Axiomatic basis for spaces with noninteger dimension

TLDR

For integer D, 𝒮D behaves like a conventional Euclidean vector space, whereas for noninteger D it exhibits nonvector character. The authors propose five axioms that generate a space 𝒮D with noninteger dimension D and outline rudiments for extending quantum and classical mechanics to such spaces, including discussion of experimental measurement of D. They introduce five axioms—four topological and one defining an integration measure—to construct 𝒮D, and outline rudiments for extending Schrödinger wave mechanics and classical statistical mechanics to noninteger dimensions. The constructed spaces 𝒮D satisfy the informal use of continuously variable D in recent physical contexts, allow more mutually perpendicular lines than the nominal dimension D, and admit derived integration rules and a generalized Laplacian operator.

Abstract

Five structural axioms are proposed which generate a space 𝒮D with ’’dimension’’ D that is not restricted to the positive integers. Four of the axioms are topological; the fifth specifies an integration measure. When D is a positive integer, 𝒮D behaves like a conventional Euclidean vector space, but nonvector character otherwise occurs. These 𝒮D conform to informal usage of continuously variable D in several recent physical contexts, but surprisingly the number of mutually perpendicular lines in 𝒮D can exceed D. Integration rules for some classes of functions on 𝒮D are derived, and a generalized Laplacian operator is introduced. Rudiments are outlined for extension of Schrödinger wave mechanics and classical statistical mechanics to noninteger D. Finally, experimental measurement of D for the real world is discussed.