Hyperoctahedral Chen calculus for effective Hamiltonians

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 or quantum field theory. We show, among other things, 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. Introduction We start with a short overview of the classical theory of Chen calculus, that is, iterated integral computations. The subject is classical but is rarely presented from the suitable theoretical perspective -that is, emphasizing the role of the convolution product on the direct sum of the symmetric groups group algebras. We give therefore a brief account of the theory that takes into account this point of view -this will be useful later in the article. Then, we recall the construction of effective Hamiltonians in the time-dependent interaction representation. The third section is devoted to the investigation of the structure of the hyperoctahedral group algebras. Although we are really interested in the applications of these objects to the study of effective Hamiltonians, and although the definitions we introduce are motivated by the behavior of the iterated integrals showing up in this setting, we postpone the description of the way the two theories interact to a later stage of the article. Roughly stated, we show that the descent algebra approach to Lie calculus, as emphasized in Reutenauer’s [25] can be lifted to the hyperoctahedral setting. This extends previous works [14, 1, 5, 2, 19] on the subject and shows that these results (focussing largely on Solomon’s algebras of hyperoctahedral groups and other wreath product group algebras) are naturally connected to the study of physical systems through the properties of their Hamiltonians and of the corresponding differential equations, very much as the classical theory of free Lie algebras relates naturally to the study of differential equations and topological groups. Notice however that the statistics we introduce here on hyperoctahedral groups seems to be new –and is different from the statistics naturally associated to the noncommutative representation theoretic approach to hyperoctahedral groups, as it appears in these works. The fourth section studies the effective adiabatic evolution operator and shows that it can be expanded 1 as a generalized Picard series by means of the statistics introduced on hyperoctahedral groups. As a corollary, we derive in the last section an exponential expansion for the evolution operator. Such expansions are particularly useful in view of numerical computations, since they usually lead to approximating series converging much faster than the ones obtained from the Picard series. Acknowledgements. We thank Kurusch Ebrahimi-Fard, Jean-Yves Thibon and an anonymous referee. Their comments and suggestions resulted into several improvements with respect to a preliminary version of this article. 1 The algebra of iterated integrals Let us recall the basis of Chen’s iterated integrals calculus, starting with a first order linear differential equation (with, say, operator or matrix coefficients): A(t) = H(t)A(t), A(0) = 1 The solution can be expanded as the Picard series: