The Point-to-Loop Degrees-of-Freedom Mismatch in Loop Quantum Gravity

Loop Quantum Gravity (LQG) replaces point-particles with loops as the fundamental objects of quantum geometry. This transition is geometrically significant: a point in three-dimensional space has three translational degrees of freedom (DoF), whereas a loop has six — three translational and three rotational. However, LQG imports SU(2) spinor representations that were originally designed for point-particles, without correcting for this increase in degrees of freedom. We demonstrate that this omission constitutes a structural mismatch: the SU(2) representation used in LQG encodes only 3 DoF, while the loops it is meant to describe require 6. We formalize this as a no-go theorem and show that the Barbero–Immirzi parameter γ appears precisely as a symptom of this uncorrected mismatch. No resolution within a new physical framework is proposed; the sole claim of this paper is that the mismatch exists and has not been addressed in the LQG literature.