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I am interested in the calculation of a correlation function in the Fock space of a system of $N$ bosons. As a trace it would be convenient for me to sum over the elements of the occupation numbers basis
\begin{equation} |n_1\cdots n_N>=\frac 1 {\sqrt{\Pi_{i=1}^N}n_i!}({a_1}^\dagger)^{n_1}\cdots ({a_N}^\dagger)^{n_N}|0> \end{equation} In order to compute the trace, I would like to check beforehand that the elements of the basis are orthogonal. Is it possible to prove it? If so, how?

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    $\begingroup$ They are eigenstates of commuting hermitian operators, and their individual spectra are non-degenerate so yes they will automatically be orthogonal. $\endgroup$ Dec 10, 2018 at 12:44

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