Extended Generalized Uncertainty Principle (EGUP)

• EGUP is a class of deformations of the Heisenberg uncertainty principle that incorporates both ultraviolet and infrared corrections to establish minimum measurable length and momentum scales.
• It is realized through diverse frameworks such as noncommutative phase space, modified operator algebras, and smeared space constructions, affecting black-hole thermodynamics and quantum-spacetime models.
• EGUP modifies the phase-space measure and density of states, leading to observable impacts in statistical mechanics, gravitational optics, and cosmological phenomena.

The Extended Generalized Uncertainty Principle (EGUP) denotes a class of deformations of the Heisenberg uncertainty principle that incorporate both ultraviolet and infrared corrections, usually by combining the momentum-dependent term of the Generalized Uncertainty Principle (GUP) with the position-dependent term of the Extended Uncertainty Principle (EUP). In much of the recent literature, the combined product-type relation is called either EGUP or GEUP, whereas other work reserves “EGUP” for a distinct duality-based construction; in phase-space formulations, the same idea is realized through noncommutativity of both coordinates and momenta (Pachoł, 2024, Carr et al., 5 Feb 2026, Lake, 2021). The subject therefore comprises not a single universally fixed equation, but a family of closely related UV+IR uncertainty structures with applications to noncommutative phase space, statistical mechanics, black-hole thermodynamics, gravitational optics, and quantum-spacetime phenomenology.

1. Definitions and nomenclature

The standard starting point is the Heisenberg relation

$\Delta x\,\Delta p \ge \frac{\hbar}{2},$

or, in heuristic form, $\Delta x \ge \hbar/(2\Delta p)$. In EGUP research, the GUP contribution is associated with a minimum length or Planck-scale correction, while the EUP contribution is associated with a minimum momentum or large-scale curvature correction. A widely used combined form is

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

with normalization conventions varying across papers through factors of $\ell_P$, $L_*$, or $\Lambda$ (Lake, 2021).

Formulation	Representative relation	Characteristic statement
Product-type combined EGUP/GEUP	$$\Delta x\,\Delta p \ge \frac{1}{2}\left(1+\alpha^2(\Delta x)^2+\beta^2(\Delta p)^2\right)$$	Combines EUP and GUP in one multiplicative relation (Pachoł, 2024)
Duality-symmetric EGUP	$$\Delta x + \frac{\beta Pl^2}{\Delta x} \ge \frac{\hbar}{2 \Delta p} + \frac{\alpha \Delta p \, Pl^2}{2 \hbar}$$	Reduces to GUP for $\beta=0$, but not to standard EUP for $\alpha=0$ (Carr et al., 5 Feb 2026)
Asymptotic curvature-combined form	$\Delta x \ge \hbar/(2\Delta p)$0	Curvature-derived EUP combined with GUP in an asymptotic regime (Dabrowski et al., 2020)

Within this literature, the deformation parameters are not purely symbolic. The Planck scale controls the short-distance sector, while the cosmological constant, de Sitter radius, or a large fundamental length $\Delta x \ge \hbar/(2\Delta p)$1 controls the long-distance sector. Several papers explicitly interpret EGUP as a framework in which both a minimum measurable length and a minimum momentum scale coexist (Hassanabadi et al., 2020, Liu et al., 8 Sep 2025).

A recurrent source of ambiguity is nomenclature. Some authors use “GEUP” and “EGUP” interchangeably for the product-type combined relation, while one recent study distinguishes GEUP from EGUP by reserving “EGUP” for a duality-symmetric alternative with different branch structure and limiting behavior (Carr et al., 5 Feb 2026). This suggests that the acronym alone does not uniquely determine the underlying model.

2. Algebraic and geometric realizations

One major line of development derives EGUP from modified operator algebras or from genuinely noncommutative phase space. A general phase-space deformation can be written as

$\Delta x \ge \hbar/(2\Delta p)$2

with Jacobi identities imposed; in the classical limit these become deformed Poisson brackets, and the phase-space Jacobian need no longer be unity under time evolution (Pachoł, 2024). In this setting, EGUP is not merely a modification of the uncertainty product but a deformation of the underlying symplectic geometry.

