Improved Generalized Uncertainty Principle (GUP)

• Improved GUP is a deformation of the Heisenberg algebra that adds linear, quadratic, and higher-order corrections to establish cutoffs like minimal length and maximal momentum.
• It encompasses various formulations—including exact square-root, linear-plus-quadratic, and infrared deformations—to refine quantum, cosmological, and gravitational models.
• The framework yields actionable predictions and phenomenological constraints, influencing studies in phase-space methods, quantum optics, and black hole thermodynamics.

Searching arXiv for recent and foundational papers on improved/generalized uncertainty principles and related variants. Search query: "improved generalized uncertainty principle maximal momentum minimal length arXiv" Improved generalized uncertainty principle denotes a family of deformations of the Heisenberg algebra that extend the quadratic Kempf–Mangano–Mann relation in order to realize additional Planck-scale structures—minimal measurable length, maximal observable momentum, minimum uncertainty in momentum, maximal measurable length, or exact nonperturbative behavior—and to propagate those structures into quantum mechanics, black-hole thermodynamics, cosmology, phase-space methods, and relativistic kinematics. In the literature considered here, the expression does not denote a single canonical formula: it includes linear-plus-quadratic deformations associated with Ali, Das, and Vagenas, exact square-root commutators with bounded momentum, higher-order infrared deformations, doubly-special-relativity constructions, and entropy-derived commutators whose sign depends on the underlying non-extensive statistics (Petruzziello, 2020, Majumder, 2011, Berkowitz, 2020, Bizet et al., 2022).

1. Baseline structure and the sense of “improvement”

The standard reference point for much of the literature is the quadratic GUP

$$\Delta x\,\Delta p \ge \frac{\hbar}{2}\left(1+\beta (\Delta p)^2\right),$$

or, at the operator level,

$[x,p]=i\hbar(1+\beta p^2),$

with $\beta$ proportional to an inverse Planck-momentum scale. In the works surveyed here, this baseline is associated with a minimal measurable length but not, in its basic form, with a maximal momentum (Pedram, 2011, 0901.1768).

“Improved” GUP is then used in several related senses. One common meaning is a deformation that adds a term linear in momentum or in Planck length, as in the Ali–Das–Vagenas-type commutator

$$[x_i,p_j] = i\hbar \left[ \delta_{ij} - l\left(p\delta_{ij}+\frac{p_ip_j}{p}\right) + l^2\left(p^2\delta_{ij}+3p_ip_j\right) \right],$$

or the equivalent phenomenological uncertainty relation

$$\delta x\,\delta p \ge 1 + \beta l_p \delta p + \alpha^2 l_p^2 (\delta p)^2.$$

In that formulation the deformation implies both a minimum measurable length and a maximum measurable momentum, and is described as approximately compatible with doubly special relativity (Majumder, 2011).

A second meaning of “improvement” is exact completion beyond perturbation theory. The paper “Generalized uncertainty principle with maximal observable momentum and no minimal length indeterminacy” constructs the exact algebra

$[\hat x,\hat p]=i\hbar\sqrt{1-2\beta \hat p^2},$

which is presented as an exact GUP valid at all energy scales. Its low-energy expansion reproduces the quadratic deformation with the opposite sign of the usual KMM/string-inspired case, while the exact square root enforces a finite momentum cutoff (Petruzziello, 2020).

A third meaning is extension beyond ultraviolet minimal-length logic. Infrared GUPs deform the algebra through position rather than momentum, for example

$[\hat x,\hat p]=i\hbar(1+a\hat x^2),$

and imply a minimum uncertainty in momentum rather than a minimum uncertainty in position. Higher-order IR completions such as

$$[\hat x,\hat p]=i\hbar e^{a\hat x^2}, \qquad [\hat x,\hat p]=i\hbar\sqrt{1+2a\hat x^2},$$

and the maximal-length model

$$[\hat\alpha,\hat p_\alpha]=i\hbar\frac{1}{1-z\hat\alpha^2}$$

are explicitly described as higher-order or improved infrared generalizations (Berkowitz, 2020).

2. Algebraic constructions and characteristic cutoff structures

The improved-GUP literature organizes itself largely through the structure of the deformed commutator. Several recurrent constructions are summarized below.

