scholarly journals A New Solution to the Structure Equation in Noncommutative Spacetime

2014 ◽  
Vol 24 (1) ◽  
pp. 21 ◽  
Author(s):  
Nguyen Ai Viet
Keyword(s):  
Noncommutative Geometry ◽  
Minimal Set ◽  
Full Spectrum ◽  
Physical Content ◽  
Common Foundation ◽  
Klein Theory ◽  
Structure Equations ◽  
Noncommutative Spacetime ◽  
The Common ◽  
Kaluza Klein

In this paper, starting from the common foundation of Connes' noncommutative geometry ( NCG)\cite{Connes1, Connes2, CoLo, Connes3}, various possible alternatives in the formulation of atheory of gravity in noncommutative spacetime are discussed indetails. The diversity in the final physical content of the theory is shown to be the consequence of the arbitrary choices in each construction steps. As an alternative in the last step, when the structure equations are to be solved, a minimal set of constraints on the torsion and connection is found to determine all the geometric notions in terms of the metric. In the Connes-Lott model of noncommutative spacetime, in order to keep the full spectrum of the discretized Kaluza-Klein theory \cite{VW2}, it is necessary to include the torsion in the generalized Einstein-Hilbert-Cartan action.

scholarly journals A DISCRETIZED VERSION OF KALUZA–KLEIN THEORY WITH TORSION AND MASSIVE FIELDS

1996 ◽  
Vol 11 (13) ◽  
pp. 2403-2418 ◽  
Author(s):  
NGUYEN AI VIET ◽  
KAMESHWAR C. WALI
Keyword(s):  
Noncommutative Geometry ◽  
Vector Fields ◽  
Complex Structure ◽  
Scalar Fields ◽  
Type Model ◽  
Internal Space ◽  
Tensor Fields ◽  
Fifth Dimension ◽  
Klein Theory ◽  
Kaluza Klein

We consider an internal space of two discrete points in the fifth dimension of the Kaluza–Klein theory by using the formalism of noncommutative geometry — developed in a previous paper1 — of a spacetime supplemented by two discrete points. With the non-vanishing internal torsion two-form there are no constraints implied on the vielbeins. The theory contains a pair of tensor fields, a pair of vector fields and a pair of scalar fields. Using the generalized Cartan structure equation we are able to uniquely determine not only the Hermitian and metric-compatible connection one-forms, but also the nonvanishing internal torsion two-form in terms of vielbeins. The resulting action has a rich and complex structure, a particular feature being the existence of massive modes. Thus the nonvanishing internal torsion generates a Kaluza–Klein type model with zero and massive modes.

scholarly journals DYNAMICS OF THE SCALAR FIELD IN FIVE-DIMENSIONAL KALUZA–KLEIN THEORY

2004 ◽  
Vol 19 (27) ◽  
pp. 4671-4685 ◽  
Author(s):  
INGUNN KATHRINE WEHUS ◽  
FINN RAVNDAL
Keyword(s):  
Scalar Field ◽  
Ad Hoc ◽  
Differential Forms ◽  
Casimir Energy ◽  
Jordan Frame ◽  
Physical Content ◽  
Four Dimensions ◽  
Compact Dimension ◽  
Klein Theory ◽  
Kaluza Klein

Using the language of differential forms, the Kaluza–Klein theory in 4+1 dimensions is derived. This theory unifies electromagnetic and gravitational interactions in four dimensions when the extra space dimension is compactified. Without any ad hoc assumptions about the five-dimensional metric, the theory also contains a scalar field coupled to the other fields. By a conformal transformation the theory is transformed from the Jordan frame to the Einstein frame where the physical content is more manifest. Including a cosmological constant in the five-dimensional formulation, it is seen to result in an exponential potential for the scalar field in four dimensions. A similar potential is also found from the Casimir energy in the compact dimension. The resulting scalar field dynamics mimics realistic models recently proposed for cosmological quintessence.

scholarly journals NONCOMMUTATIVE GEOMETRY AND A DISCRETIZED VERSION OF KALUZA-KLEIN THEORY WITH A FINITE FIELD CONTENT

1996 ◽  
Vol 11 (03) ◽  
pp. 533-551 ◽  
Author(s):  
NGUYEN AI VIET ◽  
KAMESHWAR C. WALI
Keyword(s):  
Finite Field ◽  
Noncommutative Geometry ◽  
Dimensional Space ◽  
Complex Structure ◽  
Special Cases ◽  
Klein Theory ◽  
Dimensional Space Time ◽  
The Rich ◽  
Torsion Free ◽  
Kaluza Klein

