InfoIn mathematics and physics, the spectral asymmetry is the asymmetry in the distribution of the spectrum of eigenvalues of an operator.more...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.

The entries in Grandi's series can be paired to the eigenvalues of an infinite-dimensional operator on Hilbert space.Giving the series this interpretation gives rise to the idea of spectral asymmetry, which occurs widely in physics.The value that the series sums to depends on the asymptotic behaviour of the eigenvalues of the operator.…

Other regulators, such as the zeta function regulator, may be used.The need for both a positive and negative spectrum in the definition is why the spectral asymmetry usually occurs in the study of Dirac operators.…

This resulted in a series of papers on spectral asymmetry, which were later unexpectedly used in theoretical physics, in particular in Witten's work on anomalies.The fundamental solutions of linear hyperbolic partial differential equations often have Petrovsky lacunas: regions where they vanish identically.…

The general series is discussed in greater detail in the article on spectral asymmetry, whereas methods used to sum it are discussed in the articles on regularization and, in particular, the zeta function regulator.

In the chiral bag model of the nucleon, the Casimir energy plays an important role in showing the mass of the nucleon is independent of the bag radius.In addition, the spectral asymmetry is interpreted as a non-zero vacuum expectation value of the baryon number, cancelling the topological winding number of the pion field surrounding the nucleon.The dynamical Casimir effect is the production of particles and energy from an accelerated moving mirror.…