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.

1. Introduction

In its canonical formulation, quantum theory is constructed by promoting classical dynamical variables to operators acting on a Hilbert space [1,2], foreshadowing the idea that the classical description logically precedes the quantum one as its semantic and structural framework. The canonical construction of quantum theory relies on classical phase-space structures, which provide the conceptual and mathematical scaffolding for the quantization procedure. At the same time, quantum mechanics is widely regarded as the more fundamental framework, from which classical behavior is expected to emerge in an appropriate limit. The coexistence of these two claims, i.e., structural reliance in formulation and ontological primacy in interpretation, reveals a tension that has never been fully clarified. Such considerations do not change if one follows, for example, path integral quantization.

The discussion becomes even more subtle when one considers axiomatic quantum field theory [3,4]. In this framework there is no quantization procedure: the theory is formulated directly in terms of quantum operators and the axioms they must satisfy. Nevertheless, space-time variables remain classical. In order to recover the standard model of particle physics, space-time is typically treated as a fixed background. However, if stronger couplings between quantum fields and space-time are considered, the classical geometry may itself evolve. In such situations a quantum–classical description naturally emerges once again. A useful analogy arises in the Born–Oppenheimer approximation of molecular quantum mechanics, where electrons move, on relatively shorter time scales, in the nearly static field generated by the nuclei [5]. When nonadiabatic effects become important, this approximation breaks down and the nuclear degrees of freedom necessarily enter into play in the overall dynamics of the system [5].

Thus, the standard hierarchy assumes a unidirectional relation,

$$\mathrm{quantum}\to \mathrm{classical}\mathrm{limit},$$

(1)

according to which classical mechanics arises as an approximation to an underlying autonomous quantum dynamics. Here, by autonomy we mean the assumption that quantum theory forms a logically closed and universally valid dynamical framework, from which all classical phenomena can, in principle, be derived without introducing additional classical structures. However, this autonomy is not anchored to a solid theoretical basis; rather, it is a guiding assumption about the scope and completeness of the formalism.

Quantum predictions require classical reference frames and measurement apparatuses for their empirical articulation, and without such structures the operational meaning of the theory becomes ill-defined. Lev Landau has emphasized [6] that quantum mechanics occupies a peculiar logical position among physical theories, insofar as it presupposes the classical domain for its own formulation. Measurement theory epitomizes this asymmetry [7,8]: although macroscopic instruments are composed of quantum constituents, a fully closed account of how definite classical outcomes arise from unitary many-body evolution remains elusive. Decoherence clarifies the suppression of interference through environmental monitoring [9,10], yet the emergence of definite outcomes still relies, at some stage, on classical structures [11,12]. The disappearance of the off-diagonal elements of the density matrix in a selected basis does not by itself remove the probabilistic content of the theory; from a classical perspective, an account is still needed of how one definite outcome is registered among the possible alternatives [12]. This is different from the deterministic structure of classical mechanics.

At this point, it is worth emphasizing that the present study does not aim to take a position on the different interpretations of quantum mechanics. Nevertheless, the question of the quantum–classical boundary is closely connected with the measurement problem, decoherence, and macro-objectivation. General reviews and discussions of these issues can be found, for example, in Refs. [13,14,15,16].

Evidence for the relevance of quantum–classical dynamics emerges from several independent directions, including measurement theory, the algebraic structure of hybrid dynamics, and the empirical modeling of complex physical systems. Together, these considerations motivate a re-examination of the assumed hierarchy between the two theories. Rather than viewing classicality merely as a limiting case of quantum mechanics, one may ask whether the relation between the two domains is structurally more balanced. Intrinsic angular momentum and other distinctly quantum properties, such as exchange symmetry, have no classical analogue, while classical chaos does not straightforwardly arise from unitary quantum dynamics, appearing instead through spectral statistics only in particular limits or under coarse-graining [17,18]. Such phenomena suggest a form of mutual irreducibility: neither theory can be reduced to the other without leaving residual structures unexplained [19]. In this paper, we argue that a consistent quantum–classical framework provides a more faithful structural account of empirically accessible physical systems in regimes where quantum and classical degrees of freedom must be coupled within a single dynamical scheme. We focus in particular on the quantum–classical Liouville equation (QCLE) [20,21] and its quasi–Lie bracket (QL) structure, which furnish a mathematically coherent effective realization of the structural relation between quantum and classical mechanics. Within this setting, the relation between quantum and classical theories is not framed as a strict hierarchy, but as a hybrid formal framework applicable to measurement-motivated models, chemical dynamics, and complex open systems [22,23,24].

