Is there an uncertainty relationship that can be constructed in special relativity similar to $[x,p]=i$? For example if I know the position of a clock with zero uncertainty then I do not have any information about its momentum and hence its propertime?

Would using a quantized Klein Gordon field in position space offer some insight, or is the question not well-defined to begin with. I did find something about, mass-propertime uncertainty which follows from looking at conjugate variables in the action of SR-tic action, are there other types of uncertainty relationships also?


It all depends on which system you are describing.

First-quantized system

For example, it is possible to quantize the following (first quantized) action for the relativistic point particle:

$$ S[X] = -m c \int d{\lambda} \sqrt{g_{\mu \nu} \dot{X}^{\mu} \dot{X}^{\nu}}, $$

where $\dot{X}$ stands for $dX/d\lambda$ and $\lambda$ is some arbitrary unphysical parameter used to label the points of the worldline with real numbers.

When you quantize this system, your kinematical (unconstrained) Hilbert space $\mathcal{K}$ is a space of rapidly decreasing spacetime functions:

$$ \Psi(t,\mathbf{r}) \in \mathcal{K}. $$

Just like in the nonrelativistic case, the uncertainty relations read

$$ \Delta{x} \Delta{p} \le \frac{\hbar}{2}, $$

but there's also another relation

$$ \Delta{t} \Delta{E} \le \frac{c \hbar}{2}$$

which we encountered earlier as the time-energy uncertainty relation, only in this description it is a first-class citizen.

However, the picture is obscured by the existence of constraints. The physical Hilbert space $\mathcal{H}$ is given by those elements of $\mathcal{K}$ which are solutions of the Klein-Gordon equation. (To be precise, by elements of $\mathcal{K}^{*}$ which vanish when evaluated on solutions of Klein-Gordon equation).

Second-quantized system

Or you can jump right to Quantum Field Theory and inspect the Klein-Gordon lagrangian as a Lagrangian for the quantum field. You will then have wavefunctionals

$$ \Psi[\phi(\mathbf{r})] $$

as states, and the uncertainty relation reads

$$ \Delta \phi \Delta \pi \le \frac{h}{2} $$

with $\pi(\mathbf{r})$ the canonical momenta for $\phi(\mathbf{r})$.

This description is considered more fundamental. The position-momenta and time-energy uncertainty relations for elementary particles follow from this if one considers the fluctuations of the vacuum state of the field (elementary particle states).


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