⚛️ general relativity

Duru-Kleinert Path Integral in Unimodular Quantum Cosmology

This paper employs the Duru-Kleinert path integration technique to establish a correspondence between unimodular quantum cosmology and the hydrogen atom, enabling the derivation of a quantized negative cosmological constant and the calculation of both the quantum tunneling rate from "nothing" and the spectral density with Krylov complexity for positive cosmological constants.

Original authors: Xiao-Kan Guo

Published 2026-08-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Xiao-Kan Guo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

To understand the earliest moments of our universe, physicists often turn to a set of rules that attempt to merge the smooth, curved geometry of space and time with the jittery, uncertain nature of quantum particles. In the very beginning, before stars or galaxies existed, the universe was likely a tiny, dense point where the familiar laws of gravity break down. To describe this era, scientists use a framework called quantum cosmology, which treats the entire universe as a single quantum object with a wave function. A central equation in this field, known as the Wheeler-DeWitt equation, describes how this universal wave function behaves. However, solving this equation is notoriously difficult because it involves complex mathematics and the strange concept of time itself, which may not exist in the way we experience it at the quantum level. One specific approach, called unimodular gravity, offers a way to fix the scale of space and time, making the equations more manageable. Within this framework, researchers have recently discovered a surprising link between the birth of the universe and the behavior of a simple atom, specifically the hydrogen atom. This connection suggests that the chaotic, high-energy physics of the early cosmos might be mathematically identical to the well-understood physics of an electron orbiting a proton.

Building on this intriguing connection, a new study by Xiao-Kan Guo at the Yancheng Institute of Technology revisits the relationship between the universe and the hydrogen atom using a powerful mathematical tool called the Duru-Kleinert path integral. This technique allows physicists to calculate the probability of a system moving from one state to another by summing up every possible path it could take. In this case, the researcher applied this method to the universe itself, treating it as a particle moving through a landscape of possibilities. By carefully reshaping the mathematical description of time and space, the study transformed the complex problem of the universe's quantum birth into the simpler problem of a particle moving under the influence of a single force, much like an electron feeling the pull of a nucleus. This transformation revealed that the universe, when filled with a specific type of invisible dust and a cosmological constant, behaves exactly like a one-dimensional version of a hydrogen atom.

The study confirms that this mathematical mapping leads to a specific, quantized set of possibilities for the universe's cosmological constant, a value that drives the expansion of space. Just as an electron in a hydrogen atom can only exist at specific energy levels, the universe in this model can only exist with specific values for its cosmological constant. These allowed values are negative and follow a precise pattern, similar to the famous Rydberg formula used in atomic physics. The research shows that the universe can only take on these discrete values, meaning the cosmological constant is not a random number but a quantized property determined by the amount of dust-like matter in the early universe. This finding provides a rigorous derivation of a result that was previously known only through direct wave function solutions, offering a deeper understanding of how the universe's fundamental parameters are set.

Beyond the static properties of the universe, the paper also explores the dynamic process of the universe coming into existence from "nothing." In quantum cosmology, this is described as a tunneling event, where the universe spontaneously appears by passing through a barrier of probability. Using the simplified model derived from the Duru-Kleinert technique, the researcher calculated the likelihood of this event for different quantum states. The calculation revealed that the probability of the universe appearing is highest for the lowest energy state, or the ground state, and decreases rapidly for higher, more excited states. The study found a precise mathematical relationship for this probability, showing that the chance of the universe appearing in a specific state drops exponentially as the state becomes more complex. This suggests that the simplest, most stable version of the universe is the most likely to emerge from the quantum vacuum.

Finally, the study looked at what happens when the cosmological constant is positive, a scenario that corresponds to an expanding universe like our own. In this case, the universe does not settle into a fixed state but instead expands indefinitely. The researcher calculated the density of possible states for this expanding universe and used it to measure a quantity called Krylov complexity, which tracks how much information is needed to describe the system as it evolves. The results showed that the presence of the dust-like matter is essential for the mathematics to work correctly. Without this matter, the description of the universe's evolution becomes undefined and breaks down, mirroring the classical singularity at the beginning of time. However, with the dust included, the complexity grows in a smooth, predictable way, suggesting that the quantum nature of the dust resolves the initial singularity and provides a complete, finite description of the universe's birth and evolution. This work demonstrates that by using advanced path integral techniques, physicists can unlock exact solutions to some of the most difficult problems in quantum gravity, revealing a deep and unexpected harmony between the physics of the very small and the very large.

Technical Summary: Duru-Kleinert Path Integral in Unimodular Quantum Cosmology

Problem Statement
The paper addresses the quantization of a flat Friedmann-Lemaitre-Robertson-Walker (FLRW) universe filled with pressureless dust and a cosmological constant ( $\Lambda$ ) within the framework of unimodular gravity. While recent work established a formal equivalence between the Wheeler-DeWitt (WDW) equation for this system and the radial Schrödinger equation of a hydrogen atom, this study seeks to revisit that correspondence using the Duru-Kleinert (DK) path integral technique. The primary objectives are to derive the quantized spectrum of the cosmological constant, compute the quantum tunneling rate from "nothing" to a universe, and analyze the spectral density and Krylov complexity for unbounded states ( $\Lambda > 0$ ).

