On a few new quantization recipes using \mathcal{P}\mathcal{T}-symmetry

A review of a few recent developments in our analysis and applications of the coupled-channel version of the formalism of -symmetric quantum mechanics is given.     

Parity anomaly in a \mathcal{P}\mathcal{T}-symmetric quartic Hamiltonian

In this paper, two independent methods are used to show that the non-Hermitian -symmetric wrong-sign quartic Hamiltonian H = (1/2m)p ² − gx ⁴ is exactly equivalent to the conventional Hermitian Hamiltonian . First, this equivalence is demonstrated by using elementary differential-equation techniques and second, it is demonstrated by using functional-integration methods. As the linear term in the Hermitian Hamiltonian is proportional to ℏ, this term is anomalous; that is, the linear term in the potential has no classical analog. The anomaly is a consequence of the broken parity symmetry of the original non-Hermitian -symmetric Hamiltonian. The anomaly term in remains unchanged if an x ² term is introduced into H. When such a quadratic term is present in H, this Hamiltonian possesses bound states. The corresponding bound states in are a direct physical measure of the anomaly. If there were no anomaly term, there would be no bound states.     

Krein-space formulation of \mathcal{P}\mathcal{T} symmetry, \mathcal{C}\mathcal{P}\mathcal{T}-inner products, and pseudo-Hermiticity

Emphasizing the physical constraints on the formulation of the quantum theory, based on the standard measurement axiom and the Schrödinger equation, we comment on some conceptual issues arising in the formulation of the $\mathcal{P}\mathcal{T}$ -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 $\mathcal{C}\mathcal{P}\mathcal{T}$ -inner product, that was widely known to mathematicians since 1950’s. We show how this derivation is linked with the pseudo-Hermitian formulation of the $\mathcal{P}\mathcal{T}$ -symmetric quantum mechanics.     

Pseudo-Hermitian Systems with $\mathcal {P}\mathcal {T}$-Symmetry: Degeneracy and Krein Space

We show in the present paper that pseudo-Hermitian Hamiltonian systems with even \(\mathcal {P}\mathcal {T}\)-symmetry \((\mathcal {P}^{2}=1,\mathcal {T}^{2}=1)\) admit a degeneracy structure. This kind of degeneracy is expected traditionally in the odd \(\mathcal {P}\mathcal {T}\)-symmetric systems \((\mathcal {P}^{2}=1,\mathcal {T}^{2}=-1)\) which is appropriate to the fermions (Scolarici and Solombrino, Phys. Lett. A 303, 239 2002; Jones-Smith and Mathur, Phys. Rev. A 82, 042101 2010). We establish that the pseudo-Hermitian Hamiltonians with even \(\mathcal {P}\mathcal {T}\)-symmetry admit a degeneracy structure if the operator \(\mathcal {PT}\) anticommutes with the metric operator η _σ which is necessarily indefinite. We also show that the Krein space formulation of the Hilbert space is the convenient framework for the implementation of unbroken \(\mathcal {P}\mathcal {T}\)-symmetry. These general results are illustrated with great details for four-level pseudo-Hermitian Hamiltonian with even \(\mathcal {P}\mathcal {T}\) -symmetry.     

Tunable $\chi /\mathcal {P}\mathcal {T}$ Symmetry in Noisy Graphene

We investigate the resonant regime of a mesoscopic cavity made of graphene or a doped beam splitter. Using Non-Hermitian Quantum Mechanics, we consider the Bender-Boettcher assumption that a system must obey parity and time reversal symmetry. Therefore, we describe such system by coupling chirality, parity, and time reversal symmetries through the scattering matrix formalism and apply it in the shot noise functions, also derived here. Finally, we show how to achieve the resonant regime only by setting properly the parameters concerning the chirality and the PT symmetry.     

