Spectral asymmetry

In mathematics, the spectral asymmetry arises in the study of elliptic operators on compact manifolds, and is given a deep meaning by the Atiyah-Singer index theorem.In physics, it has numerous applications, typically resulting in a fractional charge due to the asymmetry of the spectrum of a Dirac operator.For example, the vacuum expectation value of the baryon number is given by the spectral asymmetry of the Hamiltonian operator.The spectral asymmetry of the confined quark fields is an important property of the chiral bag model., an equal number of which are positive and negative, the spectral asymmetry may be defined as the sum whereThe need for both a positive and negative spectrum in the definition is why the spectral asymmetry usually occurs in the study of Dirac operators.

mathematicsphysicsspectrumeigenvaluesoperatorelliptic operatorscompact manifoldsAtiyah-Singer index theoremchargeDirac operatorvacuum expectation valuebaryon numberHamiltonian operatorchiral bag modelfermionsWitten indexCasimir effectsign functionregulatorszeta function regulatorDirac operatorsCasimir energyAtiyah, M. F.Patodi, V. K.Singer, I. M.Proceedings of the Cambridge Philosophical SocietyBibcodeFunctional analysistopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledCompleteLocally convexReflexiveSeparableHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointBoundedCompactHilbert–SchmidtNormalTrace classTransposeUnboundedUnitaryBanach algebraC*-algebraSpectrum of a C*-algebraOperator algebraGroup algebra of a locally compact groupVon Neumann algebraInvariant subspace problemMahler's conjectureHardy spaceSpectral theory of ordinary differential equationsHeat kernelIndex theoremCalculus of variationsFunctional calculusIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisDistributionGeneralized functionsApproximation propertyBalanced setChoquet theoryWeak topologyBanach–Mazur distanceTomita–Takesaki theorySpectral theory*-algebrasInvolution/*-algebraB*-algebraNoncommutative topologyProjection-valued measureSpectral radiusOperator spaceGelfand–Mazur theoremGelfand–Naimark theoremGelfand representationPolar decompositionSingular value decompositionSpectral theoremSpectral theory of normal C*-algebrasIsospectralHermitian/Self-adjointKrein–Rutman theoremNormal eigenvalueSpectral gapDecomposition of a spectrumContinuousDirect integralDiscreteSpectral abscissaBorel functional calculusMin-max theoremPositive operator-valued measureRiesz projectorRigged Hilbert spaceSpectral theory of compact operatorsAmenable Banach algebraApproximate identityBanach function algebraDisk algebraNuclear C*-algebraUniform algebraAlon–Boppana boundBauer–Fike theoremNumerical rangeSchur–Horn theoremDirac spectrumEssential spectrumPseudospectrumStructure spaceShilov boundaryAbstract index groupBanach algebra cohomologyCohen–Hewitt factorization theoremExtensions of symmetric operatorsFredholm theoryLimiting absorption principleSchröder–Bernstein theorems for operator algebrasSherman–Takeda theoremUnbounded operatorWiener algebraAlmost Mathieu operatorCorona theoremHearing the shape of a drumDirichlet eigenvalueKuznetsov trace formulaLax pairProto-value functionRamanujan graphRayleigh–Faber–Krahn inequalitySpectral geometrySpectral methodSturm–Liouville theorySuperstrong approximationTransfer operatorTransform theoryWeyl lawWiener–Khinchin theorem