Real-Valued Spacetime Dimensions, Black Hole Dualities, and the Riemann Zeta Function

Multidimensional spaces are widely used in mathematics and physics. It is well known that we live in a 3+1 spacetime. Models in lower-dimensional spacetimes are often employed as toy models to explore specific processes under simplified conditions.
In physics, models in higher-dimensional spacetimes are not merely mathematical generalizations of our observed reality; they can encode essential physical information. In this context, it is worth mentioning the Kaluza-Klein theory, superstring theory, and others. A special role is played by five-dimensional models, known as holographic models, in which the fifth coordinate serves as an auxiliary dimension representing the energy scale.
This talk will also address the emergence of negative dimensions, which appear when attempting to describe black hole thermodynamics using quantum statistical models. The origin of this phenomenon lies in the violation of the third law of thermodynamics (in Planck’s formulation) by the Schwarzschild black hole. To assign mathematical meaning to geometrical objects in negative dimensions, we use analytical continuation and properties of the Riemann zeta function.
The talk is based on:
I. Y. Aref'eva et al,“Holographic QCD running coupling for light quarks in strong magnetic field,”Phys. Rev. D 110, 086021 (2024)
I. Y. Aref'eva and I. V. Volovich,“Bose gas modeling of the Schwarzschild black hole thermodynamics,”Theor. Math. Phys. 218, 192-204 (2024)
I. Aref'eva and I.Volovich,“Violation of the Third Law of Thermodynamics by Black Holes, Riemann Zeta Function and Bose Gas in Negative Dimensions,”Eur. Phys. J. Plus 139, 300 (2024)