${ \mathcal P }{ \mathcal T }$-symmetry berry phases,topology and ${ \mathcal P }{ \mathcal T }$-symmetry breaking

We study the complex Berry phases in non-Hermitian systems with parity- and time-reversal $\left({ \mathcal P }{ \mathcal T }\right)$ symmetry. We investigate a kind of two-level system with ${ \mathcal P }{ \mathcal T }$ symmetry. We find that the real part of the the complex Berry phases have two quantized values and they are equal to either 0 or π, which originates from the topology of the Hermitian eigenstates. We also find that if we change the relative parameters of the Hamiltonian from the unbroken-${ \mathcal P }{ \mathcal T }$-symmetry phase to the broken-${ \mathcal P }{ \mathcal T }$-symmetry phase, the imaginary part of the complex Berry phases are divergent at the exceptional points. We exhibit two concrete examples in this work, one is a two-level toys model, which has nontrivial Berry phases; the other is the generalized Su–Schrieffer–Heeger (SSH) model that has physical loss and gain in every sublattice. Our results explicitly demonstrate the relation between complex Berry phases, topology and ${ \mathcal P }{ \mathcal T }$-symmetry breaking and enrich the field of the non-Hermitian physics.     

Impact of P T $\mathcal {P}\mathcal {T}$ -Symmetric Operation on Concurrence and the First-Order Coherence

International Journal of Theoretical Physics - In this paper, we lock the focus in effect of $\mathcal {P}\mathcal {T}$ -symmetric operation on the dynamics of concurrence and the first-order...     

Effective Potential for \mathcal{P}\mathcal{T}-Symmetric Quantum Field Theories

Recently, a class of -invariant scalar quantum field theories described by the non-Hermitian Lagrangian = () ² +g ² (i) was studied. It was found that there are two regions of . For <0 the -invariance of the Lagrangian is spontaneously broken, and as a consequence, all but the lowest-lying energy levels are complex. For 0 the -invariance of the Lagrangian is unbroken, and the entire energy spectrum is real and positive. The subtle transition at =0 is not well understood. In this paper we initiate an investigation of this transition by carrying out a detailed numerical study of the effective potential V _eff (_c) in zero-dimensional spacetime. Although this numerical work reveals some differences between the <0 and the >0 regimes, we cannot yet see convincing evidence of the transition at =0 in the structure of the effective potential for -symmetric quantum field theories.     

Understanding Partial P T $\mathcal {P}\mathcal {T}$ Symmetry as Weighted Composition Conjugation in Reproducing Kernel Hilbert Space: An application to Non-hermitian Bose-Hubbard Type Hamiltonian in Fock space

International Journal of Theoretical Physics - A new kind of symmetry behaviour introduced as partial $\mathcal {P}\mathcal {T}$ -symmetry(henceforth $\partial _{\mathcal {P}\mathcal {T}}$ ) is...     

$$\mathcal{P}\mathcal{T}$$ -supersymmetric partner of a short-range square well

In a box of size L, a spatially antisymmetric square-well potential of a purely imaginary strength ig and size l < L is interpreted as an initial element of the SUSY hierarchy of solvable Hamiltonians, the energies of which are all real for g < g _c (l). The first partner potential is constructed in closed form and discussed. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

Comparative analysis of real and \mathcal{P}\mathcal{T}-symmetric Scarf potentials

The Scarf I and Scarf II potentials are discussed within a common mathematical framework, which is then specified to handle the two potentials separately both in the conventional Hermitian and in the -symmetric setting. The physically admissible solutions are identified in each case together with the corresponding energy eigenvalues. Several main differences between the -symmetric Scarf I and II potentials are pointed out. These include the presence and absence of the quasi-parity quantum number, the sign of the pseudo-norm, the mechanism of the spontaneous breakdown of symmetry and the non- orthogonality of otherwise admissible solutions in the Scarf I potential. Similarities and differences with respect to the corresponding Hermitian systems are also pointed out.     