Construction	Representative algebra	Primary cutoff structure
Quadratic KMM form	$[x,p]=i\hbar(1+\beta p^2)$	Minimal length
Linear-plus-quadratic form	$[x,p]=i\hbar(1+\beta p^2),$0	Minimal length and maximal momentum
Exact square-root form	$[x,p]=i\hbar(1+\beta p^2),$1	Maximal momentum, no nonzero minimal $[x,p]=i\hbar(1+\beta p^2),$2
DSR-motivated bounded momentum	$[x,p]=i\hbar(1+\beta p^2),$3	Minimal length and maximal momentum
Relativistic covariant form	$[x,p]=i\hbar(1+\beta p^2),$4	Lorentz-covariant minimal-length sector

The linear-plus-quadratic algebra plays a central role in phenomenology. In the formulation used to constrain GUP with GW150914, the deformation parameter is

$[x,p]=i\hbar(1+\beta p^2),$5

and the model implies both

$[x,p]=i\hbar(1+\beta p^2),$6

The same structure appears in harmonic-oscillator and cosmological applications, where the linear term is responsible for qualitatively new corrections, including $[x,p]=i\hbar(1+\beta p^2),$7 contributions to horizon entropy in FRW thermodynamics (Feng et al., 2016, Majumder, 2011, Bosso et al., 2017).

The exact square-root model differs sharply from positive-$[x,p]=i\hbar(1+\beta p^2),$8 KMM-type GUPs. From

$[x,p]=i\hbar(1+\beta p^2),$9

one derives

$\beta$0

and, for mirror-symmetric states,

$\beta$1

Reality of the square root yields

$\beta$2

while saturation gives

$\beta$3

The model therefore enforces a maximal momentum without a nonzero minimal position uncertainty; at the cutoff the commutator vanishes, $\beta$4, which the paper interprets as a classical-like regime at the endpoint (Petruzziello, 2020).

A distinct DSR realization is obtained by keeping the coordinate undeformed and deforming only the physical momentum: $\beta$5 This construction saturates at $\beta$6, thereby implementing a maximal physical momentum directly at the operator level. The corresponding uncertainty relation contains both $\beta$7 and $\beta$8, and the model is presented as a DSR-consistent alternative to quadratic GUPs in which the physical momentum remains unbounded (Chung et al., 2018).

Relativistic extension changes the algebraic setting rather than merely adding terms. In the Lorentz-covariant model of Bosso and Das, the preferred commutator sector is

$\beta$9

with

$$[x_i,p_j] = i\hbar \left[ \delta_{ij} - l\left(p\delta_{ij}+\frac{p_ip_j}{p}\right) + l^2\left(p^2\delta_{ij}+3p_ip_j\right) \right],$$0

and $$[x_i,p_j] = i\hbar \left[ \delta_{ij} - l\left(p\delta_{ij}+\frac{p_ip_j}{p}\right) + l^2\left(p^2\delta_{ij}+3p_ip_j\right) \right],$$1. In that sector the Poincaré algebra remains undeformed, while the phase-space algebra and the mass-shell relation are modified (Farutin et al., 2018).

3. Infrared, higher-order, and entropy-derived variants

Infrared-modified GUPs reverse the usual ultraviolet logic. Instead of momentum-dependent deformation leading to a minimal length, they use position-dependent deformation leading to a minimum uncertainty in momentum. The first-order IR relation

$$[x_i,p_j] = i\hbar \left[ \delta_{ij} - l\left(p\delta_{ij}+\frac{p_ip_j}{p}\right) + l^2\left(p^2\delta_{ij}+3p_ip_j\right) \right],$$2

gives

$$[x_i,p_j] = i\hbar \left[ \delta_{ij} - l\left(p\delta_{ij}+\frac{p_ip_j}{p}\right) + l^2\left(p^2\delta_{ij}+3p_ip_j\right) \right],$$3

with

$$[x_i,p_j] = i\hbar \left[ \delta_{ij} - l\left(p\delta_{ij}+\frac{p_ip_j}{p}\right) + l^2\left(p^2\delta_{ij}+3p_ip_j\right) \right],$$4

