Fractional Quantum Cosmology

• Fractional quantum cosmology is a framework where nonlocal fractional operators modify standard quantum gravitational equations, enabling novel inflation and bounce dynamics.
• It employs operators like Riesz, Caputo, and conformable derivatives to fractionalize the Wheeler–DeWitt equation and adjust minisuperspace behavior.
• The approach yields testable predictions such as power-law inflation, nonsingular bounces, and modified tunnelling probabilities, with potential observational fits in cosmology.

Searching arXiv for recent and foundational papers on fractional quantum cosmology. Fractional quantum cosmology is a family of minisuperspace programs in which the Wheeler–DeWitt equation, the Hamiltonian constraint, or the semiclassical evolution law is modified by fractional operators, fractional powers of momenta, or effective memory kernels. In the literature, these modifications are implemented through several distinct constructions: Riesz fractional derivatives motivated by Lévy-flight quantum mechanics, fractional Hamiltonian constraints on minisuperspace, conformable or Riemann–Liouville/Caputo operators in matter sectors, and canonical transformations that convert noninteger momentum powers into ordinary differential operators. Across these formulations, recurring consequences include power-law inflation in place of de Sitter expansion, nonsingular bounces, modified tunnelling probabilities for quantum creation, effective fractal dimensions in horizon or matter sectors, and late-time cosmological fits with a fractional index very close to the standard limit (Moniz et al., 2020, Costa et al., 2023, Jalalzadeh et al., 2022).

1. Origins and scope of the program

The modern formulation of fractional quantum cosmology emerged by analogy with fractional quantum mechanics. In the latter, Laskin’s space-fractional framework replaces the Laplacian by the Riesz fractional operator $(-\hbar^2\Delta)^{\alpha/2}$, with free Hamiltonian $H_\alpha(p)=D_\alpha |p|^\alpha$ for $1<\alpha\leq 2$. In minisuperspace, this analogy motivates either a direct fractionalization of the Wheeler–DeWitt kinetic term or a fractional deformation of the Hamiltonian constraint itself. The resulting theories inherit nonlocality in configuration or momentum space and are commonly parametrized by a Lévy index $\alpha$ (Costa et al., 2023).

The subject was explicitly proposed as an open research program in “From Fractional Quantum Mechanics to Quantum Cosmology: An Overture” (Moniz et al., 2020). That work emphasized that fractional quantum cosmology is not a single formalism but a research direction with multiple unresolved technical choices, notably the selection of fractional derivatives and the preservation of minisuperspace covariance. It also framed fractional quantization as a possible new mechanism for addressing standard cosmological problems, including the flatness problem, through modified Wheeler–DeWitt spectra.

A useful way to organize the literature is to distinguish the fractional ingredient that is being modified.

Formulation	Fractional ingredient	Representative settings
Riesz-fractional Wheeler–DeWitt	$(-\hbar^2\Delta)^{\alpha/2}$ or $(-\partial_a^2)^{\alpha/2}$	de Sitter, inflationary FLRW, FRW tunnelling, AdS dust
Fractional Hamiltonian constraint	$$\mathcal H_\alpha=\tfrac12 D_\alpha |f^{\mu\nu}\Pi_\mu\Pi_\nu|^{\alpha/2}+U(q)$$	fractional $\Lambda$CDM minisuperspace
Matter-induced fractional power	$\Pi_\phi^\beta$, with $\beta=\tfrac{2\alpha}{2\alpha-1}$	FLRW and Bianchi I $H_\alpha(p)=D_\alpha |p|^\alpha$0-essence
Noninteger momentum Hamiltonian	$H_\alpha(p)=D_\alpha |p|^\alpha$1	Fab Four John minisuperspace

This classification suggests that “fractional quantum cosmology” is best understood as a common umbrella over several non-equivalent quantization strategies rather than as a single canonical theory.

2. Minisuperspace fractionalization mechanisms

One route starts from a general ADM minisuperspace action,

$H_\alpha(p)=D_\alpha |p|^\alpha$2

with canonical Hamiltonian $H_\alpha(p)=D_\alpha |p|^\alpha$3. The fractional extension replaces the quadratic form $H_\alpha(p)=D_\alpha |p|^\alpha$4 by $H_\alpha(p)=D_\alpha |p|^\alpha$5, producing

$H_\alpha(p)=D_\alpha |p|^\alpha$6

In homogeneous and isotropic minisuperspace this yields a fractional Friedmann equation and an effective minisuperspace dimension

$H_\alpha(p)=D_\alpha |p|^\alpha$7

The standard Wheeler–DeWitt/FLRW limit is recovered at $H_\alpha(p)=D_\alpha |p|^\alpha$8, where $H_\alpha(p)=D_\alpha |p|^\alpha$9 (Costa et al., 2023).