Section 2 discusses the structural relation between classical and quantum mechanics and motivates a consistent hybrid quantum–classical (HQC) framework by examining physical regimes in which classical and quantum degrees of freedom coexist. Section 3 analyzes the commonly assumed hierarchy between quantum and classical mechanics and argues that both frameworks possess structural features that are not reducible to the other. Section 4 develops the formal framework of HQC dynamics, introducing the QL bracket that consistently couples quantum and classical degrees of freedom. Section 5 explores the case of gravity as a critical test of the relation between quantum and classical descriptions, reviewing recent postquantum frameworks in which quantum matter couples consistently to a classical spacetime geometry. Section 6 reviews examples from condensed matter physics showing that the coexistence of quantum and classical degrees of freedom reflects the multiscale structure of realistic many-body systems. Section 7 examines living systems as an example of quantum–classical coexistence, discussing how biological functions emerge from the interplay between quantum processes and structured classical environments. Appendix A presents the Hamiltonian formulation for continuous fields. Appendix B develops the HQC formulation for systems involving quantum particles or quantum fields interacting with classical fields.

2. The Structural Relation Between Quantum and Classical Mechanics

Quantum mechanics is often described as a complete and self-contained theory of microscopic phenomena. Yet, at its operational core, it has an intrinsically statistical structure. Both preparation and measurement are formulated in classical terms: definite pointer values, records, and spatial locations [7]. Quantum observables acquire empirical meaning only with respect to classical reference frames [25]. This reliance on classical structures is therefore not merely practical. We claim that it is structural, i.e., without classical variables, outcomes, positions, and macroscopic records, the operational content of the quantum formalism remains indeterminate. As emphasized by Landau [6], quantum mechanics occupies a peculiar logical position among physical theories, in that its formulation and interpretation presuppose the classical domain. If by “autonomy” one means logical closure and the universal derivability of classical phenomena from purely quantum dynamics, then such autonomy is not borne out at the empirical level.

HQC theories take this dual character quite seriously [26,27,28,29,30]. Rather than attempting to derive the classical exclusively from the quantum, or conversely reducing quantum effects to classical statistical models, they treat the interaction between quantum and classical degrees of freedom as a primitive feature of physical description. The goal is not to introduce ad hoc collapse mechanisms, but to construct a dynamically consistent effective framework in which measurement-like correlations and registration processes can be modeled through structured coupling. The relevance of such coexistence is not confined to interpretational questions. There exist concrete physical domains in which a purley quantum or classical description is insufficient.

General relativity treats space-time geometry as a classical dynamical field [31,32,33]. Despite major efforts, a fully consistent quantum theory of gravity remains elusive. Semiclassical gravity, that is, quantum matter evolving on a classical background, works in many regimes [34,35,36]. Proposals connecting macroscopic definiteness or state reduction to gravitational effects continue to be investigated [37,38]. Quantum fluctuations during inflation are likewise described as evolving on a classical background that seeds large-scale structure [34,35,36]. Thermodynamic and entanglement-based arguments further connect classical gravitational dynamics to quantum theory [39,40]. These developments indicate that quantum–classical dynamics is not merely a provisional approximation, but an active structural component of current theoretical frameworks.

Most realistic systems are neither closed and purely quantum nor fully classical. Decoherence explains the suppression of interference via environmental monitoring [9,10], yet practical modeling frequently treats the environment classically or stochastically [41]. In molecular and materials modeling, nuclei are often described classically while electrons are treated quantum mechanically; collective modes such as phonons are handled semiclassically. Methods including surface hopping, Ehrenfest dynamics, and the QCLE [42,43] succeed precisely because many systems inhabit a mixed regime. The effectiveness of these approaches suggests that hybrid dynamics captures structural features of real systems rather than serving as a mere computational convenience.

