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arXiv:hep-th/0611287

Relativistic vector bosons and PT-symmetry

Jaroslav Smejkal, Vít Jakubský, Miloslav Znojil

Published Nov 27, 2006• 2 citations
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Abstract

Relativistic massive bosons with spin one are considered in several quantization schemes. In all of them the system is shown described by a non-Hermitian Hamiltonian and helicity operator. Constructively we show that in all of the contemplated schemes both these operators prove simultaneously PT-symmetric, i.e., pseudo-Hermitian with respect to a certain not too complicated indefinite pseudo-metric operator.

DOI
10.48550/arxiv.hep-th/0611287

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Relativistic massive bosons with spin one are considered in several quantization schemes. In all of them the system is shown described by a non-Hermitian Hamiltonian and helicity operator. Constructively we show that in all of the contemplated schemes both these operators prove simultaneously PT-symmetric, i.e., pseudo-Hermitian with respect to a certain not too complicated indefinite pseudo-metric operator.
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We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler-DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.
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The family of metric operators, constructed by Musumbu et al (2007 J. Phys. A: Math. Theor. 40 F75), for Swanson’s PT-symmetric Hamiltonian, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized Swanson’s non-Hermitian (not necessarily PT-symmetric) Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed. Short title: Swanson’s non-Hermitian Hamiltonian and su(1,1)
Two non-Hermitian PT-symmetric Hamiltonian systems are reconsidered by means of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the PT-symmetric ones. Compared with the way converting a non-Hermitian Hamiltonian to its Hermitian counterpart, this method has the merit that keeps the Hilbert space of the non-Hermitian PT-symmetric Hamiltonian unchanged. In order to give the positive definite inner product for the PT-symmetric systems, a new operator V, instead of C, can be introduced. The operator V has the similar function to the operator C adopted normally in the PT-symmetric quantum mechanics, however, it can be constructed, as an advantage, directly in terms of Hamiltonians. The spectra of the two non-Hermitian PT-symmetric systems are obtained, which coincide with that given in literature, and in particular, the Hilbert spaces associated with positive definite inner products are worked out.