Multidimensional spaces are widely used in mathematics and physics. This talk will discuss the emergence of negative dimensions in black hole thermodynamics and quantum cosmology. Most black holes violate the third law of thermodynamics. We are looking for quantum statistical models with such exotic thermodynamic behavior. It is shown that a Schwarzschild black hole in D=4 spacetime dimensions corresponds to a Bose gas in space with d=-4 spatial dimensions. The Riemann zeta function is used to determine the entropy of a Bose gas in negative dimensions. An approach to compute the cosmological constant in the early universe is proposed. Assuming that the creation and evolution of the universe obey the second law of thermodynamics, and the "pre-Big Bang" state of the universe was a sphere of negative dimension, then in the semiclassical approximation to the Wheeler-DeWitt equation, the entropy reaches its maximum value at the de Sitter spacetime with the observed dimension D = 4 with cosmological constant $\Lambda= 3\pi\,\exp\{-\psi(3/2)\}\approx 9.087$, where $\psi$ is the digamma function.
The talk is based on:
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,'' arXiv:2304.04695.
I.Aref'eva and I.Volovich, ''Bose Gas Modeling of the Schwarzschild Black Hole Thermodynamics,'' arXiv:2305.19827.