In the traditional theoretical descriptions of microscopic physical systems （typically, atoms and molecules） people strongly relied upon analogies between the classical mechanics and quantum theory. Naturally, such ...In the traditional theoretical descriptions of microscopic physical systems （typically, atoms and molecules） people strongly relied upon analogies between the classical mechanics and quantum theory. Naturally, such a methodical framework proved limited as it excluded, up to the recent past, multiple, less intuitively accessible phenomenological models from the serious consideration. For this reason, the classical-quantum parallels were steadily weakened, preserving still the basic and robust abstract version of the key Copenhagen-school concept of treating the states of microscopic systems as elements of a suitable linear Hilbert space. Less than 20 years ago, finally, powerful innovations emerged on mathematical side. Various less standard representations of the Hilbert space entered the game. Pars pro toto, one might recall the Dyson＇s representation of the so-called interacting boson model in nuclear physics, or the steady increase of popularity of certain apparently non-Hermitian interactions in field theory. In the first half of the author＇s present paper the recent heuristic progress as well as phenomenologieal success of the similar use of non-Hermitian Ham iltonians will be reviewed, being characterized by their self-adjoint form in an auxiliary Krein space K. In the second half of the author＇s text a further extension of the scope of such a mathematically innovative approach to the physical quantum theory is proposed. The author＇s key idea lies in the recommendation of the use of the more general versions of the indefinite metrics in the space of states （note that in the Krein-space case the corresponding indefinite metric P is mostly treated as operator of parity）. Thus, the author proposes that the operators P should be admitted to represent, in general, the indefinite metric in a Pontryagin space. A constructive version of such a generalized quantization strategy is outlined and found feasible.展开更多
文摘In the traditional theoretical descriptions of microscopic physical systems （typically, atoms and molecules） people strongly relied upon analogies between the classical mechanics and quantum theory. Naturally, such a methodical framework proved limited as it excluded, up to the recent past, multiple, less intuitively accessible phenomenological models from the serious consideration. For this reason, the classical-quantum parallels were steadily weakened, preserving still the basic and robust abstract version of the key Copenhagen-school concept of treating the states of microscopic systems as elements of a suitable linear Hilbert space. Less than 20 years ago, finally, powerful innovations emerged on mathematical side. Various less standard representations of the Hilbert space entered the game. Pars pro toto, one might recall the Dyson＇s representation of the so-called interacting boson model in nuclear physics, or the steady increase of popularity of certain apparently non-Hermitian interactions in field theory. In the first half of the author＇s present paper the recent heuristic progress as well as phenomenologieal success of the similar use of non-Hermitian Ham iltonians will be reviewed, being characterized by their self-adjoint form in an auxiliary Krein space K. In the second half of the author＇s text a further extension of the scope of such a mathematically innovative approach to the physical quantum theory is proposed. The author＇s key idea lies in the recommendation of the use of the more general versions of the indefinite metrics in the space of states （note that in the Krein-space case the corresponding indefinite metric P is mostly treated as operator of parity）. Thus, the author proposes that the operators P should be admitted to represent, in general, the indefinite metric in a Pontryagin space. A constructive version of such a generalized quantization strategy is outlined and found feasible.