A second route arises dynamically from specific matter sectors. In FLRW and Bianchi I $1<\alpha\leq 2$0-essence cosmologies with $1<\alpha\leq 2$1, canonical reduction yields a Wheeler–DeWitt equation in which the scalar-field operator appears at noninteger order

$1<\alpha\leq 2$2

The literature relates this to the barotropic parameter through distinct parametrizations; in particular, the order $1<\alpha\leq 2$3 falls in $1<\alpha\leq 2$4 for $1<\alpha\leq 2$5 and in $1<\alpha\leq 2$6 for $1<\alpha\leq 2$7. In these models, fractional differentiation is not added ad hoc after quantization but appears from the noncanonical matter Hamiltonian itself (Socorro et al., 2023, Socorro et al., 2023).

A third mechanism is more singular: the classical Fab Four John minisuperspace Hamiltonian contains explicit fractional powers of the momenta,

$1<\alpha\leq 2$8

Here the fractional structure resides in the Hamiltonian, not initially in a chosen fractional differential operator. This distinction is central to later quantization choices (Torres et al., 2018).

3. Quantization strategies and Wheeler–DeWitt equations

The most direct quantization strategy replaces the ordinary minisuperspace Laplacian by a Riesz fractional operator. In closed de Sitter minisuperspace the resulting Wheeler–DeWitt equation takes the form

$1<\alpha\leq 2$9

while in compact flat inflationary minisuperspace one obtains a corresponding Riesz-fractional Wheeler–DeWitt equation whose WKB reduction replaces $\alpha$0 by $\alpha$1. Related Riesz constructions also appear in the Schrödinger-like FRW tunnelling problem and in AdS Brown–Kuchař dust quantization (Jalalzadeh et al., 2022, Rasouli et al., 2022, Canedo et al., 19 Mar 2025, Júnior et al., 2 Jan 2025).

A distinct treatment of the Fab Four John model avoids selecting any fractional derivative operator at all. The authors state that “there are, in fact, several definitions of fractional derivatives. To avoid this ambiguity we perform a canonical transformation.” After this transformation the secondary constraint reduces to

$\alpha$2

and standard canonical quantization gives the mixed Wheeler–DeWitt equation

$\alpha$3

Thus a Hamiltonian with fractional momentum powers is mapped to an equation involving only ordinary second-order derivatives (Torres et al., 2018).

The same Fab Four sector was later quantized by another route, using the conformable fractional derivative. In that approach,

$\alpha$4

and the corresponding Wheeler–DeWitt equation becomes

$\alpha$5

This formulation supports a Bohm–de Broglie interpretation and introduces a quantum potential $\alpha$6 as a criterion for whether a quantum solution is acceptable for further study (Torres et al., 2020).

Time-fractionalization produces a different structure. For a stiff-matter, spatially flat FLRW model, the space–time fractional Schutz–Wheeler–DeWitt equation is

$\alpha$7

with Caputo time derivative and Riesz spatial derivative. Its momentum-space solution is governed by the Mittag–Leffler function. In this setting, for $\alpha$8, total probability is not conserved, reflecting non-unitary and non-Markovian evolution (Rasouli et al., 2021).

4. Cosmological dynamics: inflation, bounces, and quantum creation

A major recurring result is the replacement of de Sitter inflation by accelerated power-law expansion. In the compact flat FLRW pre-inflation model, the slow-roll constant-potential regime yields

$\alpha$9

so that the exponent depends on the effective fractal dimension rather than on the height of the inflaton potential. In the closed de Sitter Riesz model, the late-time solution is

$(-\hbar^2\Delta)^{\alpha/2}$0

and acceleration occurs only for $(-\hbar^2\Delta)^{\alpha/2}$1. These results are repeatedly interpreted as a signature of nonlocal or fractal minisuperspace structure (Rasouli et al., 2022, Jalalzadeh et al., 2022).

The Fab Four John literature provides a second major class of dynamics: bouncing, cyclic, singular, and de Sitter branches within a single minisuperspace framework. In the canonical-transformation treatment, plane-wave solutions

$(-\hbar^2\Delta)^{\alpha/2}$2

have constant amplitude, so the Bohmian quantum potential vanishes and the guidance equations reproduce the classical first-order system. Depending on the choice of $(-\hbar^2\Delta)^{\alpha/2}$3, $(-\hbar^2\Delta)^{\alpha/2}$4, and the initial data for $(-\hbar^2\Delta)^{\alpha/2}$5 and $(-\hbar^2\Delta)^{\alpha/2}$6, the model admits singular power-law universes, de Sitter solutions, nonsingular bounces,

$(-\hbar^2\Delta)^{\alpha/2}$7

and cyclic oscillations $(-\hbar^2\Delta)^{\alpha/2}$8 (Torres et al., 2018).

