be a homogeneous canonical transformation. Use the convention that an operator $U: C_c^\infty(Y) \to \mathcal{D}'(X)$ quantising $\kappa$ has twisted canonical relation
be a microlocal inverse for $U$ over $\Omega_Y$ and $\Omega_X$, so that $QU$ is microlocally the identity on $\Omega_Y$ and $UQ$ is microlocally the identity on $\Omega_X$. If
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\begin{align*}
P \in \Psi^r(X)
\end{align*}
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is a classical pseudodifferential operator of order $r \in \mathbb{R}$ with principal symbol
microlocally on $\Omega_Y$, and its principal symbol on $\Omega_Y$ is
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\begin{align*}
\sigma_r(QPU) = p \circ \kappa.
\end{align*}
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With a fixed choice of quantisation, half-density convention, and elliptic amplitudes for $U$ and $Q$, the full classical symbol expansion of $QPU$ is obtained from the full symbol of $P$ by transport through $\kappa$, with lower-order correction terms determined by the symbolic composition formula and by those choices.