On Padmanabhan's duality invariance and the quantum of length

Abstract: We provide a field-theoretic construction of Padmanabhan's duality-invariant Feynman propagator for a massive point particle in Euclidean space $\mathbb{R}^D$. Padmanabhan's propagator includes quantum-gravity effects due to the existence of a quantum of length $\ell$. Including $O(\ell^2)$ corrections, the corresponding field-theory model turns out to be a free, massive scalar defined in $\mathbb{R}^{D+2}$. The two additional dimensions with respect to the original $\mathbb{R}^D$ provide the necessary room, so to speak, for quantum-gravity fluctuations.

• The paper presents a field-theoretic construction that reproduces Padmanabhan’s duality-invariant point-particle propagator up to O(ℓ²) using a D+2 dimensional formulation.
• It employs a dimensional extension to encode quantum gravity effects as a local free scalar theory, ensuring the standard QFT propagator is recovered as ℓ → 0.
• The approach circumvents higher-derivative issues and outperforms naive lattice discretization in capturing essential nonlocal quantum gravitational corrections.

Field-Theoretic Construction of Padmanabhan's Duality-Invariant Propagator and the Quantum of Length

Overview and Motivation

This work addresses a fundamental question in quantum gravity (QG): how to consistently incorporate the notion of a minimal length—modeled as a quantum of length $\ell$—into the framework of propagators for relativistic particles and scalar fields. The approach is based on Padmanabhan's hypothesis of path integral duality, which modifies the standard Feynman propagator via a duality-invariant prescription that encodes leading QG effects by enforcing invariance under the transformation $\mathcal{S}_D \to L^2/\mathcal{S}_D$, where $\mathcal{S}_D$ is the proper-time action and $L$ is the zero-point length.

The main result is a field-theoretic construction that matches, up to $\mathcal{O}(\ell^2)$, the duality-invariant point-particle propagator by localizing the QG corrections via a dimensional extension to $D+2$ spacetime dimensions. The physical $D$-dimensional propagator with QG corrections is shown to correspond, to this order, to a free scalar theory in $D+2$ dimensions, with the quantum of length realized as a geometric constraint on the coordinates.

Duality Invariance and QG-Corrected Propagators

Padmanabhan's path-integral approach leads to a duality-invariant Euclidean propagator for a massive point particle: $$\mathcal{G}^{\text{(QG)}}_D(s) = \frac{m^{D-2}}{(2\pi)^{D/2}} \frac{K_{D/2-1}(m\sqrt{s^2+\ell^2})}{(m\sqrt{s^2+\ell^2})^{D/2-1}},$$ where the shift $s \to \sqrt{s^2+\ell^2}$ incorporates the quantum of length (Equation (98) in the paper). In the $\mathcal{S}_D \to L^2/\mathcal{S}_D$0 limit this reduces to the standard propagator. The corresponding field-theory propagator computed via the Schwinger proper-time formalism matches the standard particle propagator. However, with QG corrections, there is no a priori guarantee of such an equivalence, motivating the search for a consistent field-theoretic realization of the duality-invariant prescription.

Field-Theoretic Realization via Dimensional Extension

By expanding the QG-corrected propagator $\mathcal{S}_D \to L^2/\mathcal{S}_D$1 in powers of $\mathcal{S}_D \to L^2/\mathcal{S}_D$2, it is shown (Equation (99)-(101)) that: $\mathcal{S}_D \to L^2/\mathcal{S}_D$3 where $\mathcal{S}_D \to L^2/\mathcal{S}_D$4 is the free scalar field propagator in $\mathcal{S}_D \to L^2/\mathcal{S}_D$5 dimensions, and $\mathcal{S}_D \to L^2/\mathcal{S}_D$6 the analogous object in $\mathcal{S}_D \to L^2/\mathcal{S}_D$7 dimensions. This form strongly suggests that the $\mathcal{S}_D \to L^2/\mathcal{S}_D$8 effects induced by the quantum of length—instead of generating higher-derivative or nonlocal modifications in $\mathcal{S}_D \to L^2/\mathcal{S}_D$9 dimensions—can be reproduced by a field theory in an effectively higher-dimensional setting. The action functional proposed is: $\mathcal{S}_D$0 with a geometric constraint $\mathcal{S}_D$1 on the auxiliary dimensions. Importantly, this construction is not interpreted as physical propagation in $\mathcal{S}_D$2 dimensions but as an effective mathematical representation of nonlocal (duality-invariant) QG-induced features, avoiding the proliferation of higher-derivative terms and retaining locality in the augmented space.

Failure of Naive Discretization

The authors analyze an alternative approach where QG is modeled as a naive lattice discretization of $\mathcal{S}_D$3, positing a duality-invariant action that exchanges $\mathcal{S}_D$4 (Eqs. (81)-(82)). However, explicit calculation of the corresponding heat kernel and propagator on the full lattice theory reveals a mismatch at $\mathcal{S}_D$5 between the point-particle duality-invariant and the lattice field theory propagators. Specifically, the higher-derivative momentum terms (e.g., $\mathcal{S}_D$6) appearing in the lattice dispersion relation are not present in the duality-invariant propagator expansion. This demonstrates that the naive discretization fails to capture the correct QG corrections, further substantiating the necessity for the dimensional extension framework.

Theoretical and Practical Implications

This construction yields several insights:

• Equivalence up to $\mathcal{S}_D$7: The duality-invariant, QG-corrected propagator for a relativistic point particle in $\mathcal{S}_D$8 dimensions is exactly matched, up to order $\mathcal{S}_D$9, by a local, free scalar field theory in $L$0 dimensions with a particular kinematical constraint.
• Nonlocality by Dimensional Extension: The dimensional extension realizes QG-induced nonlocality without the computational or conceptual complications of infinite-derivative terms or explicit nonlocal operators in the action (contrast with higher-derivative constructions such as (Kan et al., 2020)).
• Reduction to Conventional QFT: For $L$1, the standard field-theoretic formalism and propagators are exactly recovered, ensuring consistency with the QG-free limit.
• Obstructions to Lattice Modeling: Naive discretization strategies, even when duality-invariant, do not yield the correct continuum limit for the QG-corrected propagator, highlighting the limitations of minimal-length models based solely on lattice regularization.

Prospects and Connections

The result establishes that, at least up to the first nontrivial QG correction, a higher-dimensional field theory can encode QG-induced duality invariance and the quantum of length in an efficient, local, and computationally tractable way. This method parallels the extension of the heat kernel approach in QG and may link to dimensional reduction scenarios at the Planck scale (Padmanabhan et al., 2015).

The paper remarks that the dimensional uplift is not to be interpreted as a literal change in physical spacetime dimensionality but as an effective device for capturing quantum gravitational fluctuations. In this respect, the approach is consistent with effective field theory perspectives and provides a path for systematically constructing higher-order corrections in $L$2.

Future work should address the generalization to interactions, the structure of higher-order QG corrections in this framework, and the possible extension to curved background geometries. Furthermore, deriving the field-theoretic action directly from a more fundamental (possibly microscopic QG) path integral remains an open target.

Conclusion

The authors have systematically constructed a field-theoretic model, valid up to $L$3, whose propagator reproduces the duality-invariant QG-corrected point-particle propagator predicted by Padmanabhan's framework. The essential innovation is the localization of QG effects in two auxiliary dimensions, providing a pragmatic and computationally advantageous realization of minimal-length physics within quantum field theory. This method circumvents the limitations of naive lattice models and avoids nonlocality at the level of field equations, offering a robust and extensible strategy for incorporating quantum gravitational effects in effective field theories.

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