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Toral Diffeomorphisms Induce Quantum Superoperators via TAQS

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 343)

Abstract

We propose a new method for adapting (perturbing) models of quantum observables. The method is dubbed TAQS as it is based on toral automorphisms (diffeomorphisms) and the Q-transform, which together induce superoperators acting on observables. We demonstrate via examples that TAQS perturbations often lead to radical changes in the observables’ structure and spectra. This is a preliminary exploration in which emphasis is put on connections with some exciting canonical topics (the almost Mathieu operators), and with recent trends in the study of quantum metamaterials (fractal-structured operators).

Keywords

  • Quantum theory
  • Superoperators
  • Q-transform
  • Automorphisms of a torus

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Fig. 1

Notes

  1. 1.

    It is important that the basis index set consist of integers (rather than natural numbers). The finite-dimensional version of the Q-transform is also easy to interpret but requires an odd number of indices, [9].

  2. 2.

    More precisely, the direct sum of a discrete family of Hamiltonians is needed to model electron hopping on \(\mathbb {Z}^{2}\) lattice.

  3. 3.

    It is easily seen that \(I \in \mathcal {O}\), as is the case for other operators considered in this section.

  4. 4.

    Also, the problem of similarity of matrices over \(GL(2, \mathbb {Z})\) is nontrivial but well understood, [6].

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Acknowledgements

My thinking about the matters presented here has been influenced by the following people: John-Carl Bermodes, Robert Green, Natalia Janson, Bing-Zhao Li, Robert Moody, Raymond Spiteri, and Alexandre Zagoskin.

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Correspondence to Artur Sowa .

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Sowa, A. (2021). Toral Diffeomorphisms Induce Quantum Superoperators via TAQS. In: Kilgour, D.M., Kunze, H., Makarov, R., Melnik, R., Wang, X. (eds) Recent Developments in Mathematical, Statistical and Computational Sciences. AMMCS 2019. Springer Proceedings in Mathematics & Statistics, vol 343. Springer, Cham. https://doi.org/10.1007/978-3-030-63591-6_41

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