Padmanabhan's Duality-Invariant Feynman Propagator

• The propagator introduces a zero-point length by replacing the geodesic distance s with sqrt(s² + ell²), effectively regularizing short-distance divergences.
• It employs a duality transformation at the worldline level, making the propagator invariant under the exchange of short and long proper lengths and aligning with standard scalar field theory.
• The method yields an effective higher-dimensional representation that underpins UV softening in scalar, gauge, and gravitational models, and facilitates nonlocal corrections.

Padmanabhan's duality-invariant Feynman propagator is a modification of the Euclidean propagator designed to encode a fundamental quantum of length. In the formulation analyzed in "On Padmanabhan's duality invariance and the quantum of length" (Cordoba et al., 28 Jun 2026), the ordinary geodesic distance $s$ is replaced by the smeared interval $\sqrt{s^2+\ell^2}$, with $\ell=2L$, so that the quantum-gravity-corrected propagator takes the compact form

$${\cal G}_D^{\rm (QG)}(s)={\cal G}_D\!\left(\sqrt{s^2+\ell^2}\right).$$

Equivalently, at the point-particle level the construction is based on a duality transformation of the worldline action, ${\cal S}_D\to L^2/{\cal S}_D$. The central consequence is a zero-point-length effect: the propagator no longer probes arbitrarily short distances. This structure has been used as an effective implementation of minimal-length or T-duality effects in scalar field theory, nonlocal electrodynamics, and regular black-hole models (Cordoba et al., 28 Jun 2026, Gaete et al., 2022, Gaete et al., 2022).

1. Duality principle and the modified point-particle propagator

For a free massive relativistic point particle in Euclidean space $\mathbb{R}^D$, the classical action along a path $p({\bf x},{\bf y})$ is

$$m{\cal S}_D\!\left[p({\bf x},{\bf y})\right]=m\int_{\bf x}^{\bf y} ds,$$

and the Euclidean Feynman propagator is obtained by summing over paths with weight $\exp[-m{\cal S}_D]$. In the standard case the result depends only on the Euclidean distance $s^2=({\bf x}-{\bf y})^2$. Padmanabhan’s modification replaces the ordinary path-integral weight by a duality-invariant one,

$\sqrt{s^2+\ell^2}$0

which is invariant under the exchange of short and long proper lengths in the specific sense encoded by $\sqrt{s^2+\ell^2}$1 (Cordoba et al., 28 Jun 2026).

The corrected propagator is known exactly and preserves the functional form of the ordinary Euclidean propagator, but with the distance shifted from $\sqrt{s^2+\ell^2}$2 to $\sqrt{s^2+\ell^2}$3, or equivalently to $\sqrt{s^2+\ell^2}$4 after introducing $\sqrt{s^2+\ell^2}$5. This is the hallmark of the construction: quantum-gravity effects are not represented by an explicit cutoff inserted by hand, but by a modified notion of distance. In this sense, the zero-point length is built directly into the Green function.

A common oversimplification is to view the proposal merely as a regularized propagator. That characterization is incomplete. In the formulation above, the regularization is tied to a definite duality prescription at the worldline level, and the short-distance smoothing is a consequence of that prescription rather than an independent assumption (Cordoba et al., 28 Jun 2026).

2. Relation to ordinary scalar field theory in $\sqrt{s^2+\ell^2}$6

The uncorrected point-particle propagator has the standard field-theoretic counterpart of a free massive scalar in Euclidean $\sqrt{s^2+\ell^2}$7,

$\sqrt{s^2+\ell^2}$8

with momentum-space propagator

$\sqrt{s^2+\ell^2}$9

Using the Schwinger proper-time representation,

$\ell=2L$0

one recovers the same Bessel-function form as in the point-particle computation, so that $\ell=2L$1 (Cordoba et al., 28 Jun 2026).

This equivalence is structurally important. It shows that, before minimal-length effects are introduced, the worldline and field-theory descriptions are exactly aligned. The duality-invariant propagator therefore raises a sharply defined problem: whether the replacement $\ell=2L$2 can be reproduced by an ordinary field theory rather than only by a modified point-particle sum over paths.

The 2026 construction answers that question perturbatively in $\ell=2L$3. It does not claim a fundamental reformulation of quantum gravity. Instead, it identifies a field-theoretic representation of the leading duality-invariant correction. This distinction matters because it separates the exact propagator-level statement from the effective field-theory model used to realize it (Cordoba et al., 28 Jun 2026).

