Prospecting black hole thermodynamics with fractional quantum mechanics

Abstract

This paper investigates whether the framework of fractional quantum mechanics can broaden our perspective of black hole thermodynamics. Concretely, we employ a space-fractional derivative (Riesz in Acta Math 81:1, 1949) as our main tool. Moreover, we restrict our analysis to the case of a Schwarzschild configuration. From a subsequently modified Wheeler–DeWitt equation, we retrieve the corresponding expressions for specific observables. Namely, the black hole mass spectrum, M, its temperature T, and entropy, S. We find that these bear consequential alterations conveyed through a fractional parameter, $$\alpha $$ α . In particular, the standard results are recovered in the specific limit $$\alpha =2$$ α = 2 . Furthermore, we elaborate how generalizations of the entropy-area relation suggested by Tsallis and Cirto (Eur Phys J C 73:2487, 2013) and Barrow (Phys Lett B 808:135643, 2020) acquire a complementary interpretation in terms of a fractional point of view. A thorough discussion of our results is presented.

Quantum Cosmology in $f(Q)$ theory

“…What is more interesting, is the emergence of the need for the theory of fractional Quantum Mechanics in the description of the system. In a recent work [68], a connection between quantum modifications in terms of fractional derivatives in the Hamiltonian constraint operator and generalizations of the black hole entropy, like the Barrow entropy [69] (for further applications see [70][71][72][73]), has been explored. It is rather intriguing that in the case of a perfect fluid in non-linear f (Q) theory, such a modification appears in a natural manner already from the classical constraint.…”

Inflation and Fractional Quantum Cosmology

“…, N, and F denotes Fourier transformation. Hence, the fractional counterpart of the WDW Equation ( 23) will be [14,[33][34][35]59]…”

“…Hence the final step of our procedure here is to obtain a particular fractional WDW equation similar to that proposed above for the SE case. Consequently, we can replace the ordinary derivative with the fractional Riesz derivative [14,[33][34][35]59]:…”

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]…”