The hypothesis of path integral duality II: corrections to quantum field theoretic results

TL;DR

This paper applies the path integral duality principle to quantum field theory, introducing a zero-point length to regularize divergences and explore quantum gravitational corrections in flat and curved spacetimes.

Contribution

It extends Padmanabhan's duality principle to Schwinger's proper time formalism, providing a novel method to incorporate quantum gravitational effects as a natural regulator.

Findings

The modified weightage factor acts as a Planck-scale regulator.

Quantum gravitational corrections are derived for flat and curved spacetimes.

Divergences in quantum field theory are effectively removed by the duality-inspired modification.

Abstract

In the path integral expression for a Feynman propagator of a spinless particle of mass $m$, the path integral amplitude for a path of proper length ${\cal R}(x,x'| g_{\mu\nu})$ connecting events $x$ and $x'$ in a spacetime described by the metric tensor $g_{\mu\nu}$ is $\exp-[m {\cal R}(x,x'| g_{\mu\nu})]$. In a recent paper, assuming the path integral amplitude to be invariant under the duality transformation ${\cal R} \to (L_P^2/{\cal R})$, Padmanabhan has evaluated the modified Feynman propagator in an arbitrary curved spacetime. He finds that the essential feature of this `principle of path integral duality' is that the Euclidean proper distance $(\Delta x)^2$ between two infinitesimally separated spacetime events is replaced by $[(\Delta x)^2 + 4L_P^2 ]$. In other words, under the duality principle the spacetime behaves as though it has a `zero-point length' $L_P$, a feature that is expected to arise in a quantum theory of gravity. In the Schwinger's proper time description of the Feynman propagator, the weightage factor for a path with a proper time $s$ is $\exp-(m^2s)$. Invoking Padmanabhan's `principle of path integral duality' corresponds to modifying the weightage factor $\exp-(m^2s)$ to $\exp-[m^2s + (L_P^2/s)]$. In this paper, we use this modified weightage factor in Schwinger's proper time formalism to evaluate the quantum gravitational corrections to some of the standard quantum field theoretic results in flat and curved spacetimes. We find that the extra factor $\exp-(L_P^2/s)$ acts as a regulator at the Planck scale thereby `removing' the divergences that otherwise appear in the theory. Finally, we discuss the wider implications of our analysis.