The higher-order exponential and square-root forms,

$$[x_i,p_j] = i\hbar \left[ \delta_{ij} - l\left(p\delta_{ij}+\frac{p_ip_j}{p}\right) + l^2\left(p^2\delta_{ij}+3p_ip_j\right) \right],$$5

resum the $$[x_i,p_j] = i\hbar \left[ \delta_{ij} - l\left(p\delta_{ij}+\frac{p_ip_j}{p}\right) + l^2\left(p^2\delta_{ij}+3p_ip_j\right) \right],$$6 dependence and yield

$$[x_i,p_j] = i\hbar \left[ \delta_{ij} - l\left(p\delta_{ij}+\frac{p_ip_j}{p}\right) + l^2\left(p^2\delta_{ij}+3p_ip_j\right) \right],$$7

respectively. A separate IR deformation,

$$[x_i,p_j] = i\hbar \left[ \delta_{ij} - l\left(p\delta_{ij}+\frac{p_ip_j}{p}\right) + l^2\left(p^2\delta_{ij}+3p_ip_j\right) \right],$$8

predicts both a minimum momentum uncertainty and a maximal measurable scale factor

$$[x_i,p_j] = i\hbar \left[ \delta_{ij} - l\left(p\delta_{ij}+\frac{p_ip_j}{p}\right) + l^2\left(p^2\delta_{ij}+3p_ip_j\right) \right],$$9

which is interpreted as a fixed maximum cosmological horizon (Berkowitz, 2020).

Another strand of improved-GUP research derives the deformation from non-extensive statistics rather than postulating it. In “Modified entropies as the origin of generalized uncertainty principles”, the non-extensive entropies

$$\delta x\,\delta p \ge 1 + \beta l_p \delta p + \alpha^2 l_p^2 (\delta p)^2.$$0

lead, via generalized exponentials and an effective Hamiltonian, to an effective momentum

$$\delta x\,\delta p \ge 1 + \beta l_p \delta p + \alpha^2 l_p^2 (\delta p)^2.$$1

and hence to the commutator

$$\delta x\,\delta p \ge 1 + \beta l_p \delta p + \alpha^2 l_p^2 (\delta p)^2.$$2

The deformation coefficients are

$$\delta x\,\delta p \ge 1 + \beta l_p \delta p + \alpha^2 l_p^2 (\delta p)^2.$$3

so $$\delta x\,\delta p \ge 1 + \beta l_p \delta p + \alpha^2 l_p^2 (\delta p)^2.$$4 produces a negative deformation parameter and $$\delta x\,\delta p \ge 1 + \beta l_p \delta p + \alpha^2 l_p^2 (\delta p)^2.$$5 a positive one. In the paper’s interpretation, the $$\delta x\,\delta p \ge 1 + \beta l_p \delta p + \alpha^2 l_p^2 (\delta p)^2.$$6 branch corresponds to minimal-length behavior, whereas the $$\delta x\,\delta p \ge 1 + \beta l_p \delta p + \alpha^2 l_p^2 (\delta p)^2.$$7 branch corresponds to maximal-momentum behavior (Bizet et al., 2022).

The same statistical mechanism has also been extended to the energy-time sector. Introducing the tempus operator $$\delta x\,\delta p \ge 1 + \beta l_p \delta p + \alpha^2 l_p^2 (\delta p)^2.$$8 and an entropy-deformed energy $$\delta x\,\delta p \ge 1 + \beta l_p \delta p + \alpha^2 l_p^2 (\delta p)^2.$$9, the generalized commutator becomes

$[\hat x,\hat p]=i\hbar\sqrt{1-2\beta \hat p^2},$0

with uncertainty relation

$[\hat x,\hat p]=i\hbar\sqrt{1-2\beta \hat p^2},$1

The $[\hat x,\hat p]=i\hbar\sqrt{1-2\beta \hat p^2},$2 branch yields a minimum measurable time interval of order the Planck time, while the $[\hat x,\hat p]=i\hbar\sqrt{1-2\beta \hat p^2},$3 branch yields a maximum energy uncertainty of order the Planck energy. The same framework also produces the modified dispersion relation

$[\hat x,\hat p]=i\hbar\sqrt{1-2\beta \hat p^2},$4

and sign-dependent corrections to black-hole temperature (Bizet et al., 2024).

4. Quantum-mechanical realizations, states, and phase-space structure

Improved GUP models are not only algebraic deformations; they are accompanied by concrete Hilbert-space realizations. For the exact square-root model, the momentum-space representation is

$[\hat x,\hat p]=i\hbar\sqrt{1-2\beta \hat p^2},$5

with

$[\hat x,\hat p]=i\hbar\sqrt{1-2\beta \hat p^2},$6

The inner product is modified to

$[\hat x,\hat p]=i\hbar\sqrt{1-2\beta \hat p^2},$7

reflecting the finite momentum interval. Position eigenstates are generalized plane waves in $[\hat x,\hat p]=i\hbar\sqrt{1-2\beta \hat p^2},$8, and they are not orthogonal. Because the model has no nonzero lower bound on $[\hat x,\hat p]=i\hbar\sqrt{1-2\beta \hat p^2},$9, it does not require the construction of “maximally localized states” familiar from minimal-length GUPs (Petruzziello, 2020).

