Quantum spacetime
In mathematical physics, the concept of quantum spacetime is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra.The germ of the idea passed from Heisenberg to Rudolf Peierls, who noted that electrons in a magnetic field can be regarded as moving in a quantum spacetime, and to Robert Oppenheimer, who carried it to Hartland Snyder, who published the first concrete example.Because of the Heisenberg uncertainty relations, greater energy is needed to probe smaller distances.Ultimately, according to gravity theory, the probing particles form black holes that destroy what was to be measured.This limited measurability led many to expect that the usual picture of continuous commutative spacetime breaks down at Planck scale distances, if not sooner.Any noncommutative algebra with at least four generators could be interpreted as a quantum spacetime, but the following desiderata have been suggested: This would permit wave equations for particles and fields and facilitate predictions for experimental deviations from classical spacetime physics that can then be tested experimentally.The bicrossproduct model spacetime was introduced by Shahn Majid and Henri Ruegg[3] and has Lie algebra relations for the spatial variablesMeanwhile, the Lorentz group generators enjoy their usual relations among themselves but act non-linearly on the momentum space.The on-shell region describing particles in the upper center of the image would normally be hyperboloids but these are now 'squashed' into the cylinder in simplified units.This squashing comes from the non-linearity of the Lorentz boost and is an endemic feature of bicrossproduct quantum groups known since their introduction in 1988.[4] Some physicists dub the bicrossproduct model doubly special relativity, since it sets an upper limit to both speed and momentum.Arguments for this identification were provided in 1999 by Giovanni Amelino-Camelia and Majid[5] through a study of plane waves for a quantum differential calculus in the model.At the moment, such wave analysis represents the best hope to obtain physically testable predictions from the model.Prior to this work there were a number of unsupported claims to make predictions from the model based solely on the form of the Poincaré quantum group.This model was introduced independently by a team[7] working under Julius Wess in 1990 and by Shahn Majid and coworkers in a series of papers on braided matrices starting a year later.In quantum group theory and using braided monoidal category methods, a natural q-version of this is defined here for real values ofas linear combinations of these, in particular, time is given by a natural braided trace of the matrix and commutes with the other generators (so this model is different from the bicrossproduct one).returns the usual Minkowski distance (this translates to a metric in the quantum differential geometry).That is, there are indications that this model relates to quantum gravity with a non-zero cosmological constant, the choice ofThe momentum space for the theory is another copy of the same algebra and there is a certain 'braided addition' of momentum on it expressed as the structure of a braided Hopf algebra or quantum group in a certain braided monoidal category).[9] In the process it was discovered that the Poincaré group not only had to be deformed but had to be extended to include dilations of the quantum spacetime.If current thinking in cosmology is correct, then this model is more appropriate, but it is significantly more complicated and for this reason its physical predictions have yet to be worked out.[how?]These relations were proposed by Roger Penrose in his earliest spin network theory of space.The idea was revived in a modern context by Sergio Doplicher, Klaus Fredenhagen and John Roberts in 1995,[12] by letting, any nondegenerate such theta can be transformed to a normal form in which this really is just the Heisenberg algebra but the difference that the variables are being proposed as those of spacetime.When posited as quantum spacetime, it is hard to obtain physical predictions and one reason for this is that ifInstead of invisible curled up extra dimensions as in string theory, Alain Connes and coworkers have argued that the coordinate algebra of this extra part should be replaced by a finite-dimensional noncommutative algebra.For a certain reasonable choice of this algebra, its representation and extended Dirac operator, the Standard Model of elementary particles can be recovered.In this point of view, the different kinds of matter particles are manifestations of geometry in these extra noncommutative directions.
mathematical physicsspacetimecommuteLie algebraHeisenbergIvanenkoRudolf PeierlsRobert OppenheimerHartland SnyderC. N. Yangquantum mechanicsnoncommutativeHeisenberg uncertainty principleblack holesPlanck scalegeneral relativityquantum gravityPlanck timenoncommutative geometryquantum geometryquantum groupsLorentz groupPoincaré groupquantum groupPlanck lengthquantum differential calculussemisimpleShahn Majidbicrossproductnon-abelian groupdoubly special relativityvariable speed of lightGiovanni Amelino-CameliaJulius WessPauli matricesbraided monoidal categoryMinkowski distancecosmological constantbraided Hopf algebraPlanck massangular momentumRoger Penrosespin networkGerardus 't Hooftfuzzy spheresSergio DoplicherKlaus FredenhagenHeisenberg algebranoncommutative quantum field theoryAlain ConnesStandard ModelAnabelian topologyQuantum reference frameBibcodeCiteSeerXConnes, A.Marcolli, M.IntroductionHistoryTimelineClassical mechanicsOld quantum theoryGlossaryBorn ruleBra–ket notation ComplementarityDensity matrixEnergy levelGround stateExcited stateDegenerate levelsZero-point energyEntanglementHamiltonianInterferenceDecoherenceMeasurementNonlocalityQuantum stateSuperpositionTunnellingScattering theorySymmetry in quantum mechanicsUncertaintyWave functionCollapseWave–particle dualityFormulationsInteractionMatrix mechanicsSchrödingerPath integral formulationPhase spaceKlein–GordonMajoranaRarita–SchwingerRydbergInterpretationsBayesianConsistent historiesCopenhagende Broglie–BohmEnsembleHidden-variableSuperdeterminismMany-worldsObjective collapseQuantum logicRelationalTransactionalVon Neumann–WignerBell testDavisson–GermerDelayed-choice quantum eraserDouble-slitFranck–HertzMach–Zehnder interferometerElitzur–VaidmanPopperQuantum eraserStern–GerlachWheeler's delayed choiceScienceQuantum biologyQuantum chemistryQuantum chaos