= Parity-twisted oscillator thermal trace
{title2=$\operatorname{Tr}(P e^{-\beta H})$}
For two equal-frequency <quantum harmonic oscillators>, the ordinary <thermal trace> is $r/(1-r)^2$ and the <trace> with the <parity operator> is $r/(1+r)^2$, where $r=e^{-\beta\omega}$. Total excitation $N$ has degeneracy $N+1$ and parity $(-1)^N$. Therefore the traces expand as $\sum_{n\ge1}(\pm1)^{n-1}nr^n$. The alternating weights come from parity insertion in a bosonic state space, not fermionic statistics.
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