The Holst Action by the Spectral Action Principle

We investigate the Holst action for closed Riemannian 4-manifolds with orthogonal connections. For connections whose torsion has zero Cartan type component we show that the Holst action can be recovered from the heat asymptotics for the natural Dirac operator acting on left-handed spinor fields.     

The dirac operator and gravitation

We give a brute-force proof of the fact, announced by Alain Connes, that the Wodzicki residue of the inverse square of the Dirac operator is proportional to the Einstein-Hilbert action of general relativity. We show that this also holds for twisted (e. g. by electrodynamics) Dirac operators, and more generally, for Dirac operators pertaining to Clifford connections of general Clifford bundles.     

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.     

Should There Be a Spin-Rotation Coupling for a Dirac Particle?

It was argued by Mashhoon that a spin-rotation coupling term should add to the Hamiltonian operator in a rotating frame, as compared with the one in an inertial frame. For a Dirac particle, the Hamiltonian and energy operators H and E in a given reference frame were recently proved to depend on the tetrad field. We argue that this non-uniqueness of H and E really is a physical problem. We show that a tetrad field contains two informations about local rotation, which usually do not coincide. We compute the energy operator in the inertial and the rotating frame, using three different tetrad fields. We find that Mashhoon’s term is there if the spatial triad rotates as does the reference frame—but then it is also there in the energy operator for the inertial frame. In fact, if one uses the same given tetrad field, the Dirac Hamiltonian operators in two reference frames in relative rotation differ only by the angular momentum term. If the Mashhoon effect is to exist for a Dirac particle, the tetrad field must be selected in a specific way for each reference frame.     

General chiral gauge theories on the lattice

We still extend the large class of Dirac operators decribing massless fermions on the lattice found recently, only requiring that such operators decompose into Weyl operators. After deriving general relations and constructions of operators, we study the basis representations of the chiral projections. We then investigate correlation functions of Weyl fermions for any value of the index, stressing the related conditions for basis transformations and their consequences, and getting the precise behaviors under gauge transformations and CP transformations. Various further developments include considerations of the explicit form of the effective action and of a representation of the general correlation functions in terms of alternating multilinear forms. For comparison we also consider gauge-field variations and their respective applications. Finally we compare with continuum perturbation theory.     

Finite 1D-Lattice Physics as Induced by Dirac–Markov Operators

Finite 1D-lattice physics as induced by Dirac operators was examined. We identified the Dirac operators with Bi-Graded Markovian matrices. The Dirac operators dictate the time evolution of states in Markovian-like processes. By applying these Dirac operators to finite 1D-lattices, we find differences between the vacuum physical spinorial states over lattices with an even number of sites as compared to an odd number of sites. Solitonic states that are created by particle pairing appear on lattices with an even number of sites. On lattices with an odd number of sites, we find global solitonic states and global spin wave states, as well as a global steady oscillation of the spinorial wave function. This demonstrates how the lattice world, in a few number of sites, dramatically affects the vacuum physical states. All these vacuum states can be realized as entangled local particles over the lattice.     

Energy states of colored particle in a chromomagnetic field

The unitary transformation which diagonalizes the squared Dirac equation in a constant chromomagnetic field is found. Applying this transformation, we find the eigenfunctions of the diagonalized Hamiltonian, that describes the states with a definite value of energy, and we call them energy states. It is pointed out that the energy states are determined by the color interaction term of the particle with the background chromofield, and this term is responsible for the splitting of the energy spectrum. We construct supercharge operators for the diagonal Hamiltonian that ensure the superpartner property of the energy states. PACS 03.65.-w An erratum to this article can be found at     

Generally covariant quantization and the Dirac field

Canonical Hamiltonian field theory in curved spacetime is formulated in a manifestly covariant way. Second quantization is achieved invoking a correspondence principle between the Poisson bracket of classical fields and the commutator of the corresponding quantum operators. The Dirac theory is investigated and it is shown that, in contrast to the case of bosonic fields, in curved spacetime, the field momentum does not coincide with the generators of spacetime translations. The reason is traced back to the presence of second class constraints occurring in Dirac theory. Further, it is shown that the modification of the Dirac Lagrangian by a surface term leads to a momentum transfer between the Dirac field and the gravitational background field, resulting in a theory that is free of constraints, but not manifestly hermitian.     

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.     

Z2 topological term, the global anomaly, and the two-dimensional symplectic symmetry class of Anderson localization

We discuss, for a two-dimensional Dirac Hamiltonian with a random scalar potential, the presence of a Z2 topological term in the nonlinear sigma model encoding the physics of Anderson localization in the symplectic symmetry class. The Z2 topological term realizes the sign of the Pfaffian of a family of Dirac operators. We compute the corresponding global anomaly, i.e., the change in the sign of the Pfaffian by studying a spectral flow numerically. This Z2 topological effect can be relevant to graphene when the impurity potential is long ranged and, also, to the two-dimensional boundaries of a three-dimensional lattice model of Z2 topological insulators in the symplectic symmetry class.     

