Four formalisms;
Defining formalism manifestly symmetric between space and time; other three as well.
With laboratory time defined as geodesic time, all pieces manifestly symmetric between time and space
First question: correct reduction in various limits:
_nonrel
Semi-classical (temporal quantization gives in many cases merely temporal fuzz around standard quantum theory solution)
Stationary states (when base frequency high only stationary part reinforces)
Bohr condition
Second question: If fluctutations in time are real, if time is fuzzy: why have they not been seen?
Two part answer: first, both halves of apparatus have, in general, to vary in time. Particle not beam, chopper not slit.
Second, hard to see what you are not looking for.
Complement of consistent with known: we want temporal quantization consistent, but not too consistent.
300 foundational experiments
Experiments strictly in time also possible
Simplicity important in picking the Lagrangian quadratic in four velocity
Also a product:
Most complete symmetry between time and space possible, by construction
As a side-effect, has no free parameters.
Manifestly unitary
Requires no technical tricks: no factors or , or rotations to Euclidean time.
Gives a simple Schrödinger equation in four dimensions.
Fully relativistic single particle treatment.