The Snyder–de Sitter and Yang models provide explicit examples in which both coordinates and momenta fail to commute. In the author’s interpretation, this is the natural phase-space framework for quantum-gravity-inspired deformations: noncommuting coordinates encode the Planck length, while noncommuting momenta encode the cosmological scale or de Sitter radius (Pachoł, 2024). In one-dimensional limits, these models reproduce quadratic or square-root uncertainty relations, including bounded positions, bounded momenta, or bounded combinations of them, depending on the sign choices and realization.

A second major line of work rejects modified commutators as fundamental. In the “smeared space” formalism, the background geometry is treated as a genuinely quantum object with its own state vector, and each classical point is mapped to a smeared state,

$\Delta x \ge \hbar/(2\Delta p)$3

so that the measured distribution becomes a convolution of matter and geometric fluctuations (Lake, 2021). In this approach, the canonical Heisenberg algebra remains unchanged,

$\Delta x \ge \hbar/(2\Delta p)$4

or is affected only by an extremely small rescaling, while the EGUP arises from variance addition in the enlarged state space rather than from a nonlinear deformation of the algebra (Lake, 2019, Lake, 2020).

This alternative construction is presented as a way to evade several well-known pathologies attributed to modified-commutator models, including equivalence-principle violation, Lorentz-invariance problems in relativistic limits, reference-frame dependence of the minimum length, and the soccer-ball problem for multiparticle states (Lake, 2021). In the 2020 formulation, the approach further implies two explicit claims: space must be quantized on a different scale than matter, and the fundamental quanta of geometry are fermions (Lake, 2020).

Constraint-based derivations form a related precursor literature. In particular, extended GUP commutators can be generated from an ordinary canonical Hamiltonian system by enlarging phase space, imposing second-class constraints, and computing Dirac brackets, so that the deformed bracket structure emerges from constrained symplectic reduction rather than from postulation (Eune et al., 2013). Although this construction is presented for GUP-type algebras, it is part of the broader effort to place generalized uncertainty relations on a dynamical footing.

3. Duality, curvature, and alternative formulations

The most explicit alternative to the standard product-type combination is the duality-symmetric EGUP proposed in 2026:

$\Delta x \ge \hbar/(2\Delta p)$5

Its defining feature is an explicit $\Delta x \ge \hbar/(2\Delta p)$6 duality: the left-hand side contains $\Delta x \ge \hbar/(2\Delta p)$7 and $\Delta x \ge \hbar/(2\Delta p)$8, while the right-hand side contains $\Delta x \ge \hbar/(2\Delta p)$9 and $$\Delta x\,\Delta p \ge \frac{\hbar}{2}\left(1+\alpha(\Delta p)^2+\eta(\Delta x)^2\right),$$0 (Carr et al., 5 Feb 2026). The authors emphasize that it reduces to the GUP when $$\Delta x\,\Delta p \ge \frac{\hbar}{2}\left(1+\alpha(\Delta p)^2+\eta(\Delta x)^2\right),$$1, but is not identical to the EUP when $$\Delta x\,\Delta p \ge \frac{\hbar}{2}\left(1+\alpha(\Delta p)^2+\eta(\Delta x)^2\right),$$2.

Solving this relation for $$\Delta x\,\Delta p \ge \frac{\hbar}{2}\left(1+\alpha(\Delta p)^2+\eta(\Delta x)^2\right),$$3 yields two branches,

$$\Delta x\,\Delta p \ge \frac{\hbar}{2}\left(1+\alpha(\Delta p)^2+\eta(\Delta x)^2\right),$$4