On the geometrization of matter by exotic smoothness

In this paper we discuss the question how matter may emerge from space. For that purpose we consider the smoothness structure of spacetime as underlying structure for a geometrical model of matter. For a large class of compact 4-manifolds, the elliptic surfaces, one is able to apply the knot surgery of Fintushel and Stern to change the smoothness structure. The influence of this surgery to the Einstein–Hilbert action is discussed. Using the Weierstrass representation, we are able to show that the knotted torus used in knot surgery is represented by a spinor fulfilling the Dirac equation and leading to a Dirac term in the Einstein–Hilbert action. For sufficient complicated links and knots, there are “connecting tubes” (graph manifolds, torus bundles) which introduce an action term of a gauge field. Both terms are genuinely geometrical and characterized by the mean curvature of the components. We also discuss the gauge group of the theory to be U(1) × SU(2) ×?SU(3).     

Dirac Induction for Loop Groups

Using a coset version of the cubic Dirac operators for affine Lie algebras, we give an algebraic construction of the Dirac induction homomorphism for loop group representations. With this, we prove a homogeneous generalization of the Weyl–Kac character formula and show compatibility with Dirac induction for compact Lie groups.     

Versal Deformations of a Dirac Type Differential Operator

Abstract

If we are given a smooth differential operator in the variable x ∈ R/2πZ, its normal form, as is well known, is the simplest form obtainable by means of the Diff(S ¹)-group action on the space of all such operators. A versal deformation of this operator is a normal form for some parametric infinitesimal family including the operator. Our study is devoted to analysis of versal deformations of a Dirac type differential operator using the theory of induced Diff(S ¹)-actions endowed with centrally extended Lie-Poisson brackets. After constructing a general expression for tranversal deformations of a Dirac type differential operator, we interpret it via the Lie-algebraic theory of induced Diff(S ¹)-actions on a special Poisson manifold and determine its generic moment mapping. Using a Marsden-Weinstein reduction with respect to certain Casimir generated distributions, we describe a wide class of versally deformed Dirac type differential operators depending on complex parameters.     

Wess-Zumino terms and the geometry of the determinant line bundle

The determinant line bundle L of a family of Dirac operators coupled to Yang-Mills (YM) in any dimension is constructed from the corresponding Wess-Zumino (WZ) term. The equivalence between the algebraic and topological approaches to anomalies is established by straightforward computation. As a by-product the first Chern class of L is expressed through the WZ term and the integrated anomaly is explicitly seen to play the role of a functioonal magnetic field on the gauge-orbit space.     

Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum

In this work we apply the Dirac method in order to obtain the classical relations for a particle on an ellipsoid. We also determine the quantum mechanical form of these relations by using Dirac quantization. Then by considering the canonical commutation relations between the position and momentum operators in terms of curved coordinates, we try to propose the suitable representations for momentum operator that satisfy the obtained commutators between position and momentum in Euclidean space. We see that our representations for momentum operators are the same as geometric one.     

Axial anomalies and index theorems on open spaces

Using an approach inspired by the theory of the anomalous divergence of the axial vector current, we derive trace formulas for the resolvents of Dirac operators on open spaces of odd dimension. These formulas readily yield index theorems for these operators. As applications we determine the index of the Dirac operator for a particle of arbitrary isospin in the background field of a static system of SU(2) monopoles; and we find formulas in essentially closed form for certain determinants involving these operators.This work is supported in part through funds provided by the U.S. Department of Energy (DOE) under contract EY-76-C-02-3069     

Induced Rigid String Action from Fermions

By adding the Dirac action on the world sheet, an effective action is obtained by integrating over the four-dimensional fermion fields pulled back to the world sheet. This action consists of the Nambu–Goto area term with a right dimensionful constant in front, extrinsic curvature action, and the topological Euler characteristic term. The divergent coefficients in front of these terms are absorbed in the rigid string action without fermions.     

The analysis of elliptic families

In this paper we specialize the results obtained in [BF1] to the case of a family of Dirac operators. We first calculate the curvature of the unitary connection on the determinant bundle which we introduced in [BF1].We also calculate the odd Chern forms of Quillen for a family of self-adjoint Dirac operators and give a simple proof of certain results of Atiyah-Patodi-Singer on êta invariants.We finally give a heat equation proof of the holonomy theorem, in the form suggested by Witten [W 1, 2].     

Dirac structures of integrable evolution equations

An algebraic theory of Dirac structures is presented, enclosing finite-dimensional pre-symplectic and Poisson structures, as well as their infinite-dimensional analogs determined by local operators. The generalized Lenard scheme of integrability is considered together with examples of its action.     

A Hardy’s Uncertainty Principle Lemma in Weak Commutation Relations of Heisenberg-Lie Algebra

In this article we consider linear operators satisfying a generalized commutation relation of a type of the Heisenberg-Lie algebra. It is proven that a generalized inequality of the Hardy’s uncertainty principle lemma follows. Its applications to time operators and abstract Dirac operators are also investigated.