Rethinking the Hierarchy: On the Structural Relation Between Quantum and Classical Theories

Abstract

Quantum mechanics is among the most successful physical theories, yet its formulation and empirical testing rely on classical structures. Following Lev Landau and Niels Bohr, this reliance is not merely pragmatic: quantum observables acquire empirical meaning only relative to classical reference frames, and, in practice, quantization starts from classical models. At the same time, the two domains display forms of mutual irreducibility: intrinsically quantum features (that is, spin and exchange statistics) have no counterpart in the phase-space ontology of classical point-particle mechanics, while classical trajectory chaos does not arise straightforwardly from unitary quantum evolution in closed systems. A hierarchy is commonly established between classical and quantum theories, namely, a claim of ontological and explanatory priority according to which quantum mechanics is fundamental and classical mechanics is only a limiting case. This claim is less secure than is often assumed; therefore, the traditional hierarchy deserves to be examined. In this paper, we argue that a quantum–classical framework provides an effective and structurally faithful representation of empirically accessible physical systems in regimes where quantum and classical degrees of freedom coexist within a single, consistent effective dynamical description. To give this point of view a firm theoretical basis, we discuss the quasi-Lie formal structure underlying quantum–classical hybrid dynamics, with applications ranging from gravity and condensed matter to open, driven systems in biology and complex media.