Dirac spectrum
Encyclopedia
In mathematics
, a Dirac spectrum, named after Paul Dirac
, is the spectrum of eigenvalues of a Dirac operator
on a Riemannian manifold
with a spin structure
. The isospectral problem for the Dirac spectrum asks whether two Riemannian spin manifolds have identical spectra. The Dirac spectrum depends on the spin structure
in the sense that there exists a Riemannian manifold with two different spin structures that have different Dirac spectra.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Dirac spectrum, named after Paul Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
, is the spectrum of eigenvalues of a Dirac operator
Dirac operator
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian...
on a Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
with a spin structure
Spin structure
In differential geometry, a spin structure on an orientable Riemannian manifold \,allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry....
. The isospectral problem for the Dirac spectrum asks whether two Riemannian spin manifolds have identical spectra. The Dirac spectrum depends on the spin structure
Spin structure
In differential geometry, a spin structure on an orientable Riemannian manifold \,allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry....
in the sense that there exists a Riemannian manifold with two different spin structures that have different Dirac spectra.
See also
- Can you hear the shape of a drum?
- Dirichlet eigenvalueDirichlet eigenvalueIn mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape. The problem of whether one can hear the shape of a drum is: given the Dirichlet eigenvalues, what features of the shape of the drum can one deduce. Here a "drum" is thought of...
- Spectral asymmetrySpectral asymmetryIn mathematics and physics, the spectral asymmetry is the asymmetry in the distribution of the spectrum of eigenvalues of an operator. 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...
- Angle-resolved photoemission spectroscopyARPESAngle-resolved photoemission spectroscopy , also known as ARUPS , is a direct experimental technique to observe the distribution of the electrons in the reciprocal space of solids...