Zero-Point Length in Quantum Gravity

• Zero-point length is a universal minimal scale arising in quantum gravity frameworks via duality arguments, ensuring that spacetime intervals remain finite.
• It modifies standard propagators and geometric measures by introducing a nonlocal deformation to the geodesic interval, which regularizes black hole interiors and avoids classical singularities.
• The concept induces effective dimensional reduction at Planck scales, influencing cosmological dynamics by producing nonsingular bouncing universes and altered inflationary signatures.

A zero-point length is a universal, minimal length scale that emerges in quantum gravity frameworks as a nonzero lower bound on physically meaningful spacetime intervals. It functions as a phenomenological parameter—denoted $L_0$ or %%%%1%%%%—that encapsulates quantum-gravitational nonlocality, universally regulates short-distance divergences in field theory, modifies the structure of black holes and cosmological models, and enforces a fundamental limit to localizability in spacetime. The existence and role of a zero-point length have been extensively developed through path-integral duality arguments, string-theoretic T-duality, effective geometry constructions, and thermodynamic considerations in gravitational settings (Nicolini, 2022, Jusufi et al., 2023, Sooraki et al., 2 Dec 2025).

1. Path Integral Duality and Emergence of Zero-Point Length

The zero-point length was first identified as a universal feature arising from a duality property in the quantum-mechanical path integral for a relativistic particle. Considering the Euclidean heat kernel (Schwinger representation) for the scalar propagator in $D$-dimensional flat space,

$$G_0(x,y) = \int_0^\infty ds ~ \exp\left[-m^2s - \frac{(x-y)^2}{4s}\right],$$

Padmanabhan introduced an invariance under the exchange $s \leftrightarrow L_0^2/s$, leading to a modified kernel

$$G(x,y) = \int_0^\infty ds ~ \exp\left[-m^2s - \frac{(x-y)^2}{4s} - \frac{L_0^2}{s}\right].$$

This duality enforces a cutoff for $s\to0$, ensuring no path segment can probe sub–$L_0$ distances. In turn, the two-point correlation functions and resulting physics are regularized at the scale $L_0$, which is naturally identified with the Planck length, $L_0 \sim \ell_P$, on fundamental quantum gravity grounds (Nicolini, 2022, Padmanabhan, 2020).

2. Minimal Length and T-Duality in String Theory

String-theoretic T-duality provides independent evidence for a minimal zero-point length. Closed string spectra on a circle of radius $R$ are invariant under $R \leftrightarrow \alpha'/R$ (where $\alpha'$ is the inverse string tension), implying a minimal observable length $\sqrt{\alpha'}$. This duality prevents probes of sub–string-length distances: in the low-energy effective theory, the propagation amplitudes are modified to encode a regularization at $l_0\simeq 2\pi\sqrt{\alpha'}$ (Nicolini, 2022, Jusufi et al., 2023, Wani et al., 2020). The same duality emerges in particle path integrals and effective geometries, suggesting the universality of the zero-point length across different quantum gravity approaches.

3. Geometric Implementation: The q-Metric and Nonlocal Deformations

The zero-point length can be encoded directly at the geometric level via a nonlocal deformation of the background metric or interval. The central prescription is to replace the squared geodesic interval $\sigma^2(x,x')$ by $\sigma^2(x,x') + L_0^2$. Kothawala and others have shown that the corresponding metric deformation (the "q-metric") is

$$\tilde{g}_{ab}(p;P) = A\,g_{ab}(p) - \varepsilon \left[A - A^{-1}\right] t_{a} t_{b}$$

with $A = 1 + L_0^2/\sigma^2(p,P)$ and $t_a = \nabla_a\sigma(p,P)$. The resultant geometry is everywhere regular and enforces that physical intervals cannot shrink below $L_0$; for large separations, standard Riemannian geometry is recovered (Kothawala, 2013, Padmanabhan et al., 2015).

4. Dimensional Reduction and UV Finiteness

A robust consequence of introducing a zero-point length is dimensional reduction in the UV. The effective spectral, thermodynamic, and potential-based measures of spacetime dimension demonstrate a universal flow toward lower values as one probes Planckian distances. For example, the effective Euclidean volume of a geodesic ball scales as

$$V_D(\ell, L_0) = \frac{\Omega_{D-1}}{D} \left[(\ell^2 + L_0^2)^{D/2} - L_0^D\right]$$

so that $V_D \propto L_0^{D-2} \ell^2$ for $\ell \sim L_0$, implying $D_{\rm eff} \rightarrow 2$ in the deep UV (Padmanabhan et al., 2015). Correspondingly, heat kernel analyses yield a spectral dimension $d_S \to 3.5$ at the Planck scale, and the thermodynamic dimension runs from $4$ (IR) to $1.5$ (Planck) to $1$ in the deep UV, supporting the scenario of effective two-dimensionality near $L_0$ (Mondal, 2021).

This dimensional reduction is believed to ameliorate UV divergences in quantum field theory and suppress classical singularities—features confirmed by direct computation of propagators, potentials, and curvature invariants in the zero-point–length–deformed frameworks (Kothawala, 2013, Nicolini, 2022).

