I was trying to naively draw a parallel between special relativity and the Heisenberg uncertainty principle. I try to understand uncertainty principle as a consequence of 4-position and 4-momentum being conjugate variables in phase space, and this arises from a Lagrangian that looks like this: $p_\mu dx^\mu$. If in special relativity, the Lagrangian looks like this: $mc^2 d\tau$, then could I say $\Delta m\Delta\tau\ge\frac{\hbar}{2c^2}$?
I know that ultimately I would need a formalism in terms of mass and proper time operators to get a formalism of quantum mechanics this way, but I know that you can derive the uncertainty principle just from Fourier analysis and assuming that position (or time) and momentum (or energy) are conjugate variables. If mass and proper time are conjugate variables in special relativity, could one write an uncertainty relation between $mc^2$ and proper time?
If this is correct, would it imply a particle's mass and proper time cannot be known simultaneously? What would be some consequences of this?
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