126 research outputs found
Noncommutative Symmetries and Gravity
Spacetime geometry is twisted (deformed) into noncommutative spacetime
geometry, where functions and tensors are now star-multiplied. Consistently,
spacetime diffeomorhisms are twisted into noncommutative diffeomorphisms. Their
deformed Lie algebra structure and that of infinitesimal Poincare'
transformations is defined and explicitly constructed.
This allows to construct a noncommutative theory of gravity.Comment: 26 pages. Lectures given at the workshop `Noncommutative Geometry in
Field and String Theories', Corfu Summer Institute on EPP, September 2005,
Corfu, Greece. Version 2: Marie Curie European Reintegration Grant
MERG-CT-2004-006374 acknowledge
Star Product Geometries
We consider noncommutative geometries obtained from a triangular Drinfeld
twist. This allows to construct and study a wide class of noncommutative
manifolds and their deformed Lie algebras of infinitesimal diffeomorphisms.
This way symmetry principles can be implemented. We review two main examples
[15]-[18]: a) general covariance in noncommutative spacetime. This leads to a
noncommutative gravity theory. b) Symplectomorphims of the algebra of
observables associated to a noncommutative configuration space. This leads to a
geometric formulation of quantization on noncommutative spacetime, i.e., we
establish a noncommutative correspondence principle from *-Poisson brackets to
*-commutators.
New results concerning noncommutative gravity include the Cartan structural
equations for the torsion and curvature tensors, and the associated Bianchi
identities. Concerning scalar field theories the deformed algebra of classical
and quantum observables has been understood in terms of a twist within the
algebra.Comment: 27 pages. Based on the talk presented at the conference "Geometry and
Operators Theory," Ancona (Italy), September 200
N-enlarged Galilei Hopf algebra and its twist deformations
The N-enlarged Galilei Hopf algebra is constructed. Its twist deformations
are considered and the corresponding twisted space-times are derived.Comment: 8 pages, no figure
Noncommutative Solitons of Gravity
We investigate a three-dimensional gravitational theory on a noncommutative
space which has a cosmological constant term only. We found various kinds of
nontrivial solutions, by applying a similar technique which was used to seek
noncommutative solitons in noncommutative scalar field theories. Some of those
solutions correspond to bubbles of spacetimes, or represent dimensional
reduction. The solution which interpolates and Minkowski metric
is also found. All solutions we obtained are non-perturbative in the
noncommutative parameter , therefore they are different from solutions
found in other contexts of noncommutative theory of gravity and would have a
close relation to quantum gravity.Comment: 29 pages, 5 figures. v2: minor corrections done in Section 3.1 and
Appendix, references added. v3, v4: typos correcte
Twisted Rindler space-times
The (linearized) noncommutative Rindler space-times associated with
canonical, Lie-algebraic and quadratic twist-deformed Minkowski spaces are
provided. The corresponding deformed Hawking spectra detected by Rindler
observers are derived as well.Comment: 13 pages, no figures, keywords: quantum space-times, Hawking
radiatio
An Unfolded Quantization for Twisted Hopf Algebras
In this talk I discuss a recently developed "Unfolded Quantization
Framework". It allows to introduce a Hamiltonian Second Quantization based on a
Hopf algebra endowed with a coproduct satisfying, for the Hamiltonian, the
physical requirement of being a primitive element. The scheme can be applied to
theories deformed via a Drinfeld twist. I discuss in particular two cases: the
abelian twist deformation of a rotationally invariant nonrelativistic Quantum
Mechanics (the twist induces a standard noncommutativity) and the Jordanian
twist of the harmonic oscillator. In the latter case the twist induces a Snyder
non-commutativity for the space-coordinates, with a pseudo-Hermitian deformed
Hamiltonian. The "Unfolded Quantization Framework" unambiguously fixes the
non-additive effective interactions in the multi-particle sector of the
deformed quantum theory. The statistics of the particles is preserved even in
the presence of a deformation.Comment: 9 pages. Talk given at QTS7 (7th Int. Conf. on Quantum Theory and
Symmetries, Prague, August 2011
Crossed Module Bundle Gerbes; Classification, String Group and Differential Geometry
We discuss nonabelian bundle gerbes and their differential geometry using
simplicial methods. Associated to any crossed module there is a simplicial
group NC, the nerve of the 1-category defined by the crossed module and its
geometric realization |NC|. Equivalence classes of principal bundles with
structure group |NC| are shown to be one-to-one with stable equivalence classes
of what we call crossed module gerbes bundle gerbes. We can also associate to a
crossed module a 2-category C'. Then there are two equivalent ways how to view
classifying spaces of NC-bundles and hence of |NC|-bundles and crossed module
bundle gerbes. We can either apply the W-construction to NC or take the nerve
of the 2-category C'. We discuss the string group and string structures from
this point of view. Also a simplicial principal bundle can be equipped with a
simplicial connection and a B-field. It is shown how in the case of a
simplicial principal NC-bundle these simplicial objects give the bundle gerbe
connection and the bundle gerbe B-field
Dynamical noncommutativity and Noether theorem in twisted phi^*4 theory
A \star-product is defined via a set of commuting vector fields X_a = e_a^\mu
(x) \partial_\mu, and used in a phi^*4 theory coupled to the e_a^\mu (x)
fields. The \star-product is dynamical, and the vacuum solution phi =0, e_a^\mu
(x)=delta_a^\mu reproduces the usual Moyal product. The action is invariant
under rigid translations and Lorentz rotations, and the conserved
energy-momentum and angular momentum tensors are explicitly derived.Comment: 15 pages LaTeX, minor typos, added reference
Twists in U(sl(3)) and their quantizations
The solution of the Drinfeld equation corresponding to the full set of
different carrier subalgebras in sl(3) are explicitly constructed. The obtained
Hopf structures are studied. It is demonstrated that the presented twist
deformations can be considered as limits of the corresponding quantum analogues
(q-twists) defined for the q-quantized algebras.Comment: 31 pages, Latex 2e, to be published in Journ. Phys. A: Math. Ge
Twisting all the way: from Classical Mechanics to Quantum Fields
We discuss the effects that a noncommutative geometry induced by a Drinfeld
twist has on physical theories. We systematically deform all products and
symmetries of the theory. We discuss noncommutative classical mechanics, in
particular its deformed Poisson bracket and hence time evolution and
symmetries. The twisting is then extended to classical fields, and then to the
main interest of this work: quantum fields. This leads to a geometric
formulation of quantization on noncommutative spacetime, i.e. we establish a
noncommutative correspondence principle from *-Poisson brackets to
*-commutators. In particular commutation relations among creation and
annihilation operators are deduced.Comment: 32 pages. Added references and details in the introduction and in
Section
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