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arXiv:quant-ph/0308028 · Czechoslovak Journal of Physics

Is Pseudo-Hermitian Quantum Mechanics an Indefinite-Metric Quantum Theory?

Ali Mostafazadeh

Published Nov 1, 2003• 36 citations
arXivHTMLPDF
DOI
10.1023/b:cjop.0000010537.23790.8c

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We show that the metric operator for a pseudo-supersymmetric Hamiltonian that has at least one negative real eigenvalue is necessarily indefinite. We introduce pseudo-Hermitian fermion (phermion) and abnormal phermion algebras and provide a pair of basic realizations of the algebra of N = 2 pseudo-supersymmetric quantum mechanics in which pseudo-supersymmetry is identified with either a boson-phermion or a boson-abnormal-phermion exchange symmetry. We further establish the physical equivalence (non-equivalence) of phermions (abnormal phermions) with ordinary fermions, describe the underlying Lie algebras and study multi-particle systems of abnormal phermions. The latter provides a certain bosonization of multi-fermion systems.
The structure of quantum theories with indefinite metric is studied with the aid of several simple models. It is shown that the pseudo-unitary scattering matrices entering such a theory are not inconsistent with physical interpretation provided a suitable invariant projection of physical state is carried out from among all the states. A relativistic quantum theory of interacting fields is outlined and is suggested as a basis for a dynamical theory of elementary particles. It is argued that the formal introduction of an indefinite metric together with supplementary interpretive postulates may help to reinstate the principle of simplicity in a consistent theory of elementary particles.