3. Expansion in even powers and the $\ell=2L$4 representation

A central observation is the identity

$\ell=2L$5

written after setting $\ell=2L$6 for notational convenience. The expansion organizes the zero-point-length correction into contributions from dimensions $\ell=2L$7, and only even powers appear. Truncating to the first nontrivial order gives

$\ell=2L$8

This is the key clue behind the field-theoretic reconstruction (Cordoba et al., 28 Jun 2026).

The proposed effective action is

$\ell=2L$9

where the correction is represented by a free scalar theory in two additional dimensions. The extra coordinates are constrained by

$${\cal G}_D^{\rm (QG)}(s)={\cal G}_D\!\left(\sqrt{s^2+\ell^2}\right).$$0

With this choice, the resulting field-theory propagator reproduces the duality-invariant point-particle propagator up to $${\cal G}_D^{\rm (QG)}(s)={\cal G}_D\!\left(\sqrt{s^2+\ell^2}\right).$$1,

$${\cal G}_D^{\rm (QG)}(s)={\cal G}_D\!\left(\sqrt{s^2+\ell^2}\right).$$2

The paper is explicit that this is not a fundamental higher-dimensional theory. The two additional dimensions are an effective mathematical representation of the minimal-length fluctuation. Their role is to localize, in a higher-dimensional auxiliary description, a correction that would otherwise appear as nonlocality in $${\cal G}_D^{\rm (QG)}(s)={\cal G}_D\!\left(\sqrt{s^2+\ell^2}\right).$$3. The stated advantages are that the construction preserves second-order field equations and avoids introducing an infinite tower of higher derivatives directly in $${\cal G}_D^{\rm (QG)}(s)={\cal G}_D\!\left(\sqrt{s^2+\ell^2}\right).$$4 (Cordoba et al., 28 Jun 2026).

The authors compare this mechanism with the use of the Schwinger parameter $${\cal G}_D^{\rm (QG)}(s)={\cal G}_D\!\left(\sqrt{s^2+\ell^2}\right).$$5 in heat-kernel methods. The comparison does not identify the extra coordinates with a physical proper-time direction; rather, it indicates that both serve as auxiliary structures through which nontrivial propagator behavior can be encoded geometrically.

4. Ultraviolet softening, T-duality, and nonlocal electrodynamics

A distinct but closely related realization appears in "Finite electrodynamics from T-duality" (Gaete et al., 2022). There the propagator is used to implement stringy T-duality effects in a $${\cal G}_D^{\rm (QG)}(s)={\cal G}_D\!\left(\sqrt{s^2+\ell^2}\right).$$6 gauge theory, with a minimal length $${\cal G}_D^{\rm (QG)}(s)={\cal G}_D\!\left(\sqrt{s^2+\ell^2}\right).$$7. The motivating duality is presented as

$${\cal G}_D^{\rm (QG)}(s)={\cal G}_D\!\left(\sqrt{s^2+\ell^2}\right).$$8

together with the statement that the path integral should be invariant under the exchange of proper lengths $${\cal G}_D^{\rm (QG)}(s)={\cal G}_D\!\left(\sqrt{s^2+\ell^2}\right).$$9. In momentum space the propagator is written as

${\cal S}_D\to L^2/{\cal S}_D$0

with ${\cal S}_D\to L^2/{\cal S}_D$1 the modified Bessel function of the second kind. Because ${\cal S}_D\to L^2/{\cal S}_D$2 has exponential behavior for large ${\cal S}_D\to L^2/{\cal S}_D$3, the propagator suppresses high-momentum modes and regularizes the ultraviolet.

The corresponding nonlocal gauge-invariant action is

${\cal S}_D\to L^2/{\cal S}_D$4

In the infrared, ${\cal S}_D\to L^2/{\cal S}_D$5 for ${\cal S}_D\to L^2/{\cal S}_D$6, so Maxwell theory is recovered at long distances. At short distances the same kernel yields a finite static interaction energy, a regularized Coulomb profile, and an electric field that vanishes linearly at the origin. The paper further verifies the potential by a gauge-invariant, path-dependent Hamiltonian formalism, showing that the result is not a gauge artifact.

