Extended Uncertainty Principle Overview

• Extended Uncertainty Principle is a modification of Heisenberg's uncertainty relation that introduces infrared, curvature-sensitive corrections, establishing a minimum momentum scale.
• It employs frameworks like smeared-space formalisms and finite-resolution measurements to incorporate position-dependent deformations while preserving core quantum commutators.
• EUP applications span black hole thermodynamics, effective spacetime metrics, and cosmological models, offering novel insights into horizon structures and gravitational lensing.

The Extended Uncertainty Principle (EUP) denotes a family of infrared, large-scale, or curvature-sensitive deformations of the Heisenberg uncertainty principle in which the uncertainty product acquires a position-dependent correction. In one widely used de Sitter form, the momentum uncertainty is written as

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

with $\Lambda$ the cosmological constant and $\eta$ a dimensionless constant of order unity; in other conventions one finds

$$\Delta X\,\Delta P \ge \hbar\left(1+\beta(\Delta X)^2\right),$$

or exact operational bounds such as

$$\sigma_p\Delta x \ge \pi\hbar\,\Phi_\alpha(\Delta x/2), \qquad \Phi_\alpha(z)=\frac{\sqrt{\alpha}\,z}{\arctan(\sqrt{\alpha}\,z)}.$$

Across this literature, the EUP is usually interpreted as the infrared counterpart of the GUP: it is associated with minimum momentum, maximal length, or large-distance curvature effects, although some higher-order variants instead preserve a minimum length. The subject now spans superpositions-of-geometries models, finite-accuracy measurement theory, black-hole thermodynamics, effective black-hole metrics, cosmology, and many-body quantum systems (Lake, 2019, Schürmann, 10 Jan 2025, Luo et al., 2023).

1. Definitions, scales, and representative formulas

A central theme of the EUP literature is that the deformation becomes relevant when $\Delta x$ is large rather than small. In the smeared-space treatment, the EUP appears as the “large-scale” analogue of the GUP: the GUP is tied to Planck-scale physics and a minimum length,

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

whereas the EUP is tied to cosmological curvature and a minimum momentum,

$$\Delta p \ge \frac{\hbar}{2\Delta x}+\eta\,\Lambda\,\Delta x.$$

The combined extended generalized uncertainty principle (EGUP) is then written as

$$\Delta x\,\Delta p \ge \hbar+\tilde{\alpha}(\Delta p)^2+\tilde{\eta}(\Delta x)^2,$$

with the EUP recovered as the infrared limit (Lake, 2019).

The physical interpretation adopted in several papers is that the EUP encodes a nonzero minimum momentum scale set by cosmological curvature. In the de Sitter discussion of smeared space, this scale is of order the de Sitter momentum $\sim m_{\rm dS}c$, with $\Lambda$0. In the large-fundamental-length formulation,

$\Lambda$1

the relevant scale is a new infrared length $\Lambda$2, and the GR or HUP limit is recovered as $\Lambda$3 (Mureika, 2018, Ökcü et al., 2022).

Operational work has sharpened this picture by replacing wavefunction-based $\Lambda$4 with apparatus-centred confinement lengths. For a slit of width $\Lambda$5, the theorem

$\Lambda$6

provides a rigorous lower bound, and the same structure extends to three dimensions and to $\Lambda$7-balls of radius $\Lambda$8. In this formulation $\Lambda$9, so the EUP strengthens the confinement-induced momentum lower bound relative to the ordinary operational inequality $\eta$0 (Schürmann, 10 Jan 2025).

2. Derivational frameworks and operator realizations

One major line of development derives the EUP without modifying the Heisenberg algebra in any substantive way. In the smeared-space formalism, a classical point $\eta$1 is replaced by a superposition of nearby points $\eta$2, the quantum state becomes entangled with geometry, and the observed probability distribution is a convolution. In momentum space, an analogous smearing function produces an enlarged momentum variance; after using the ordinary HUP and a first-order Taylor expansion, one obtains an EUP term proportional to $\eta$3. In this construction the canonical commutator survives up to a tiny rescaling,

$\eta$4

no nonlinear momentum-addition law is required, and the framework is presented as compatible with the equivalence principle and with local Poincaré invariance in the relativistic limit (Lake, 2019).

