= Gaussian projection in a Weyl representation
{title2=$P=\pi^{-1}\int e^{-|a|^2/2}W(a)\,da$}
The integrated Gaussian in any nonzero regular representation of the <Weyl relations> is an orthogonal projection and obeys $PW(a)P=e^{-|a|^2/2}P$. Gaussian integration with the Weyl phase proves both identities. It cannot be zero: conjugating it by all Weyl operators would otherwise make the Fourier transform of every Gaussian-weighted matrix coefficient vanish, including its nonzero value at the identity. In an irreducible representation its range has dimension one, and the orbit of a unit vector there gives a dense cyclic space with universal Gaussian inner products.
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