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Article

From Fractional Quantum Mechanics to Quantum Cosmology: An Overture

1
Departamento de Física and Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, Rua Marquês d’Ávila e Bolama, 6200 Covilhã, Portugal
2
Departmento de Física, Universidade Federal de Pernambuco, Pernambuco, PE 52171-900, Brazil
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(3), 313; https://doi.org/10.3390/math8030313
Received: 27 December 2019 / Revised: 19 February 2020 / Accepted: 24 February 2020 / Published: 1 March 2020
(This article belongs to the Special Issue Mathematical and Computational Cosmology)
Fractional calculus is a couple of centuries old, but its development has been less embraced and it was only within the last century that a program of applications for physics started. Regarding quantum physics, it has been only in the previous decade or so that the corresponding literature resulted in a set of defying papers. In such a context, this manuscript constitutes a cordial invitation, whose purpose is simply to suggest, mostly through a heuristic and unpretentious presentation, the extension of fractional quantum mechanics to cosmological settings. Being more specific, we start by outlining a historical summary of fractional calculus. Then, following this motivation, a (very) brief appraisal of fractional quantum mechanics is presented, but where details (namely those of a mathematical nature) are left for literature perusing. Subsequently, the application of fractional calculus in quantum cosmology is introduced, advocating it as worthy to consider: if the progress of fractional calculus serves as argument, indeed useful consequences will also be drawn (to cite from Leibnitz). In particular, we discuss different difficulties that may affect the operational framework to employ, namely the issues of minisuperspace covariance and fractional derivatives, for instance. An example of investigation is provided by means of a very simple model. Concretely, we restrict ourselves to speculate that with minimal fractional calculus elements, we may have a peculiar tool to inspect the flatness problem of standard cosmology. In summary, the subject of fractional quantum cosmology is herewith proposed, merely realised in terms of an open program constituted by several challenges. View Full-Text
Keywords: quantum cosmology; fractional calculus; early universe quantum cosmology; fractional calculus; early universe
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MDPI and ACS Style

Moniz, P.V.; Jalalzadeh, S. From Fractional Quantum Mechanics to Quantum Cosmology: An Overture. Mathematics 2020, 8, 313. https://doi.org/10.3390/math8030313

AMA Style

Moniz PV, Jalalzadeh S. From Fractional Quantum Mechanics to Quantum Cosmology: An Overture. Mathematics. 2020; 8(3):313. https://doi.org/10.3390/math8030313

Chicago/Turabian Style

Moniz, Paulo V., and Shahram Jalalzadeh. 2020. "From Fractional Quantum Mechanics to Quantum Cosmology: An Overture" Mathematics 8, no. 3: 313. https://doi.org/10.3390/math8030313

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