An exact formulation of the time-ordered exponential using path-sums

Abstract

We present the path-sum formulation for OE [H](t ′ , t) = T exp t ′ t H(τ ) dτ , the time-ordered exponential of a time-dependent matrix H(t). The path-sum formulation gives OE[H] as a branched continued fraction of finite depth and breadth. The terms of the path-sum have an elementary interpretation as selfavoiding walks and self-avoiding polygons on a graph. Our result is based on a representation of the time-ordered exponential as the inverse of an operator, the mapping of this inverse to sums of walks on graphs and the algebraic structure of sets of walks. We give examples demonstrating our approach. We establish a super-exponential decay bound for the magnitude of the entries of the timeordered exponential of sparse matrices. We give explicit results for matrices with commonly encountered sparse structures.Keywords: Time-ordered exponential, path-ordered exponential, path-sum, differential equation, finite continued fraction, super-exponential decay ContextThe time-ordered exponential function OE[H](t ′ , t) = T exp t ′ t H(τ ) dτ , also known as path-ordered exponential, is the unique solution of the system of differential equationssuch that OE[H](t ′ , t ′ ) = I is the identity at all times. We take H ∈ C n×n [I], n ∈ N\{0}, to be a matrix depending smoothly on the continuous variable t ∈ I, which we call time without loss of generality. In spite of the importance of the

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