Common Origin of Quantum Regression and Quantum Fluctuation Dissipation Theorems

It is shown that the quantum fluctuation dissipation theorem can be considered as a mathematical formulation in the spectral representation of Onsager hypothesis on the regression of fluctuations in physical systems. It is shown that the quantum fluctuation dissipation theorem can be generalized to an arbitrary stationary state. 1 Introduction. Under thermal equilibrium conditions the behavior of fluctuations of macroscopic observables of a physical system is governed by relationships which are formulated usually in terms of the regression theorem (the so called Onsager hypothesis [1]) and the fluctuation-dissipation theorem [2-4]. The former pertains to the time domain and states that the relaxation of a correlation of fluctuations is described by the same law governing the irreversible processes of the observable quantity itself. The latter pertains to the frequency domain and interrelates in some universal way the spectral characteristics of fluctuations and linear response (i.e. dissipation) of an observable of the physical system. In the framework of a classical approach, the two theorems complement each other providing a closed description of fluctuations and linear response in thermal equilibrium. By contrast, within the more general quantum approach there appears a conflict between these two theorems. This conflict is usually interpreted as a violation of the quantum regression theorem (QRT) (see, for example, Refs. [5-7]). In the most evident form such a violation is demonstrated in Ref. [7], where the conclusion " there is no quantum regression theorem " is announced. The proof of the general character of such a statement is based on the fact that the violation of QRT follows from the quantum fluctuation dissipation theorem (QFDT). A proof that QRT is valid independently of QFDT was given by Lax [8] on the basis of the general principles of quantum statistics (see also Refs. [9-11]). Since from Ref. [7] it is claimed that QFDT contradicts the validity of QRT, we argue that the origin of such a conflict is related with QFDT and its interpretation. The aim of this letter is to establish the origin of such a conflict and to provide arguments to reconcile the two theorems. For this sake, the conditions necessary for QRT to be fullfilled are firstly considered in the framework of the general formalism of linear response functions. Then, the equivalence between QFDT and QRT is proven by revisiting the derivation of QFDT. Finally, an extension of QFDT to an arbitrary stationary state which can be …