and the qualitative behavior depends on the sign and relative magnitude of $$\Delta x\,\Delta p \ge \frac{\hbar}{2}\left(1+\alpha(\Delta p)^2+\eta(\Delta x)^2\right),$$5 and $$\Delta x\,\Delta p \ge \frac{\hbar}{2}\left(1+\alpha(\Delta p)^2+\eta(\Delta x)^2\right),$$6 (Carr et al., 5 Feb 2026). The same study analyzes negative parameter values, notes a transition at $$\Delta x\,\Delta p \ge \frac{\hbar}{2}\left(1+\alpha(\Delta p)^2+\eta(\Delta x)^2\right),$$7, and finds connected or disconnected particle/black-hole regions depending on whether $$\Delta x\,\Delta p \ge \frac{\hbar}{2}\left(1+\alpha(\Delta p)^2+\eta(\Delta x)^2\right),$$8 or $$\Delta x\,\Delta p \ge \frac{\hbar}{2}\left(1+\alpha(\Delta p)^2+\eta(\Delta x)^2\right),$$9. By contrast, the authors regard the more standard GEUP as the better-motivated model for black-hole correspondence and reserve for it the emergence of a “strong-gravity black hole phase” (Carr et al., 5 Feb 2026).

A different route to UV+IR uncertainty relations is the asymptotic curvature program. There the EUP is derived perturbatively on arbitrary curved spatial slices from the lowest eigenvalue of the Laplace–Beltrami operator on a geodesic ball, giving

$\ell_P$0

with the leading second-order correction controlled by the Ricci scalar at the expectation position $\ell_P$1 (Dabrowski et al., 2020). Fourth-order terms involve the Cartan invariant $\ell_P$2 and the curved-space Laplacian of the Ricci scalar. Combining this curvature-derived EUP with a quadratic GUP produces the Asymptotic Generalized Extended Uncertainty Principle (AGEUP), which makes explicit that the infrared sector can be local, observer dependent, and curvature sensitive rather than reducible to a single global de Sitter scale (Dabrowski et al., 2020).

These variants show that EGUP is not exhausted by the simple addition of a $\ell_P$3 term and a $\ell_P$4 term. The literature includes algebraic, geometric, duality-based, and asymptotic-curvature realizations, each preserving different aspects of symmetry, locality, or interpretability.

4. Phase-space measure, Liouville theorem, and density of states

A central result of the phase-space-noncommutative program is that EGUP changes the elementary phase-space cell and therefore the density of states. In ordinary statistical mechanics, each state occupies a uniform cell $\ell_P$5; under EGUP deformations, the cell volume becomes position and momentum dependent, so the standard Liouville measure is no longer invariant (Pachoł, 2024).

For the Snyder–de Sitter model, the classical Poisson brackets are

$\ell_P$6

$\ell_P$7

and the infinitesimal Jacobian differs from unity under Hamiltonian time evolution (Pachoł, 2024). The invariant weighted measure is then

$\ell_P$8

which restores the classical analogue of Liouville’s theorem. The parameters $\ell_P$9 and $L_*$0 respectively control large-scale/curvature and small-scale/Planckian corrections, while the mixed $L_*$1 term couples the two sectors (Pachoł, 2024).

Special limits recover familiar one-sector deformations. In the Snyder limit $L_*$2, one obtains the quadratic GUP and invariant measure

$L_*$3

In the (anti-)de Sitter limit $L_*$4, one obtains the quadratic EUP and invariant measure

$L_*$5

The same paper further notes that Snyder–de Sitter can be mapped to Snyder by a noncanonical linear transformation (Pachoł, 2024).

For the Yang model, the classical realization gives a higher-order square-root relation; in $L_*$6,

$L_*$7

leading to

$L_*$8

Expanding to leading order yields the invariant weighted measure

$L_*$9

while the limits $\Lambda$0 and $\Lambda$1 produce higher-order GUP and EUP forms, respectively (Pachoł, 2024).

The physical consequence is direct: the density of states is modified by position- and momentum-dependent weights, which can alter partition functions, entropy, equations of state, and other statistical-mechanical observables (Pachoł, 2024). By contrast, the fuzzy-sphere model with commuting momenta,

$\Lambda$2

admits an invariant standard phase-space volume, so no weighted measure is required and the density of states is unchanged (Pachoł, 2024). This provides a useful counterexample: noncommutativity alone does not automatically imply a modified state-counting measure; the simultaneous structure of coordinate and momentum brackets matters.

