TY - JOUR
T1 - Dirac operators on noncommutative principal circle bundles
Y1 - 2014
A1 - Andrzej Sitarz
A1 - Alessandro Zucca
A1 - Ludwik Dabrowski
AB - We study spectral triples over noncommutative principal U(1)-bundles of arbitrary dimension and a compatibility condition between the connection and the Dirac operator on the total space and on the base space of the bundle. Examples of low-dimensional noncommutative tori are analyzed in more detail and all connections found that are compatible with an admissible Dirac operator. Conversely, a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection is exhibited. These examples are extended to the theta-deformed principal U(1)-bundle S 3 θ → S2.
PB - World Scientific Publishing
UR - http://urania.sissa.it/xmlui/handle/1963/35125
U1 - 35363
U2 - Mathematics
U4 - 1
ER -
TY - JOUR
T1 - Quantum gauge symmetries in noncommutative geometry
Y1 - 2014
A1 - Jyotishman Bhowmick
A1 - Francesco D'Andrea
A1 - Biswarup Krishna Das
A1 - Ludwik Dabrowski
AB - We discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite-dimensional C*-algebra, ii) gauge transformations and iii) (real) automorphisms in the framework of compact quantum group theory and spectral triples. The quantum analogue of these groups are defined as universal (initial) objects in some natural categories. After proving the existence of the universal objects, we discuss several examples that are of interest to physics, as they appear in the noncommutative geometry approach to particle physics: in particular, the C*-algebras M n(R), Mn(C) and Mn(H), describing the finite noncommutative space of the Einstein-Yang-Mills systems, and the algebras A F = C H M3 (C) and Aev = H H M4(C), that appear in Chamseddine-Connes derivation of the Standard Model of particle physics coupled to gravity. As a byproduct, we identify a "free" version of the symplectic group Sp.n/ (quaternionic unitary group).
PB - European Mathematical Society Publishing House
UR - http://urania.sissa.it/xmlui/handle/1963/34897
U1 - 35182
U2 - Mathematics
U4 - 1
ER -
TY - JOUR
T1 - Curved noncommutative torus and Gauss--Bonnet
JF - Journal of Mathematical Physics. Volume 54, Issue 1, 22 January 2013, Article number 013518
Y1 - 2013
A1 - Ludwik Dabrowski
A1 - Andrzej Sitarz
KW - Geometry
AB - 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.
PB - American Institute of Physics
UR - http://hdl.handle.net/1963/7376
N1 - The article is composed of 13 pages and is recorded in PDF format
U1 - 7424
U2 - Mathematics
U4 - 1
U5 - MAT/07 FISICA MATEMATICA
ER -
TY - JOUR
T1 - Dirac operator on spinors and diffeomorphisms
JF - Classical and Quantum Gravity. Volume 30, Issue 1, 7 January 2013, Article number 015006
Y1 - 2013
A1 - Ludwik Dabrowski
A1 - Giacomo Dossena
KW - gravity
AB - The issue of general covariance of spinors and related objects is reconsidered. Given an oriented manifold $M$, to each spin structure $\sigma$ and Riemannian metric $g$ there is associated a space $S_{\sigma, g}$ of spinor fields on $M$ and a Hilbert space $\HH_{\sigma, g}= L^2(S_{\sigma, g},\vol{M}{g})$ of $L^2$-spinors of $S_{\sigma, g}$. The group $\diff{M}$ of orientation-preserving diffeomorphisms of $M$ acts both on $g$ (by pullback) and on $[\sigma]$ (by a suitably defined pullback $f^*\sigma$). Any $f\in \diff{M}$ lifts in exactly two ways to a unitary operator $U$ from $\HH_{\sigma, g} $ to $\HH_{f^*\sigma,f^*g}$. The canonically defined Dirac operator is shown to be equivariant with respect to the action of $U$, so in particular its spectrum is invariant under the diffeomorphisms.
PB - IOP Publishing
UR - http://hdl.handle.net/1963/7377
N1 - This article is composed of 13 pages and is recorded in PDF format
U1 - 7425
U2 - Mathematics
U4 - 1
U5 - MAT/07 FISICA MATEMATICA
ER -
TY - JOUR
T1 - Noncommutative circle bundles and new Dirac operators
JF - Communications in Mathematical Physics. Volume 318, Issue 1, 2013, Pages 111-130
Y1 - 2013
A1 - Ludwik Dabrowski
A1 - Andrzej Sitarz
KW - Quantum principal bundles
AB - We study spectral triples over noncommutative principal U(1) bundles. Basing on the classical situation and the abstract algebraic approach, we propose an operatorial definition for a connection and compatibility between the connection and the Dirac operator on the total space and on the base space of the bundle. We analyze in details the example of the noncommutative three-torus viewed as a U(1) bundle over the noncommutative two-torus and find all connections compatible with an admissible Dirac operator. Conversely, we find a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection.
