A perspective on non-commutative quantum gravity

2015 ◽  
Vol 12 (08) ◽  
pp. 1560021
Author(s):  
Rachel A. D. Martins
Keyword(s):  
Quantum Gravity ◽  
Mass Matrix ◽  
Spin Manifold ◽  
Fermion Mass ◽  
Spectral Triples ◽  
Bounded Linear ◽  
Topological Quantum ◽  
Novel Approach ◽  
Linear Maps ◽  
Einstein's Equivalence Principle

In this paper, we present some of the concepts underlying a program of non-commutative quantum gravity and recall some of the results. This program includes a novel approach to spectral triple categorification and also a precise connection between Fell bundles and Connes' non-commutative geometry. Motivated by topics in quantization of the non-commutative standard model and introduction of algebraic techniques and concepts into quantum gravity (following for example Crane, Baez and Barrett), we define spectral C*-categories, which are deformed spectral triples in a sense made precise. This definition gives to representations of a C*-category on a small category of Hilbert spaces and bounded linear maps, the interpretation of a topological quantum field theory. The construction passes two mandatory tests: (i) there is a classical limit theorem reproducing a Riemannian spin manifold manifesting Connes' and Schücker's non-commutative counterpart of Einstein's equivalence principle, and (ii) there is consistency with the experimental fermion mass matrix. We also present an algebra invariant taking the form of a partition function arising from a C*-bundle dynamical system in connection with C*-subalgebra theory.

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).

scholarly journals NONCOMMUTATIVE FERMION MASS MATRIX AND GRAVITY

2013 ◽  
Vol 28 (25) ◽  
pp. 1350120 ◽  
Author(s):  
RACHEL A. D. MARTINS
Keyword(s):  
Quantum Gravity ◽  
Dirac Operator ◽  
Cross Section ◽  
Mass Matrix ◽  
Fermion Mass ◽  
Algebraic Characterization ◽  
Small Cross Section ◽  
Fermion Mass Matrix ◽  
Noncommutative Quantum

The first part is an introductory description of a small cross-section of the literature on algebraic methods in nonperturbative quantum gravity with a specific focus on viewing algebra as a laboratory in which to deepen understanding of the nature of geometry. This helps to set the context for the second part, in which we describe a new algebraic characterization of the Dirac operator in noncommutative geometry and then use it in a calculation on the form of the fermion mass matrix. Assimilating and building on the various ideas described in the first part, the final part consists of an outline of a speculative perspective on (noncommutative) quantum spectral gravity. This is the second of a pair of papers so far on this project.

Topology knots and hierarchy problem

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Quantum Theory ◽  
String Theory ◽  
Quantum Gravity ◽  
Dimensional Space ◽  
Space Time ◽  
Elementary Particles ◽  
Hierarchy Problem ◽  
Topological Quantum ◽  
Dimensional Space Time ◽  
New Formula

Many approaches to quantum gravity consider the revision of the space-time geometry and the structure of elementary particles. One of the main candidates is string theory. It is possible that this theory will be able to describe the problem of hierarchy, provided that there is an appropriate Calabi-Yau geometry. In this paper we will proceed from the traditional view on the structure of elementary particles in the usual four-dimensional space-time. The only condition is that quarks and leptons should have a common emerging structure. When a new formula for the mass of the hierarchy is obtained, this structure arises from topological quantum theory and a suitable choice of dimensional units.

scholarly journals A SOLUTION TO THE STRONG CP PROBLEM TRANSFORMING THE THETA ANGLE TO THE KM CP-VIOLATING PHASE

2010 ◽  
Vol 25 (32) ◽  
pp. 5897-5911 ◽  
Author(s):  
JOSÉ BORDES ◽  
HONG-MO CHAN ◽  
SHEUNG TSUN TSOU
Keyword(s):  
Mass Matrix ◽  
Fermion Mass ◽  
Mass Hierarchy ◽  
Order Unity ◽  
Ckm Matrix ◽  
Cp Violations ◽  
Fermion Mass Matrix

It is shown that in the scheme with a rotating fermion mass matrix (i.e. one with a scale-dependent orientation in generation space) suggested earlier for explaining fermion mixing and mass hierarchy, the theta angle term in the QCD action of topological origin can be eliminated by chiral transformations, while giving still nonzero masses to all quarks. Instead, the effects of such transformations get transmitted by the rotation to the CKM matrix as the KM phase giving, for θ of order unity, a Jarlskog invariant typically of order 10-5, as experimentally observed. Strong and weak CP violations appear then as just two facets of the same phenomenon.

