Understanding Continuous Space, Spectra, and Planck Units in Quantum Spacetime

[denisv]

How are continuous space and continuous spectra of operators mathematically consistent with Planck units?

Shouldn't the quantum spacetime be a lattice (or what have you) of integer multiples of the Planck length?

 

[muppet]

Probably.
In ordinary QM treating spacetime as continuous has the distinct advantage that the wavefunction is a differentiable function. The level at which this is an approximation is really, really small, much smaller than the levels at which one uses ordinary QM. If you're probing those kind of distance scales you probably at least want to be using QFT anyway, if not BSM (beyond the standard model) physics; nobody claims that QM alone is a complete and accurate description of nature (or if they do, they're wrong).
Afraid I can't tell you much about anything more advanced than QM though (soon ... once I've done these exams :( ) Perhaps someone else on here can help. If not, try the dedicated BSM forum.