Real Commutants and Positive-Energy Complex Structures in Real Hilbert Representations
Abstract
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 $C$, the Real commutant \[\AR(U)=\{T\in B(H_{\C}):TV(g)=V(g)T\text{ for all }g,\ TC=CT\}\] is the intrinsic real $C^*$-algebra controlling this choice. We prove that symmetry-compatible orthogonal complex structures are exactly the skew-adjoint square roots of $-\Id$ in $\AR(U)$. In fixed central/disjoint type~I direct-integral realizations, canonical conjugation determines a multiplicity-side antiunitary transport gauge class, yielding a decomposable formula for $\AR(U)$. Abelian spectral orientations and compact Frobenius--Schur decompositions appear as scalar and atomic cases. For strongly continuous orthogonal flows, the sign of the Stone generator selects a canonical complex structure on the nonzero-energy sector and factors the real generator as $A=JB=BJ$ with $B\ge0$ on its natural domain; this positive-energy factorization is unique. With a compatible invariant symplectic form, the selected structure yields the complex one-particle space, a pure gauge-invariant quasifree Weyl state, and, under a norm-equivalence hypothesis, the ground-state Fock representation. The stationary Klein--Gordon formula $J(q,p)=(\omega^{-1}p,-\omega q)$ is obtained as…
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