Remarks on the zeta-regularized determinant of differential operators

Abstract

It is shown that the equalities det $\left( {\not D + im} \right) = \det \left( {\not D - im} \right) = \sqrt {\det \left( {{{\not D}^2} + {m^2}} \right)}$, where D̸ is the massless Dirac operator, hold if and only if: (i) det(D̸ + im) and det(D̸ - im) are defined by using the symmetry property of the spectrum of D̸; (ii) Z(D̸ ² + m ², 0) is an even integer (the sign of the square root above is positive if Z(D̸ ² + m ²,0) is divisible by 4, and negative otherwise). On the other hand, the equality $\left| {\det \left( {\not D \pm im} \right)} \right| = \sqrt {\det \left( {{{\not D}^2} + {m^2}} \right)}$ is always holding. It is also shown, by applying the standard definition of zeta-determinant, that det(D̸ + im) is not equal to det(D̸ - im) in general. In order to show this fact, a variational formula for det(D + m) with respect to m is derived for general self-adjoint operators.