Positive Decompositions of Exponential Operators

The solution of many physical evolution equations can be expressed as an exponential of two or more operators. Approximate solutions can be systematically derived by decomposing the exponential in a product form. For time-reversible equations, such as the Hamilton or the Schrödinger equation, it is immaterial whether the decomposition coefficients are positive or negative. For timeirreversible systems, such as the Fokker-Planck equation or the quantum statistical propagator, only positive-coefficient decompositions which respect the time-irreversibility of the diffusion kernel can yield practical algorithms. These positive time step, or forward decompositions, are a highly restrictive class of factorization algorithms. This work proves a fundamental theorem: in order for a 2nth order forward algorithm to become (2n + 2)th order, one must include a new, higher order commutator in the decomposition process. Since these higher order commuatators are highly complex, it seems difficult to produce practical forward algorithms beyond fourth order. This proof generalize the Sheng-Suzuki theorem for the case of n = 1. In particular, this work shows that it is not possible to have a sixth order forward algorithm by including just the [V, [T, V ]] commutator.