319 research outputs found
Noncommutative differential geometry with higher order derivatives
We build a toy model of differential geometry on the real line, which
includes derivatives of the second order. Such construction is possible only
within the framework of noncommutative geometry. We introduce the metric and
briefly discuss two simple physical models of scalar field theory and gauge
theory in this geometry.Comment: 10 page
Dynamical noncommutativity
We present a model of Moyal-type noncommutativity with time-depending
noncommutativity parameter and the exact gauge invariant action for the U(1)
noncommutative gauge theory. We briefly result the results of the analysis of
plane-wave propagation in a regime of a small but rapidly changing
noncommutativity.Comment: 10 pages, JHEP styl
Noncommutative Geometry and The Ising Model
The main aim of this work is to present the interpretation of the Ising type
models as a kind of field theory in the framework of noncommutative geometry.
We present the method and construct sample models of field theory on discrete
spaces using the introduced tools of discrete geometry. We write the action for
few models, then we compare them with various models of statistical physics. We
construct also the gauge theory with a discrete gauge group.Comment: 12 pages, LaTeX, TPJU - 18/92, December 199
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.Comment: 11 pages of Late
-deformation, affine group and spectral triples
A regular spectral triple is proposed for a two-dimensional
-deformation. It is based on the naturally associated affine group ,
a smooth subalgebra of , and an operator \caD defined by two
derivations on this subalgebra. While \caD has metric dimension two, the
spectral dimension of the triple is one. This bypasses an obstruction described
in \cite{IochMassSchu11a} on existence of finitely-summable spectral triples
for a compactified -deformation.Comment: 29 page
Curved noncommutative torus and Gauss--Bonnet
We study perturbations of the flat geometry of the noncommutative
two-dimensional torus T^2_\theta (with irrational \theta). They are described
by spectral triples (A_\theta, \H, D), with the Dirac operator D, which is a
differential operator with coefficients in the commutant of the (smooth)
algebra A_\theta of T_\theta. We show, up to the second order in perturbation,
that the zeta-function at 0 vanishes and so the Gauss-Bonnet theorem holds. We
also calculate first two terms of the perturbative expansion of the
corresponding local scalar curvature.Comment: 13 pages, LaTe
Spectral action on noncommutative torus
The spectral action on noncommutative torus is obtained, using a
Chamseddine--Connes formula via computations of zeta functions. The importance
of a Diophantine condition is outlined. Several results on holomorphic
continuation of series of holomorphic functions are obtained in this context.Comment: 57 page
Higgs field as the gauge field corresponding to parity in the usual space-time
We find that the local character of field theory requires the parity degree
of freedom of the fields to be considered as an additional dicrete fifth
dimension which is an artifact emerging due to the local description of
space-time. Higgs field arises as the gauge field corresponding to this
discrete dimension. Hence the noncommutative geometric derivation of the
standard model follows as a manifestation of the local description of the usual
space-time.Comment: 14 pages, latex, no figure
Discrete Differential Manifolds and Dynamics on Networks
A `discrete differential manifold' we call a countable set together with an
algebraic differential calculus on it. This structure has already been explored
in previous work and provides us with a convenient framework for the
formulation of dynamical models on networks and physical theories with discrete
space and time. We present several examples and introduce a notion of
differentiability of maps between discrete differential manifolds. Particular
attention is given to differentiable curves in such spaces. Every discrete
differentiable manifold carries a topology and we show that differentiability
of a map implies continuity.Comment: 26 pages, LaTeX (RevTex), GOET-TP 88/9
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