00961nas a2200121 4500008004300000245005200043210005100095260001300146520059700159100002200756700002500778856003600803 2011 en_Ud 00aPoincaré covariance and κ-Minkowski spacetime0 aPoincaré covariance and κMinkowski spacetime bElsevier3 aA 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\\\".1 aDabrowski, Ludwik1 aPiacitelli, Gherardo uhttp://hdl.handle.net/1963/389302121nas a2200145 4500008004100000245007900041210006900120260001300189520164700202100002001849700002201869700002301891700002501914856003601939 2011 en d00aQuantum Geometry on Quantum Spacetime: Distance, Area and Volume Operators0 aQuantum Geometry on Quantum Spacetime Distance Area and Volume O bSpringer3 aWe develop the first steps towards an analysis of geometry on the quantum\\r\\nspacetime proposed in Doplicher et al. (Commun Math Phys 172:187–220, 1995). The homogeneous elements of the universal differential algebra are naturally identified with operators living in tensor powers of Quantum Spacetime; this allows us to compute their spectra. In particular, we consider operators that can be interpreted as distances, areas, 3- and 4-volumes. The Minkowski distance operator between two independent events is shown to have pure Lebesgue spectrum with infinite multiplicity. The Euclidean distance operator is shown to have spectrum bounded below by a constant of the order of the Planck length. The corresponding statement is proved also for both the space-space and space-time area operators, as well as for the Euclidean length of the vector representing the 3-volume operators. However, the space 3-volume operator (the time component of that vector) is shown to have spectrum equal to the whole complex plane. All these operators are normal, while the distance operators are also selfadjoint. The Lorentz invariant spacetime volume operator, representing the 4- volume spanned by five\\r\\nindependent events, is shown to be normal. Its spectrum is pure point with a\\r\\nfinite distance (of the order of the fourth power of the Planck length) away\\r\\nfrom the origin. The mathematical formalism apt to these problems is developed and its relation to a general formulation of Gauge Theories on Quantum Spaces is outlined. As a byproduct, a Hodge Duality between the absolute differential and the Hochschild boundary is pointed out.1 aBahns, Dorothea1 aDoplicher, Sergio1 aFredenhagen, Klaus1 aPiacitelli, Gherardo uhttp://hdl.handle.net/1963/520300564nas a2200109 4500008004100000245005700041210005700098260001000155520022800165100002500393856003600418 2010 en d00aAspects of Quantum Field Theory on Quantum Spacetime0 aAspects of Quantum Field Theory on Quantum Spacetime bSISSA3 aWe provide a minimal, self-contained introduction to the covariant DFR flat\\r\\nquantum spacetime, and to some partial results for the corresponding quantum field theory. Explicit equations are given in the Dirac notation.1 aPiacitelli, Gherardo uhttp://hdl.handle.net/1963/417101351nas a2200109 4500008004300000245003600043210003500079520104400114100002501158700002201183856003601205 2010 en_Ud 00aCanonical k-Minkowski Spacetime0 aCanonical kMinkowski Spacetime3 aA 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.1 aPiacitelli, Gherardo1 aDabrowski, Ludwik uhttp://hdl.handle.net/1963/386300901nas a2200121 4500008004300000245004400043210004300087520054400130100002200674700002200696700002500718856003600743 2010 en_Ud 00aLorentz Covariant k-Minkowski Spacetime0 aLorentz Covariant kMinkowski Spacetime3 aIn recent years, different views on the interpretation of Lorentz covariance of non commuting coordinates were discussed. Here, by a general procedure, we construct the minimal canonical central covariantisation of the k-Minkowski spacetime. We then show that, though the usual k-Minkowski spacetime is covariant under deformed (or twisted) Lorentz action, the resulting framework is equivalent to taking a non covariant restriction of the covariantised model. We conclude with some general comments on the approach of deformed covariance.1 aDabrowski, Ludwik1 aGodlinski, Michal1 aPiacitelli, Gherardo uhttp://hdl.handle.net/1963/382901210nas a2200097 4500008004300000245004000043210003900083520092900122100002501051856003601076 2010 en_Ud 00aQuantum Spacetime: a Disambiguation0 aQuantum Spacetime a Disambiguation3 aWe review an approach to non-commutative geometry, where models are constructed by quantisation of the coordinates. In particular we focus on the full DFR model and its irreducible components; the (arbitrary) restriction to a particular irreducible component is often referred to as the \\\"canonical quantum spacetime\\\". The aim is to distinguish and compare the approaches under various points of view, including motivations, prescriptions for quantisation, the choice of mathematical objects and concepts, approaches to dynamics and to covariance. Some incorrect statements as \\\"universality of Planck scale conflicts with Lorentz-Fitzgerald contraction and requires a modification of covariance\\\", or \\\"stability of the geometric background requires an absolute lower bound of (\\\\Delta x^\\\\mu)\\\", or \\\"violations of unitarity are due to time/space non-commutativity\\\" are put in context, and discussed.1 aPiacitelli, Gherardo uhttp://hdl.handle.net/1963/386401480nas a2200097 4500008004300000245008700043210006900130520112200199100002501321856003601346 2010 en_Ud 00aTwisted Covariance as a Non Invariant Restriction of the Fully Covariant DFR Model0 aTwisted Covariance as a Non Invariant Restriction of the Fully C3 aWe discuss twisted covariance over the noncommutative spacetime algebra generated by the relations [q_theta^mu,q_theta^nu]=i theta^{mu nu}, where the matrix theta is treated as fixed (not a tensor), and we refrain from using the asymptotic Moyal expansion of the twists. We show that the tensor nature of theta is only hidden in the formalism: in particular if theta fulfils the DFR conditions, the twisted Lorentz covariant model of the flat quantum spacetime may be equivalently described in terms of the DFR model, if we agree to discard a huge non invariant set of localisation states; it is only this last step which, if taken as a basic assumption, severely breaks the relativity principle. We also will show that the above mentioned, relativity breaking, ad hoc rejection of localisation states is an independent, unnecessary assumption, as far as some popular approaches to quantum field theory on the quantum Minkowski spacetime are concerned. The above should raise some concerns about speculations on possible observable consequences of arbitrary choices of theta in arbitrarily selected privileged frames.1 aPiacitelli, Gherardo uhttp://hdl.handle.net/1963/360500591nas a2200097 4500008004300000245004400043210004400087520030100131100002500432856003600457 2009 en_Ud 00aTwisted Covariance vs Weyl Quantisation0 aTwisted Covariance vs Weyl Quantisation3 aIn this letter we wish to clarify in which sense the tensor nature of the commutation relations [x^mu,x^nu]=i theta ^{mu nu} underlying Minkowski spacetime quantisation cannot be suppressed even in the twisted approach to Lorentz covariance. We then address the vexata quaestio \\\"why theta\\\"?1 aPiacitelli, Gherardo uhttp://hdl.handle.net/1963/3451