In the conformable-derivative treatment of the same sector, the bounce is explicitly driven by the quantum potential. For the perfect-fluid-like ansatz

$(-\hbar^2\Delta)^{\alpha/2}$9

the constructed $(-\partial_a^2)^{\alpha/2}$0 is finite for all $(-\partial_a^2)^{\alpha/2}$1, peaks around $(-\partial_a^2)^{\alpha/2}$2, and falls off to zero as $(-\partial_a^2)^{\alpha/2}$3. This supports singularity avoidance together with late-time recovery of the classical expansion law $(-\partial_a^2)^{\alpha/2}$4 (Torres et al., 2020).

Quantum creation problems are also altered. In the FRW tunnelling model with radiation, positive cosmological constant, and an ad hoc potential, the WKB barrier penetration factor depends on the fractional index through

$(-\partial_a^2)^{\alpha/2}$5

The numerical conclusion is that decreasing $(-\partial_a^2)^{\alpha/2}$6 decreases the tunnelling probability, so the standard local limit $(-\partial_a^2)^{\alpha/2}$7 is the most likely within that setup. The same work points out a quantitative trade-off between $(-\partial_a^2)^{\alpha/2}$8 and $(-\partial_a^2)^{\alpha/2}$9: for example, the transition probability for $$\mathcal H_\alpha=\tfrac12 D_\alpha |f^{\mu\nu}\Pi_\mu\Pi_\nu|^{\alpha/2}+U(q)$$0 is close to that for $$\mathcal H_\alpha=\tfrac12 D_\alpha |f^{\mu\nu}\Pi_\mu\Pi_\nu|^{\alpha/2}+U(q)$$1 (Canedo et al., 19 Mar 2025).

The closed de Sitter Riesz model yields a different but related creation scenario. The tunnelling proposal favors $$\mathcal H_\alpha=\tfrac12 D_\alpha |f^{\mu\nu}\Pi_\mu\Pi_\nu|^{\alpha/2}+U(q)$$2, hence nearly smooth horizons and accelerated power-law expansion, whereas the no-boundary proposal peaks toward $$\mathcal H_\alpha=\tfrac12 D_\alpha |f^{\mu\nu}\Pi_\mu\Pi_\nu|^{\alpha/2}+U(q)$$3, corresponding to highly fractal horizons and decelerated expansion. In compact flat and open cases the classically forbidden region disappears, so creation is unsuppressed relative to the closed case (Jalalzadeh et al., 2022).

5. Phenomenology and observational confrontation

The most explicit background-data fit in this literature appears in the fractional $$\mathcal H_\alpha=\tfrac12 D_\alpha |f^{\mu\nu}\Pi_\mu\Pi_\nu|^{\alpha/2}+U(q)$$4CDM construction based on a fractional minisuperspace Hamiltonian constraint. Using Pantheon Type Ia supernovae, BAO distance ratios, CMB shift parameters, a Big Bang Nucleosynthesis constraint, and 31 cosmic chronometer $$\mathcal H_\alpha=\tfrac12 D_\alpha |f^{\mu\nu}\Pi_\mu\Pi_\nu|^{\alpha/2}+U(q)$$5 points, the joint $$\mathcal H_\alpha=\tfrac12 D_\alpha |f^{\mu\nu}\Pi_\mu\Pi_\nu|^{\alpha/2}+U(q)$$6-minimization gives, at $$\mathcal H_\alpha=\tfrac12 D_\alpha |f^{\mu\nu}\Pi_\mu\Pi_\nu|^{\alpha/2}+U(q)$$7 CL,

$$\mathcal H_\alpha=\tfrac12 D_\alpha |f^{\mu\nu}\Pi_\mu\Pi_\nu|^{\alpha/2}+U(q)$$8

together with $$\mathcal H_\alpha=\tfrac12 D_\alpha |f^{\mu\nu}\Pi_\mu\Pi_\nu|^{\alpha/2}+U(q)$$9 and $\Lambda$0. In this model, vacuum domination no longer gives $\Lambda$1 but the exact power law

$\Lambda$2

and the age of the universe is

$\Lambda$3

The asymptotic dimensionless age remains finite, with $\Lambda$4 for the best-fit case (Costa et al., 2023).

