Surprisingly Long Length Scales for Semiclassical Loop Quantum Gravity and Their Physical Consequences

When gauge field theory coherent states for loop quantum gravity (LQG) were introduced, an optimized semiclassical proper length emerged, corresponding to the edge length $\epsilon$ of a graph embedded in a given classical geometry. Here $\epsilon$ is explored in more detail. $\epsilon$ at the Earth's surface is found to lie between 100 $\mu$m and 0.7 m. The implied quantum fluctuating space-time strain amplitude and noise spectrum are estimated to be $4\; 1/2$ orders smaller than the current experimental detectability. However, such a macroscopic $\epsilon$ makes regularization of the semiclassical electromagnetic Hamiltonian problematic for photon wavelengths shorter than $\epsilon$. The origin of a large $\epsilon$ is traced to an edge-wise tensor product of independent edge-based coherent states for the whole graph state. This provides physical grounds for recently proposed collective coherent states, where $\epsilon$ acquires the interpretation of a sliding scale. A new proper distance $\xi$ emerges as the characteristic length of semiclassical LQG. $\xi$ will affect the LQG photon vacuum dispersion relations, and is also accessible to current measurements of space-time strain. Matter interactions may also affect $\xi$.