A further foundational point concerns systems in which both sectors comprise fields rather than particles. Quantum fluctuations typically reside at microscopic or mesoscopic scales, whereas classical organization manifests as extended structures such as polarization patterns, elastic deformations, biochemical concentration fields, or, in the case of gravity, the space-time metric itself. The coexistence and consistent interaction of these sectors requires a framework in which quantum fields evolve under commutators while classical fields evolve under functional Poisson brackets. The quantum–classical bracket suggested in this work defines a QL algebra governing the consistent interaction of these sectors. Its field-theoretic formulation, including the functional Poisson bracket and the corresponding hybrid Liouville equation, is presented in the Appendices. The field-theoretical formulation is extended in Ref. [44], where the proposed bracket does not conserve the energy, and is closer in spirit to the energy-conserving bracket proposed in Ref. [45]. The bracket from Ref. [45] is also presented in Refs. [46,47], considering statistical systems.

Across these domains, classical and quantum degrees of freedom coexist in the operative description. This coexistence is not merely an artifact of approximation; it reflects a structural feature of many empirically accessible systems. Therefore, the assumption that quantum theory forms a universally autonomous and logically closed framework requires critical and constructive reconsideration.

To clarify what is at stake, one must examine more carefully the standard hierarchical picture itself and ask whether it is structurally justified. To achieve this goal, in what follows we move from operational evidence to a direct comparison of the internal architectures of classical and quantum theory.

3. A Misleading Hierarchy

The standard hierarchy between quantum and classical theories rests on the assumption that quantum mechanics constitutes a logically closed and structurally complete framework, from which classical mechanics can be recovered as a limiting case. In this view, classical physics is conceptually subordinate: it is expected to arise in the limit in which the reduced Planck constant $\hslash \to 0$, or through decoherence and coarse-graining, while quantum theory itself requires no external structural completion. However, several features of both theories indicate that this hierarchy is not as straightforward as often assumed.

A first complication concerns intrinsically quantum structures that possess no natural precursor in the ontology of classical point-particle mechanics. Among the most striking examples is the spin dynamical variable. While orbital angular momentum can be obtained through quantization of classical phase-space variables, intrinsic spin, such as spin-$1/2$, does not emerge from a classical phase-space coordinate in the same way. Its introduction is required to account for experimental phenomena including the Stern–Gerlach experiment and fine-structure splittings. Spin is therefore not just a quantized classical observable, although classical and semiclassical spin models can reproduce selected algebraic or phenomenological aspects.

Equally fundamental are the symmetry properties of the wavefunction under exchange of identical particles. Fermionic antisymmetry under particle exchange gives rise to the Pauli exclusion principle, while bosonic symmetry underlies collective phenomena such as Bose–Einstein condensation. Classical field models may reproduce certain macroscopic aspects of collective behavior [48,49,50], for instance through coherent wave occupation of a single mode. However, classical fields can mimic the effects of exchange interaction only to a limited extent. Typically, these fields can simulate statistical macroscopic features, but they do not reflect the fundamental role of the (anti)symmetrization of many-body wavefunctions at the microscopic level. In fact, the (anti)symmetrization of many-body wavefunctions in quantum theory is a fundamental postulate that constrains the admissible state space. Such exchange symmetry cannot be generated by quantizing any classical model; it represents independent structural content.

This evident lack of perfect correspondence between the classical and quantum worlds is further confirmed by noting that classical physics exhibits dynamical features that do not arise directly from the standard formulation of closed quantum mechanics. The canonical example is deterministic chaos. In classical Hamiltonian systems, nonlinear equations of motion can lead to exponential sensitivity to initial conditions, characterized by positive Lyapunov exponents in phase space. This mechanism provides a structural explanation for complex and unpredictable behavior. In quantum mechanics, by contrast, the Schrödinger equation is linear, and the notion of exponentially diverging trajectories in phase space has no direct analogue in closed Hilbert-space evolution. What is commonly referred to as “quantum chaos” manifests instead through semiclassical, spectral, phase-space, or coarse-grained signatures, such as level-spacing distributions [17,18]. These signatures encode traces of classical chaotic structures, but they do not reproduce classical trajectory divergence as such within the unitary formalism itself.

Taken together, the above considerations reveal a form of mutual irreducibility. Quantum mechanics contains structural elements such as intrinsic spin and exchange symmetry that have no direct classical origin. Classical mechanics contains dynamical structures such as deterministic chaos that are not generated as such by closed quantum evolution. This mutual irreducibility challenges a strictly reductionist interpretation in which one framework is wholly contained within the other. The point is not that the two theories are incompatible, nor that one should replace the other. Rather, the evidence suggests that quantum and classical descriptions capture distinct but indispensable structural aspects of physical systems. Their relationship is therefore better characterized not as a strict inclusion hierarchy, but as a reciprocal overlap: each framework contributes structural ingredients that the other does not supply.

