Continuous reformulation of Planck Quantization

Jaafar Gaber

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# Continuous reformulation of Planck Quantization

Jaafar GABER

## Abstract

In this article, we propose to replace classical discrete quantization, as formulated by Planck, with a continuous exponential structure. We establish the mathematical foundations of this reformulation by introducing exponential spectral series and interpreting them geometrically within the framework of hyperbolic geometry. This association between energy levels and a negatively curved metric space provides a new perspective on the structure of quantum spectra. While we focus in this paper on the exponential form $E_n = E_0 e^{\lambda n}$, we also outline a more general class of continuous spectra of the form $E_n = E_0 f(n)$, where $f(n)$ is any strictly increasing function adapted to the system. This opens the way for future extensions to more complex quantum systems.

**Keywords:** quantization, exponential spectrum, Mellin transform, zeta function, hyperbolic geometry, quantum mechanics, energy levels.

## 1 Introduction

Max Planck (1900) introduced the concept of energy quanta to explain blackbody radiation. He proposed that the energy of an oscillator took discrete values $E = hu$, where $h$ is Planck's constant and $u$ is the frequency [1].

Quantum mechanics has since been built on the notion of discrete quantization, initiated by Planck with the formula $E_n = nhf$. This discretization has successfully explained atomic energy levels, spin, and various spectroscopic phenomena. However, this hypothesis may not be fundamental.

We propose here a continuous reformulation of quantization, in which the energy levels follow a continuous exponential law. This idea leads naturally to deep mathematical structures such as Mellin transforms, zeta functions, and hyperbolic geometry.

Remarkably, since Planck's introduction of the quantum, the discreteness of energy levels has been treated as a fundamental axiom and has rarely been questioned in its formulation. Most theoretical efforts have focused on extending the quantum framework (towards general relativity, quantum gravity, or gauge theories), without considering a direct reformulation of quantization itself.

The present approach offers an alternative reading, in which the discrete structure is replaced by a continuous exponential progression, interpreted through a geometric lens. It aims to revisit the foundations of quantum mechanics via a continuous, analytic, and metric modeling of spectra.

To our knowledge, no previous work has proposed a reformulation of Planck's quantization based on a continuous exponential structure embedded in a hyperbolic geometric framework. This work seeks to enrich the foundational analysis of quantum mechanics through a continuous and geometric perspective.

## 2 Mathematical Reformulation

In this section, we establish the analytical foundations of the continuous reformulation: exponential quantization, logarithmic structure, and Mellin transform.
## 2.1 Exponential Quantization

We define the energy levels as:

$$E_n = E_0 e^{\lambda n}, \qquad (1)$$

where $E_0 > 0$ and $\lambda$ is a continuous regularization parameter. This expression naturally generalizes Planck's linear formula.

This exponential expression introduces a continuous spectral structure whose analysis requires the functional inverse of the exponential: the continuous logarithm. Indeed, if one wishes to recover a continuous index from a given energy, we obtain:

$$n(E) = \frac{1}{\lambda} \ln \left( \frac{E}{E_0} \right),$$

which reveals the logarithmic nature of the spectral scale. This transformation is a key step toward a hyperbolic geometry, where distances and levels are distributed according to nonlinear patterns.

## 2.2 Thermodynamic Link

Let us recall that Planck introduced energy quantization to solve a problem arising from statistical physics: explaining the spectrum of blackbody radiation. He combined ideas from thermodynamics, Boltzmann statistics, and a discretization hypothesis for energy levels. In particular, he used the expression $z = e^{-E/kT}$, which represents the weighting of an energy level $E$ in a Boltzmann distribution.

In the following, we reuse this thermodynamic change of variable to remain within the fundamental framework introduced by Planck, but we apply it to a continuous energy structure. To this end, we introduce the indexed variable:

$$z_n = e^{-E_n/kT},$$

where $E_n = E_0 e^{\lambda n}$ represents our exponential reformulation of energy levels. This variable directly connects each level to a classical exponential quantity in statistical physics. It reformulates the energy scale into a convergent exponential series, paving the way for continuous spectral analysis using tools such as the Mellin transform. It thus forms a natural bridge between the energy structure and the language of analytic series.

It should be noted that in Planck's original model, the Planck constant $h$ appears in the expression $E = h\nu$. In our reformulation, the information carried by $h$ is transferred to the constants $E_0$ and $\lambda$, which respectively describe the scale and the growth of energy levels. Thus, even though the constant $h$ does not appear explicitly, its physical significance is implicitly encoded in the exponential structure of the spectrum.

By setting:

$$z_n = e^{-E_n/kT} = e^{-E_0 e^{\lambda n}/kT}, \qquad (2)$$

we obtain a convergent exponential series that admits a Mellin transform.

It is worth emphasizing that the Mellin transform, which will be used in the next section, serves here as an analytical tool to handle the series $z_n^s$. It facilitates spectral analysis and establishes a formal link with the zeta function. However, this reformulation could also be considered from a more geometric or abstract perspective, without necessarily relying on the Mellin transform, or by replacing it with other equivalent integral structures.

