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Phase spaceCorrelation dimensionDimensional analysisRenormalization groupSectional curvatureCodimensionDimensionParameter spaceConfiguration spaceFractal dimensionQuantum geometryCritical exponentTangent spaceLebesgue covering dimensionSpacetimeHigh-dimensional statisticsGeometric quantizationMultidimensional transformDensity of statesEuclidean topologyHigher-dimensional gamma matricesTrace distanceQuantum field theoryHopf bifurcationFractalMultifractal systemOrthogonal groupTranscritical bifurcationDynamic structure factorPseudo-euclidean spaceBraneDirac equation in curved spacetimeHausdorff measureQuantum topologyCausal structureLattice field theoryFunctional renormalization groupGeneral covarianceSpatial scaleLow-dimensional topologyCalculus on euclidean spaceGeometric multiplicityQuantum operationRenormalizationStein manifoldHilbert spaceSpace groupDynamical systems theoryDifferential structureBifurcationQuantum heisenberg modelGeometric phaseTopological quantum field theoryCatastrophe theoryStefan problemSymmetry breakingComplex coordinate spaceExact diagonalizationConformal symmetryLorentz groupTopological propertyGroup actionFundamental representationTopological quantum numberCorrelation functionSelf-organized criticalityCanonical coordinatesMultiple-scale analysisQuantum stateHomotopical connectivitySelection ruleBifurcation theoryGauge theoryFunctional analysisCenter of massQuantum chaosCartan decompositionTopological manifoldFour-momentumBare massQuantum groupWavefunctionRelative interiorGeometrodynamicsQuantum phase estimation algorithmLiouville field theoryMapping class groupCauchy horizonContinuum theoryQuantum complexity theoryHigher-dimensional algebraState spaceGeometric analysisPartition functionDimensionality reductionSize theoryMatter power spectrumPhase problemFractal derivativeRadial distribution function