arXiv subjects

J. Madore

Publications and source records attributed to J. Madore.

At least 37 records · Page 2Linked to original sources

arXiv1999

Finite Field Theory on Noncommutative Geometries

The propagator is calculated on a noncommutative version of the flat plane and the Lobachevsky plane with and without an extra (euclidean) time parameter. In agreement with the general idea of noncommutative geometry it is found that the limit when the two `points' coincide is finite and diverges only when the geometry becomes commutative. The flat 4-dimensional case is also considered. This is at the moment less interesting since there has been no curved case developed with which it can be compared.

hep-th↗

arXiv1998

The Geometry of a $q$-Deformed Phase Space

The geometry of the $q$-deformed line is studied. A real differential calculus is introduced and the associated algebra of forms represented on a Hilbert space. It is found that there is a natural metric with an associated linear connection which is of zero curvature. The metric, which is formally defined in terms of differential forms, is in this simple case identifiable as an observable.

math.QA↗

arXiv1998

A Calculus Based on a q-deformed Heisenberg Algebra

We show how one can construct a differential calculus over an algebra where position variables x and momentum variables p have be defined. As the simplest example we consider the one-dimensional q-deformed Heisenberg algebra. This algebra has a subalgebra generated by x and its inverse which we call the coordinate algebra. A physical field is considered to be an element of the completion of this algebra. We can construct a derivative which leaves invariant the coordinate algebra and so takes physical fields into physical fields. A generalized Leibniz rule for this algebra can be found. Based on this derivative differential forms and an exterior differential calculus can be constructed.

math.QA↗

arXiv1997

Noncommutative Geometry of the h-deformed Quantum Plane

The $h$-deformed quantum plane is a counterpart of the $q$-deformed one in the set of quantum planes which are covariant under those quantum deformations of GL(2) which admit a central determinant. We have investigated the noncommutative geometry of the $h$-deformed quantum plane. There is a 2-parameter family of torsion-free linear connections, a 1-parameter sub-family of which are compatible with a skew-symmetric non-degenerate bilinear map. The skew-symmetric map resembles a symplectic 2-form and induces a metric. It is also shown that the extended $h$-deformed quantum plane is a noncommutative version of the Poincaré half-plane, a surface of constant negative Gaussian curvature.

q-alg↗

arXiv1997

Gravity on Fuzzy Space-Time

A review is made of recent efforts to add a gravitational field to noncommutative models of space-time. Special emphasis is placed on the case which could be considered as the noncommutative analog of a parallelizable space-time. It is argued that, at least in this case, there is a rigid relation between the noncommutative structure of the space-time on the one hand and the nature of the gravitational field which remains as a `shadow' in the commutative limit on the other.

gr-qc↗

arXiv1997

Topology at the Planck Length

A basic arbitrariness in the determination of the topology of a manifold at the Planck length is discussed. An explicit example is given of a `smooth' change in topology from the 2-sphere to the 2-torus through a sequence of noncommuting geometries. Applications are considered to the theory of D-branes within the context of the proposed $M$(atrix) theory.

gr-qc↗

arXiv1997

Fuzzy Surfaces of Genus Zero

A fuzzy version of the ordinary round 2-sphere has been constructed with an invariant curvature. We here consider linear connections on arbitrary fuzzy surfaces of genus zero. We shall find as before that they are more or less rigidly dependent on the differential calculus used but that a large number of the latter can be constructed which are not covariant under the action of the rotation group. For technical reasons we have been forced to limit our considerations to fuzzy surfaces which are small perturbations of the fuzzy sphere.

gr-qc↗

arXiv1997

On Poisson Structure and Curvature

We consider a curved space-time whose algebra of functions is the commutative limit of a noncommutative algebra and which has therefore an induced Poisson structure. In a simple example we determine a relation between this structure and the Riemann tensor.

gr-qc↗

arXiv1996

Classical Gravity on Fuzzy Space-Time

A review is made of recent efforts to find relations between the commutation relations which define a noncommutative geometry and the gravitational field which remains as a shadow in the commutative limit.

gr-qc↗

arXiv1996

Fuzzy Space-Time

A review is made of recent efforts to define linear connections and their corresponding curvature within the context of noncommutative geometry. As an application it is suggested that it is possible to identify the gravitational field as a phenomenological manifestation of space-time commutation relations and to thereby clarify its role as an ultraviolet regularizer.

gr-qc↗

arXiv1996

Quantum Space-time and Classical Gravity

A method has been recently proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example a version of quantized space-time is considered here. It is found that there is a natural differential calculus using which the space-time is necessarily flat Minkowski space-time. Perturbations of this calculus are shown to give rise to non-trivial gravitational fields.

gr-qc↗

arXiv1996

Noncommutative Kaluza-Klein Theory

Efforts have been made recently to reformulate traditional Kaluza-Klein theory by using a generalized definition of a higher-dimensional extended space-time. Both electromagnetism and gravity have been studied in this context. We review some of the models which have been proposed, with a special effort to keep the mathematical formalism to a very minimum.

hep-th↗

arXiv1996

Differential Calculi and Linear Connections

A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a one-to-one correspondence, between the module structure of the 1-forms and the metric torsion-free connections on it. In the commutative limit the connection remains as a shadow of the algebraic structure of the 1-forms.

q-alg↗

arXiv1996

Deformations of Differential Calculi

It has been suggested that quantum fluctuations of the gravitational field could give rise in the lowest approximation to an effective noncommutative version of Kaluza-Klein theory which has as extra hidden structure a noncommutative geometry. It would seem however from the Standard Model, at least as far as the weak interactions are concerned, that a double-sheeted structure is the phenomenologically appropriate one at present accelerator energies. We examine here to what extent this latter structure can be considered as a singular limit of the former.

hep-th↗

arXiv1995

On Curvature in Noncommutative Geometry

A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra. The corresponding curvatures are also compared.

q-alg↗

arXiv1995

On The non-commutative Riemannian geometry of GL_q(n)

A recently proposed definition of a linear connection in non-commutative geometry, based on a generalized permutation, is used to construct linear connections on GL_q(n). Restrictions on the generalized permutation arising from the stability of linear connections under involution are discussed. Candidates for generalized permutation on GL_q(n) are found. It is shown that, for a given generalized permutation, there exists one and only one associated linear connection. Properties of the linear connection are discussed, in particular its bicovariance, torsion and commutative limit.

q-alg↗

arXiv1995

Linear Connections on Fuzzy Manifolds

Linear connections are introduced on a series of noncommutative geometries which have commutative limits. Quasicommutative corrections are calculated.

hep-th↗

arXiv1995

ON THE ORIGIN OF KALUZA-KLEIN STRUCTURE

It is suggested that quantum fluctuations of the light cone are at the origin of what appears at low energy to be a higher-dimensional structure over space-time. A model is presented which has but a finite number of Yang-Mills fields although the supplementary algebraic structure is of infinite dimension.

hep-th↗