PT-Symmetric Quantum Theory Defined in a Krein Space

We provide a mathematical framework for PT-symmetric quantum theory, which is applicable irrespective of whether a system is defined on R or a complex contour, whether PT symmetry is unbroken, and so on. The linear space in which PT-symmetric quantum theory is naturally defined is a Krein space constructed by introducing an indefinite metric into a Hilbert space composed of square integrable complex functions in a complex contour. We show that in this Krein space every PT-symmetric operator is P-Hermitian if and only if it has transposition symmetry as well, from which the characteristic properties of the PT-symmetric Hamiltonians found in the literature follow. Some possible ways to construct physical theories are discussed within the restriction to the class K(H).Comment: 8 pages, no figures; Refs. added, minor revisio

General Aspects of PT-Symmetric and P-Self-Adjoint Quantum Theory in a Krein Space

In our previous work, we proposed a mathematical framework for PT-symmetric quantum theory, and in particular constructed a Krein space in which PT-symmetric operators would naturally act. In this work, we explore and discuss various general consequences and aspects of the theory defined in the Krein space, not only spectral property and PT symmetry breaking but also several issues, crucial for the theory to be physically acceptable, such as time evolution of state vectors, probability interpretation, uncertainty relation, classical-quantum correspondence, completeness, existence of a basis, and so on. In particular, we show that for a given real classical system we can always construct the corresponding PT-symmetric quantum system, which indicates that PT-symmetric theory in the Krein space is another quantization scheme rather than a generalization of the traditional Hermitian one in the Hilbert space. We propose a postulate for an operator to be a physical observable in the framework.Comment: 32 pages, no figures; explanation, discussion and references adde

On Existence of a Biorthonormal Basis Composed of Eigenvectors of Non-Hermitian Operators

We present a set of necessary conditions for the existence of a biorthonormal basis composed of eigenvectors of non-Hermitian operators. As an illustration, we examine these conditions in the case of normal operators. We also provide a generalization of the conditions which is applicable to non-diagonalizable operators by considering not only eigenvectors but also all root vectors.Comment: 6 pages, no figures; (v2) minor revisions based on the comment quant-ph/0603096; (v3) presentation improved, final version to appear in Journal of Physics

On elements of the Lax-Phillips scattering scheme for PT-symmetric operators

Generalized PT-symmetric operators acting an a Hilbert space $\mathfrak{H}$ are defined and investigated. The case of PT-symmetric extensions of a symmetric operator $S$ is investigated in detail. The possible application of the Lax-Phillips scattering methods to the investigation of PT-symmetric operators is illustrated by considering the case of 0-perturbed operators

PT symmetry, Cartan decompositions, Lie triple systems and Krein space related Clifford algebras

Gauged PT quantum mechanics (PTQM) and corresponding Krein space setups are studied. For models with constant non-Abelian gauge potentials and extended parity inversions compact and noncompact Lie group components are analyzed via Cartan decompositions. A Lie triple structure is found and an interpretation as PT-symmetrically generalized Jaynes-Cummings model is possible with close relation to recently studied cavity QED setups with transmon states in multilevel artificial atoms. For models with Abelian gauge potentials a hidden Clifford algebra structure is found and used to obtain the fundamental symmetry of Krein space related J-selfadjoint extensions for PTQM setups with ultra-localized potentials.Comment: 11 page

Schroedinger equation for joint bidirectional motion in time

The conventional, time-dependent Schroedinger equation describes only unidirectional time evolution of the state of a physical system, i.e., forward or, less commonly, backward. This paper proposes a generalized quantum dynamics for the description of joint, and interactive, forward and backward time evolution within a physical system. [...] Three applications are studied: (1) a formal theory of collisions in terms of perturbation theory; (2) a relativistically invariant quantum field theory for a system that kinematically comprises the direct sum of two quantized real scalar fields, such that one field evolves forward and the other backward in time, and such that there is dynamical coupling between the subfields; (3) an argument that in the latter field theory, the dynamics predicts that in a range of values of the coupling constants, the expectation value of the vacuum energy of the universe is forced to be zero to high accuracy. [...]Comment: 30 pages, no figures. Related material is in quant-ph/0404012. Differs from published version by a few added remarks on the possibility of a large-scale-average negative energy density in spac

JJ-self-adjoint operators with C\mathcal{C}-symmetries: extension theory approach

A well known tool in conventional (von Neumann) quantum mechanics is the self-adjoint extension technique for symmetric operators. It is used, e.g., for the construction of Dirac-Hermitian Hamiltonians with point-interaction potentials. Here we reshape this technique to allow for the construction of pseudo-Hermitian ($J$-self-adjoint) Hamiltonians with complex point-interactions. We demonstrate that the resulting Hamiltonians are bijectively related with so called hypermaximal neutral subspaces of the defect Krein space of the symmetric operator. This symmetric operator is allowed to have arbitrary but equal deficiency indices . General properties of the $\cC$ operators for these Hamiltonians are derived. A detailed study of $\cC$-operator parametrizations and Krein type resolvent formulas is provided for $J$-self-adjoint extensions of symmetric operators with deficiency indices . The technique is exemplified on 1D pseudo-Hermitian Schr\"odinger and Dirac Hamiltonians with complex point-interaction potentials

Krein-Space Formulation of PT-Symmetry, CPT-Inner Products, and Pseudo-Hermiticity

Emphasizing the physical constraints on the formulation of a quantum theory based on the standard measurement axiom and the Schroedinger equation, we comment on some conceptual issues arising in the formulation of PT-symmetric quantum mechanics. In particular, we elaborate on the requirements of the boundedness of the metric operator and the diagonalizability of the Hamiltonian. We also provide an accessible account of a Krein-space derivation of the CPT-inner product that was widely known to mathematicians since 1950's. We show how this derivation is linked with the pseudo-Hermitian formulation of PT-symmetric quantum mechanics.Comment: published version, 17 page

Statistical Origin of Pseudo-Hermitian Supersymmetry and Pseudo-Hermitian Fermions

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.Comment: 20 pages, to appear in J.Phys.