In mathematics and

physics Physics is the natural science that studies matter, its 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 which rel ...

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

spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ...

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...

s of an

operator Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...

. 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 without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example i ...

s, and is given a deep meaning by the Atiyah-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 Aqua ...

due to the asymmetry of the spectrum of a

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. The original case which concerned Paul Dirac was to factorise formally ...

. For example, the

vacuum expectation value In quantum field theory the vacuum expectation value (also called condensate or simply 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. ...

of the

baryon number In particle physics, the baryon number is a strictly conserved additive quantum number of a system. It is defined as ::B = \frac\left(n_\text - n_\bar\right), where ''n''q is the number of quarks, and ''n'' is the number of antiquarks. Baryo ...

is given by the spectral asymmetry of the

Hamiltonian operator Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonia ...

. 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 nuclei. All common ...

fields is an important property of the

chiral bag model In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number (nucleon number). Until the 1960s, nucleons we ...

. For

fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...

s, it is known as the

Witten index In quantum field theory and statistical mechanics, the Witten index at the inverse temperature β is defined as a modification of the standard partition function: :\textrm -1)^F e^/math> Note the (-1)F operator, where F is the fermion number ...

, and can be understood as describing the

Casimir effect In quantum field theory, the Casimir effect is a physical force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of the field. It is named after the Dutch physicist Hendrik Casimir, who pred ...

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 an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To a ...

. 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

* MF Atiyah, VK Patodi and IM Singer, ''Spectral asymmetry and Riemannian geometry I'', Proc. Camb. Phil. Soc., 77 (1975), 43-69. * Linas Vepstas, A.D. Jackson, A.S. Goldhaber, ''Two-phase models of baryons and the chiral Casimir effect'', Physics Letters B140 (1984) p. 280-284. * Linas Vepstas, A.D. Jackson, ''Justifying the Chiral Bag'', Physics Reports, 187 (1990) p. 109-143. {{SpectralTheory Spectral theory Asymmetry