Generalised uncertainty relations from finite-accuracy measurements

In this short note we show how the Generalised Uncertainty Principle (GUP) and Extended (EUP), two of most common generalised uncertainty relations proposed in quantum gravity literature, can be derived within context canonical theory, without need for modified commutation relations. A GUP-type relation naturally emerges when standard position operator is replaced by an appropriate Positive Operator Valued Measure (POVM), representing a finite-accuracy measurement that localises wave packet to spatial region $\sigma_g > 0$. This length scale deviation envelope function, $g$, defines POVM elements. Similarly, EUP-type momentum $\tilde{\sigma}_g 0$ space. The usual GUP EUP are recovered setting \simeq \sqrt{\hbar G/c^3}$, Planck length, \hbar\sqrt{\Lambda/3}$, where $\Lambda$ cosmological constant. Crucially, Hamiltonian relations, and, hence, Schr{\" o}dinger Heisenberg equations, remain unchanged. demonstrates phenomenology obtained commutators, which known lead various pathologies, including violation equivalence principle, Lorentz invariance relativistic limit, reference frame-dependence `minimum' so-called soccer ball problem multi-particle states.