Spectra of $$ \mathcal{P}\mathcal{T} $$-symmetric Hamiltonians on tobogganic contours

A non-standard generalization of the Bender potentials x ²(ix ^ɛ) is suggested. The spectra are obtained numerically and some of their particular properties are discussed.     

Particles versus fields in $$ \mathcal{P}\mathcal{T} $$-symmetrically deformed integrable systems

The elastic scattering and the ⁶He angular distributions were measured in ⁷Li + ⁷Li reaction at two energies, E _lab = 20 and 25 MeV. FRDWBA calculations have been performed to explain the measured ⁶He data. The calculations were very sensitive to the choice of the optical model potentials in entrance and exit channels. The one-step proton transfer was found to be the dominant reaction mechanism in ⁶He production.      

Analytic wave functions and energies for two-dimensional \mathcal{P}\mathcal{T} -symmetric quartic potentials

Analytic wave functions and the corresponding energies for a class of the $ \mathcal{P}\mathcal{T} $ -symmetric two-dimensional quartic potentials are found. The general form of the solutions is discussed.     

\mathcal{P}\mathcal{T}-symmetric quartic anharmonic oscillator and position-dependent mass in a perturbative approach

To lowest order of perturbation theory we show that an equivalence can be established between a -symmetric generalized quartic anharmonic oscillator model and a Hermitian position-dependent mass Hamiltonian h. An important feature of h is that it reveals a domain of couplings where the quartic potential could be attractive, vanishing or repulsive. We also determine the associated physical quantities.     

A search for an \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \mathcal{N} =2 $\end{document} inflaton potential

We consider $ \mathcal{N} =2 $ supergravity theories that have the same spectrum as the R + R² supergravity, as predicted from the off‐shell counting of degrees of freedom. These theories describe standard $ \mathcal{N} =2 $ supergravity coupled to one or two long massive vector multiplets. The central charge is not gauged in these models and they have a Minkowski vacuum with $ \mathcal{N} =2 $ unbroken supersymmetry. The gauge symmetry, being non‐compact, is always broken. α‐deformed inflaton potentials are obtained, in the case of a single massive vector multiplet, with α = 1/3 and 2/3. The α = 1 potential (i.e. the Starobinsky potential) is also obtained, but only at the prize of having a single massive vector and a residual unbroken gauge symmetry. The inflaton corresponds to one of the Cartan fields of the non‐compact quaternionic‐Kähler cosets.     

Pseudo-Hermiticity, $$\mathcal{P}\mathcal{T}$$ symmetry,and the metric operator

The main achievements of Pseudo-Hermitian Quantum Mechanics and its distinction from the indefinite-metric quantum theories are reviewed. The issue of the non-uniqueness of the metric operator and its consequences for defining the observables are discussed. A systematic perturbative expression for the most general metric operator is offered and its application for a toy model is outlined. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

Spontaneous breakdown of $$ \mathcal{P}\mathcal{T} $$ symmetry in the complex Coulomb potential

The $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} symmetry of the Coulomb potential and its solutions are studied along trajectories satisfying the $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} symmetry requirement. It is shown that with appropriate normalization constant the general solutions can be chosen $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric if the L parameter that corresponds to angular momentum in the Hermitian case is real. $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} symmetry is spontaneously broken, however, for complex L values of the form L = −1/2 + iλ. In this case the potential remains $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric, while the two independent solutions are transformed to each other by the $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} operation and at the same time, the two series of discrete energy eigenvalues turn into each other’s complex conjugate.     

Scarcity of real discrete eigenvalues in non-analytic complex $$ \mathcal{P}\mathcal{T} $$-symmetric potentials

We find that a non-differentiability occurring whether in real or imaginary part of a complex $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric potential causes a scarcity of the real discrete eigenvalues despite the real part alone possessing an infinite spectrum. We demonstrate this by perturbing the real potentials x ² and |x| by imaginary $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric potentials ix/it|x| and ix, respectively.