A NONCOMMUTATIVE EXTENSION OF GRAVITY

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.

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

International Journal of Modern Physics A ◽

10.1142/s0217751x96001206 ◽

1996 ◽

Vol 11 (13) ◽

pp. 2403-2418 ◽

Cited By ~ 7

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.

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Spherically symmetric T-solution of the equations of 5-dimensional Kaluza–Klein theory

Journal of Physics and Electronics ◽

10.15421/332020 ◽

2020 ◽

Vol 28 (2) ◽

pp. 51-56

Author(s):

V. D. Gladush

Keyword(s):

Cauchy Problem ◽

Cosmological Model ◽

Configuration Space ◽

Jacobi Equation ◽

Spherically Symmetric ◽

Klein Theory ◽

The Cauchy Problem ◽

Hamilton Jacobi Equation ◽

Hilbert Action ◽

Kaluza Klein

A geometrodynamical approach to the five-dimensional (5D) spherically symmetric cosmological model in the Kaluza–Klein theory is constructed. After dimensional reduction, the 5D Hilbert action is reduced to the Einstein form describing the gravitational, electromagnetic, and scalar interacting fields. The subsequent transition to the configuration space leads to the supermetric and the Einstein–Hamilton–Jacobi equation, with the help of which the trajectories in the configuration space are found. Then the evolutionary coordinate is restored, and the Cauchy problem is solved to find the time dependence of the metric and fields. The configuration corresponds to a cosmological model of the Kantovsky–Sachs type, which has a hypercylinder topology and includes scalar and electromagnetic fields with contact interaction.

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FUZZY COMPLEX GRASSMANNIAN SPACES AND THEIR STAR PRODUCTS

International Journal of Modern Physics A ◽

10.1142/s0217751x03014113 ◽

2003 ◽

Vol 18 (11) ◽

pp. 1935-1958 ◽

Cited By ~ 18

Author(s):

BRIAN P. DOLAN ◽

OLIVER JAHN

Keyword(s):

Commutative Algebra ◽

Star Product ◽

Matrix Multiplication ◽

Noncommutative Algebra ◽

Infinite Dimensional ◽

Complex Projective Spaces ◽

Finite Dimensional ◽

Algebra Of Functions ◽

Finite Sum ◽

Complex Projective

We derive an explicit expression for an associative star product on noncommutative versions of complex Grassmannian spaces, in particular for the case of complex two-planes. Our expression is in terms of a finite sum of derivatives. This generalizes previous results for complex projective spaces and gives a discrete approximation for the Grassmannians in terms of a noncommutative algebra, represented by matrix multiplication in a finite-dimensional matrix algebra. The matrices are restricted to have a dimension which is precisely determined by the harmonic expansion of functions on the commutative Grassmannian, truncated at a finite level. In the limit of infinite-dimensional matrices we recover the commutative algebra of functions on the complex Grassmannians.

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MASSLESS SCALAR FIELDS IN NON-ABELIAN KALUZA-KLEIN THEORY

International Journal of Modern Physics A ◽

10.1142/s0217751x94000248 ◽

1994 ◽

Vol 09 (04) ◽

pp. 507-515 ◽

Cited By ~ 1

Author(s):

M. ARIK ◽

V. GABAY

Keyword(s):

Cosmological Constant ◽

Direct Product ◽

Scalar Fields ◽

Gauge Fields ◽

Internal Space ◽

Massless Scalar ◽

Klein Theory ◽

Euler Form ◽

Odd Dimensions ◽

Kaluza Klein

We investigate the presence of massless scalar fields in a Kaluza—Klein theory based on a dimensionally continued Euler-form action. We show that massless scalar fields exist provided that the internal space is a direct product of two irreducible manifolds. The condition of a vanishing effective four-dimensional cosmological constant and the presence of a graviton, gauge fields and massless scalar fields can be satisfied if both irreducible manifolds have odd dimensions and the sum of these dimensions is equal to the dimension of the Euler form.

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KALUZA-KLEIN THEORY

International Journal of Modern Physics A ◽

10.1142/s0217751x86000022 ◽

1986 ◽

Vol 01 (01) ◽

pp. 1-37 ◽

Cited By ~ 7

Author(s):

J. STRATHDEE

Keyword(s):

Internal Space ◽

Coset Space ◽

Spontaneous Compactification ◽

Zero Modes ◽

Coset Spaces ◽

Klein Theory ◽

Recent Developments ◽

Ground State Geometry ◽

Energy Physics ◽

Kaluza Klein

Recent developments in Kaluza-Klein theory are reviewed. Starting with the concept of spontaneous compactification, the problem of determining the ground state geometry and its symmetry is discussed. While it is generally believed that only the zero modes can be relevant for low energy physics, it is possible in some cases to deduce the entire excitation spectrum. This is true when the internal space is a coset space. A technique is described for setting up harmonic expansions on coset spaces. Consistency in chiral Kaluza-Klein theories demands freedom from both gauge and gravitational anomalies. General features of the chiral anomalies are reviewed.

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A New Solution to the Structure Equation in Noncommutative Spacetime

Communications in Physics ◽

10.15625/0868-3166/24/1/3606 ◽

2014 ◽

Vol 24 (1) ◽

pp. 21 ◽

Cited By ~ 1

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.

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

International Journal of Modern Physics A ◽

10.1142/s0217751x96000249 ◽

1996 ◽

Vol 11 (03) ◽

pp. 533-551 ◽

Cited By ~ 14

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.

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

Modern Physics Letters A ◽

10.1142/s021773231750095x ◽

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.

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