Lorentz Invariance in Loop Quantum Gravity

From the principle of relativity, where the laws of physics is presumed to be the same in all inertial reference frames, we have Lorentz invariance. This invariance leads to rotational and boost invariance. These invariances are most recognizable in the theory of special relativity. For decades, physicists have been attempting to develop a theory of quantum gravity where all of general relativity, special relativity, and quantum mechanics can come together. Two of the biggest developments so far have been that of String Theory and Loop Quantum Gravity. In the case of Loop Quantum Gravity, Lorentz invariance has not emerged so smoothly. This is because of the postulates of Loop Quantum Gravity, postulating that a discrete structure of spacetime exists near the Planck scale, where there is a minimum length and minimum time. The minimum length and minimum time are the Planck length and Planck time, respectively. The crucial role of the Planck scale is that it is the scale at which gravitational effects become relevant in a quantum setting, unlike in quantum mechanics where gravitational effects are too weak to play much of a role. The minimum length, however, appears to be a contradiction to that of Lorentz invariance since a boost in one frame of reference would lead to a length contraction. By considering relative locality and the postulates of Loop Quantum Gravity, we develop two ways in which we could resolve this contradiction.