Hyperoctahedral Chen calculus for effective Hamiltonians

Details Hyperoctahedral Chen calculus for effective Hamiltonians Hyperoctahedral Chen calculus for effective Hamiltonians

2008-11-29

arxiv.org/abs/0812.0061v3

Journal of Algebra 322, 11 (2009) 4105-4120

Physics

Mathematical Physics

Minor corrections. Some misleading notations and typos in the first version have been fixed

Scientific paper

10.1016/j.jalgebra.2009.07.017

The algebraic structure of iterated integrals has been encoded by Chen. Formally, it identifies with the shuffle and Lie calculus of Lyndon, Ree and Sch\"utzenberger. It is mostly incorporated in the modern theory of free Lie algebras. Here, we tackle the problem of unraveling the algebraic structure of computations of effective Hamiltonians. This is an important subject in view of applications to chemistry, solid state physics, quantum field theory or engineering. We show, among others, that the correct framework for these computations is provided by the hyperoctahedral group algebras. We define several structures on these algebras and give various applications. For example, we show that the adiabatic evolution operator (in the time-dependent interaction representation of an effective Hamiltonian) can be written naturally as a Picard-type series and has a natural exponential expansion.

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