Recognizing this overlap has direct implications for foundational physics. If neither domain is fully derivable from the other without remainder, then the autonomy of quantum theory cannot be assumed as a trivial consequence of its formalism. Instead, it becomes a substantive structural hypothesis. This motivates the search for a consistent framework in which quantum and classical degrees of freedom coexist within a unified dynamical description, without privileging either as universally more fundamental. The development of such a framework is the objective of this study.

4. Formal Structure of Quantum–Classical Dynamics

If the more fundamental status of quantum theory relative to classical theory is not a structural theorem but a working assumption, then it is reasonable to seek a consistent dynamical framework in which quantum and classical degrees of freedom can be coupled without reducing either sector to the other. HQC theory helps establish the foundation for precisely such a framework. An HQC theory defined by means of QL brackets provides the dynamics of the statistical operator (a hybrid density operator) $\tilde{\rho }(X,t)$ coupled to the classical phase-space flow of $X=(R,P)$ (where R and P denote the coordinates and momenta, respectively), while conserving the energy of the combined HQC system, the normalization of the density matrix, and covariance under linear canonical transformations [45,46,47]. Such a theoryalso provides the correct classical and quantum limits. These properties show that the hybrid theory is not merely an ad hoc interpolation, but a structurally controlled effective extension that reproduces both quantum and classical mechanics in their respective domains of validity. Among the various existing formulations of hybrid dynamics, we adopt the QL formulation.

Following Ref. [21], the time evolution of the statistical operator is written in terms of the QL bracket

$${\left[A,B\right]}_{\mathrm{QL}}={\displaystyle \frac{i}{\hslash }}\left[\tilde{A},\tilde{B}\right]-\overrightarrow{\aleph }\sum _{I,J}\tilde{A}{\overleftarrow{\nabla }}_I{\Omega }_{I,J}{\overrightarrow{\nabla }}_J\tilde{B},$$

(2)

where [$·,·]$ denotes the commutator, ${\Omega }_{I,J}$ is the symplectic matrix, defined as

$$\Omega =\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right],$$

(3)

and ℵ is the antisymmetrization operator. The bracket in Equation (2) is antisymmetric but, in general, violates the Jacobi identity [21]. HQC dynamics suggests that the Lie algebraic structures underlying classical and quantum mechanics may represent special limits of a more general QL dynamical framework. This feature is not a defect of the construction. Rather, it reflects a structural property: a genuinely hybrid dynamics cannot, in general, be embedded within a single Lie algebra while simultaneously reproducing the correct quantum and classical limits [26,27,28,29,30]. Therefore, the departure from a strict Lie structure signals the coexistence of two distinct algebraic sectors within a unified dynamical scheme.

Equation (2) provides a unified and operationally consistent effective evolution for quantum and classical observables, without treating either sector as merely derivative of the other within the model’s domain of validity. The extension to field–theoretic degrees of freedom, including the functional Poisson bracket and the corresponding hybrid Liouville equation, is presented in Appendix A. This extension shows that the structural relation between quantum and classical mechanics is not limited to finite-dimensional systems but applies naturally to interacting quantum and classical fields.

As a simple quantitative illustration of the QL bracket formalism, let us consider a composite system consisting of light quantum degrees of freedom, represented by the position and momentum operators $(\widehat{q},\widehat{p})$, coupled to heavy degrees of freedom described by the phase-space variables $X=(R,P)$. A standard partially Wigner-transformed Hamiltonian can be written as

$$\tilde{H}\left(X\right)={\displaystyle \frac{P^2}{2M}}+\widehat{h}\left(R\right),$$

(4)

where

$$\widehat{h}\left(R\right)={\displaystyle \frac{{\widehat{p}}^2}{2m}}+\widehat{V}(\widehat{q},R),$$

(5)

$\widehat{V}$ is potential energy operator, m and M denote light and heavy masses, respectively.

In the derivation of the QCLE, Equation (2), the full quantum Liouville equation is partially Wigner transformed with respect to the heavy variables. The resulting Moyal operator is then expanded in the small parameter

$$\mu =\sqrt{{\displaystyle \frac{m}{M}}}\ll 1.$$

(6)

To leading order in $\mu$, the higher-order Moyal terms associated with the heavy variables are neglected. The heavy degrees of freedom are therefore represented by classical phase-space coordinates, whereas the light subsystem retains its quantum operator character. The resulting equation is precisely the QCLE [51].

