On the concept of determinant for the differential operators of Quantum Physics

Abstract

The concept of determinant for a linear operator in an infinite-dimensional space is addressed, by using the derivative of the operator's zeta-function (following Ray and Singer) and, eventually, through its zeta-function trace. A little play with operators as simple as $\pm I$ ($I$ being the identity operator) and variations thereof, shows that the presence of a non-commutative anomaly (i.e., the fact that det $(AB) \neq$ det $A$ det $B$), is unavoidable, even for commuting and, remarkably, also for almost constant operators. In the case of Dirac-type operators, similarly basic arguments lead to the conclusion ---contradicting common lore--- that in spite of being $\det (\slash D +im) = \det (\slash D -im)$ (as follows from the symmetry condition of the $\slash D$-spectrum), it turns out that these determinants may {\it not} be equal to $\sqrt{\det (\slash D^2 +m^2)}$, simply because $\det [(\slash D +im) (\slash D -im)] \neq \det (\slash D +im) \det (\slash D -im)$. A proof of this fact is given, by way of a very simple example, using operators with an harmonic-oscillator spectrum and fulfilling the symmetry condition. This anomaly can be physically relevant if, in addition to a mass term (or instead of it), a chemical potential contribution is added to the Dirac operator.