Time-Ordering and a Generalized Magnus Expansion |
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Authors: | Michel Bauer Raphael Chetrite Kurusch Ebrahimi-Fard Frédéric Patras |
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Institution: | 1. Institut de Physique Théorique de Saclay, CEA-Saclay, 91191, Gif-sur-Yvette, France 2. Laboratoire J.-A. Dieudonné UMR 7351, CNRS, Parc Valrose, 06108, Nice Cedex 02, France 3. ICMAT, C/Nicolás Cabrera, no. 13-15, 28049, Madrid, Spain
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Abstract: | Both the classical time-ordering and the Magnus expansion are well known in the context of linear initial value problems. Motivated by the noncommutativity between time-ordering and time derivation, and related problems raised recently in statistical physics, we introduce a generalization of the Magnus expansion. Whereas the classical expansion computes the logarithm of the evolution operator of a linear differential equation, our generalization addresses the same problem, including, however, directly a non-trivial initial condition. As a by-product we recover a variant of the time-ordering operation, known as ${\mathsf{T}^\ast}$ -ordering. Eventually, placing our results in the general context of Rota–Baxter algebras permits us to present them in a more natural algebraic setting. It encompasses, for example, the case where one considers linear difference equations instead of linear differential equations. |
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