I How to define expectation value in relativistic quantum mechanics?

Foracle
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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##?
 
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Go to the momentum space (via Fourier transform) and then define scalar product, probability and expectation value as in "ordinary" QM.
 
Foracle said:
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,
$$\langle \hat O(x)\rangle=i\int\left(\psi^*\frac{\partial \hat O(x)\psi}{\partial t}-\frac{\partial \psi^*}{\partial t}\hat O(x)\psi\right)dx.$$
 
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