--- title: Extended Generalized Uncertainty Principle (EGUP) url: https://www.emergentmind.com/topics/extended-generalized-uncertainty-principle-egup type: topic --- # Extended Generalized Uncertainty Principle (EGUP) 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 [2409.05110][2602.06189][2112.13938]. 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 $\Lambda$ [2112.13938]. | 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 [2409.05110] | | 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 $\alpha=0$ [2602.06189] | | Asymptotic curvature-combined form | $\sigma_P\rho\simeq \pi\hbar\left(1-\frac{R|_{p_0}\rho^2}{12\pi^2}+\alpha\frac{l_p^2\sigma_p^2}{\hbar^2}+O(\rho^4)\right)$ | Curvature-derived EUP combined with GUP in an asymptotic regime [2006.02188] | 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 $L_*$ 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 [2009.03627][2509.08005]. 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 [2602.06189]. 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 $$ [\hat x_i,\hat x_j] = i\hbar\,a_{ij}(\hat x,\hat p),\qquad [\hat p_i,\hat p_j] = i\hbar\,b_{ij}(\hat x,\hat p),\qquad [\hat x_i,\hat p_j] = i\hbar\,c_{ij}(\hat x,\hat p), $$ 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 [2409.05110]. 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 [2409.05110]. 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, $$ S: |x\rangle \to |x\rangle \otimes |g_x\rangle, $$ so that the measured distribution becomes a convolution of matter and geometric fluctuations [2112.13938]. In this approach, the canonical Heisenberg algebra remains unchanged, $$ [\hat{x}^i,\hat{p}_j]=i\hbar\,\delta^i{}_j\,I, $$ 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 [1912.07093][2008.13183]. 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 [2112.13938]. 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 [2008.13183]. 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 [1309.0588]. 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: $$ \Delta x + \frac{\beta Pl^2}{\Delta x} \geq \frac{\hbar}{2 \Delta p} + \frac{\alpha \Delta p \, Pl^2}{2 \hbar}. $$ Its defining feature is an explicit $\Delta x \leftrightarrow \Delta p$ duality: the left-hand side contains $\Delta x$ and $1/\Delta x$, while the right-hand side contains $\Delta p$ and $1/\Delta p$ [2602.06189]. The authors emphasize that it reduces to the GUP when $\beta=0$, but is not identical to the EUP when $\alpha=0$. Solving this relation for $\Delta x$ yields two branches, $$ \Delta x = \frac{1}{4} \left(\frac{\hbar}{\Delta p}+\frac{\alpha\Delta p \ell_P^2}{\hbar}\right)\left(1\pm\sqrt{1-\frac{8\beta\ell_P^2}{\left(\frac{\hbar}{\Delta p}+\frac{\alpha\Delta p \ell_P^2}{\hbar}\right)^2}}\right), $$ and the qualitative behavior depends on the sign and relative magnitude of $\alpha$ and $\beta$ [2602.06189]. The same study analyzes negative parameter values, notes a transition at $\alpha=\beta$, and finds connected or disconnected particle/black-hole regions depending on whether $\alpha>\beta$ or $\alpha<\beta$. 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” [2602.06189]. 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 $$ \sigma_p\rho\gtrsim \pi\hbar\sqrt{1-\frac{R|_{p_0}}{6\pi^2}\rho^2}, $$ with the leading second-order correction controlled by the Ricci scalar at the expectation position $p_0$ [2006.02188]. Fourth-order terms involve the Cartan invariant $\Psi_2^{\,2}$ 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 [2006.02188]. These variants show that EGUP is not exhausted by the simple addition of a $(\Delta p)^2$ term and a $(\Delta x)^2$ 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 $(2\pi\hbar)^D$; under EGUP deformations, the cell volume becomes position and momentum dependent, so the standard Liouville measure is no longer invariant [2409.05110]. For the Snyder–de Sitter model, the classical Poisson brackets are $$ \{x_i,x_j\} = \beta^2(x_i p_j - x_j p_i),\qquad \{p_i,p_j\} = \alpha^2(x_i p_j - x_j p_i), $$ $$ \{x_i,p_j\}=\delta_{ij}+\alpha^2 x_i x_j+\beta^2 p_i p_j+2\alpha\beta p_i x_j, $$ and the infinitesimal Jacobian differs from unity under Hamiltonian time evolution [2409.05110]. The invariant weighted measure is then $$ d^D x\,d^D p\,(1+\alpha^2 x^2+\beta^2 p^2+2\alpha\beta\,x\!\cdot\! p), $$ which restores the classical analogue of Liouville’s theorem. The parameters $\alpha$ and $\beta$ respectively control large-scale/curvature and small-scale/Planckian corrections, while the mixed $\alpha\beta$ term couples the two sectors [2409.05110]. Special limits recover familiar one-sector deformations. In the Snyder limit $\alpha\to 0$, one obtains the quadratic GUP and invariant measure $$ d^D x\,d^D p\,(1+\beta^2 p^2). $$ In the (anti-)de Sitter limit $\beta\to 0$, one obtains the quadratic EUP and invariant measure $$ d^D x\,d^D p\,(1+\alpha^2 x^2). $$ The same paper further notes that Snyder–de Sitter can be mapped to Snyder by a noncanonical linear transformation [2409.05110]. For the Yang model, the classical realization gives a higher-order square-root relation; in $D=1$, $$ \{x,p\}= \sqrt{1-\alpha^2 x^2-\beta^2 p^2}, $$ leading to $$ \Delta x\,\Delta p \ge \frac{1}{2}\sqrt{1-\alpha^2(\Delta x)^2-\beta^2(\Delta p)^2}. $$ Expanding to leading order yields the invariant weighted measure $$ dx\,dp\left(1-\frac{1}{2}(\alpha^2 x^2+\beta^2 p^2)\right), $$ while the limits $\alpha\to 0$ and $\beta\to 0$ produce higher-order GUP and EUP forms, respectively [2409.05110]. 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 [2409.05110]. By contrast, the fuzzy-sphere model with commuting momenta, $$ [\hat p_i,\hat p_j]=0, $$ admits an invariant standard phase-space volume, so no weighted measure is required and the density of states is unchanged [2409.05110]. 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 [2009.03627]. In one representative treatment, the modified Unruh temperature takes the form $$ T^{EGUP} = T_U\left[1 + \frac{ \left| \Lambda \right|}{27} \left(\frac{ c}{k_BT_U}\right)^2 + \frac{9\beta_0 l_P^2 }{4} \left(\frac{k_BT_U}{ c}\right)^2 + \frac{\beta_0 l_P^2 \left| \Lambda \right| }{6}\right], $$ so that the EUP-like term dominates at low Unruh temperature and the GUP-like term dominates at high Unruh temperature [2009.03627]. 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, $$ G'=\frac{G}{1+\frac{\alpha_{0}|\Lambda|}{3}X^2+\frac{\beta_{0}\ell_P^2}{\hbar^2}P^2}, $$ which is then inserted into a modified Schwarzschild metric [2102.13466]. 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 [2102.13466]. 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 [2205.00232][2405.15558][2508.15467][2509.08005][2606.23756]. 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 $\omega=-1/3$, 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 [2606.23756]. 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 $P