How to define expectation value in relativistic quantum mechanics?

[Foracle]

TL;DR Summary

How to define expectation value in relativistic quantum mechanics?

In non relativistic quantum mechanics, the expectation value of an operator $\hat{O}$ in state $\psi$ is defined as $$<\psi |\hat{O}|\psi>=\int\psi^* \hat{O} \psi dx$$.
Since the scalar product in relativistic quantum has been altered into $$|\psi|^2=i\int\left(\psi^*\frac{\partial \psi}{\partial t}-\frac{\partial \psi^*}{\partial t}\psi\right)dx$$
how do we define expectation value of an operator $\hat{O}$ in state $\psi$?

 

[Demystifier]

Go to the momentum space (via Fourier transform) and then define scalar product, probability and expectation value as in "ordinary" QM.

 

[A. Neumaier]

Since the scalar product in relativistic quantum has been altered into $$|\psi|^2=i\int\left(\psi^*\frac{\partial \psi}{\partial t}-\frac{\partial \psi^*}{\partial t}\psi\right)dx$$
how do we define expectation value of an operator $\hat{O}$ in state $\psi$?

$$\langle \hat O\rangle=i\int\left(\psi^*\hat O\frac{\partial \psi}{\partial t}-\frac{\partial \psi^*}{\partial t}\hat O\psi\right)dx.$$
works if $O$ does not depend on $x$. In general,
$$\langle \hat O(x)\rangle=i\int\left(\psi^*\frac{\partial \hat O(x)\psi}{\partial t}-\frac{\partial \psi^*}{\partial t}\hat O(x)\psi\right)dx.$$