## 2.3 Mellin Transform and Zeta Function

We now explore the spectral analysis of the series previously defined by employing a classical analytical tool: the Mellin transform. This method allows us to extract global properties of the reformulated energy distribution.
Let us consider the spectral series built from the variable $z_n$ defined earlier:

$$Z(s) = \sum_{n=0}^{\infty} z_n^s = \sum_{n=0}^{\infty} e^{-sE_0e^{\lambda n}/kT}. \qquad (3)$$

This expression defines a series with rapidly growing exponents, but rapidly decaying terms, ensuring convergence for all $s > 0$.

This series can be interpreted as a Mellin transform of a spectral density. Under certain conditions, depending on parameter choices and normalization, it admits an asymptotic expression of the form:

$$\mathcal{M}[I](s) \sim \Gamma(s+3)\zeta(s+3), \qquad (4)$$

where $\Gamma$ denotes the Euler gamma function, and $\zeta$ the Riemann zeta function [2].

This type of expression reflects a regular spectral organization, which is commonly found in many thermodynamic structures and quantum models, and it illustrates the coherence of our approach with classical analytical tools.

This observation leads us to clarify the scope and limitations of using the zeta function in our framework. It should be emphasized that the Riemann zeta function, which emerges naturally in this analysis via the Mellin transform, is not essential to our reformulation. It appears here as an analytical consequence of the exponential structure of the studied series. However, other special functions (such as polylogarithms, Dirichlet $L$-functions, or modular functions) could also be considered depending on the underlying mathematical structure of the energy distribution. Likewise, alternative integral methods, such as Laplace or Fourier transforms, could offer a complementary or substitute approach depending on the chosen analytical or modeling framework. It is important to emphasize that the exponential growth of energy levels is not introduced here as an arbitrary assumption, but rather emerges naturally from Planck's original thermodynamic framework via the Boltzmann distribution. Moreover, it integrates coherently into a geometric interpretation based on negative curvature. From a mathematical standpoint, however, variations in the growth law are entirely conceivable. The relation $E_n = E_0e^{\lambda n}$ represents a particular case within a broader family of continuous spectral structures, of the form $E_n = E_0f(n)$, where $f(n)$ is a strictly increasing function tailored to the spectral regime under consideration. In such generalizations, special functions such as polylogarithms or Dirichlet $L$-functions could be employed depending on the analytic properties of the resulting spectral series. While our use of an exponential growth model in this article is motivated by its analytic consistency and spectral regularity, the formulation remains open to generalizations that may better model specific energetic behaviors across different physical contexts.

## 3 Towards a unified Generalization of Spectral Structures

We propose here a general formulation of the energy spectrum in the form:

$$E_n = E_0 f(n), \qquad (5)$$

where $f(n)$ is a strictly increasing function defined on $\mathbb{N}$ or on $\mathbb{R}_+$ for a continuous extension. This structure encompasses a range of known spectral laws while allowing for a reformulation adaptable to various physical contexts.

Among the notable special cases:

* Linear quantization: $f(n) = n$, historically used in early quantum models;

* Exponential quantization: $f(n) = e^{\lambda n}$, developed in this article;

* Logarithmic or inverse quantizations, used in certain diffusion or ionization models.
This generalization establishes a unified spectral framework, in which each choice of $f(n)$ defines an analytical and physical signature of the modeled system. It paves the way for a functional classification of spectra according to their growth properties, regularity, convergence, or asymptotic density.

In this context, exponential quantization emerges as a privileged case due to:

* the rapid convergence of its associated series,

* its compatibility with the thermodynamic formalism,

* and its natural interpretation within a negatively curved geometric framework.

The general structure may also serve as a foundation for future extensions, whether to explore discrete or quasi-continuous spectra, or those strongly coupled to environmental effects or external fields.

## 3.1 Visualization and Convergence

To visually confirm the validity of the previous analysis, we illustrate here the rapid decay of the terms in the series $z_n^s = e^{-sE_0e^{\lambda n}/kT}$. This exponential decay ensures the absolute convergence of the series for all values $s > 0$, which is an essential condition for the spectral interpretation based on the Mellin transform.

<figure><img src="image_2.png" /><figcaption>Figure 1: Convergence of the exponential series terms $z_n^s = e^{-sE_0e^{\lambda n}/kT}$ for various values of $s$. The rapid decay confirms the convergence of the spectral series.</figcaption></figure>

The figure above highlights the fast decay of the series terms for different values of $s$, which fully supports the analytical approach adopted. This representation provides an intuitive view of the mathematical stability of our reformulation, while foreshadowing the continuous geometric structure explored in the following section.

# 4 Connection with Hyperbolic Geometry

We now show how the energy levels arising from exponential quantization can be interpreted geometrically within the broader framework of hyperbolic geometry. This exponential growth is naturally related to a negative curvature of the state space, where the Poincaré disk model provides both a precise metric and an intuitive visual representation of this structure.
In the Poincaré disk [3], the hyperbolic distance between two points $z_1$ and $z_2$ is given by:

$$d(z_1, z_2) = \operatorname{arcosh} \left( 1 + \frac{2|z_1 - z_2|^2}{(1 - |z_1|^2)(1 - |z_2|^2)} \right). \qquad (6)$$

This metric allows us to relate the exponential growth of the energy levels to a radial displacement in a curved space. The level $n$ then corresponds to a circle of increasing energy in the disk.

