111 research outputs found
PT Symmetric, Hermitian and P-Self-Adjoint Operators Related to Potentials in PT Quantum Mechanics
In the recent years a generalization of the
harmonic oscillator using a complex deformation was investigated, where
\epsilon\ is a real parameter. Here, we will consider the most simple case:
\epsilon even and x real. We will give a complete characterization of three
different classes of operators associated with the differential expression H:
The class of all self-adjoint (Hermitian) operators, the class of all PT
symmetric operators and the class of all P-self-adjoint operators.
Surprisingly, some of the PT symmetric operators associated to this expression
have no resolvent set
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
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
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
The superfield quantisation of a superparticle action with an extended line element
A massive superparticle action based on the generalised line element in N = 1 global superspace is quantised canonically. A previous method of quantising this action, based on a Fock space analysis, showed that states existed in three supersymmetric multiplets, each of a different mass. The quantisation procedure presented uses the single first class constraint as an operator condition on a general N = 1 superwavefunction. The constraint produces coupled equations of motion for the component wavefunctions. Transformations of the component wavefunctions are derived that decouple the equations of motion and partition the resulting wavefunctions into three separate supermultiplets. Unlike previous quantisations of superparticle actions in N = 1 global superspace, the spinor wavefunctions satisfy the Dirac equation and the vector wavefunctions satisfy the Proca equation. The off-shell closure of the commutators of the supersymmetry transformations, that include mass parameters, are derived by the introduction of auxiliary wavefunctions. To avoid the ghosts arising in a previous Fock space quantisation an alternative conjugation is used in the definition of the current, based on a Krein space approach
On elements of the Lax-Phillips scattering scheme for PT-symmetric operators
Generalized PT-symmetric operators acting an a Hilbert space
are defined and investigated. The case of PT-symmetric extensions of a
symmetric operator 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
Space of State Vectors in PT Symmetrical Quantum Mechanics
Space of states of PT symmetrical quantum mechanics is examined. Requirement
that eigenstates with different eigenvalues must be orthogonal leads to the
conclusion that eigenfunctions belong to the space with an indefinite metric.
The self consistent expressions for the probability amplitude and average value
of operator are suggested. Further specification of space of state vectors
yield the superselection rule, redefining notion of the superposition
principle. The expression for the probability current density, satisfying
equation of continuity and vanishing for the bound state, is proposed.Comment: Revised version, explicit expressions for average values and
probability amplitude adde
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
Hilbert Space Structures on the Solution Space of Klein-Gordon Type Evolution Equations
We use the theory of pseudo-Hermitian operators to address the problem of the
construction and classification of positive-definite invariant inner-products
on the space of solutions of a Klein-Gordon type evolution equation. This
involves dealing with the peculiarities of formulating a unitary quantum
dynamics in a Hilbert space with a time-dependent inner product. We apply our
general results to obtain possible Hilbert space structures on the solution
space of the equation of motion for a classical simple harmonic oscillator, a
free Klein-Gordon equation, and the Wheeler-DeWitt equation for the
FRW-massive-real-scalar-field models.Comment: 29 pages, slightly revised version, accepted for publication in
Class. Quantum Gra
The PT-symmetric brachistochrone problem, Lorentz boosts and non-unitary operator equivalence classes
The PT-symmetric (PTS) quantum brachistochrone problem is reanalyzed as
quantum system consisting of a non-Hermitian PTS component and a purely
Hermitian component simultaneously. Interpreting this specific setup as
subsystem of a larger Hermitian system, we find non-unitary operator
equivalence classes (conjugacy classes) as natural ingredient which contain at
least one Dirac-Hermitian representative. With the help of a geometric analysis
the compatibility of the vanishing passage time solution of a PTS
brachistochrone with the Anandan-Aharonov lower bound for passage times of
Hermitian brachistochrones is demonstrated.Comment: 12 pages, 2 figures, strongly extended versio
- …