A related but conceptually different line derives fractional-fractal Friedmann and Raychaudhuri equations from a Riesz-fractional Wheeler–DeWitt equation, a fractal horizon entropy law, and Padmanabhan’s emergent-space paradigm. There the horizon fractal dimension is

$\Lambda$5

and the flat Friedmann equation becomes $\Lambda$6. The model is claimed to accommodate standard background evolution while reinterpreting the cold matter sector: for a tiny deviation from $\Lambda$7, the measured total cold-matter density can correspond entirely to baryons. The cosmic age varies between $\Lambda$8 and $\Lambda$9 Gyr as $\Pi_\phi^\beta$0 runs from $\Pi_\phi^\beta$1 (Junior et al., 2023).

A more recent reinterpretation does not introduce fractional derivatives directly at the fundamental level. Instead, a causal memory kernel is inserted at subleading order in the semiclassical Wheeler–DeWitt expansion, and an effective Caputo time derivative emerges,

$\Pi_\phi^\beta$2

In de Sitter perturbation theory this induces a primordial power-spectrum correction

$\Pi_\phi^\beta$3

which predominantly affects high-$\Pi_\phi^\beta$4 CMB anisotropies rather than the largest angular scales. The same framework also predicts scale-dependent corrections to the bispectrum and motivates the bound $\Pi_\phi^\beta$5 if memory-induced power is to remain subdominant (Shah et al., 11 Jun 2026).

6. Extensions, controversies, and broader theoretical relations

Fractional quantum cosmology extends beyond isotropic FLRW. In anisotropic Bianchi I $\Pi_\phi^\beta$6-essence models, the Wheeler–DeWitt equation contains a scalar-sector fractional derivative of order $\Pi_\phi^\beta$7, while the gravitational sectors remain Bessel-type. The scalar mode satisfies a linear fractional differential equation with variable coefficients, solved by fractional power series or by Mittag–Leffler functions in appropriate limits. The same literature reports that anisotropy parameters freeze out in the dust limit, suggesting isotropization for large fractional order (Socorro et al., 2023).

Noncommutative deformations have also been combined with fractional minisuperspace dynamics. In the flat FRW and Bianchi I $\Pi_\phi^\beta$8-essence models, the minisuperspace coordinates are deformed by a Moyal-type symplectic structure, implemented through Bopp shifts such as

$\Pi_\phi^\beta$9

The quantum probability density then develops a $\beta=\tfrac{2\alpha}{2\alpha-1}$0-dependent shift in the scalar-field sector. In the flat FRW case this shift pushes wave-packet support toward larger scale factors and causes classical evolution to arise earlier than in the commutative world; in the Bianchi I case the peak is shifted and sharpened in the $\beta=\tfrac{2\alpha}{2\alpha-1}$1-plane, which is interpreted as earlier classical emergence (Socorro et al., 2024, Socorro et al., 2024).

The literature also contains extensions away from standard cosmological backgrounds. In fractional AdS quantum gravity with Brown–Kuchař dust, the Riesz-fractional Wheeler–DeWitt eigenvalue problem yields a fractional mass spectrum,

$\beta=\tfrac{2\alpha}{2\alpha-1}$2

and a fractal mass dimension

$\beta=\tfrac{2\alpha}{2\alpha-1}$3

The associated entropy scales fractionally with the mass spectrum, and the nonlocal fractional Laplacian is interpreted as encoding heavy-tailed “Lévy-flight” behavior in quantum-gravitational dynamics (Júnior et al., 2 Jan 2025).

Several conceptual issues remain unsettled. The “Overture” stresses that a naïve fractional replacement of the minisuperspace d’Alembertian generally breaks covariance under field redefinitions because fractional derivatives are nonlocal and single out preferred differences in configuration space. It also notes that time-fractional derivatives can spoil unitarity unless the inner-product structure is correspondingly modified. A further point of method concerns the meaning of “fractional”: in some models it denotes genuinely nonlocal Riesz operators, in others Riemann–Liouville or Caputo derivatives, in others conformable derivatives, and in yet others merely fractional momentum powers that can be removed by canonical transformation. The Fab Four John literature is particularly instructive here, because one treatment explicitly states that no new fractional derivative operator is needed after a suitable canonical transformation (Moniz et al., 2020, Torres et al., 2018).

A broader, more top-down development is classical fractional gravity with self-adjoint fractional d’Alembertian operators. Its cosmological reduction yields exact de Sitter and bouncing solutions, with de Sitter an exact stable solution and bounces sustained by phantom or ghost fluids. This is not itself a minisuperspace quantum cosmology, but it suggests a route by which fractional quantum cosmology could be embedded in a nonlocal ultraviolet-complete gravitational theory rather than introduced only as a bottom-up deformation of the Wheeler–DeWitt equation (Salvador-García et al., 30 Apr 2026).