The exponential increase of energy levels can thus be interpreted geometrically as a radial expansion in a hyperbolic space. The figure below shows the Poincaré disk, where each level $n$ is represented by a concentric circle. This visualization reflects the logarithmic nature of distances and provides a natural geometric reading of the continuous spectral structure.

<figure><img src="image_3.png" /><figcaption>Figure 2: Geometric interpretation of energy levels as concentric circles in the Poincaré disk. The exponential increase of energy levels can be visualized as moving radially outward in a curved space with hyperbolic geometry.</figcaption></figure>

This representation illustrates geometrically the structure of the energy spectrum. Each concentric circle in the disk corresponds to a higher energy level, and the radial distance reflects this growth within an abstract space endowed with hyperbolic geometry.

# 5 Application to Atomic Structure

We propose here an alternative reading of electronic energy levels based on exponential quantization. The classical Bohr model assigns to the energy levels of the hydrogen atom the following law:

$$E_n^{\text{Bohr}} = -\frac{13.6 \text{ eV}}{n^2},$$

where $n$ is a positive integer. Although effective in describing the first levels, this discrete expression imposes a rapid decay and a strict discretization.

In our approach, we replace this structure with a continuous exponential quantization:

$$E_n = E_0 e^{-\lambda n}, \qquad (6)$$
where $E_0$ is an initial negative energy, and $\lambda$ is a tunable parameter.

This reformulation allows for:

* **A more flexible modeling of energy decay:** Unlike the sharp $1/n^2$ drop of the Bohr model, the exponential scale offers a more gradual and adjustable variation, which facilitates adaptation to intermediate energy regimes.

* **A more regular level density near ionization:** The level spacing obtained through this construction remains regular as the system approaches ionization. The difference between two consecutive levels is proportional to a decaying exponential, avoiding the abrupt transitions observed in purely discrete models.

* **A natural transition to a quasi-continuous spectrum at high energies:** This continuous structure leads naturally to a quasi-continuous spectrum at high energy, by allowing $n$ to tend toward real values. It thus enables a smooth transition between discrete quantization and continuous spectral modeling, which is especially useful for the study of excited states or open systems.

It may also serve as a foundation for generalizations to other atomic or molecular systems, where the energy structure does not strictly follow a $1/n^2$ law but exhibits a smooth, exponential deformation.

# 6 Discussion and Implications

In this section, we discuss the physical, mathematical, and theoretical consequences of our approach. Although grounded in classical analytical tools, our reformulation offers a renewed perspective on quantum energy structures by questioning the postulate of fundamental discretization.

The logarithmic substitution $z = e^{-E/kT}$ enables the construction of a rigorously convergent exponential spectral series, compatible with thermodynamic analysis. The use of the Mellin transform provides a solid analytical framework, connected to special functions such as the Riemann zeta function, while remaining open to other formal settings (Fourier, Laplace, polylogarithms).

Furthermore, the introduction of hyperbolic geometry via the Poincaré disk model allows energy levels to be interpreted as distances in a negatively curved state space. This yields a geometric interpretation of quantization, based on exponential growth and the metric structure of levels.

Our approach lays the groundwork for a new interpretation of quantization—continuous, geometric, and analytic—which may shed complementary light on the foundational structures of quantum mechanics.

Moreover, the generalized formulation $E_n = E_0 f(n)$, with $f(n)$ a strictly increasing function, extends the applicability of our approach. It provides a flexible modeling tool for describing quasi-continuous spectra, energy structures influenced by external fields, or systems with environment-induced spectral deformation. This reinforces the robustness and adaptability of the analytical framework introduced in this work.

# 7 Conclusion

We have proposed a continuous reformulation of Planck's quantization based on an exponential structure defined within a hyperbolic state space. This construction relies on rigorous analytical tools (thermodynamic variable substitution, Mellin transform, and geometric interpretation) which enable the reconstruction of energy levels in a regular and convergent manner.
This reformulation provides an alternative modeling of quantum spectra, compatible with
atomic observations, and offers a conceptual foundation for a geometric interpretation of quan-
tization, linked to the negative curvature of the state space.

Finally, the general structure $E_n = E_0 f(n)$, introduced as a natural extension of the exponential case, lays the groundwork for future developments in quantum modeling, particularly for systems exhibiting more complex or environment-dependent spectral behavior.

References

[1] M. Planck, "Zur Theorie des Gesetzes der Energieverteilung im Normalspectrum", *Verhandlungen der Deutschen Physikalischen Gesellschaft*, vol. 2, p. 237, 1900.

[2] B. Riemann, "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse", *Monatsberichte der Königlich Preusschen Akademie der Wissenschaften zu Berlin*, 1859.

[3] H. Poincaré, "Les méthodes nouvelles de la mécanique céleste", vol. 1, Gauthier-Villars, 1892.