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- W2047079413 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." @default.
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- W2047079413 date "1999-07-14" @default.
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- W2047079413 title "On the concept of determinant for the differential operators of quantum physics" @default.
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- W2047079413 doi "https://doi.org/10.1088/1126-6708/1999/07/015" @default.
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