This example makes explicit the hybrid structure of the dynamics. The commutator term governs the quantum evolution of the light subsystem, while the Poisson-bracket terms generate the classical phase-space flow of the heavy degrees of freedom. Thus, the quantum–classical bracket does not describe a direct reduction of quantum mechanics to classical mechanics. Rather, it provides a leading-order dynamical scheme for a regime in which quantum and classical variables coexist within a single evolution equation.

For finite but sufficiently small $\mu$, this equation describes an overlap regime: the heavy variables are sufficiently classical to be represented in phase space, but they remain dynamically coupled to the quantum subsystem through $\tilde{h}\left(R\right)$. In the limiting case in which the interaction between the two sectors is removed, $\tilde{V}(\widehat{q},R)=0$, the two dynamics separate. The light subsystem evolves according to the ordinary quantum Liouville equation, while the heavy subsystem follows a classical Liouville flow. In this restricted sense, the quantum and classical sectors can become dynamically independent. The hybrid formalism therefore supports the view that the classical and quantum descriptions are not simply arranged in a one-way hierarchy, but can appear as distinct dynamical sectors whose coupling or decoupling depends on the physical regime considered.

HQC dynamics also provides a natural description of measurement-like processes, in which the pointer is treated as a classical degree of freedom dynamically correlated with quantum observables, so that aspects of outcome registration can be modeled continuously without claiming to solve the measurement problem in its strongest form [7,8,29,30]. This observation already suggests that the relation between the two theories is not merely hierarchical, since the construction of quantum mechanics itself presupposes classical structures at the formal level. At the same time, it respects the operational asymmetry emphasized by Niels Bohr and Lev Landau, in which classical variables supply the reference structure required to define quantum measurements [6,25]. In this sense, the framework incorporates the quantum–classical interface as a dynamical feature rather than as an external interpretive addition.

In HQC, decoherence-like phenomena may arise from classical fluctuations and noise, yielding phase randomization and mixture formation in the quantum subsystem [41]. At the algebraic level, the QL bracket provides a flexible antisymmetric structure for coupling quantum and classical variables [21,26,27]. This makes it possible to describe transitions from quantum coherence toward classical-like statistical mixtures within an effective hybrid framework. The quantum–classical framework, therefore, challenges the assumption that a closed purely quantum description is always sufficient for modeling real measurement processes.

The decoupled case clarifies the meaning of the two limiting sectors. When the interaction between the quantum and classical variables vanishes, quantum observables no longer depend on the classical phase-space coordinates, and their evolution is generated solely by the quantum Liouville operator. Conversely, purely classical phase-space variables evolve according to the classical Liouville operator. In this case, the quantum and classical sectors become dynamically independent from each other. However, such an independence is not obtained by performing a quantum-to-classical limit. Rather, it follows from the absence of coupling between two already distinct sectors.

This distinction is relevant to the argument developed in the present study. The quantum-to-classical limit, understood as the emergence of classical definiteness from an underlying quantum description, remains a separate foundational problem and can still be regarded as one of the unresolved issues of quantum theory [52]. This issue differs from many-worlds accounts [53], which face difficulties in accounting for the emergence of single, definite outcomes [53], and from objective-collapse models, which introduce explicit dynamical modifications [28,37,38]. In this sense, HQC theory provides a minimal and dynamically consistent effective extension of quantum mechanics for situations in which classical degrees of freedom play an essential operational and structural role.

5. Postquantum Theory of Classical Gravity

If the independence and more fundamental status of quantum mechanics relative to classical theory is not structurally guaranteed, then the status of gravity provides one of its most stringent tests. The coupling between quantum matter and space-time geometry forces the question of whether universal quantization is a necessity or an assumption.

Besides QL brackets [21,43], several distinct approaches have been developed to formulate consistent quantum–classical dynamics Oppenheim’s postquantum theory of classical gravity [54] is particularly notable because it addresses the problem of coupling directly quantum matter to a classical space-time geometry. In line with the perspective developed in this paper, the postquantum theory of classical gravity treats the gravitational field as classical while matter fields remain quantum, thereby rejecting the premise that consistency requires quantization of all degrees of freedom. Oppenheim develops a consistent theory in which matter is quantum while the space-time metric remains classical. The coupled dynamics is linear in the density operator, completely positive and trace preserving, and reduces to general relativity in the appropriate classical limit. Fundamental stochasticity is required, which in turn produces objective decoherence of quantum matter through its interaction with the classical metric. This framework is designed to avoid the pathologies of naive semiclassical couplings based on expectation values and yields testable phenomenology at low energies [54,55,56].

