Minimal-length kinematics of coherent light and a proposed two-color vacuum chronoscopy

Abstract

The unification of quantum theory and general relativity suggests the existence of a minimal measurable length, leading to a generalized uncertainty principle (GUP) and an energy-dependent vacuum group velocity. Testing these effects with macroscopic objects typically encounters the ``soccer-ball problem,'' where applying Planck-scale deformations to the total momentum predicts observationally ruled-out velocity deviations. We adopt the QFT-natural prescription in which the GUP modifies single-mode commutators, and verify by explicit calculation that the macroscopic group velocity of a multimode coherent pulse reduces to the single-photon GUP velocity evaluated at the central wavelength, multiplied by the classical Gaussian-beam factor, with no dependence on the total pulse energy. The energy--momentum relation $E=pc$ itself is unmodified; the deformation enters through the velocity--momentum relation and is equivalent to a modified dispersion relation in the wave vector, with a quadratic coefficient that depends on which of the two variables is used. The soccer-ball problem $\,$---$\,$ which arises only when the GUP is applied to the center-of-mass momentum of a composite system $\,$---$\,$ is therefore absent for coherent light, and macroscopic petawatt pulses probe the same parameter space as single high-energy photons. Building on this framework, we propose a two-color vacuum chronoscopy experiment using a short-pulse petawatt laser and its high-harmonic generation. By systematically varying the macroscopic beam waist, our setup separates Planck-scale kinematic shifts from classical diffraction delays, establishing a controlled, source-independent terrestrial methodology whose sensitivity we assess against current astrophysical bounds and against the dominant waist-independent systematics that limit it, residual-gas dispersion and the harmonic birth-time offset.