A closely related canonical route replaces ideal observables by finite-resolution POVMs. For momentum measurements the smeared variance becomes

$\eta$5

and identifying the momentum-space resolution scale with the de Sitter scale yields

$\eta$6

Because the canonical Hamiltonian, Schrödinger equation, Heisenberg equation, and commutators remain unchanged, this approach explicitly interprets EUP phenomenology as a finite-accuracy measurement effect rather than a deformation of quantum dynamics (Lake et al., 2023).

A different strand keeps a deformed commutator but insists on Hermitian momentum operators and well-posed boundary conditions. In one dimension the operational paper uses

$\eta$7

whose extra $\eta$8 term is essential for self-adjointness. The associated eigenfunctions reduce smoothly to plane waves as $\eta$9, and in three dimensions the Hermitian operator

$$\Delta X\,\Delta P \ge \hbar\left(1+\beta(\Delta X)^2\right),$$0

leads to a deformed spherical Bessel problem inside a ball of radius $$\Delta X\,\Delta P \ge \hbar\left(1+\beta(\Delta X)^2\right),$$1. The lower bound on $$\Delta X\,\Delta P \ge \hbar\left(1+\beta(\Delta X)^2\right),$$2 then comes from the smallest allowed eigenvalue in a bounded domain, not from heuristic commutator manipulations alone (Schürmann, 10 Jan 2025).

3. Horizons, black holes, and entropy

Black-hole thermodynamics is one of the most active arenas for EUP applications. A prominent result is the identification of the EUP parameter with the Rényi non-extensivity parameter. Starting from

$$\Delta X\,\Delta P \ge \hbar\left(1+\beta(\Delta X)^2\right),$$3

and using a Bekenstein-type area-change argument, one obtains

$$\Delta X\,\Delta P \ge \hbar\left(1+\beta(\Delta X)^2\right),$$4

with $$\Delta X\,\Delta P \ge \hbar\left(1+\beta(\Delta X)^2\right),$$5. In the Bohr-like quantization picture for Schwarzschild black holes,

$$\Delta X\,\Delta P \ge \hbar\left(1+\beta(\Delta X)^2\right),$$6

so the entropy, temperature, heat capacity, and evaporation time all become explicit functions of the principal quantum number $$\Delta X\,\Delta P \ge \hbar\left(1+\beta(\Delta X)^2\right),$$7. The heat capacity changes sign at

$$\Delta X\,\Delta P \ge \hbar\left(1+\beta(\Delta X)^2\right),$$8

producing stable and unstable phases, and the entropy can vanish at the top excitation level while the temperature remains finite and positive (Moradpour et al., 2019).

A complementary program derives exact EUP formulas from horizon geometry itself. For Rindler and Friedmann horizons, the uncertainty relation is obtained by solving a Laplace-Beltrami eigenvalue problem on a compact domain with Dirichlet boundary conditions. The resulting exact formulas reduce asymptotically to familiar EUP-like expansions, and when these are translated into black-hole thermodynamics they lower the Hawking temperature and increase the Bekenstein entropy. The paper interprets this behavior as similar to canonical thermal-fluctuation corrections and as signaling further loss of information (Dabrowski et al., 2019).

EUP-deformed black-hole thermodynamics has also been studied for nonstandard black holes. For Van der Waals black holes, the working deformation

$$\Delta X\,\Delta P \ge \hbar\left(1+\beta(\Delta X)^2\right),$$9

implies

$$\sigma_p\Delta x \ge \pi\hbar\,\Phi_\alpha(\Delta x/2), \qquad \Phi_\alpha(z)=\frac{\sqrt{\alpha}\,z}{\arctan(\sqrt{\alpha}\,z)}.$$0

modifies the Hawking temperature through a factor $$\sigma_p\Delta x \ge \pi\hbar\,\Phi_\alpha(\Delta x/2), \qquad \Phi_\alpha(z)=\frac{\sqrt{\alpha}\,z}{\arctan(\sqrt{\alpha}\,z)}.$$1, and yields the entropy

$$\sigma_p\Delta x \ge \pi\hbar\,\Phi_\alpha(\Delta x/2), \qquad \Phi_\alpha(z)=\frac{\sqrt{\alpha}\,z}{\arctan(\sqrt{\alpha}\,z)}.$$2

