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An extension of the scope of quantum theory is proposed in a way inspired bythe recent heuristic as well as phenomenological success of the use ofnon-Hermitian Hamiltonians which are merely required self-adjoint in a Kreinspace with an indefinite metric (chosen, usually, as the operator of parity).In nuce, the parity-like operators are admitted to represent the mereindefinite metric in a Pontryagin space. A constructive version of such ageneralized quantization strategy is outlined and, via a non-numericalillustrative example, found feasible.

PT-symmetric quantum models living in an auxiliary Pontryagin space