mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and

physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

, the spectral asymmetry is the asymmetry in the distribution of the

spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...

eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

s of an operator. In mathematics, the spectral asymmetry arises in the study of

elliptic operator In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which im ...

s on

compact manifold In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The onl ...

s, and is given a deep meaning by the Atiyah-Patodi-Singer index theorem. In physics, it has numerous applications, typically resulting in a fractional

charge Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * '' Charge!!'', an album by The Aqu ...

due to the asymmetry of the spectrum of a

Dirac operator In mathematics and in quantum mechanics, a Dirac operator is a first-order differential operator that is a formal square root, or half-iterate, of a second-order differential operator such as a Laplacian. It was introduced in 1847 by William Ham ...

. For example, the

vacuum expectation value In quantum field theory, the vacuum expectation value (VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle. One of the most widely used exa ...

of the

baryon number In particle physics, the baryon number (B) is an additive quantum number of a system. It is defined as B = \frac(n_\text - n_), where is the number of quarks, and is the number of antiquarks. Baryons (three quarks) have B = +1, mesons (one q ...

is given by the spectral asymmetry of the

Hamiltonian operator In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's ''energy spectrum'' or its set of ''energy eigenvalu ...

. The spectral asymmetry of the confined

quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nucleus, atomic nuclei ...

fields is an important property of the chiral bag model. For

fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...

s, it is known as the Witten index, and can be understood as describing the

Casimir effect In quantum field theory, the Casimir effect (or Casimir force) is a physical force (physics), force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of a field (physics), field. The term Casim ...

for fermions.

Definition

Given an operator with

\omega_n

, an equal number of which are positive and negative, the spectral asymmetry may be defined as the sum :

B=\lim_ \frac\sum_n \sgn(\omega_n) \exp (-t, \omega_n, )

where

\sgn(x)

is the

sign function In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is a function that has the value , or according to whether the sign of a given real number is positive or negative, or the given number is itself zer ...

. 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

Example

As an example, consider an operator with a spectrum :

\omega_n=n+\alpha

where ''n'' is an integer, ranging over all positive and negative values. One may show in a straightforward manner that in this case

B(\alpha)

obeys

B(\alpha)= B(\alpha +m)

for any integer

m

, and that for

0<\alpha<1

we have

B(\alpha)=1/2-\alpha

. The graph of

B(\alpha)

is therefore a periodic sawtooth curve.

Discussion

Related to the spectral asymmetry is the vacuum expectation value of the energy associated with the operator, the Casimir energy, which is given by :

E=\lim_ \frac\sum_n , \omega_n,  \exp (-t, \omega_n, )

This sum is formally divergent, and the divergences must be accounted for and removed using standard regularization techniques.

References

* * * {{SpectralTheory Spectral theory Asymmetry