Introduction: The development of adaptive resolution simulation schemes is a subject of growing interest within the community of condensed matter, material and chemical physics. By adaptive resolution it is meant that the space is partitioned in regions characterized by different molecular resolutions where molecules can freely diffuse changing their representa- tion according to the region where they are instantaneously located. In the last few years several approaches have been presented and they are characterized by different levels of theoretical sophistication and computational complexity [1–6]. The interest in this kind of approach arises from the fact that it may efficiently tackle the problem of interplay between different scales. However while the adaptive process can be described in a reasonable way according to the basic principles of classical dynamics and thermodynamics, the same can- not be said when quantum mechanics enters into the game. In general, the proper coupling of quantum and classical mechanics is known to be a non trivial (and open) problem (see e.g.[7]) and here the adaptive character adds up as a further major difficulty [8]. In this work we aim to develop an approach where the idea of coupling a classical and quantum molecular model in an adaptive fashion can be made in a way that the ”probabilistic-deterministic” discontinuity is removed and yet the adaptivity takes place in a smooth controlled numerical way. Path Integral representation of atoms and molecules: In this work our quantum sys- tems are composed by atoms described within the path integral formalism. This approach is, by now, a standard tool in molecular dynamics and well established in literature (see e.g. [9]). For this reason here we do not present a technical description of the method, instead we explain why this approach is optimal to adaptively couple quantum systems with a classical bath. The important aspect of the path integral approach for this work is that an atom, which is usually represented as a sphere in classical force fields, becomes, within the path integral formalism, a classical “polymer ring” so that the interaction site is delocalized into the beads of such a polymer; each bead of the polymer is linked to its next neighbors along the chain by a harmonic potential. The elastic constant depends on the temperature, T , of the system and the number of beads, n , used to represent each atom, k = m ( k$_{B}$T ) 2 n $^{2}$¯ h 2 , with k$_{B}$ being , as usual, the Boltzmann constant, m the mass of the atom, ¯ h Planck’s constant. Such a delocalization characterizes the quantum nature of the atom. This approach has been extensively used in molecular dynamics to account for basic quantum effects otherwise not described by standard classical force fields [10–15]. For the adaptive process the classical 2