scholarly journals DIFFERENTIAL CALCULUS ON FUZZY SPHERE AND SCALAR FIELD

1998 ◽  
Vol 13 (19) ◽  
pp. 3235-3243 ◽  
Author(s):  
URSULA CAROW-WATAMURA ◽  
SATOSHI WATAMURA
Keyword(s):  
Scalar Field ◽  
Dirac Operator ◽  
Differential Calculus ◽  
Spectral Triple ◽  
Fuzzy Sphere ◽  
Alternative Possibility ◽  
Operator Ordering

We find that there is an alternative possibility to define the chirality operator on the fuzzy sphere, due to the ambiguity of the operator ordering. Adopting this new chirality operator and the corresponding Dirac operator, we define Connes' spectral triple on the fuzzy sphere and the differential calculus. The differential calculus based on this new spectral triple is simplified considerably. Using this formulation the action of the scalar field is derived.

scholarly journals Missing the point in noncommutative geometry

Synthese ◽  
2021 ◽  
Author(s):  
Nick Huggett ◽  
Fedele Lizzi ◽  
Tushar Menon
Keyword(s):  
Scalar Field ◽  
Noncommutative Geometry ◽  
Smooth Manifold ◽  
Operational Definition ◽  
Spectral Triple ◽  
Fundamental Quantity

AbstractNoncommutative geometries generalize standard smooth geometries, parametrizing the noncommutativity of dimensions with a fundamental quantity with the dimensions of area. The question arises then of whether the concept of a region smaller than the scale—and ultimately the concept of a point—makes sense in such a theory. We argue that it does not, in two interrelated ways. In the context of Connes’ spectral triple approach, we show that arbitrarily small regions are not definable in the formal sense. While in the scalar field Moyal–Weyl approach, we show that they cannot be given an operational definition. We conclude that points do not exist in such geometries. We therefore investigate (a) the metaphysics of such a geometry, and (b) how the appearance of smooth manifold might be recovered as an approximation to a fundamental noncommutative geometry.

scholarly journals Effective action and phase transitions of scalar field on the fuzzy sphere

2013 ◽  
Vol 88 (6) ◽  
Author(s):  
Alexios P. Polychronakos
Keyword(s):  
Phase Transitions ◽  
Scalar Field ◽  
Effective Action ◽  
Fuzzy Sphere

scholarly journals On the Koszul formula in noncommutative geometry

2020 ◽  
Vol 32 (10) ◽  
pp. 2050032 ◽  
Author(s):  
Jyotishman Bhowmick ◽  
Debashish Goswami ◽  
Giovanni Landi
Keyword(s):  
Scalar Curvature ◽  
Noncommutative Geometry ◽  
Spectral Triple ◽  
Fuzzy Sphere ◽  
Civita Connection ◽  
Canonical Metric ◽  
Levi Civita Connection

We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul formula, we show that our Levi-Civita connection is a bimodule connection. We construct a spectral triple on a fuzzy sphere and compute the scalar curvature for the Levi-Civita connection associated to a canonical metric.

scholarly journals Simulating the scalar field on the fuzzy sphere

2005 ◽  
Author(s):  
Fernando Garcia Flores ◽  
Denjoe O'Connor ◽  
X. Martin
Keyword(s):  
Scalar Field ◽  
Fuzzy Sphere

scholarly journals "SCHWINGER MODEL" ON THE FUZZY SPHERE

2010 ◽  
Vol 25 (37) ◽  
pp. 3151-3167 ◽  
Author(s):  
E. HARIKUMAR
Keyword(s):  
Partition Function ◽  
Field Strength ◽  
Dirac Operator ◽  
Path Integral ◽  
Integral Method ◽  
Gauge Fields ◽  
Fuzzy Sphere ◽  
Schwinger Model ◽  
Spinor Fields ◽  
Path Integral Method

In this paper, we construct a model of spinor fields interacting with specific gauge fields on the fuzzy sphere and analyze the chiral symmetry of this "Schwinger model". In constructing the theory of gauge fields interacting with spinors on the fuzzy sphere, we take the approach that the Dirac operator Dq on the q-deformed fuzzy sphere [Formula: see text] is the gauged Dirac operator on the fuzzy sphere. This introduces interaction between spinors and specific one-parameter family of gauge fields. We also show how to express the field strength for this gauge field in terms of the Dirac operators Dq and D alone. Using the path integral method, we have calculated the 2n-point functions of this model and show that, in general, they do not vanish, reflecting the chiral non-invariance of the partition function.

Simulating the scalar field using the fuzzy sphere

2003 ◽  
Vol 18 (33n35) ◽  
pp. 2389-2396 ◽  
Author(s):  
XAVIER MARTIN
Keyword(s):  
Scalar Field ◽  
Numerical Scheme ◽  
Approximation Scheme ◽  
Fuzzy Sphere ◽  
Continuous Space ◽  
The Real ◽  
Matrix Algebras ◽  
Real Scalar ◽  
Fuzzy Space ◽  
Commutative Matrix

Fuzzy spaces provide a new approximation scheme using (non–commutative) matrix algebras to approximate the algebra of function of the continuous space. This paper describes how to implement a numerical scheme based on a fuzzy space approximation. In this first attempt, the simplest fuzzy space and field theory, respectively the fuzzy two–sphere and the real scalar field, are used to simulate the real scalar field on the plane. Along the way, this method is compared to its traditional lattice discretisation equivalent.

scholarly journals The dirac operator on the fuzzy sphere

1995 ◽  
Vol 33 (2) ◽  
pp. 171-181 ◽  
Author(s):  
H. Grosse ◽  
P. Prešnajder
Keyword(s):  
Dirac Operator ◽  
Fuzzy Sphere

scholarly journals Dirac operator on the q-deformed fuzzy sphere and its spectrum

2006 ◽  
Vol 2006 (09) ◽  
pp. 037-037 ◽  
Author(s):  
E Harikumar ◽  
Amilcar R Queiroz ◽  
Paulo Teotonio-Sobrinho
Keyword(s):  
Dirac Operator ◽  
Fuzzy Sphere
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