PB - Springer
UR - http://hdl.handle.net/1963/7384
N1 - This article is composed of 25 pages and is recorded in PDF format
U1 - 7432
U2 - Mathematics
U4 - 1
U5 - MAT/07 FISICA MATEMATICA
ER -
TY - JOUR
T1 - Poincaré covariance and κ-Minkowski spacetime
JF - Physics Letters A 375 (2011) 3496-3498
Y1 - 2011
A1 - Ludwik Dabrowski
A1 - Gherardo Piacitelli
AB - A fully Poincaré covariant model is constructed out of the k-Minkowski spacetime. Covariance is implemented by a unitary representation of the Poincaré group, and thus complies with the original Wigner approach to quantum symmetries. This provides yet another example (besides the DFR model), where Poincaré covariance is realised á la Wigner in the presence of two characteristic dimensionful parameters: the light speed and the Planck length. In other words, a Doubly Special Relativity (DSR) framework may well be realised without deforming the meaning of \\\"Poincaré covariance\\\".
PB - Elsevier
UR - http://hdl.handle.net/1963/3893
U1 - 816
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Product of real spectral triples
JF - International Journal of Geometric Methods in Modern Physics 8 (2011) 1833-1848
Y1 - 2011
A1 - Ludwik Dabrowski
A1 - Giacomo Dossena
AB - We construct the product of real spectral triples of arbitrary finite dimension (and arbitrary parity) taking into account the fact that in the even case there are two possible real structures, in the odd case there are two inequivalent representations of the gamma matrices (Clifford algebra), and in the even-even case there are two natural candidates for the Dirac operator of the product triple.
PB - World Scientific
UR - http://hdl.handle.net/1963/5510
N1 - Based on the talk given at the conference \\\"Noncommutative Geometry and Quantum Physics, Vietri sul Mare, Aug 31 - Sept 5, 2009\\\"
U1 - 5345
U2 - Mathematics
U3 - Mathematical Physics
U4 - -1
ER -
TY - JOUR
T1 - Quantum Isometries of the finite noncommutative geometry of the Standard Model
JF - Commun. Math. Phys. 307:101-131, 2011
Y1 - 2011
A1 - Jyotishman Bhowmick
A1 - Francesco D'Andrea
A1 - Ludwik Dabrowski
AB - We compute the quantum isometry group of the finite noncommutative geometry F describing the internal degrees of freedom in the Standard Model of particle physics. We show that this provides genuine quantum symmetries of the spectral triple corresponding to M x F where M is a compact spin manifold. We also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries.
PB - Springer
UR - http://hdl.handle.net/1963/4906
U1 - 4688
U2 - Mathematics
U3 - Mathematical Physics
U4 - -1
ER -
TY - RPRT
T1 - Canonical k-Minkowski Spacetime
Y1 - 2010
A1 - Gherardo Piacitelli
A1 - Ludwik Dabrowski
AB - A complete classification of the regular representations of the relations [T,X_j] = (i/k)X_j, j=1,...,d, is given. The quantisation of RxR^d canonically (in the sense of Weyl) associated with the universal representation of the above relations is intrinsically \\\"radial\\\", this meaning that it only involves the time variable and the distance from the origin; angle variables remain classical. The time axis through the origin is a spectral singularity of the model: in the large scale limit it is topologically disjoint from the rest. The symbolic calculus is developed; in particular there is a trace functional on symbols. For suitable choices of states localised very close to the origin, the uncertainties of all spacetime coordinates can be made simultaneously small at wish. On the contrary, uncertainty relations become important at \\\"large\\\" distances: Planck scale effects should be visible at LHC energies, if processes are spread in a region of size 1mm (order of peak nominal beam size) around the origin of spacetime.
UR - http://hdl.handle.net/1963/3863
U1 - 846
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Dirac Operators on Quantum Projective Spaces
JF - Comm. Math. Phys. 295 (2010) 731-790
Y1 - 2010
A1 - Francesco D'Andrea
A1 - Ludwik Dabrowski
AB - We construct a family of self-adjoint operators D_N which have compact resolvent and bounded commutators with the coordinate algebra of the quantum projective space CP_q(l), for any l>1 and 0