The same paper studies the one-dimensional harmonic oscillator in the exact square-root algebra and maps the momentum-space Schrödinger equation to a quantum-pendulum/Mathieu problem. No closed analytic spectrum is given, but two structural results are emphasized: the spectrum remains discrete, and because momentum is bounded, the harmonic-oscillator energy spectrum is finite and bounded from above (Petruzziello, 2020).

Phase-space formulations make the deformation visually explicit. In “Generalized Uncertainty Principle Corrections to the Simple Harmonic Oscillator in Phase Space”, Wigner functions are constructed numerically from GUP-corrected momentum-space eigenfunctions. For the quadratic model

$[\hat x,\hat p]=i\hbar(1+a\hat x^2),$0

the Wigner functions lose the circular symmetry of the ordinary oscillator while preserving evenness in $[\hat x,\hat p]=i\hbar(1+a\hat x^2),$1 and $[\hat x,\hat p]=i\hbar(1+a\hat x^2),$2. For the linear-plus-quadratic model

$[\hat x,\hat p]=i\hbar(1+a\hat x^2),$3

$[\hat x,\hat p]=i\hbar(1+a\hat x^2),$4 symmetry remains, but $[\hat x,\hat p]=i\hbar(1+a\hat x^2),$5 symmetry is broken, and the momentum marginal becomes asymmetric. The energy spectrum is shifted upward in both models, with corrections growing rapidly with $[\hat x,\hat p]=i\hbar(1+a\hat x^2),$6 (Das et al., 2014).

The operator-algebraic treatment of the GUP-perturbed harmonic oscillator was pushed further in “Planck scale Corrections to the Harmonic Oscillator, Coherent and Squeezed States”, which uses the one-dimensional commutator

$[\hat x,\hat p]=i\hbar(1+a\hat x^2),$7

The resulting Hamiltonian contains cubic and quartic momentum terms,

$[\hat x,\hat p]=i\hbar(1+a\hat x^2),$8

and the level spacing depends on the combination $[\hat x,\hat p]=i\hbar(1+a\hat x^2),$9. The paper constructs modified ladder operators $$[\hat x,\hat p]=i\hbar e^{a\hat x^2}, \qquad [\hat x,\hat p]=i\hbar\sqrt{1+2a\hat x^2},$$0 acting on perturbed oscillator states, defines coherent states by $$[\hat x,\hat p]=i\hbar e^{a\hat x^2}, \qquad [\hat x,\hat p]=i\hbar\sqrt{1+2a\hat x^2},$$1, and shows that these coherent states still saturate the Schrödinger–Robertson bound. By contrast, squeezed states generally cease to be minimum-uncertainty states when the linear term is present (Bosso et al., 2017).

Quantum-optical realizations exhibit similar structure. In the GUP-corrected Jaynes–Cummings model, the initial field commutator again contains both linear and quadratic momentum terms, but after the rotating-wave approximation the surviving correction is effectively quadratic. The resonant Rabi frequency becomes

$$[\hat x,\hat p]=i\hbar e^{a\hat x^2}, \qquad [\hat x,\hat p]=i\hbar\sqrt{1+2a\hat x^2},$$2

while in the dispersive regime the evolution of a coherent field produces one- and two-photon-added coherent-state components. The paper identifies the corresponding Wigner-function change as the most plausible optical signature of the deformation (Khanna et al., 2022).

Improved GUPs also modify entanglement criteria. For continuous-variable systems, the variance-based analyses of Rigolin, Duan, and van Loock–Furusawa acquire positive GUP corrections, raising the relevant lower or upper bounds. In the Shannon-entropic formulation, the deformed momentum variable

$$[\hat x,\hat p]=i\hbar e^{a\hat x^2}, \qquad [\hat x,\hat p]=i\hbar\sqrt{1+2a\hat x^2},$$3

changes the probability measure and adds the positive term

$$[\hat x,\hat p]=i\hbar e^{a\hat x^2}, \qquad [\hat x,\hat p]=i\hbar\sqrt{1+2a\hat x^2},$$4

to the entropy. The result is a higher separability threshold in the entropic uncertainty relation (Blado et al., 2017, Gadea et al., 2018).

