In textbook quantum mechanics, one deals with expectation values of the form $$\langle O \rangle = \text{tr}(\rho O)$$ where $\rho$ is assumed to be trace-class (in particular, $\text{tr}\rho = 1$). Hence, $\rho O$ is also trace-class and the right hand side is finite. However, isn't this constraint too restrictive? In real life, certain common distributions (e.g. power law distributions) do not have certain finite moments. So, my question is: Can we extend the framework of quantum mechanics to model observables that do not have certain finite moments?
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