From Real Hilbert Symmetry to Complex One-Particle Structure: Real Commutants, Type I Descent, and Positive Energy

Complex one-particle Hilbert spaces are often obtained from real phase or solution spaces by choosing a compatible imaginary unit. We treat this choice as a representation-theoretic invariant. For an orthogonal representation $U$ on a real Hilbert space $H_{\R}$, with complexification $V$ on $H_{\C}$ and canonical conjugation $C$, define the Real commutant
\[
\AR(U)=\{T\in B(H_{\C}):TV(g)=V(g)T\text{ for all }g,\ TC=CT\}.
\]
We prove that the symmetry-compatible orthogonal complex structures are exactly the skew-adjoint square roots of $-\Id$ in this real $C^*$-algebra. For central, equivalently disjoint, type~I direct-integral realizations of $V$, canonical conjugation determines a multiplicity-side antiunitary transport gauge class, and $\AR(U)$ is the algebra of decomposable multiplicity operators satisfying the corresponding transport relation. Abelian spectral orientations and compact Frobenius–Schur decompositions are recovered as scalar and atomic cases. For strongly continuous orthogonal flows, the sign of the Stone generator selects a canonical complex structure $J$ on the nonzero-energy sector and factors the real generator as $A=JB=BJ$ with $B\ge0$; this positive-energy factorization characterizes $J$ uniquely. With a compatible invariant symplectic form, the selected $J$ yields the complex one-particle space, the pure gauge-invariant quasifree Weyl state, and, under a norm-equivalence hypothesis, the corresponding ground-state Fock representation. The stationary Klein–Gordon formula $J(q,p)=(\omega^{-1}p,-\omega q)$ is obtained as a direct example