5. Implications for Black Holes and Quantum Gravity Phenomenology

The introduction of a zero-point length regularizes the interiors of black holes. In both four and three dimensions, the modified energy-momentum tensor for a point source acquires a Gaussian smearing of scale $L_0$, and the corresponding spherically symmetric spacetime metric is

$$ds^2 = -V(r)dt^2 + V^{-1}(r)dr^2 + r^2 d\Omega^2, \quad V(r) = 1 - \frac{2M r^2}{(r^2 + L_0^2)^{3/2}}$$

with $V(r)$ everywhere regular and a de Sitter core at $r=0$. Black-hole thermodynamics is qualitatively altered: the Hawking temperature rises to a maximum then falls to zero at extremality, implying the evaporation process halts with a remnant of mass $M \sim M_P$ and size $\sim L_0$ (Nicolini, 2022, Jusufi et al., 2023, Jusufi, 2022).

Furthermore, the minimal horizon area aligns with the Bekenstein area quantization condition $A_{\min} = 8\pi L_0^2$, and the existence of a minimal throat size gives a concrete geometric basis for the ER=EPR conjecture, connecting entanglement and wormhole topology at the Planck scale (Jusufi et al., 2023).

6. Cosmological Consequences and Constraints

Zero-point length corrections universally modify Friedmann equations via additional quartic terms in the Hubble parameter:

$H^2 - \alpha H^4 = \frac{8\pi G}{3} \rho$

with $\alpha \propto L_0^2$, or, more generally,

$$H^2 + \frac{k}{a^2} = \frac{8\pi G}{3} \rho \big(1 + \Gamma \rho\big), \qquad \Gamma \sim L_0^2$$

This slows the early universe expansion at high energy densities, extends the hot early phase (for fixed $t$ the temperature is higher than in standard cosmology), and generically removes classical singularities: for suitable initial conditions, the scale factor bounces at a nonzero minimum $a_0 \sim L_0$, and all curvature invariants remain finite (Sooraki et al., 2 Dec 2025, Jusufi et al., 2022, Sheykhi et al., 2024, Luciano et al., 2024, Bhuyan et al., 2024). The second law of thermodynamics continues to hold with the corrected entropy, and current observational data from baryogenesis and inflation constrain $L_0$ to be within $\lesssim 440$ times the Planck length (Sooraki et al., 2 Dec 2025, Luciano et al., 2024).

In the context of inflation, the zero-point length induces calculable $O(\alpha)$ modifications in the tensor-to-scalar ratio $r$ and the scalar tilt $n_s$, and predicts a broken power-law power spectrum, as well as potential signatures for primordial gravitational waves if $L_0$ exhibits scale dependence (Luciano et al., 2024).

7. Operational and Thermodynamic Role: Gravity as Emergent from Zero-Point Length

The requirement of a nonzero zero-point length modifies geometric objects such as the Ricci biscalar. In the "q-metric" framework, the coincidence limit of the Ricci biscalar in the presence of a zero-point length picks out $(D-1)R_{ab}l^a l^b$ instead of the Ricci scalar. This structure is central in thermodynamic derivations of gravity, where the balance between gravitational and matter "heat densities" along null surfaces yields the Einstein equations as a macroscopic consequence of an underlying quantum-spacetime zero-point length (Pesci, 2019, Pesci, 2020, Kothawala et al., 2014). The surface term $K\sqrt{h}$ in the gravitational action, when evaluated using the q-metric, recovers the gravitational heat density used in emergent gravity formulations, providing a direct link from quantum discreteness to macroscopic gravitational dynamics.

---

References:

• (Nicolini, 2022) Quantum gravity and the zero point length
• (Kothawala, 2013) Minimal Length and Small Scale Structure of Spacetime
• (Padmanabhan et al., 2015) Spacetime with zero point length is two-dimensional at the Planck scale
• (Mondal, 2021) Ultraviolet dimensional reduction of spacetime with zero-point length
• (Padmanabhan, 2020) Principle of Equivalence at Planck scales, QG in locally inertial frames and the zero-point-length of spacetime
• (Jusufi et al., 2023) Einstein-Rosen bridge from the minimal length
• (Sooraki et al., 2 Dec 2025) Constraining Zero-Point Length from Gravitational Baryogenesis
• (Kothawala et al., 2014) Entropy density of spacetime from the zero point length
• (Luciano et al., 2024) Cosmology from String T-duality and zero-point length
• (Jusufi et al., 2022) Entropic corrections to Friedmann equations and bouncing universe due to the zero-point length
• (Sheykhi et al., 2024) Thermodynamical properties of nonsingular universe
• (Bhuyan et al., 2024) Stability of the Einstein Static Universe in Zero-Point Length Cosmology with Topological Defects
• (Jusufi, 2022) Regular black holes in three dimensions and the zero point length
• (Wani et al., 2020) Compactification, T-Duality and Quantum Erasers
• (Pesci, 2019) Minimum-length Ricci scalar for null separated events
• (Pesci, 2020) Zero-point gravitational field equations
• (Cordoba et al., 2023) Thermality of the zero-point length and gravitational selfduality