Spectral asymmetry

Given an operator with eigenvalues ${\displaystyle \omega _{n}}$, an equal number of which are positive and negative, the spectral asymmetry may be defined as the sum

${\displaystyle B=\lim _{t\to 0}{\frac {1}{2}}\sum _{n}\operatorname {sgn}(\omega _{n})\exp(-t|\omega _{n}|)}$

where ${\displaystyle \operatorname {sgn}(x)}$ is the sign function. 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.

As an example, consider an operator with a spectrum

${\displaystyle \omega _{n}=n+\alpha }$

where n is an integer, ranging over all positive and negative values. One may show in a straightforward manner that in this case ${\displaystyle B(\alpha )}$ obeys ${\displaystyle B(\alpha )=B(\alpha +m)}$ for any integer ${\displaystyle m}$, and that for ${\displaystyle 0<\alpha <1}$ we have ${\displaystyle B(\alpha )=1/2-\alpha }$. The graph of ${\displaystyle B(\alpha )}$ is therefore a periodic sawtooth curve.

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

${\displaystyle E=\lim _{t\to 0}{\frac {1}{2}}\sum _{n}|\omega _{n}|\exp(-t|\omega _{n}|)}$

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

• 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.