5. Black holes, cosmology, and gravitational extensions

Black-hole thermodynamics is one of the most developed application areas. In the exact square-root model, identifying $$[\hat x,\hat p]=i\hbar e^{a\hat x^2}, \qquad [\hat x,\hat p]=i\hbar\sqrt{1+2a\hat x^2},$$5 with the Schwarzschild radius yields the modified Hawking temperature

$$[\hat x,\hat p]=i\hbar e^{a\hat x^2}, \qquad [\hat x,\hat p]=i\hbar\sqrt{1+2a\hat x^2},$$6

with $$[\hat x,\hat p]=i\hbar e^{a\hat x^2}, \qquad [\hat x,\hat p]=i\hbar\sqrt{1+2a\hat x^2},$$7 fixed by the $$[\hat x,\hat p]=i\hbar e^{a\hat x^2}, \qquad [\hat x,\hat p]=i\hbar\sqrt{1+2a\hat x^2},$$8 limit. Instead of diverging as $$[\hat x,\hat p]=i\hbar e^{a\hat x^2}, \qquad [\hat x,\hat p]=i\hbar\sqrt{1+2a\hat x^2},$$9, the temperature approaches a finite maximum,

$$[\hat\alpha,\hat p_\alpha]=i\hbar\frac{1}{1-z\hat\alpha^2}$$0

The evaporation rate

$$[\hat\alpha,\hat p_\alpha]=i\hbar\frac{1}{1-z\hat\alpha^2}$$1

tends to zero as $$[\hat\alpha,\hat p_\alpha]=i\hbar\frac{1}{1-z\hat\alpha^2}$$2, the heat capacity remains negative and diverges asymptotically, and the total evaporation time diverges. The endpoint is therefore an asymptotic or metastable remnant rather than a finite nonzero remnant mass. The entropy acquires a logarithmic correction with positive sign, implying $$[\hat\alpha,\hat p_\alpha]=i\hbar\frac{1}{1-z\hat\alpha^2}$$3 in the reported expression (Petruzziello, 2020).

Cosmological horizon thermodynamics provides another arena in which improved GUPs are operationalized. Using the generalized relation

$$[\hat\alpha,\hat p_\alpha]=i\hbar\frac{1}{1-z\hat\alpha^2}$$4

and applying it to the apparent horizon of an FRW universe, the entropy-area relation becomes

$$[\hat\alpha,\hat p_\alpha]=i\hbar\frac{1}{1-z\hat\alpha^2}$$5

The leading correction is proportional to $$[\hat\alpha,\hat p_\alpha]=i\hbar\frac{1}{1-z\hat\alpha^2}$$6, a distinctive effect of the linear Planck-length term. Substituting the corrected entropy into the Clausius relation produces modified Friedmann equations with half-integer powers of the apparent-horizon area (Majumder, 2011).

Infrared GUPs have been applied directly to minisuperspace quantum cosmology. In the LRS Bianchi I model, only the $$[\hat\alpha,\hat p_\alpha]=i\hbar\frac{1}{1-z\hat\alpha^2}$$7-sector of phase space is deformed, leading to Wheeler–DeWitt equations such as

$$[\hat\alpha,\hat p_\alpha]=i\hbar\frac{1}{1-z\hat\alpha^2}$$8

for the exponential higher-order IR deformation. Numerical wave packets show suppression of non-central peaks, with the amount of suppression tracking the size of the corresponding minimum momentum uncertainty. The dominant central peak is interpreted as a preferred cosmic geometry selected by infrared quantum-gravity effects (Berkowitz, 2020).

In gravitational theory proper, GUP has also been implemented through a nonlocal deformation of the gravitational action. The resulting metric has the Schwarzschild form with an effective mass function

$$[\hat\alpha,\hat p_\alpha]=i\hbar\frac{1}{1-z\hat\alpha^2}$$9

so the point source is replaced by a smeared profile. That construction yields a two-horizon geometry and an extremal zero-temperature remnant. Extending such GUP scenarios to higher-dimensional spacetimes is, however, nontrivial. The review “Generalized uncertainty principle and extra dimensions” argues that common higher-dimensional extensions do not suppress ultraviolet modes strongly enough to improve the short-distance Newtonian potential for $[x,p]=i\hbar(1+\beta p^2)$0, and proposes instead the stronger measure

$[x,p]=i\hbar(1+\beta p^2)$1

as a more plausible higher-dimensional implementation below the compactification scale (Köppel et al., 2017).

