On the tensor-algebra over the basic space C2, a P-functional is constructed. Using methods which are due to J.P. Antoine and S. Ota, a Krein-space theory (i.e., a *-algebra of operators which are defined on a common, dense, and invariant domain in a Krein-space K) is obtained via the GNS construction. It is shown that K does not contain any π-invariant dual pairs. This gives an answer to a problem first posed by J.P. Antoine and S. Ota. The theory so obtained describes the complex superposition of two harmonic oscillators. With this in mind, the annihilation and creation operators, the operator of total electric charge, and the gauge group are explicitly given.
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Gerald Hofmann, An Explicit Realization of a GNS Representation in a Krein-Space. Publ. Res. Inst. Math. Sci. 29 (1993), no. 2, pp. 267–287DOI 10.2977/PRIMS/1195167273