The electrodynamic construction also introduces a scale-dependent effective dimension for the static potential,

${\cal S}_D\to L^2/{\cal S}_D$7

Accordingly, ${\cal S}_D\to L^2/{\cal S}_D$8 for ${\cal S}_D\to L^2/{\cal S}_D$9, $\mathbb{R}^D$0 for $\mathbb{R}^D$1, and $\mathbb{R}^D$2 as $\mathbb{R}^D$3. The paper interprets this as dimensional fractalization induced by T-duality rather than by scale invariance.

5. Static potentials, smeared sources, and regular black holes

In "Charged black holes from T-duality" (Gaete et al., 2022), Padmanabhan’s propagator is used as the technical input that replaces the standard Feynman propagator in the derivation of static potentials. The modified Newtonian potential is

$\mathbb{R}^D$4

so the singular $\mathbb{R}^D$5 behavior is softened by the zero-point length $\mathbb{R}^D$6. Acting with the Newton-Poisson operator yields a smooth source density,

$\mathbb{R}^D$7

which is then used as the matter distribution sourcing Einstein’s equations.

For the charged sector, the same propagator-driven smearing modifies the electrostatic potential and field,

$\mathbb{R}^D$8

The electromagnetic energy density is finite everywhere, vanishes at the origin, and falls off as $\mathbb{R}^D$9 at infinity. The resulting metric is regular, de Sitter-like near the core,

$p({\bf x},{\bf y})$0

and asymptotically Reissner–Nordström at large radii. The ADM mass is shifted by the regularized self-energy of the electromagnetic field,

$p({\bf x},{\bf y})$1

The paper further states that the static charged solution resembles the Ayón-Beato–García spacetime provided the T-duality scale is redefined as $p({\bf x},{\bf y})$2. In the neutral limit, the analogous replacement relates the T-duality metric to the Bardeen geometry. Thermodynamically, the de Sitter core implied by the propagator leads to a temperature profile that rises to a maximum and then decreases toward an extremal remnant, with positive heat capacity in the final phase. Within the framework of the paper, the propagator therefore functions not only as a UV-softened Green function but as the mechanism that generates smooth matter and electromagnetic profiles.

6. Scope, misconceptions, and limitations

Several interpretive points recur across the recent literature. First, the minimal length encoded by the propagator is not presented as a hard cutoff in momentum space. In the scalar construction it appears as the distance replacement $p({\bf x},{\bf y})$3; in the gauge-theory and black-hole applications it appears through Bessel-function kernels that exponentially suppress large momenta (Cordoba et al., 28 Jun 2026, Gaete et al., 2022, Gaete et al., 2022). The regularization is therefore nonlocal in origin even when it admits a local auxiliary representation.

Second, the two extra dimensions of the 2026 field-theory model are not to be interpreted as ordinary propagating spacetime directions. The paper states that they are an effective bookkeeping device, that they localize the nonlocality caused by the minimal length, and that they preserve second-order field equations. This directly addresses a likely misconception that the construction is a Kaluza–Klein-type extension of the original theory. It is instead a local representation of the $p({\bf x},{\bf y})$4 correction (Cordoba et al., 28 Jun 2026).

Third, duality invariance by itself is not sufficient to guarantee equivalence with Padmanabhan’s propagator. The 2026 paper analyzes a naive lattice discretization that is manifestly duality-invariant under

$p({\bf x},{\bf y})$5

but whose momentum-space dispersion relation contains corrections of the form

$p({\bf x},{\bf y})$6

These do not reproduce the $p({\bf x},{\bf y})$7 correction implied by the duality-invariant propagator, so the lattice model is discarded as an equivalent field-theory realization. This is an important limitation statement: formal duality symmetry does not by itself fix the correct propagator-level implementation (Cordoba et al., 28 Jun 2026).

Finally, the current field-theoretic reconstruction is perturbative in the quantum of length and explicitly established up to $p({\bf x},{\bf y})$8 for a free massive scalar in Euclidean $p({\bf x},{\bf y})$9. The gauge and gravitational applications show that the same propagator can be embedded in broader nonlocal frameworks, but they do not remove the distinction between an exact propagator prescription and a complete microscopic theory. A plausible implication is that the propagator is best regarded, at present, as a controlled minimal-length ansatz whose strength lies in calculable UV softening and in the availability of concrete realizations across several sectors.