In this model the EUP imposes an upper bound on the event horizon radius, restricts the physical range of mass and temperature, produces a finite remnant at large horizon radius, and gives parameter-dependent stability: for $$\sigma_p\Delta x \ge \pi\hbar\,\Phi_\alpha(\Delta x/2), \qquad \Phi_\alpha(z)=\frac{\sqrt{\alpha}\,z}{\arctan(\sqrt{\alpha}\,z)}.$$3 the black hole is unstable for all horizon radii, whereas for $$\sigma_p\Delta x \ge \pi\hbar\,\Phi_\alpha(\Delta x/2), \qquad \Phi_\alpha(z)=\frac{\sqrt{\alpha}\,z}{\arctan(\sqrt{\alpha}\,z)}.$$4 stable and unstable regimes both occur (Oubagha et al., 2023).

4. Effective metrics, lensing, and observational bounds

A large phenomenological literature treats the EUP as a source of explicit spacetime-metric deformations. In the EUP-inspired Schwarzschild construction,

$$\sigma_p\Delta x \ge \pi\hbar\,\Phi_\alpha(\Delta x/2), \qquad \Phi_\alpha(z)=\frac{\sqrt{\alpha}\,z}{\arctan(\sqrt{\alpha}\,z)}.$$5

the effective mass is

$$\sigma_p\Delta x \ge \pi\hbar\,\Phi_\alpha(\Delta x/2), \qquad \Phi_\alpha(z)=\frac{\sqrt{\alpha}\,z}{\arctan(\sqrt{\alpha}\,z)}.$$6

and the horizon, ISCO, and photosphere all receive corrections proportional to $$\sigma_p\Delta x \ge \pi\hbar\,\Phi_\alpha(\Delta x/2), \qquad \Phi_\alpha(z)=\frac{\sqrt{\alpha}\,z}{\arctan(\sqrt{\alpha}\,z)}.$$7. The paper argues that if $$\sigma_p\Delta x \ge \pi\hbar\,\Phi_\alpha(\Delta x/2), \qquad \Phi_\alpha(z)=\frac{\sqrt{\alpha}\,z}{\arctan(\sqrt{\alpha}\,z)}.$$8, EUP effects become relevant for black holes with $$\sigma_p\Delta x \ge \pi\hbar\,\Phi_\alpha(\Delta x/2), \qquad \Phi_\alpha(z)=\frac{\sqrt{\alpha}\,z}{\arctan(\sqrt{\alpha}\,z)}.$$9, and it further notes that the weak-field potential is enhanced by a factor $\Delta x$0, which may contribute to dark-matter-like behavior around $\Delta x$1 from the galactic center (Mureika, 2018).

Using the corresponding EUP metric in classical gravitational tests yields lower bounds on $\Delta x$2 spanning many orders of magnitude (Ökcü et al., 2022).

Test	Lower bound on $\Delta x$3
Gravitational redshift	$\Delta x$4
Geodetic precession	$\Delta x$5
Shapiro time delay	$\Delta x$6
Mercury perihelion precession	$\Delta x$7
S2 orbital precession	$\Delta x$8

The same broad metric framework has been extended to environmental and rotating backgrounds. For an EUP Schwarzschild black hole surrounded by a static spherical dark-matter shell, the inner shell radius is chosen to coincide with the horizon, so the shell does not shift the event horizon or Hawking temperature, although it does modify the ISCO, photon sphere, shadow radius, and weak deflection angle; in the diluted-shell approximation the shadow analysis requires $\Delta x$9 (Pantig et al., 2021). In a rotating Gödel background, a curvature-modified AGEUP leads to a lapse function with explicit dependence on the global rotation parameter $$\Delta x \ge \frac{\hbar}{2\Delta p}+\alpha\,G\,\Delta p,$$0 and the observer position $$\Delta x \ge \frac{\hbar}{2\Delta p}+\alpha\,G\,\Delta p,$$1; event horizon, photon sphere, shadow radius, and deflection angle all increase relative to Schwarzschild, while EHT and solar-system data imply $$\Delta x \ge \frac{\hbar}{2\Delta p}+\alpha\,G\,\Delta p,$$2 and $$\Delta x \ge \frac{\hbar}{2\Delta p}+\alpha\,G\,\Delta p,$$3 (Pantig, 30 May 2025).

