General theory of fractal path integrals with applications to many-body theories and statistical physics

Abstract

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^t_MA 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≥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.