关于久保广义累积量理论的建立。重新审视和纠正的方法
About the foundation of the Kubo Generalized Cumulants theory. A revisited and corrected approach
摘要：
五十多年前，在一对夫妇的开创性工程久保介绍了重要的思想广义累积量，扩展到随机算子这一概念，隐含介绍了拉普拉斯在1810年。久保的想法已被应用在几个分支的物理学，其中的结果，平均过程是一个刘维尔运营商或一个有效的时间演化运营商的密度矩阵的自旋系统或减少密度矩阵的玻色子费米子等尽管这一成功，理论的发展，这些久保工程提出的问题，突出了多年前由福克斯和范坎彭和从未解决。这些弱点和错误，特别是关于累积量的指数的因式分解性质和给出广义累积量的广义矩的显式表达式，反之亦然，引起了一些困惑（和混乱）关于这个过程的可能应用，限制了它的使用，在实践中。在本文中，我们给了一个健全的理由的方法，累积算子，工作在一个一般的框架，显示了潜力的老久保的想法，今天尚未充分利用。结果表明，对于相同的矩算子，可以采用不同的广义累积量定义。虽然不能得到用矩算子表示累积量算子的Kubo-Meeron封闭公式，但给出了用矩算子表示累积量的Kubo-Meeron封闭公式，而且形式上它{ \em}不依赖于矩的具体性质，而只依赖于广义累积量的定义.
英文摘要：
More than fifty years ago, in a couple of seminal works Kubo introduced the important idea of generalized cumulants, extending to stochastic operators this concept, implicitly introduced by Laplace in 1810. Kubo's idea has been applied in several branches of physics, where the result of the average process is a Lioville operator or an effective time evolution operator for the density matrix of spin systems or the reduced density matrix for boson-fermions etc. Despite this success, the theoretical developments in these Kubo works pose problems that were highlighted many years ago by Fox and van Kampen and never solved. These weaknesses and errors, in particular concerning the factorization property of exponentials of cumulants and the explicit expressions that give generalized cumulants in terms of generalized moments and vice-versa, caused some perplexity (and confusion) about the possible application of this procedure, limiting its use, in practice. In the present paper, we give a sound ground to the approach to cumulant operators, working in a general framework that shows the potentiality of the old Kubo's idea, today not yet fully exploited. It results that for the same moment operators, different definitions of generalized cumulants can be adopted. A general Kubo-Meeron closed-form formula giving cumulant operators in terms of moment operators cannot be obtained, but the reverse one, cumulants in terms of operators, is given and, noticeably, formally it {\em does not} depend on the specific nature of the moments, but just on the definition of the generalized cumulants.
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