Not all EUP-inspired metrics predict the same optical phenomenology. A 2026 construction fixes the modified metric by requiring its surface gravity to reproduce an EUP-corrected Hawking temperature,

$$\Delta x \ge \frac{\hbar}{2\Delta p}+\alpha\,G\,\Delta p,$$4

and in that model the event horizon remains at $$\Delta x \ge \frac{\hbar}{2\Delta p}+\alpha\,G\,\Delta p,$$5, the photon sphere radius increases with the EUP parameter, but the shadow radius decreases. Comparison with EHT bounds for Sgr A* gives

$$\Delta x \ge \frac{\hbar}{2\Delta p}+\alpha\,G\,\Delta p,$$6

with $$\Delta x \ge \frac{\hbar}{2\Delta p}+\alpha\,G\,\Delta p,$$7 (Zhen et al., 4 Mar 2026). This diversity suggests that “the EUP metric” is not unique; it depends on how the uncertainty relation is geometrized.

5. Cosmology and many-body applications

In cosmology, EUP corrections have been used both as dynamical modifications and as measurement-theoretic effects. A higher-order EUP with parameter adaptability,

$$\Delta x \ge \frac{\hbar}{2\Delta p}+\alpha\,G\,\Delta p,$$8

was introduced to preserve the same minimum measurable length for both signs of the deformation parameter,

$$\Delta x \ge \frac{\hbar}{2\Delta p}+\alpha\,G\,\Delta p,$$9

Applied at the apparent horizon, it produces logarithmic entropy corrections, modified Friedmann equations, and a nonzero baryon asymmetry in a radiation-dominated universe through gravitational baryogenesis. Using observational baryon-asymmetry bounds, the allowed ranges were quoted as

$$\Delta p \ge \frac{\hbar}{2\Delta x}+\eta\,\Lambda\,\Delta x.$$0

for positive deformation and

$$\Delta p \ge \frac{\hbar}{2\Delta x}+\eta\,\Lambda\,\Delta x.$$1

for negative deformation (Luo et al., 2023).

A separate thermodynamic cosmology derives modified Friedmann equations from an EUP-corrected entropy-area relation and rewrites the result as an effective fluid $$\Delta p \ge \frac{\hbar}{2\Delta x}+\eta\,\Lambda\,\Delta x.$$2. In that model negative $$\Delta p \ge \frac{\hbar}{2\Delta x}+\eta\,\Lambda\,\Delta x.$$3 is physically preferred, the effective equation of state can move from quintessence-like to phantom behavior, the system approaches a late-time de Sitter attractor, NEC/WEC/DEC are satisfied, and SEC is violated at late times. The paper presents this as a route to late-time acceleration without an explicit cosmological constant (Roushan et al., 26 Nov 2025). Another proposal interprets the Hubble tension as an infrared uncertainty effect: with an EUP-modified effective photon mass, the relative discrepancy in $$\Delta p \ge \frac{\hbar}{2\Delta x}+\eta\,\Lambda\,\Delta x.$$4 tracks the relative discrepancy in $$\Delta p \ge \frac{\hbar}{2\Delta x}+\eta\,\Lambda\,\Delta x.$$5, and the inferred EUP scale is constrained to $$\Delta p \ge \frac{\hbar}{2\Delta x}+\eta\,\Lambda\,\Delta x.$$6 or $$\Delta p \ge \frac{\hbar}{2\Delta x}+\eta\,\Lambda\,\Delta x.$$7, depending on which photon-mass upper limit is imposed (Nozari et al., 2024).