5. Black-hole thermodynamics and remnant scenarios

EGUP has been used extensively as a phenomenological input for black-hole thermodynamics. In Schwarzschild-type analyses, it modifies the relation between horizon scale and characteristic energy, yielding corrected Unruh or Hawking temperatures, mass–temperature relations, specific heats, and entropies (Hassanabadi et al., 2020). In one representative treatment, the modified Unruh temperature takes the form

$\Lambda$3

so that the EUP-like term dominates at low Unruh temperature and the GUP-like term dominates at high Unruh temperature (Hassanabadi et al., 2020). The same work reports a minimum temperature, modified mass behavior, and entropy corrections containing both inverse-temperature and logarithmic structures.

A related construction uses the Einstein–Bohr photon box to infer an effective Newton constant from the EGUP-modified uncertainty relation. In that setup,

$\Lambda$4

which is then inserted into a modified Schwarzschild metric (Farahani et al., 2021). The resulting Hawking temperature contains separate EUP, GUP, and mixed EGUP corrections, and the paper identifies a lower bound on temperature and a corresponding minimum mass in the approximation used (Farahani et al., 2021).

Charged, regular, and matter-surrounded black holes have also been studied within the same framework. For the Bardeen black hole with quintessence, the Reissner–Nordström black hole with quintessence, the RN–AdS black hole with quintessence, the RN–AdS black hole with perfect fluid dark matter, and the Kerr–Newman black hole surrounded by quintessence, the common structural result is that the square-root reality condition in the EGUP-corrected temperature imposes a nonzero lower bound on the horizon radius, so evaporation halts at a remnant rather than proceeding to zero size (Wu et al., 2022, Tan, 2024, Chen et al., 21 Aug 2025, Liu et al., 8 Sep 2025, Meitei et al., 22 Jun 2026). These papers also analyze heat-capacity sign changes, divergence points, Gibbs free energy, equation-of-state behavior, and pressure, with EGUP generally interpolating between a GUP-dominated small-scale regime and an EUP-dominated large-scale regime.

The thermodynamic phase structure is model dependent. In the Kerr–Newman-plus-quintessence case, EGUP and EUP exhibit two divergence points in the heat capacity for $\Lambda$5, indicating two second-order phase transitions and a stable–unstable–stable sequence of small, intermediate, and large black-hole phases, whereas GUP shows only one divergence point in the same setting (Meitei et al., 22 Jun 2026). In the PFDM-surrounded RN–AdS case, the heat-capacity analysis identifies stable small-black-hole and large-black-hole phases separated by an unstable intermediate branch, and the Gibbs free energy develops a swallowtail for $\Lambda$6, indicating a first-order phase transition (Liu et al., 8 Sep 2025). By contrast, one RN–AdS-plus-quintessence study emphasizes that EGUP corrections are largely quantitative rather than qualitative and that the phase-transition point shifts to almost zero relative to the RN case (Tan, 2024).

Entropy behavior is likewise not universal. Several papers obtain EGUP-corrected entropies with GUP-like logarithmic pieces and EUP-like inverse-area or large-radius contributions (Hassanabadi et al., 2020, Meitei et al., 22 Jun 2026). In the RN and Bardeen quintessence models, however, the authors explicitly state that the entropy correction does not depend on the quintessence matter, even though temperature, heat capacity, and equation-of-state properties do (Chen et al., 21 Aug 2025, Wu et al., 2022).

6. Phenomenology, cosmology, and conceptual status

Beyond equilibrium thermodynamics, EGUP-type deformations have been used to model observable black-hole signatures. For static GEUP-corrected black holes, the shadow radius and deflection angles depend on the effective mass

$\Lambda$7

so that the shadow and lensing remain close to Schwarzschild behavior unless the mass approaches the correction scales (Lobos et al., 2022). Using EHT data, one study quotes $\Lambda$8 at $\Lambda$9 for Sgr. A* and $$\Delta x\,\Delta p \ge \frac{1}{2}\left(1+\alpha^2(\Delta x)^2+\beta^2(\Delta p)^2\right)$$0 at $$\Delta x\,\Delta p \ge \frac{1}{2}\left(1+\alpha^2(\Delta x)^2+\beta^2(\Delta p)^2\right)$$1 for M87*, and concludes that weak deflection is a better probe in the micro regime whereas strong deflection is more sensitive in the astrophysical regime (Lobos et al., 2022).

