Dirac operator

E391906

differential operator elliptic operator first-order differential operator geometric operator

The Dirac operator is a fundamental first-order differential operator on spinor fields that generalizes the classical Dirac equation and plays a central role in geometry, topology, and quantum field theory.

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All labels observed (2)

Label	Occurrences
Dirac operator canonical	3
Dirac operators	1

Statements (48)

Predicate	Object
instanceOf	differential operator ⓘ elliptic operator ⓘ first-order differential operator ⓘ geometric operator ⓘ
actsOn	sections of the spinor bundle ⓘ spinor fields ⓘ
builtFrom	Clifford multiplication ⓘ Levi-Civita connection ⓘ spin connection ⓘ
centralTo	Atiyah–Singer index theorem ⓘ Lichnerowicz formula ⓘ spin geometry ⓘ
definedOn	Riemannian manifold with spin structure ⓘ spin manifold ⓘ
eigenvaluesDependOn	Riemannian metric ⓘ spin structure ⓘ
generalizes	Dirac equation ⓘ flat-space Dirac operator on Minkowski space ⓘ
hasLocalExpression	sum of Clifford matrices times covariant derivatives ⓘ
hasOrder	1 ⓘ
hasSpectrum	discrete on compact manifolds ⓘ
indexEquals	Â-genus for suitable manifolds ⓘ
introducedBy	Paul Dirac ⓘ
isElliptic	true ⓘ
isLinear	true ⓘ
isSelfAdjoint	essentially self-adjoint on complete manifolds ⓘ formally self-adjoint ⓘ
kernelDefines	harmonic spinors ⓘ
relatedTo	Clifford algebra ⓘ K-theory ⓘ noncommutative geometry ⓘ spin representation ⓘ
squareGivenBy	Lichnerowicz formula ⓘ
squareRelatesTo	Hodge Laplacian ⓘ surface form: Bochner Laplacian Hodge Laplacian ⓘ surface form: Laplace–Beltrami operator
symbolIs	Clifford multiplication by cotangent vectors ⓘ
usedIn	differential geometry ⓘ gauge theory ⓘ global analysis ⓘ index theory ⓘ quantum field theory ⓘ supersymmetry ⓘ topology ⓘ
usedToDefine	spectral triples in noncommutative geometry ⓘ
usedToStudy	Seiberg–Witten invariants ⓘ anomalies in quantum field theory ⓘ positive mass theorems ⓘ scalar curvature ⓘ

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Atiyah–Singer index theorem → concerns → Dirac operator ⓘ

Dirac matrices → usedToConstruct → Dirac operator ⓘ

noncommutative geometry → keyConcept → Dirac operator ⓘ

Noncommutative Geometry (1994 book) → subject → Dirac operator ⓘ

this entity surface form: Dirac operators