_f, K=0, 1, 2, … on (P). In this paper we calculate aa⁽⁰⁾_f, a⁽¹⁾_f, a⁽²⁾_f and study some of their properties. For instance using the 1-form a⁽²⁾_f for suitable functions f we obtain a foliation of codimension 5 of the space of G-instantons of S⁴.     

An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold admitting parallel one-form

An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold of positive scalar curvature admitting a parallel one-form is found. The possible universal covering spaces of the manifolds on which the smalles possible eigenvalue is attained are also listed. Moreover, a complete classification of the compact odd-dimensional manifolds whose universal covering space is Sⁿ⁻¹ × is given in the limiting case. All such manifolds are diffeomorphic but not necessarily isometric to Sⁿ⁻¹ × S¹.     

Riemannian curvature on the group of area-preserving diffeomorphisms (motions of fluid) of 2-sphere

Euler's equation for an incompressible fluid filled in a Riemannian manifold D is regarded as a geodesic equation on the group of volume-preserving diffeomorphisms of D provided with a one-sided invariant metric. A negative sectional curvature implies instability of the geodesic with respect to the corresponding flow and perturbation. The exponential growth of the perturbation is estimated from the values of the sectional curvatures.

This paper presents the expression of the components of Riemannian curvature tensor of the group of area-preserving diffeomorphisms of a 2-sphere in explicit formulas through 3 − j coefficients.     

Some extensions of the Einstein–Dirac equation

We considered an extension of the standard functional for the Einstein–Dirac equation where the Dirac operator is replaced by the square of the Dirac operator and a real parameter controlling the length of spinors is introduced. For one distinguished value of the parameter, the resulting Euler–Lagrange equations provide a new type of Einstein–Dirac coupling. We establish a special method for constructing global smooth solutions of a newly derived Einstein–Dirac system called the CL-Einstein–Dirac equation of type II (see Definition 3.1).     

Anisotropic and valley-resolved beam-splitter based on a tilted Dirac system

We investigate theoretically valley-resolved lateral shift of electrons traversing an n–p–n junction bulit on a typical tilted Dirac system (8-Pmmn borophene). A gauge-invariant formula on Goos–Hänchen (GH) shift of transmitted beams is derived, which holds for any anisotropic isoenergy surface. The tilt term brings valley dependence of relative position between the isoenergy surface in n region and that in the p region. Consequently, valley double refraction can occur at the n–p interface. The exiting positions of two valley-polarized beams depend on the incident angle and energy of incident beam and barrier parameters. Their spatial distance D can be enhanced to be ten to a hundred times larger than the barrier width. Due to tilting-induced high anisotropy of the isoenergy surface, D depends strongly on the barrier orientation. It is always zero when the junction is along the tilt direction of Dirac cones. Thus GH effect of transmitted beams in tilted Dirac systems can be utilized to design anisotropic and valley-resolved beam-splitter.     

Dirac Operator on the Podleś Sphere

The spectral problem for the Dirac operator on the Podle sphere is discussed and solved in one of its possible formulations. The standard constructions for the Dirac operator on classical symmetric spaces and the spectrum of the Dirac operator on the classical two-dimensional sphere are recalled. The problem of defining a spinor structure on a quantum space is discussed and the definitions of a classical spinor structure and Dirac operator according to Ðurdevich are sketched. The Dirac operator for the Podles´ quantum sphere treated as a quotient space of SU(2) is constructed using the Woronowicz left-covariant calculus over this quantum group. The spectrum of the operator is obtained. Disagreement of its asymptotic behavior with Connes' axiom of noncommutative spectral geometry is stressed.     

Spinor Derivation of Quasilocal Mean Curvature Mass in General Relativity

A spinor derivation is presented for quasilocal mean-curvature mass of spacelike 2-surfaces in General Relativity. The derivation is based on the Sen-Witten spinor identity and involves the introduction of novel nonlinear boundary conditions related to the Dirac current of the spinor at the 2-surface and the tangential flux of a boundary Dirac operator, as well the use of a spin basis adapted to the mean curvature frame of the 2-surface normal space. This setting may provide an alternative approach to a positivity proof for mean-curvature mass based on showing that Witten’s equation admits a spinor solution satisfying the proposed nonlinear boundary conditions.     

Magnetic Lieb—Thirring Inequalities with Optimal Dependence on the Field Strength

The Pauli operator describes the energy of a nonrelativistic quantum particle with spin 1/2 in a magnetic field and an external potential. Bounds on the sum of the negative eigenvalues are called magnetic Lieb–Thirring (MLT) inequalities. The purpose of this paper is twofold. First, we prove a new MLT inequality in a simple way. Second, we give a short summary of our recent proof of a more refined MLT inequality(8) and we explain the differences between the two results and methods. The main feature of both estimates, compared to earlier results, is that in the large field regime they grow with the optimal (first) power of the strength of the magnetic field. As a byproduct of the method, we also obtain optimal upper bounds on the pointwise density of zero energy eigenfunctions of the Dirac operator.     

Quarkonium and hydrogen spectra with spin-dependent relativistic wave equation

The non-linear non-perturbative relativistic atomic theory introduces spin in the dynamics of particle motion. The resulting energy levels of hydrogen atom are exactly the same as that of Dirac theory. The theory accounts for the energy due to spin-orbit interaction and for the additional potential energy due to spin and spin-orbit coupling. Spin angular momentum operator is integrated into the equation of motion. This requires modification to classical Laplacian operator. Consequently, the Dirac matrices and the k operator of Dirac’s theory are dispensed with. The theory points out that the curvature of the orbit draws on certain amount of kinetic and potential energies affecting the momentum of electron and the spin-orbit interaction energy constitutes a part of this energy. The theory is developed for spin-1/2 bound state single electron in Coulomb potential and then extended further to quarkonium physics by introducing the linear confining potential. The unique feature of this quarkonium model is that the radial distance can be exactly determined and does not have a statistical interpretation. The established radial distance is then used to determine the wave function. The observed energy levels are used as the input parameters and the radial distance and the string tension are predicted. This ensures 100% conformance to all observed energy levels for the heavy quarkonium.