6. Formal classification, phenomenology, and recurring limitations

One of the most technically explicit treatments of the formal structure is “Quantum theory of the Generalised Uncertainty Principle”. It studies the commutative $[x,p]=i\hbar(1+\beta p^2)$2-dimensional algebra

$[x,p]=i\hbar(1+\beta p^2)$3

with $[x,p]=i\hbar(1+\beta p^2)$4, and shows that, at the algebraic level, a deformed commutator with quadratic Hamiltonian is equivalent to a standard Heisenberg algebra with an aquadratic Hamiltonian, provided the boost sector is modified accordingly. The translation generators are

$[x,p]=i\hbar(1+\beta p^2)$5

and the appearance of a minimal length is shown to depend on whether these generators are bounded. In that analysis, minimal length is not an artifact of writing a deformed commutator; it is tied to the spectral relation between physical momentum and translations, and that property survives the algebraic mapping to the aquadratic formulation (Bruneton et al., 2016).

Perturbative generalizations also remain important. The class of deformed commutators

$[x,p]=i\hbar(1+\beta p^2)$6

was studied through a method that isolates physically acceptable solutions without solving the full $[x,p]=i\hbar(1+\beta p^2)$7-th order generalized Schrödinger equation. For the particle in a box, the deformed spectrum is

$[x,p]=i\hbar(1+\beta p^2)$8

while the free-particle plane wave acquires a GUP-dependent phase and normalization. The paper emphasizes that its results are perturbative to first order in $[x,p]=i\hbar(1+\beta p^2)$9 and do not address nonperturbative or full operator-domain questions for arbitrary $[x,p]=i\hbar(1+\beta p^2),$00 (Pedram, 2011).

Phenomenological constraints are model dependent. Using the gravitational-wave event GW150914, the quadratic GUP parameter and the linear-plus-quadratic parameter were bounded through modified graviton propagation: $[x,p]=i\hbar(1+\beta p^2),$01 The analysis treats the GUP-induced speed shift as a perturbative correction to the graviton dispersion relation and translates the observational upper bound on $[x,p]=i\hbar(1+\beta p^2),$02 into bounds on deformation parameters (Feng et al., 2016).

Semiclassical many-body thermodynamics has become another testing ground. In the MIT bag-model equation of state, the quadratic Kempf GUP is implemented through the deformed phase-space measure

$[x,p]=i\hbar(1+\beta p^2),$03

The resulting energy density, pressure, and baryon density all saturate at finite values as the Fermi momentum tends to infinity; there is no hard momentum cutoff, but the density of states is sufficiently suppressed that the thermodynamic integrals converge. The paper interprets the corresponding $[x,p]=i\hbar(1+\beta p^2),$04 relation as stiffer than in the undeformed MIT bag model (Netz-Marzola et al., 2024).

Recurring limitations are explicit across the literature. The exact square-root maximal-momentum model is postulated rather than derived from an underlying quantum-gravity theory, does not provide a full self-adjointness analysis, leaves the harmonic oscillator without a closed analytic spectrum, and remains largely qualitative phenomenologically (Petruzziello, 2020). The higher-order IR cosmology program deforms only the $[x,p]=i\hbar(1+\beta p^2),$05-sector of minisuperspace, gives only a sketched numerical procedure, and does not apply its maximal-length commutator to the Wheeler–DeWitt problem it proposes for future work (Berkowitz, 2020). Perturbative oscillator analyses with linear-plus-quadratic commutators develop pathologies such as negative $[x,p]=i\hbar(1+\beta p^2),$06 or a maximal oscillator number in some regimes, which are interpreted as signs that the truncated expansion is not trustworthy there (Bosso et al., 2017). The extra-dimensional review likewise emphasizes that there is no unique higher-dimensional extension and that short-distance regularization depends sensitively on the chosen deformation (Köppel et al., 2017).

Taken together, these works define improved GUP not as a fixed formula but as a research program: exact or higher-order completion of deformed commutators, inclusion of linear or infrared terms, extension to relativistic and statistical settings, and systematic tracking of how different deformations alter spectra, states, thermodynamics, geometry, and symmetry. The unifying issue is the structure of the ultraviolet or infrared cutoff—whether it appears as minimal length, maximal momentum, minimum momentum, maximal length, bounded translation generators, or entropy-induced deformation—and the extent to which that structure can be made algebraically consistent, physically interpretable, and phenomenologically testable (Petruzziello, 2020, Bruneton et al., 2016, Bizet et al., 2024).