Axiomatic basis for spaces with noninteger dimension 论文

摘要

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.