From Fractional Quantum Mechanics to Quantum Cosmology: An Overture

Abstract

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.

Revisiting Fractional Cosmology

“…TABLE VII: Eigenvalues for the equilibrium points/sets of the system ( 79), ( 80), ( 81), ( 82) that satisfy the compatibility conditions ( 75) and (83).…”

“…Specific areas of interest include fractional quantum mechanics and gravity for fractional spacetime [78,79] and fractional quantum field theory at positive temperature [80,81]. Other applications of Quantum Cosmology can be found in [82][83][84][85]. In addition, fractional calculus has recently been explored to address problems related to cosmology in .…”

“…Table VI shows the equilibrium points/sets of the system ( 79), ( 80), ( 81), ( 82) that satisfy the compatibility conditions ( 75) and (83). The analysis of the equilibrium points with A = 0 is omitted here due to physical considerations.…”

de Sitter Fractional Quantum Cosmology

“…The WDW equation ( 4) is a one-dimensional ES with zero energy. Thus, to construct a particular fractional WDW equation, we may, similarly to the SE case, replace the ordinary derivative for the fractional Riesz derivative, namely [20,21,41]…”

“…Furthermore, in the last few years, FQM has been employed as a means to explore features within quantum field theory and gravity for fractional space-time [16,17], and the fractional quantum field theory at positive temperature [18,19]. It has pointed to exciting opportunities; please see [16,20,21], for a recent survey.…”