Time-dependent fractional dynamics with memory in quantum and economic physics

Fractional dynamics of open quantum systems and sectors of national economies, where the parameters depend on time, are discussed. We show that the quantum and economic processes can demonstrate the same dynamic behavior caused by effects of power-law fading memory. In this paper, we propose generalizations of time-ordered exponential (T-exponential) and time-ordered product (T-product) for processes with power-lawmemory. The expressions of time-ordered exponential with memory and corresponding generalization time-ordered product are derived by using matrix fractional differential equations. In quantum physics, we consider equations of N-level open quantum system with memory, quantum oscillator with friction and memory. In economic physics (econophysics), we use equations of dynamic intersectoral model with power-law memory, where the matrix of direct material costs and the matrix of incremental capital intensity of production depend on time. The solutions of these equations with derivatives of non-integer orders are suggested.