We demonstrate that the quantum-classical transition can be understood as a phasetransition in the scale parameter of the fractal monad hierarchy. Within this frame-work, quantum systems correspond to regimes at small scale parameters where theregulator field P(s, t) exhibits high variance and chaotic fluctuations, while classicalsystems correspond to large scale parameters where P(s, t) becomes stable and pre-dictable. We identify a critical scale sc marking this transition and show that it dependson physical parameters including temperature, Planck’s constant, and system energy.Using both analytical arguments and numerical simulations, we calculate critical expo-nents and demonstrate that the transition belongs to the universality class of directedpercolation. This work provides a novel explanation for quantum decoherence and theemergence of classicality from quantum dynamics.
We demonstrate that the quantum-classical transition can be understood as a phasetransition in the scale parameter of the fractal monad hierarchy. Within this frame-work, quantum systems correspond to regimes at small scale parameters where theregulator field P(s, t) exhibits high variance and chaotic fluctuations, while classicalsystems correspond to large scale parameters where P(s, t) becomes stable and pre-dictable. We identify a critical scale sc marking this transition and show that it dependson physical parameters including temperature, Planck’s constant, and system energy.Using both analytical arguments and numerical simulations, we calculate critical expo-nents and demonstrate that the transition belongs to the universality class of directedpercolation. This work provides a novel explanation for quantum decoherence and theemergence of classicality from quantum dynamics.
Classical and quantum criticality In most cases the phases of quantum field theories, in particular those of interest in condensed matter physics, can be described in terms of the behavior of local observables, such as order parameters or currents that transform properly under the symmetries of the theory. Quantum and thermal phase transitions are characterized by the behavior of these observables as a function of temperature and of the coupling constants of the theory. The phase transitions themselves, quantum or thermal, are classified into universality classes, which are represented by the critical exponents which specify the scaling laws of the expectation values of the observables. Historically, the development of this approach to critical behavior goes back to the Landau theory of critical behavior. It acquired its most complete form with the development of the renormalization group (RG) in the late 1960s and early 1970s. It is the centerpiece of Wilson’s approach to quantum field theory, in which all local quantum field theories are defined by the scaling regime of a physical system near a continuous phase transition. From this point of view there is no fundamental difference between classical (or thermal) phase transitions, which are described by the theory of classical critical behavior, and quantum phase transitions. For example, the expectation value of a local order parameter M as the thermal phase transition is approached from below behaves as M ~ (Tc − T) β . Here T c is the critical temperature and β is a critical exponent that depends on the universality class of the thermal phase transition and on the dimensionality of space. While quantum mechanics can play a key role in the existence of the ordered phase, e.g. superfluidity and superconductivity are macroscopic manifestations of essentially quantum-mechanical phenomena, the thermal transition itself is governed entirely by classical statistical mechanics, and quantum mechanics plays a role in setting the value of non-universal quantities such as the critical temperature, etc. On the other hand, in the case of a quantum phase transition, the order parameter M has a similar scaling behavior as a function of the coupling constant, M~(gc−g)β˜, where g is the coupling constant, g c is the critical coupling constant, and β˜ is a critical exponent that depends on the universality class of the quantum phase transition. Here we assume that M has a non-vanishing expectation value only for g < g c .
Quantum fluctuations are usually regarded as a background effect that perturbs classical order. In this work we propose that quantum fluctuations themselves can undergo a phase transition, a phenomenon we term the Quantum Fluctuation Phase Transition (QFPT). We develop a theoretical framework that treats the amplitude of quantum fluctuations as an order parameter, allowing a direct mapping between the fluctuation strength and the free‑energy landscape. Using a transverse‑field Ising model with tunable long‑range couplings, we derive a mean‑field free energy that exhibits a discontinuity in the fluctuation amplitude at a critical field strength. We further analyze the critical scaling, demonstrating that the QFPT belongs to a new universality class distinct from conventional quantum critical points. The theory predicts observable signatures in the low‑temperature heat capacity, dynamic structure factor, and entanglement entropy, and suggests that extreme‑cold atom or solid‑state systems with controllable interaction ranges could realize QFPTs. This novel mechanism for quantum phase change offers a route to discovering new quantum phases and could have implications for quantum technologies that rely on fluctuation‑driven properties.
The quantum to classical transition for a system depends on many parameters, including a scale length for its action, variant Planck's over 2 pi, a measure of its coupling to the environment, D, and, for chaotic systems, the classical Lyapunov exponent, lambda. We propose measuring the proximity of quantum and classical evolutions as a multivariate function of (Planck's over 2 pi,lambda,D) and searching for transformations that collapse this hypersurface into a function of a composite parameter zeta= Planck's over 2 pi alpha)lambda beta D gamma. We report results for the quantum Cat Map and Duffing oscillator, showing accurate scaling behavior over a wide parameter range, indicating that this may be used to construct universality classes for this transition.
The principle of action plays a central role in both classical and quantum mechanics, yet its physical meaning is often introduced in a purely formal manner. In this work, the concept of action is examined from a scale-based perspective that clarifies its role in the quantum–classical correspondence. By emphasizing the dimensionless ratio S/ℏ, we show that the emergence of classical dynamics can be understood as a consequence of quantum phase coherence rather than as an independent postulate. Within the path integral formulation, classical trajectories arise naturally from stationary-phase conditions when the action associated with a physical process becomes large compared to Planck’s constant. This interpretation provides a unified conceptual framework in which variational principles, quantum interference, and classical determinism are linked through the scale of action. The analysis highlights the role of Planck’s constant as a universal parameter governing the physical relevance of alternative histories, analogous to the function of Boltzmann’s constant in statistical physics. Beyond its theoretical implications, the proposed perspective offers pedagogical value by providing a coherent bridge between classical and quantum mechanics suitable for advanced undergraduate and early graduate physics education.
We study how decoherence rules the quantum-classical transition of the kicked harmonic oscillator. The system presents classical dynamics that ranges from regular to strong chaotic behavior depending on the amplitude of the kicks. We show that for regular and mixed classical dynamics, and in the presence of noise, the distance between the classical and quantum phase space distributions is proportional to a single parameter chi identical to K Planck's (eff)(2)/4D(3/2) , which relates the effective Planck constant, Planck's (eff), to the kicking strength, K, and the diffusion constant, D. This relation between classical and quantum distributions is valid when chi<1 , a case that is always attainable in the semiclassical regime, independent of the value of the strength of noise given by D. Our results extend a recent study performed in the chaotic regime.
We investigate how decoherence affects the short-time separation between quantum and classical dynamics for classically chaotic systems, within the framework of a specific model. For a wide range of parameters, the distance between the corresponding phase-space distributions depends on a single parameter $\ensuremath{\chi}$ that relates an effective Planck constant ${\ensuremath{\hbar}}_{\mathrm{eff}}$, the Lyapunov coefficient, and the diffusion constant. This distance peaks at a time that depends logarithmically on ${\ensuremath{\hbar}}_{\mathrm{eff}}$, in agreement with previous estimations of the separation time for Hamiltonian systems. However, for $\ensuremath{\chi}\ensuremath{\lesssim}1$, the separation remains small, going down with ${\ensuremath{\hbar}}_{\mathrm{eff}}^{2}$, so the concept of separation time loses its meaning.