Methodology
The author employs the Duru-Kleinert path integration method, which involves a time reparametrization and a compensating canonical transformation to convert non-standard kinetic terms into canonical forms.

Model Formulation: Starting from the action for gravity, Brown-Kuchař dust, and a cosmological constant, the author reduces the system to a homogeneous, isotropic, spatially flat FLRW model. In unimodular gravity, the condition $\sqrt{-g}=1$ fixes the lapse function, promoting $\Lambda$ to a dynamical variable conjugate to a "time" variable $T$ . The resulting WDW equation is identified as structurally identical to the radial Schrödinger equation for a hydrogen atom with zero angular momentum ( $l=0$ ), where the universe volume $v=a^3$ corresponds to the radial coordinate.
Euclidean Path Integral: The Green's function (fixed- $\Lambda$ propagator) is expressed as a Euclidean path integral. The author applies the DK transformation by introducing a new time parameter $s$ via $d\tau = f(v)ds$ with $f(v) = 2v/3$ . This maps the original action to that of a one-dimensional point particle moving in a Coulomb potential $V(v) = 2\rho_0/(3v)$ with a constant energy shift.
Evaluation:
- Bound States ( $\Lambda < 0$ ): The transformed path integral is evaluated using the Mellin-Barnes integral representation. The fixed-energy amplitude is expressed in terms of Whittaker functions, which are related to the Green's function of the Coulomb system.
- Tunneling: The tunneling rate from "nothing" ( $v=0$ ) to a finite volume is calculated using the instanton (bounce) method. The author solves the Euclidean equations of motion for the effective potential, computes the classical bounce action, and determines the one-loop prefactor using the Gelfand-Yaglom theorem and collective coordinate integration.
- Unbounded States ( $\Lambda > 0$ ): The Green's function is analytically continued to positive $\Lambda$ . The spectral density is derived from the discontinuity of the Green's function's trace. This density is then used to compute moments and Lanczos coefficients to determine the Krylov complexity of the system.

Key Contributions and Results

Quantized Cosmological Constant: By analyzing the poles of the Gamma function in the Green's function, the author recovers the discrete spectrum for the cosmological constant:
$\Lambda_n = -\frac{\rho_0^2}{3n^2}, \quad n = 1, 2, 3, \dots$
This result confirms the equivalence to the hydrogen atom's Rydberg formula derived previously via wave function solutions. The bound state wave functions are identified as the residues of the Green's function at these poles.
Tunneling Rate: The paper computes the nucleation probability for a universe in the $n$ -th quantum state. The classical Euclidean bounce action is found to be exactly $B_n = 2\pi n$ . Including the one-loop prefactor (scaling as $\sqrt{n}$ ), the tunneling rate is:
$\Gamma_n \propto \sqrt{n} e^{-2\pi n}$
This result indicates that the creation of the ground state ( $n=1$ ) is exponentially favored over excited states.
Spectral Density and Krylov Complexity: For $\Lambda > 0$ , the author derives the continuum spectral density, which includes an interaction-induced correction related to the Coulomb phase shift. They demonstrate that the inclusion of the dust field ( $\rho_0 > 0$ ) is crucial for the finiteness of the Krylov complexity.
- Without dust ( $\rho_0 = 0$ ), the spectral density leads to divergent moments, rendering the Lanczos coefficients and Krylov complexity ill-defined, which the author interprets as the Krylov-space counterpart of the classical Big-Bang singularity.
- With dust, the spectral density falls off sufficiently fast ( $\sim \Lambda^{-3/2}$ ), ensuring finite moments. For broad initial states, the Lanczos coefficients grow linearly ( $b_n \sim n$ ), characteristic of integrable systems, leading to a Krylov complexity that grows as $\sinh^2(\beta t)$ .

Significance
The paper claims to provide a rigorous path integral derivation of the correspondence between unimodular quantum cosmology and the hydrogen atom, moving beyond wave function solutions to utilize Green's functions and path integral techniques. The Duru-Kleinert transformation is shown to map the cosmological model directly onto a one-dimensional Coulomb problem, allowing for exact analytical results.

The significance of the work lies in:

Exact Solvability: It demonstrates that the DK method can unlock exact results in quantum gravity models, specifically deriving the quantized $\Lambda$ spectrum and tunneling rates directly from the path integral.
Singularity Resolution: It offers an information-theoretic perspective on singularity resolution via Krylov complexity. The presence of dust regularizes the spectral density, making the quantum dynamics well-defined at $v=0$ , whereas the dust-free theory fails to provide a unique description.
Tunneling Prediction: It establishes a concrete prediction for the nucleation probability in this solvable setting, reinforcing the deep mathematical connection between atomic physics and quantum cosmology.

The author concludes that while the exact correspondence to the hydrogen atom is specific to this model, the perturbative methods based on these exact solutions may be useful for more complex scenarios, such as those with non-zero spatial curvature or inflationary potentials.

Technical Summary: Duru-Kleinert Path Integral in Unimodular Quantum Cosmology

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