GENERAL-THEORY OF FRACTAL PATH-INTEGRALS WITH APPLICATIONS TO MANY-BODY THEORIES AND STATISTICAL PHYSICS

A general scheme of fractal decomposition of exponential operators is presented in any order m. Namely, exp[x(A + B)] = S(m)(x) + O(x(m + 1)) for any positive integer m, where S(m)(x) = e(t1A) e(t2B) e(t3A) e(t4B)...e(tMA) with finite M depending on m. A general recursive scheme of construction of {t(j)} is given explicitly. It is proven that some of {t(j)} should be negative for m greater-than-or-equal-to 3 and for any finite M (nonexistence theorem of positive decomposition). General systematic decomposition criterions based on a new type of time-ordering are also formulated. The decomposition exp[x(A + B)] = [S(m)(x/n)]n + O(x(m + 1)/n(m)) yields a new efficient approach to quantum Monte Carlo simulations.