Related papers: On Dirac Operators and Spectral Geometry of Compac…
Gamma matrices for quantum Minkowski spaces are found. The invariance of the corresponding Dirac operator is proven. We introduce momenta for spin 1/2 particles and get (in certain cases) formal solutions of the Dirac equation.
Using the example of a Dirac particle in external static fields, Dirac theory is reformulated as a one-particle quantum theory in the space of normalized two-component spinors. In this formulation, the Dirac operator ``splits'' into two…
We give a direct link between description of Dirac particles in the abstract framework of unitary representation of the Poincar\'e group and description with the help of the Dirac equation. In this context we discuss in detail the spin…
An explicit construction is given of field operators satisfying the free Dirac equation. The quantum expectation of these field operators forms a spinor which satisfies the original Dirac equation. The current operators are defined as pair…
Some aspects of Dirac spinors are resumed and studied in order to interpret mathematically the P and T operations in a gravitational field.
In this lecture I will report on some recent progress in understanding the relation of Dirac operators on Clifford modules over an even-dimensional closed Riemannian manifold $M$\ and (euclidean) Einstein-Yang-Mills-Higgs models.
We study the spectral geometry of the quantum projective plane CP^2_q, a deformation of the complex projective plane CP^2, the simplest example of a spin^c manifold which is not spin. In particular, we construct a Dirac operator D which…
It is shown that the spin operator can be described by an algebra which is in between so(3) and e(2). Relativistic version of the singlet state for two Dirac electrons is discussed. It is shown that a measure of massless particle's…
Connection of the invariant Dirac equation over the de Sitter space with irreducible representations of the de Sitter group is ascertained. The set of solutions of this equation is obtained in the form of the product of two different…
A general theory of quantum spinor structures on quantum spaces is presented, within the conceptual framework of the formalism of quantum principal bundles. Quantum analogs of all basic objects of the classical theory are constructed and…
We introduce a two parameter family of Dirac operators on quantum SU(2) and analyse their properties from the point of view of non-commutative metric geometry. It is shown that these Dirac operators give rise to compact quantum metric…
We determine what should correspond to the Dirac operator on certain quantized hermitian symmetric spaces and what its properties are. A new insight into the quantized wave operator is obtained.
The relationship between spinors and Clifford (or geometric) algebra has long been studied, but little consistency may be found between the various approaches. However, when spinors are defined to be elements of the even subalgebra of some…
This paper is a mixture of expository material and current research material. Among new results are examples of generalised harmonic spinors and their gauged version, the generalised Seiberg-Witten equations.
The goal of this paper is to introduce a class of operators, which we call quantum Dirac type operators on a noncommutative sphere, by a gluing construction from copies of noncommutative disks, subject to an appropriate local boundary…
We give a construction of a Dirac operator on a quantum group based on any simple Lie algebra of classical type. The Dirac operator is an element in the vector space $U_q(\g) \otimes \mathrm{cl}_q(\g)$ where the second tensor factor is a…
The spectral action on the equivariant real spectral triple over \A(SU_q(2)) is computed explicitly. Properties of the differential calculus arising from the Dirac operator are studied and the results are compared to the commutative case of…
We find and classify possible equivariant spin structures with Dirac operators on the noncommutative torus, proving that similarly as in the classical case the spectrum of the Dirac operator depends on the spin structure.
We explain the notion of minimality for an equivariant spectral triple and show that the triple for the quantum SU(2) group constructed by Chakraborty and Pal is minimal. We also give a decomposition of the spectral triple constructed by…
Symplectic spinors form an infinite-rank vector bundle. Dirac operators on this bundle were constructed recently by K.~Habermann. Here we study the spectral geometry aspects of these operators. In particular, we define the associated…