On the fate of Lorentz symmetry in loop quantum gravity and noncommutative spacetimes

Motivated by the remarkable sensitivity levels of the Lorentz-symmetry tests at some presentlyrunning and (further improved) forthcoming experiments, I attempt a general analysis of the fate of Lorentz symmetry in quantum spacetime. In particular, I analyze the deformation of Lorentz symmetry that holds in certain noncommutative spacetimes and the way in which Lorentz symmetry is broken in other noncommutative spacetimes. I also observe that discretization of areas (and/or lengths/volumes/times) does not necessarily require departures from Lorentz symmetry, just like the discretization of angular momentum in ordinary quantum mechanics does not require departures from space-rotation symmetry. This is due to the fact that Lorentz symmetry has no implications for exclusive measurement of the area of a surface, but it governs the combined measurements of the area and the velocity of a surface. In a quantum-gravity theory Lorentz symmetry can be consistent with area discretization, but only when the observables “area of the surface” and “velocity of the surface” enjoy certain special properties. I argue that the status of Lorentz symmetry in the loop-quantumgravity approach requires careful scrutiny, since areas are discretized within a formalism that, at least presently, does not include an observable “velocity of the surface”. In general it may prove to be very difficult to reconcile Lorentz symmetry with area discretization in theories of canonical quantization of gravity, because a proper description of Lorentz symmetry appears to require that the fundamental/primary role be played by the surface’s world-sheet, whose “projection” along the space directions of a given observer describes the observable area (just like the observable “Lx” is the projection of the angular-momentum, a legitimate “space-vector observable” of nonrelativistic quantum mechanics, along the “x” axis of an observer), whereas the canonical formalism only allows the introduction as primary entities of observables defined at a fixed (common) time, and the observers that can be considered must share that time variable. I also comment on the measurability of lengths/areas/volumes in theories that quantize the fields, such as the metric, that describe spacetime geometry: for example, I show that the same conceptual ingredients that lead to the description of the area of a surface as a quantum operator should also motivate a reanalysis of the operative definition of area, and, even when formally allowed, area discretization might be unobservable, in some sense “hidden” behind a fundamental limit on the measurability of areas.