scholarly journals PRODUCT OF REAL SPECTRAL TRIPLES

2011 ◽  
Vol 08 (08) ◽  
pp. 1833-1848 ◽  
Author(s):  
LUDWIK DĄBROWSKI ◽  
GIACOMO DOSSENA
Keyword(s):  
Dirac Operator ◽  
Clifford Algebra ◽  
Finite Dimension ◽  
Spectral Triples ◽  
Inequivalent Representations ◽  
Real Spectral Triples

We construct the product of real spectral triples of arbitrary finite dimension (and arbitrary parity) taking into account the fact that in the even case there are two possible real structures, in the odd case there are two inequivalent representations of the gamma matrices (Clifford algebra), and in the even-even case there are two natural candidates for the Dirac operator of the product triple.

scholarly journals CATEGORIFIED NONCOMMUTATIVE MANIFOLDS

2009 ◽  
Vol 24 (15) ◽  
pp. 2802-2819 ◽  
Author(s):  
R. A. DAWE MARTINS
Keyword(s):  
Dirac Operator ◽  
Noncommutative Geometry ◽  
Tangent Bundle ◽  
Mass Matrix ◽  
Fermion Mass ◽  
Spectral Triples ◽  
Noncommutative Manifolds ◽  
Real Spectral Triples ◽  
Fermion Mass Matrix

We construct a noncommutative geometry with generalised 'tangent bundle' from Fell bundle C*-categories (E) beginning by replacing pair groupoid objects (points) with objects in E. This provides a categorification of a certain class of real spectral triples where the Dirac operator D is constructed from morphisms in a category. Applications for physics include quantization via the tangent groupoid and new constraints on D finite (the fermion mass matrix).

NONUNITAL SPECTRAL TRIPLES ASSOCIATED TO DEGENERATE METRICS

2004 ◽  
Vol 16 (01) ◽  
pp. 125-146
Author(s):  
A. RENNIE
Keyword(s):  
Finite Dimension ◽  
Index Theorem ◽  
Smooth Manifolds ◽  
Spectral Triples ◽  
Local Index ◽  
Dimension Spectrum ◽  
Spectrum Hypothesis ◽  
Local Index Theorem ◽  
K Theory

We show that one can define (p,∞)-summable spectral triples using degenerate metrics on smooth manifolds. Furthermore, these triples satisfy Connes–Moscovici's discrete and finite dimension spectrum hypothesis, allowing one to use the Local Index Theorem [1] to compute the pairing with K-theory. We demonstrate this with a concrete example.

scholarly journals On Twisting Real Spectral Triples by Algebra Automorphisms

2016 ◽  
Vol 106 (11) ◽  
pp. 1499-1530 ◽  
Author(s):  
Giovanni Landi ◽  
Pierre Martinetti
Keyword(s):  
Spectral Triples ◽  
Real Spectral Triples

scholarly journals CONNES DISTANCE BY EXAMPLES: HOMOTHETIC SPECTRAL METRIC SPACES

2012 ◽  
Vol 24 (09) ◽  
pp. 1250027 ◽  
Author(s):  
JEAN-CHRISTOPHE WALLET
Keyword(s):  
Dirac Operator ◽  
Harmonic Oscillator ◽  
Infinite Number ◽  
Metric Spaces ◽  
Connected Components ◽  
Spectral Triples ◽  
Metric Properties ◽  
Spectral Distance ◽  
Physics Literature ◽  
Moyal Plane

We study metric properties stemming from the Connes spectral distance on three types of non-compact non-commutative spaces which have received attention recently from various viewpoints in the physics literature. These are the non-commutative Moyal plane, a family of harmonic Moyal spectral triples for which the Dirac operator squares to the harmonic oscillator Hamiltonian and a family of spectral triples with the Dirac operator related to the Landau operator. We show that these triples are homothetic spectral metric spaces, having an infinite number of distinct pathwise connected components. The homothetic factors linking the distances are related to determinants of effective Clifford metrics. We obtain, as a by-product, new examples of explicit spectral distance formulas. The results are discussed in detail.

scholarly journals Structures of spinors fiber bundles with special relativity of Dirac operator using the Clifford algebra

2021 ◽  
Vol 54 (1) ◽  
pp. 410-424
Author(s):  
Yousif Atyeib Ibrahim Hassan
Keyword(s):  
Dirac Operator ◽  
Clifford Algebra ◽  
Special Relativity ◽  
Riemannian Manifolds ◽  
Clifford Algebras ◽  
Cohomology Theory ◽  
Fiber Bundles

Abstract The purpose of this article is to demonstrate how to use the mathematics of spinor bundles and their category. We have used the methods of principle fiber bundles obey thorough solid harmonic treatment of pseudo-Riemannian manifolds and spinor structures with Clifford algebras, which couple with Dirac operator to study important applications in cohomology theory.

scholarly journals Riemann Boundary Value Problem for Triharmonic Equation in Higher Space

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Longfei Gu
Keyword(s):  
Boundary Value Problem ◽  
Dirac Operator ◽  
Clifford Algebra ◽  
Unique Solution ◽  
Boundary Value ◽  
Riemann Boundary Value Problem ◽  
Riemann Boundary

We mainly deal with the boundary value problem for triharmonic function with value in a universal Clifford algebra:Δ3[u](x)=0,x∈Rn∖∂Ω,u+(x)=u-(x)G(x)+g(x),x∈∂Ω,(Dju)+(x)=(Dju)-(x)Aj+fj(x),x∈∂Ω,u(∞)=0, where(j=1,…,5)  ∂Ωis a Lyapunov surface inRn,D=∑k=1nek(∂/∂xk)is the Dirac operator, andu(x)=∑AeAuA(x)are unknown functions with values in a universal Clifford algebraCl(Vn,n).Under some hypotheses, it is proved that the boundary value problem has a unique solution.

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