= Complex structure on the Klein-Gordon solution space
{title2=$J^2=-I$}
On the real <Klein-Gordon field> solution space with conserved <symplectic form> $\Omega$, a compatible real-linear map $J$ obeys $J^2=-I$, preserves $\Omega$, and makes $\mu(f,g)=\Omega(f,Jg)$ positive definite. Its $+i$ eigenspace in the complexified space has positive <Klein-Gordon inner product> with the convention $(u,v)_{KG}=i\Omega(\bar u,v)$. It defines the one-particle <Hilbert space> and a free-field <Fock vacuum>.
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