We consider a four-dimensional space-time supplemented by two discrete points assigned to a Z2-algebraic structure and develop the formalism of noncommutative geometry. By setting up a generalized vielbein, we study the metric structure. Metric-compatible torsion-free connection defines a unique finite field content in the model and leads to a discretized version of Kaluza-Klein theory. We study some special cases of this model that illustrate the rich and complex structure with massive modes and the possible presence of a cosmological constant.

scholarly journals Non-Abelian gauge fields as components of gravity in the discretized Kaluza–Klein theory

2017 ◽  
Vol 32 (18) ◽  
pp. 1750095
Author(s):  
Ai Viet Nguyen ◽  
Tien Du Pham
Keyword(s):  
Gauge Group ◽  
Noncommutative Geometry ◽  
The Other ◽  
Gauge Fields ◽  
Strong Interactions ◽  
Abelian Gauge ◽  
Gauge Invariant ◽  
First Case ◽  
Klein Theory ◽  
Kaluza Klein

Discretized Kaluza–Klein theory in [Formula: see text] spacetime can be constructed based on the concepts of noncommutative geometry. In this paper, we show that it is possible to incorporate the non-Abelian gauge fields in this framework. The generalized Hilbert–Einstein action is gauge invariant only in two cases. In the first case, the gauge group must be Abelian on one sheet of spacetime and non-Abelian on the other one. In the second case, the gauge group must be the same on two sheets of spacetime. Actually, the theories of electroweak and strong interactions can fit into these two cases.

A NONCOMMUTATIVE EXTENSION OF GRAVITY

1994 ◽  
Vol 03 (01) ◽  
pp. 221-224 ◽  
Author(s):  
J. MADORE ◽  
J. MOURAD
Keyword(s):  
Tensor Product ◽  
Noncommutative Geometry ◽  
Commutative Algebra ◽  
Internal Space ◽  
Complex Matrices ◽  
Noncommutative Algebra ◽  
Algebra Of Functions ◽  
Klein Theory ◽  
Hilbert Action ◽  
Kaluza Klein

The commutative algebra of functions on a manifold is extended to a noncommutative algebra by considering its tensor product with the algebra of n×n complex matrices. Noncommutative geometry is used to formulate an extension of the Einstein-Hilbert action. The result is shown to be equivalent to the usual Kaluza-Klein theory with the manifold SUn as an internal space, in a truncated approximation.

Supersymmetry: Kaluza-Klein theory, anomalies, and superstrings

1985 ◽  
Vol 146 (8) ◽  
pp. 655 ◽  
Author(s):  
I.Ya. Aref'eva ◽  
I.V. Volovich
Keyword(s):  
Klein Theory ◽  
Kaluza Klein

Euler-Poincaré lagrangians and Kaluza-Klein theory

1987 ◽  
Vol 189 (1-2) ◽  
pp. 96-98 ◽  
Author(s):  
M. Arik ◽  
T. Dereli
Keyword(s):  
Klein Theory ◽  
Kaluza Klein

PHYSICAL ASPECTS OF SOLITONS IN (4+1) GRAVITY

1995 ◽  
Vol 04 (05) ◽  
pp. 639-659 ◽  
Author(s):  
ANDREW BILLYARD ◽  
PAUL S. WESSON ◽  
DIMITRI KALLIGAS
Keyword(s):  
Physical Properties ◽  
Extra Dimension ◽  
Dimensional Case ◽  
Schwarzschild Solution ◽  
Circular Orbits ◽  
Know How ◽  
Fifth Dimension ◽  
Klein Theory ◽  
Limiting Case ◽  
Kaluza Klein

The augmentation of general relativity’s spacetime by one or more dimensions is described by Kaluza-Klein theory and is within testable limits. Should an extra dimension be observable and significant, it would be beneficial to know how physical properties would differ from “conventional” relativity. In examining the class of five-dimensional solutions analogous to the four-dimensional Schwarzschild solution, we examine where the origin to the system is located and note that it can differ from the four-dimensional case. Furthermore, we study circular orbits and find that the 5D case is much richer; photons can have stable circular orbits in some instances, and stable orbits can exist right to the new origin in others. Finally, we derive both gravitational and inertial masses and find that they do not generally agree, although they can in a limiting case. For all three examinations, it is possible to obtain the four-dimensional results in one limiting case, that of the Schwarzschild solution plus a flat fifth dimension, and that the differences between 4D and 5D occur when the fifth dimension obtains any sort of significance.

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