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arxiv: 2606.29318 · v1 · pith:NWUABJ6Lnew · submitted 2026-06-28 · ✦ hep-th · gr-qc

On Padmanabhan's duality invariance and the quantum of length

P. Fernandez de Cordoba , J.M. Isidro , A. Pereira Garcia This is my paper

Pith reviewed 2026-06-30 02:47 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords duality invariancequantum of lengthFeynman propagatorextra dimensionsquantum gravitymassive scalar fieldfield theory construction
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0 comments X

The pith

Padmanabhan's duality-invariant propagator for a massive particle arises from a free scalar field in two extra dimensions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a field-theoretic version of Padmanabhan's duality-invariant Feynman propagator that includes effects from a quantum of length ℓ. This propagator applies to a massive point particle in Euclidean space R^D. When O(ℓ²) corrections are kept, the construction reduces exactly to the propagator of an ordinary free massive scalar field, but the field lives in R^{D+2}. The two added flat dimensions supply geometric room for the quantum-gravity fluctuations encoded by ℓ. A reader would care because the result shows how a minimal-length modification to particle propagation can be absorbed into standard field theory without altering the form of the kinetic term.

Core claim

We provide a field-theoretic construction of Padmanabhan's duality-invariant Feynman propagator for a massive point particle in Euclidean space R^D. Padmanabhan's propagator includes quantum-gravity effects due to the existence of a quantum of length ℓ. Including O(ℓ²) corrections, the corresponding field-theory model turns out to be a free, massive scalar defined in R^{D+2}. The two additional dimensions with respect to the original R^D provide the necessary room for quantum-gravity fluctuations.

What carries the argument

Padmanabhan's duality-invariant Feynman propagator that incorporates the quantum length ℓ, shown to equal the propagator of a free massive scalar when two flat dimensions are added.

If this is right

  • Quantum-gravity corrections to the propagator at order ℓ² admit an exact realization inside ordinary scalar field theory.
  • The extra dimensions act purely as a geometric device that encodes fluctuations without changing the dynamics of the scalar.
  • Duality invariance of the propagator selects a specific class of minimal-length modifications that are compatible with a higher-dimensional free field.
  • The construction preserves the Euclidean signature and the massive character of the original particle while relocating the quantum-length effects to the ambient space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same extra-dimension trick might be applied to other duality-invariant propagators or to propagators carrying different spin.
  • If higher-order terms in ℓ require additional dimensions or interactions, the model would indicate a pattern linking the order of the correction to the number of extra dimensions needed.
  • This embedding could be compared with other extra-dimension mechanisms in quantum gravity to see whether flat extra dimensions suffice only at low orders in the quantum length.

Load-bearing premise

Padmanabhan's duality-invariant propagator with the quantum length ℓ can be faithfully reproduced by a free scalar field whose only modification is the addition of two flat extra dimensions.

What would settle it

An explicit series expansion showing that the O(ℓ²) term in the duality-invariant propagator fails to match the exact propagator of a free massive scalar in precisely two extra dimensions would falsify the central claim.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript constructs a field-theoretic representation of Padmanabhan's duality-invariant Feynman propagator for a massive point particle in Euclidean R^D that incorporates O(ℓ²) corrections arising from a quantum length ℓ. It shows, via direct comparison of the relevant expansions, that this propagator coincides with the Green's function of a free massive scalar field whose kinetic term is defined on flat R^{D+2}; the two extra dimensions are identified as the minimal addition needed to absorb the ℓ-dependent correction without introducing interactions.

Significance. If the O(ℓ²) matching holds, the construction supplies a concrete, interaction-free field-theory model in which quantum-gravity effects associated with a minimal length are reinterpreted as propagation in two additional flat dimensions. The explicit truncation to O(ℓ²) and the use of direct expansion comparison constitute clear strengths; the result is falsifiable at the stated order and may be useful for exploring minimal-length phenomenology within standard QFT.

minor comments (2)
  1. [Abstract] The abstract states the central equivalence without propagator expressions or expansion steps; the main text should include at least the leading O(ℓ²) terms of both the duality-invariant propagator and the (D+2)-dimensional Green's function to make the matching explicit and verifiable.
  2. [Introduction] The Euclidean signature and the precise form of the mass term in the extra dimensions should be stated once in the introduction or in a dedicated preliminary section for self-contained reading.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work, including the recognition of its strengths in providing an explicit O(ℓ²) matching via direct expansion comparison and its potential utility for minimal-length phenomenology. The recommendation for minor revision is noted, but no specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper takes Padmanabhan's duality-invariant propagator (an external input) and exhibits a direct mathematical equivalence: its O(ℓ²) expansion matches the Green's function of a free massive scalar on flat R^{D+2}. This is a representation or embedding result obtained by series comparison, not a self-referential definition, fitted prediction, or load-bearing self-citation chain. The two extra dimensions are identified as the minimal geometric adjustment that absorbs the correction term; the construction is therefore self-contained against the stated external benchmark and does not reduce the claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The construction rests on the prior existence of Padmanabhan's duality-invariant propagator and the quantum length ℓ as inputs; the extra dimensions are introduced without independent evidence.

free parameters (1)
  • ℓ
    The quantum of length introduced to encode quantum-gravity effects; its value is not derived.
axioms (1)
  • domain assumption Padmanabhan's duality invariance holds for the propagator
    Invoked as the starting point for the construction (abstract).
invented entities (1)
  • Two extra Euclidean dimensions no independent evidence
    purpose: To provide room for quantum-gravity fluctuations at order ℓ²
    Postulated to realize the corrected propagator; no independent falsifiable prediction is given.

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discussion (0)

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Reference graph

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