Hypothesis of path integral duality. II. Corrections to quantum field theoretic results

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 $\mathcal{R}{(x,x}^{\ensuremath{'}}|{g}_{\ensuremath{\mu}\ensuremath{\nu}})$ connecting events $x$ and ${x}^{\ensuremath{'}}$ in a spacetime described by the metric tensor ${g}_{\ensuremath{\mu}\ensuremath{\nu}}$ is $\mathrm{exp}{\ensuremath{-}[m\mathcal{R}{(x,x}^{\ensuremath{'}}|{g}_{\ensuremath{\mu}\ensuremath{\nu}})]}.$ In a recent paper, assuming the path integral amplitude to be invariant under the duality transformation $\mathcal{R}\ensuremath{\rightarrow}{(L}_{P}^{2}/\mathcal{R}),$ 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 $(\ensuremath{\Delta}{x)}^{2}$ between two infinitesimally separated spacetime events is replaced by $[(\ensuremath{\Delta}{x)}^{2}{+4L}_{P}^{2}].$ 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 Schwinger's proper time description of the Feynman propagator, the weightage factor for a path with a proper time $s$ is $\mathrm{exp}[\ensuremath{-}{(m}^{2}s)].$ Invoking Padmanabhan's ``principle of path integral duality'' corresponds to modifying the weightage factor $\mathrm{exp}[\ensuremath{-}{(m}^{2}s)]$ to $\mathrm{exp}{\ensuremath{-}[{m}^{2}{s+(L}_{P}^{2}/s)]}.$ 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. In flat spacetime, we evaluate the corrections to (1) the Casimir effect, (2) the effective potential for a self-interacting scalar field theory, (3) the effective Lagrangian for a constant electromagnetic background and (4) the thermal effects in Rindler coordinates. In arbitrary curved spacetime, we evaluate the corrections to (1) the effective Lagrangian for the gravitational field and (2) the trace anomaly. In all these cases, we first briefly present the conventional result and then go on to evaluate the corrections with the modified weightage factor. We find that the extra factor $\mathrm{exp}[\ensuremath{-}{(L}_{P}^{2}/s)]$ 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.