A complementary line of research [57,58] models Newtonian gravity as a classical information channel realized by continuous measurement and feedback. The resulting master equations reproduce the Newtonian pair potential while introducing irreducible gravitational decoherence; a key prediction is that a strictly classical channel cannot generate entanglement between distant quantum systems. Refinements and dissipative extensions of this approach have subsequently been proposed [57,58,59]. These models demonstrate that consistent HQC dynamics can be constructed without embedding gravity within a fully quantum algebra.

What is structurally significant in these constructions is not merely that gravity is treated classically, but that consistency conditions, i.e., linearity, complete positivity, controlled decoherence, and correct limiting behavior, can be satisfied without postulating a fully quantum space-time. The logical possibility of such frameworks undermines the claim that universal quantum autonomy is enforced by internal consistency. Taken together, these approaches delineate the range of logically consistent ways in which quantum matter may interact with a non-quantum space-time, whether through hybrid dynamics, emergent gravity scenarios, or deeper underlying theories. They strengthen the case for hybrid and multiscale modeling, and motivate concrete criteria, such as complete positivity, consistency beyond mean-field approximations, controllable decoherence, and empirical discriminants, that any viable quantum–classical theory should satisfy. In this respect, gravity provides a paradigmatic domain in which the assumed hierarchy between quantum and classical descriptions becomes a substantive structural question rather than a settled fact.

6. Quantum–Classical Theory in Condensed Matter Physics

Condensed matter physics offers a clear empirical illustration of how the study of the dynamics of these complex many-body systems can gain a significant predictive advantage from a physically convincing approach based on the coexistence of classical and quantum variables. If quantum theory were structurally autonomous in a universal sense, one would expect that classical descriptions of many-body systems are merely provisional approximations awaiting full quantum replacement. In practice, however, the situation is more nuanced. In molecular dynamics, nuclei are routinely treated classically while electrons are described quantum mechanically. This practice is not just an approximation to be eliminated in favor of a fully quantum treatment. Rather, it reflects the physical fact that nuclear degrees of freedom often exhibit well-defined positions and momenta under experimental conditions, whereas electronic structure requires a quantum description to capture bonding, excitation, and reaction dynamics. The HQC description is therefore not arbitrary, but adapted to the distinct structural regimes occupied by different subsystems. Similarly, phonons are frequently modeled as classical or semiclassical excitations even within a quantum lattice framework. The classical treatment of collective modes provides essential insight into thermal properties and transport phenomena that would otherwise be complicated to extract directly from a purely quantum many-body wavefunction. Here again, classical variables are not external artifacts, but effective degrees of freedom that encode emergent organization at mesoscopic or macroscopic scales.

These modeling strategies illustrate that classical and quantum dynamical variables do not, in general, stand in a simple hierarchical relation. Instead, they are associated with distinct regimes determined, for example, by energies, masses, velocities, and particle numbers. These regimes may overlap, and in such regions of overlap, both descriptions are required for an accurate account of the system’s dynamics. The empirical success of quantum–classical computational methods in condensed matter, therefore, supports a structural reading of hybrid theory. It suggests that the coexistence of quantum and classical degrees of freedom is not merely a temporary calculational expedient, but reflects the multiscale organization of realistic systems. In this sense, condensed matter physics provides further evidence that the relation between quantum and classical theory is better understood as a context-dependent structural relation rather than as a strict inclusion hierarchy.

7. Modeling Living Systems

Living systems provide an extreme and highly structured example of quantum–classical coexistence. Even if biological organization can, in principle, be embedded in an underlying microscopic quantum description, its empirically effective description typically requires the structured interaction between selected quantum subsystems and unavoidable dissipative, driven, and hierarchically organized classical environments.

The term “quantum–classical biology” [22,23,24] is used to denote the study of living systems in which essential functions arise from the coupling between quantum degrees of freedom and classical ones pertaining to organized structures. This terminology is not meant merely as shorthand for the claim that “quantum effects occur in biology.” Rather, it is intended to emphasize the reciprocal interplay whereby biological architecture helps select which quantum degrees of freedom become dynamically relevant, on which timescales, and how those quantum processes in turn shape classical dynamics. Biological systems are open, out-of-equilibrium, and hierarchical; microscale processes shape macroscale organization, and vice versa [60]. In this context, classical structure is not merely an emergent by-product, but may act as an active constraint that selects and stabilizes quantum behavior in specific molecular degrees of freedom.

