Subtle inconsistencies in the straightforward definition of the logarithmic function of annihilation and creation operators and a way to avoid them
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Using the resolution of identity spanned by coherent states of the harmonic oscillator, any entire function of the creation and the annihilation operators and its action on a vector in Hilbert space can be defined directly and simply. We show that such a direct approach applied to non-entire functions ln (a) over cap and ln (a) over cap (dagger), present in the literature, may lead to errors and contradictions. We elucidate their roots and propose a way to avoid them. We discuss the obtained results.
Source:Journal of Physics. A: Mathematical and Theoretical, 2009, 42, 23
- Ministry of Science and Environment Protection of the Republic of Serbia [141041G, 141003, 141036]