Beyond gravity and cosmology, EUP deformations have been explored in many-body quantum systems. In a one-dimensional Bose-Einstein condensate with

$$\Delta p \ge \frac{\hbar}{2\Delta x}+\eta\,\Lambda\,\Delta x.$$8

the deformed Gross-Pitaevskii equation admits free dark-soliton solutions compatible with the EUP for $$\Delta p \ge \frac{\hbar}{2\Delta x}+\eta\,\Lambda\,\Delta x.$$9, whereas the free bright soliton does not yield an acceptable physical EUP state; in a harmonic trap, stable condensate states persist only for approximately $$\Delta x\,\Delta p \ge \hbar+\tilde{\alpha}(\Delta p)^2+\tilde{\eta}(\Delta x)^2,$$0 (Benhadjira et al., 2023). For harmonically trapped ideal Bose and Fermi gases, the EUP enters through a modified density of states,

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

raising the Bose condensation temperature but reducing internal energy, specific heat, and pressure. The paper concludes that these effects are relatively modest compared with GUP corrections and quotes the bound

$$\Delta x\,\Delta p \ge \hbar+\tilde{\alpha}(\Delta p)^2+\tilde{\eta}(\Delta x)^2,$$2

from comparison with experimental data (Hamil et al., 2024).

6. Sign conventions, extensions, and unresolved issues

A persistent issue in the EUP literature is the sign of the curvature correction. In a Schwinger pair-production analysis, the heuristic EUP

$$\Delta x\,\Delta p \ge \hbar+\tilde{\alpha}(\Delta p)^2+\tilde{\eta}(\Delta x)^2,$$3

reproduces the known curved-spacetime result only when $$\Delta x\,\Delta p \ge \hbar+\tilde{\alpha}(\Delta p)^2+\tilde{\eta}(\Delta x)^2,$$4 in AdS and $$\Delta x\,\Delta p \ge \hbar+\tilde{\alpha}(\Delta p)^2+\tilde{\eta}(\Delta x)^2,$$5 in dS; the paper explicitly presents this as evidence that earlier EUP literature is not fully consistent about the correct sign (Ong, 2020). Sign restrictions also appear in phenomenological metric papers: some exclude $$\Delta x\,\Delta p \ge \hbar+\tilde{\alpha}(\Delta p)^2+\tilde{\eta}(\Delta x)^2,$$6 because it leads to a maximum black-hole mass or repulsive large-scale potentials (Mureika, 2018, Ökcü et al., 2022). By contrast, the higher-order cosmological baryogenesis model was designed precisely to accommodate both $$\Delta x\,\Delta p \ge \hbar+\tilde{\alpha}(\Delta p)^2+\tilde{\eta}(\Delta x)^2,$$7 and $$\Delta x\,\Delta p \ge \hbar+\tilde{\alpha}(\Delta p)^2+\tilde{\eta}(\Delta x)^2,$$8 while retaining the same minimum length (Luo et al., 2023).

The EUP has also been embedded into broader UV/IR frameworks. One 2026 analysis contrasts GEUP,

$$\Delta x\,\Delta p \ge \hbar+\tilde{\alpha}(\Delta p)^2+\tilde{\eta}(\Delta x)^2,$$9

with an alternative EGUP that emphasizes $\sim m_{\rm dS}c$0-$\sim m_{\rm dS}c$1 duality and exhibits unusual branch structure. In the black-hole interpretation, the paper argues that the EUP alone does not provide a satisfactory particle-black-hole correspondence, whereas the GEUP admits standard black holes, sub-Planckian Compton-like states, and a new strong-gravity black-hole phase (Carr et al., 5 Feb 2026).

A further extension reverses the usual derivational direction by starting from generalized entropy and reconstructing the uncertainty relation. From MOND entropy, a “MOND EUP” is obtained,

$\sim m_{\rm dS}c$2

which reduces to the Rényi-associated EUP at $\sim m_{\rm dS}c$3, to the dual Kaniadakis EUP at $\sim m_{\rm dS}c$4, and yields the higher-order EUP as a perturbative limit when $\sim m_{\rm dS}c$5 (Sevinç et al., 20 Jan 2026). This suggests that the modern EUP should be viewed not as a single formula but as a structured family of infrared-sensitive uncertainty relations linked to horizon thermodynamics, generalized entropies, curvature scales, and concrete measurement protocols.