The absence of a universally accepted definition of life is not only a philosophical issue; it reflects the intrinsically multiscale nature of biological organization. This becomes operationally relevant, for example, in the search for extraterrestrial life, where criteria are necessarily multilevel and context-dependent. In such systems, structured classical environments provide precisely the conditions under which hybrid (quantum–classical) models become predictive [23]. Complexity and far-from-equilibrium organization further reinforce the need for multiscale, hybrid descriptions [60]. Historically, Erwin Schrödinger stressed that quantum principles underlie the stability of the genetic code [22]. Pascual Jordan highlighted the amplification of information at the quantum–to–classical interface through state reduction [23]. More recently, Jim Al-Khalili and Johnjoe McFadden suggested that tunneling in DNA may bias mutation probabilities, bridging molecular quantum events with evolutionary dynamics [23]. Electron tunneling in proteins further illustrates this interplay, with ongoing debate over whether proteins may have evolved, at least in part, to sustain such phenomena [61,62,63]. Canonical examples include electron and exciton transport in photosynthetic complexes, olfaction, and avian magnetoreception, where quantum effects have been proposed or observed in selected molecular degrees of freedom embedded in noisy classical environments. In these cases, the environment is not just a source of destructive noise, but may function as a structural element that determines which quantum processes remain dynamically relevant.

Crucially, many successful models in these domains are neither purely quantum nor purely classical. Hybrid descriptions explicitly couple a quantum subsystem to classical baths and collective variables, allowing transfer rates, yields, and coherence times to be predicted and compared with experiment. Frameworks such as the QCLE and related mixed dynamics (including QL brackets) [21,42,43] are tailored to describe this regime, preserving the correct limits and covariance while handling open, non-adiabatic evolution. Here, again, HQC theory is not just an optional numerical shortcut. It provides a mathematically consistent and empirically adequate framework for describing how classical structures may protect, select, and modulate biologically relevant quantum behavior. In this sense, HQC theory is not merely a computational convenience, but a natural effective description of the processes under consideration.

Viewed from this perspective, “quantum biology” appears as a subset of a broader program, “quantum–classical biology”, in which quantum mechanisms and classical structure co-produce function in real, out-of-equilibrium organisms. The shift from isolated quantum phenomena to their lawful embedding within classical architectures mirrors the multiscale organization already encountered in condensed matter and gravity. Biological systems, therefore, offer a vivid prospective illustration that the relation between quantum and classical theories is best understood as a context-dependent structural relation rather than as a strict inclusion hierarchy.

8. Conclusions

The analysis developed in this paper supports a re-examination of the assumed hierarchy between quantum and classical theories. Quantum mechanics relies structurally on classical reference structures for its formulation and empirical articulation [6,25], while classical dynamics exhibits features that are not straightforwardly generated by closed quantum evolution. Together, these considerations indicate that the autonomy of quantum theory is not a structural theorem of the formalism, but a substantive assumption about its scope.

The quasi-Lie formalism and the QCLE [21,26,27,29,30] provide a mathematically controlled effective realization of quantum–classical coexistence. They reproduce the correct quantum and classical limits and preserve key structural properties such as energy conservation and covariance under linear canonical transformations within their domain of validity. They also offer a dynamical description of measurement-like correlations and registration processes, without amounting to a complete solution of the measurement problem. In this sense, hybrid dynamics constitutes a minimal and structurally consistent effective extension of the standard framework.

Beyond foundational considerations, quantum–classical dynamics already plays a central role in chemistry and condensed matter physics and provides a significant conceptual framework for gravity and biological modeling [9,10,37,38,41]. These domains illustrate that mixed regimes are not exceptional but recurrent in realistic systems. The quantum–classical interface, therefore, appears not as a boundary to be eliminated, but as an intrinsic feature of multiscale physical organization.

Rethinking the hierarchy does not require abandoning quantum theory. Rather, it requires recognizing that the relation between quantum and classical descriptions may be better understood as a structural co-architecture, within which each regime captures indispensable aspects of empirically accessible physical reality. From this perspective, hybrid dynamics offers a coherent framework for extending quantum theory toward complex, interacting, and empirically accessible systems [64].