For rotating black holes, a phenomenological Kerr-like ansatz with $$\Delta x\,\Delta p \ge \frac{1}{2}\left(1+\alpha^2(\Delta x)^2+\beta^2(\Delta p)^2\right)$$2 has been used to derive modified thermodynamics, a GEUP-corrected Teukolsky equation, and eikonal quasinormal-mode shifts (Lobos, 20 Jan 2026). In that analysis, the infrared-dominated regime yields

$$\Delta x\,\Delta p \ge \frac{1}{2}\left(1+\alpha^2(\Delta x)^2+\beta^2(\Delta p)^2\right)$$3

implying rapid cooling and a greatly prolonged lifetime for supermassive black holes; the GUP parameter $$\Delta x\,\Delta p \ge \frac{1}{2}\left(1+\alpha^2(\Delta x)^2+\beta^2(\Delta p)^2\right)$$4 produces a spectral blueshift and enhanced damping, while the EUP parameter $$\Delta x\,\Delta p \ge \frac{1}{2}\left(1+\alpha^2(\Delta x)^2+\beta^2(\Delta p)^2\right)$$5 produces a spectral redshift and suppressed damping (Lobos, 20 Jan 2026). The same paper states that the shadow of M87* is approximately $$\Delta x\,\Delta p \ge \frac{1}{2}\left(1+\alpha^2(\Delta x)^2+\beta^2(\Delta p)^2\right)$$6 times more sensitive to large-scale corrections than Sgr A*, so EHT data dominantly constrain the infrared sector while gravitational-wave spectroscopy provides complementary information on the ultraviolet sector (Lobos, 20 Jan 2026).

EGUP has also been linked to cosmology at the level of physical interpretation. In the smeared-space framework, additional fluctuations of the background geometry give rise to EGUP without modifying the canonical Heisenberg algebra, and extremizing the generalized uncertainty product selects characteristic scales

$$\Delta x\,\Delta p \ge \frac{1}{2}\left(1+\alpha^2(\Delta x)^2+\beta^2(\Delta p)^2\right)$$7

from which an effective dark-energy density

$$\Delta x\,\Delta p \ge \frac{1}{2}\left(1+\alpha^2(\Delta x)^2+\beta^2(\Delta p)^2\right)$$8

is obtained (Lake, 2021). This is a model-specific proposal rather than a general consensus statement, but it shows how EGUP has been used to connect quantum-spacetime fluctuations with the observed cosmological constant scale.

Several misconceptions are addressed directly in the literature. EGUP is not always derived from modified commutators; some of the most explicit recent constructions keep the canonical algebra intact and attribute the extra uncertainty to quantum geometry degrees of freedom (Lake, 2019). EGUP is also not uniquely defined: in some papers it is the standard product-type GUP+EUP combination, whereas in others it denotes a duality-symmetric relation that does not recover the standard EUP in the $$\Delta x\,\Delta p \ge \frac{1}{2}\left(1+\alpha^2(\Delta x)^2+\beta^2(\Delta p)^2\right)$$9 limit (Carr et al., 5 Feb 2026). Finally, astrophysical effects are generally small unless the system probes the relevant UV or IR scales, a point emphasized in shadow and lensing studies (Lobos et al., 2022).

Taken together, these results place EGUP at the intersection of three research programs: deformed quantum kinematics, quantum geometry, and black-hole phenomenology. The literature supports a common structural message—simultaneous UV and IR corrections alter localization bounds, phase-space volume, and gravitational thermodynamics—but it does not support a single unique formalism. The term therefore names a research family whose internal distinctions are part of its current technical content rather than a settled terminological detail.