scholarly journals The spectrally bounded linear maps on operator algebras

2002 ◽  
Vol 150 (3) ◽  
pp. 261-271 ◽  
Author(s):  
Jianlian Cui ◽  
Jinchuan Hou
Keyword(s):  
Operator Algebras ◽  
Bounded Linear ◽  
Linear Maps ◽  
Spectrally Bounded

scholarly journals DEVELOPING THE FRAMED STANDARD MODEL

2012 ◽  
Vol 27 (17) ◽  
pp. 1250087 ◽  
Author(s):  
MICHAEL J. BAKER ◽  
JOSÉ BORDES ◽  
HONG-MO CHAN ◽  
TSOU SHEUNG TSUN
Keyword(s):  
Standard Model ◽  
Mass Matrix ◽  
Higgs Field ◽  
Fermion Mass ◽  
Ckm Matrix ◽  
Changing Scale ◽  
Rank One ◽  
Distinguishing Features ◽  
Fermion Mass Matrix

The framed standard model (FSM) suggested earlier, which incorporates the Higgs field and three fermion generations as part of the framed gauge theory (FGT) structure, is here developed further to show that it gives both quarks and leptons hierarchical masses and mixing matrices akin to what is experimentally observed. Among its many distinguishing features which lead to the above results are (i) the vacuum is degenerate under a global su(3) symmetry which plays the role of fermion generations, (ii) the fermion mass matrix is "universal," rank-one and rotates (changes its orientation in generation space) with changing scale μ, (iii) the metric in generation space is scale-dependent too, and in general nonflat, (iv) the theta-angle term in the quantum chromodynamics (QCD) action of topological origin gets transformed into the CP-violating phase of the Cabibbo–Kobayashi–Maskawa (CKM) matrix for quarks, thus offering at the same time a solution to the strong CP problem.

scholarly journals A Completely Bounded Noncommutative Choquet Boundary for Operator Spaces

2017 ◽  
Vol 2019 (22) ◽  
pp. 6819-6886 ◽  
Author(s):  
Raphaël Clouâtre ◽  
Christopher Ramsey
Keyword(s):  
Structural Properties ◽  
Extension Property ◽  
Operator Space ◽  
Operator Spaces ◽  
Unique Extension ◽  
Bounded Linear ◽  
C Algebra ◽  
Choquet Boundary ◽  
Linear Maps ◽  
Contractive Maps

Abstract We develop a completely bounded counterpart to the noncommutative Choquet boundary of an operator space. We show how the class of completely bounded linear maps is too large to accommodate our purposes. To overcome this obstacle, we isolate the subset of completely bounded linear maps admitting a dilation of the same norm that is multiplicative on the associated C*-algebra. We view such maps as analogs of the familiar unital completely contractive maps, and we exhibit many of their structural properties. Of particular interest to us are those maps that are extremal with respect to a natural dilation order. We establish the existence of extremals and show that they have a certain unique extension property. In particular, they give rise to *-homomorphisms that we use to associate to any representation of an operator space an entire scale of C*-envelopes. We conjecture that these C*-envelopes are all *-isomorphic and verify this in some important cases.

Preduals and Nuclear Operators Associated with Bounded, p-Convex, p-Concave and Positive p-Summing Operators

2007 ◽  
Vol 59 (3) ◽  
pp. 614-637 ◽  
Author(s):  
C. C. A. Labuschagne
Keyword(s):  
Banach Spaces ◽  
Positive Operators ◽  
Banach Lattices ◽  
Bounded Operators ◽  
Bounded Linear ◽  
Nuclear Operators ◽  
Linear Maps

AbstractWe use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of p-convex, p-concave and positive p-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators.

scholarly journals Circumstantial evidence for rotating mass matrix from fermion mass and mixing data

2003 ◽  
Vol 27 (2) ◽  
pp. 189-200 ◽  
Author(s):  
J. Bordes ◽  
Chan Hong-Mo ◽  
Tsou Sheung Tsun
Keyword(s):  
Mass Matrix ◽  
Circumstantial Evidence ◽  
Fermion Mass
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