Abstract. The problem of the initial conditions for the oscillator model of quantum dissipative systems is studied. It is argued that, even in the classical case, the hypothesis that the environment is in thermal equilibrium implies a statistical correlation between environment oscillators and central system. A simple form of initial conditions for the quantum problem, taking into account such a correlation in analogy with the classical ones, is derived on the base of symmetry considerations. The same symmetries also determine unambiguously the form of the Lagrangian. As a check of the new form of correlated initial conditions (and of that of the Lagrangian), the problem of a forced Brownian particle under the action of arbitrary colored noise is studied: it is shown that one obtains an average position of a quantum wave packet equal to that of the corresponding classical Brownian particle. Instead, starting from uncorrelated initial conditions based on the factorization hypothesis or from a different form of Lagrangian, non-physical results are obtained. Similar considerations apply also to the mean square displacement.
I IntroductionThe oscillator model of quantum dissipative systems [4,5,14,[22][23][24][25], through the influence functional approach pioneered by Feynman and Vernon [2], has been fruitfully applied to several problems, in which both quantum and statistical fluctuations play a significant role. Important examples are provided by quantum Brownian motion [2,7,19], dissipative tunneling [3,13], and localization-delocalization transitions in periodic potentials [1,18].In the oscillator model, the system under study, which will be assumed to have one degree of freedom x and referred to as the central system, is coupled to an infinite set of harmonic oscillators of coordinates q = {q 1 , q 2 , . . .}, representing the environment. The description of the central system at a generic time t b is made, in the coordinate representation, through the reduced density ma-where dq (. . . ) ≡ n dq n (. . . ). In order to determine the time evolution law of the reduced density matrix, one has to assign the initial conditions of the total system, that is the total density matrix ρ(. A particular kind of initial conditions, considered previously in the literature [2,5,[22][23][24][25], is based on the factorization hypothesis, Here ρ(x a , x ′ a , t a ) is the density matrix of the central system and ρ β (q a , q ′ a , t a ), that represents the initial state of the environment, describes oscillators in thermal equilibrium at an inverse temperature β = 1/k B T . According to Eq. (2), there is no statistical correlation between the central system and the environment at t = t a . Unfortunately, the simple factorization hypothesis leads to nonphysical results, as shown in Sec. V.Later on, the importance of an initial correlation for the dynamics of the central system was recognized [8] and more general forms of correlated initial conditions were studied for the quantum problem [7,[19][20][21].As discussed below, the f...