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arXiv:0707.1660

Accardi complementarity for -1/2 < mu < 0 and related results

Lenin A. Echavarria Cepeda, Claudio Pita Ruiz, Stephen Bruce Sontz

Published Jul 11, 2007• 0 citations
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Abstract

We show (an earlier conjecture of the last two authors) that the momentum and position operators of mu-deformed quantum mechanics for -1/2 < mu < 0 are not Accardi complementary. We also prove some related formulas that were conjectured by the same authors.
DOI
10.48550/arxiv.0707.1660

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In this note we show that the momentum and position operators of μμμ-deformed quantum mechanics for μ>0μ> 0μ>0 are not Accardi complementary in a sense that we will define. We conjecture that this is also true if −1/2<μ<0-1/2 < μ< 0−1/2<μ<0.
Abstract In this paper, we prove the sum and product uncertainty relations conjectured by V Dodonov for multiple observables. The uncertainty relations for linear combinations of position and momentum recently obtained by Kechrimparis and Weigert are recovered. Furthermore, the entropic uncertainty relations conjectured by the latter authors are proved for specific cases. At last, we revisit the uncertainty relation for triple canonical operators and obtain a tighter bound on real Hilbert space. A quantitative stability result is given as well.
Higher powers of qqq-deformed white noise
We introduce the renormalized powers of q-deformed white noise, for any q in the open interval (−1, 1), and we extend to them the no–go theorem recently proved by Accardi–Boukas–Franz in the Boson case. The surprising fact is that the lower bound (6.5), which defines the obstruction to the positivity of the sesquilinear form, uniquely determined by the renormalized commutation relations, is independent of q in the half-open interval (−1, 1], thus including the Boson case. The exceptional value q = −1, corresponding to the Fermion case, is dealt with in the last section of the paper where we prove that the argument used to prove the no–go theorem for q 6= 0 does not extend to this case.