QFT on homothetic Killing twist deformed curved spacetimes

We study the quantum field theory (QFT) of a free, real, massless and curvature coupled scalar field on self-similar symmetric spacetimes, which are deformed by an abelian Drinfel'd twist constructed from a Killing and a homothetic Killing vector field. In contrast to deformations solely by Killing vector fields, such as the Moyal-Weyl Minkowski spacetime, the equation of motion and Green's operators are deformed. We show that there is a *-algebra isomorphism between the QFT on the deformed and the formal power series extension of the QFT on the undeformed spacetime. We study the convergent implementation of our deformations for toy-models. For these models it is found that there is a *-isomorphism between the deformed Weyl algebra and a reduced undeformed Weyl algebra, where certain strongly localized observables are excluded. Thus, our models realize the intuitive physical picture that noncommutative geometry prevents arbitrary localization in spacetime.Comment: 23 pages, no figures; v2: extended discussion of physical consequences, compatible with version to be published in General Relativity and Gravitatio

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

Discretized Yang-Mills and Born-Infeld actions on finite group geometries

Discretized nonabelian gauge theories living on finite group spaces G are defined by means of a geometric action \int Tr F \wedge *F. This technique is extended to obtain discrete versions of the Born-Infeld action. The discretizations are in 1-1 correspondence with differential calculi on finite groups. A consistency condition for duality invariance of the discretized field equations is derived for discretized U(1) actions S[F] living on a 4-dimensional abelian G. Discretized electromagnetism satisfies this condition and therefore admits duality rotations. Yang-Mills and Born-Infeld theories are also considered on product spaces M^D x G, and we find the corresponding field theories on M^D after Kaluza-Klein reduction on the G discrete internal spaces. We examine in some detail the case G=Z_N, and discuss the limit N -> \infty. A self-contained review on the noncommutative differential geometry of finite groups is included.Comment: 31 pages, LaTeX. Improved definition of pairing between tensor products of left-invariant one-form

Dynamical generation of fuzzy extra dimensions, dimensional reduction and symmetry breaking

We present a renormalizable 4-dimensional SU(N) gauge theory with a suitable multiplet of scalar fields, which dynamically develops extra dimensions in the form of a fuzzy sphere S^2. We explicitly find the tower of massive Kaluza-Klein modes consistent with an interpretation as gauge theory on M^4 x S^2, the scalars being interpreted as gauge fields on S^2. The gauge group is broken dynamically, and the low-energy content of the model is determined. Depending on the parameters of the model the low-energy gauge group can be SU(n), or broken further to SU(n_1) x SU(n_2) x U(1), with mass scale determined by the size of the extra dimension.Comment: 27 pages. V2: discussion and references added, published versio

Translation Invariance, Commutation Relations and Ultraviolet/Infrared Mixing

We show that the Ultraviolet/Infrared mixing of noncommutative field theories with the Gronewold-Moyal product, whereby some (but not all) ultraviolet divergences become infrared, is a generic feature of translationally invariant associative products. We find, with an explicit calculation that the phase appearing in the nonplanar diagrams is the one given by the commutator of the coordinates, the semiclassical Poisson structure of the non commutative spacetime. We do this with an explicit calculation for represented generic products.Comment: 24 pages, 1 figur

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 $G_{\mu\nu}=0$ and Minkowski metric is also found. All solutions we obtained are non-perturbative in the noncommutative parameter $\theta$, 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

R-Matrix Formulation of the Quantum Inhomogeneous Groups Iso_qr(N) and Isp_qr(N)

The quantum commutations $RTT=TTR$ and the orthogonal (symplectic) conditions for the inhomogeneous multiparametric $q$-groups of the $B_n,C_n,D_n$ type are found in terms of the $R$-matrix of $B_{n+1},C_{n+1},D_{n+1}$. A consistent Hopf structure on these inhomogeneous $q$-groups is constructed by means of a projection from $B_{n+1},C_{n+1},D_{n+1}$. Real forms are discussed: in particular we obtain the $q$-groups $ISO_{q,r}(n+1,n-1)$, including the quantum Poincar\'e group.Comment: 14 pages, latex, no figure

Emergent Gravity from Noncommutative Gauge Theory

We show that the matrix-model action for noncommutative U(n) gauge theory actually describes SU(n) gauge theory coupled to gravity. This is elaborated in the 4-dimensional case. The SU(n) gauge fields as well as additional scalar fields couple to an effective metric G_{ab}, which is determined by a dynamical Poisson structure. The emergent gravity is intimately related to noncommutativity, encoding those degrees of freedom which are usually interpreted as U(1) gauge fields. This leads to a class of metrics which contains the physical degrees of freedom of gravitational waves, and allows to recover e.g. the Newtonian limit with arbitrary mass distribution. It also suggests a consistent picture of UV/IR mixing in terms of an induced gravity action. This should provide a suitable framework for quantizing gravity.Comment: 28 pages + 11 pages appendix. V2: references and discussion added. V3: minor correctio

Twisting adjoint module algebras

Transformation of operator algebras under Hopf algebra twist is studied. It is shown that that adjoint module algebras are stable under the twist. Applications to vector fields on non-commutative space-time are considered.Comment: 16 page

Noncommutative BTZ Black Hole in Polar Coordinates

Based on the equivalence between the three dimensional gravity and the Chern-Simons theory, we obtain a noncommutative BTZ black hole solution as a solution of $U(1,1)\times U(1,1)$ noncommutative Chern-Simons theory using the Seiberg-Witten map. The Seiberg-Witten map is carried out in a noncommutative polar coordinates whose commutation relation is equivalent to the usual canonical commutation relation in the rectangular coordinates up to first order in the noncommutativity parameter $\theta$. The solution exhibits a characteristic of noncommutative polar coordinates in such a way that the apparent horizon and the Killing horizon coincide only in the non-rotating limit showing the effect of noncommutativity between the radial and angular coordinates.Comment: 14 pages, V2: minor changes, v3: reduced for clarification, a reference adde