Ananya Ghatak, Bhabani Prasad Mandal
We demonstrate how to discriminate two non-orthogonal, entangled quantum state which are slightly different from each other by using pseudo-Hermitian system. The positive definite metric operator which makes the pseudo-Hermitian systems fully consistent quantum theory is used for such a state discrimination. We further show that non-orthogonal states can evolve through a suitably constructed pseudo-Hermitian Hamiltonian to orthogonal states. Such evolution ceases at exceptional points of the pseudo-Hermitian system.
http://arxiv.org/abs/1202.2413
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Sergio Albeverio, Sergii Kuzhel
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.
http://arxiv.org/abs/1202.1537
Mathematical Physics (math-ph); Quantum Physics (quant-ph)
V. Achilleos, P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González
We consider the nonlinear analogues of Parity-Time (\(\mathcal{PT}\)) symmetric linear systems exhibiting defocusing nonlinearities. We study the ground state and excited states (dark solitons and vortices) of the system and report the following remarkable features. For relatively weak values of the parameter \(\varepsilon\) controlling the strength of the \(\mathcal{PT}\)-symmetric potential, excited states undergo (analytically tractable) spontaneous symmetry breaking; as \(\varepsilon\) is further increased, the ground state and first excited state, as well as branches of higher multi-soliton (multi-vortex) states, collide in pairs and disappear in blue-sky bifurcations, in a way which is strongly reminiscent of the linear \(\mathcal{PT}\)-phase transition —thus termed the nonlinear \(\mathcal{PT}\)-phase transition. Past this critical point, initialization of, e.g., the former ground state leads to spontaneously emerging “soliton (vortex) sprinklers”.
http://arxiv.org/abs/1202.1310
Pattern Formation and Solitons (nlin.PS); Soft Condensed Matter (cond-mat.soft)
An article on a young mathematical prodigy on the 60 minutes news programme on US television shows him working on PT Symmetry with Prof YN Joglekar at IUPUI.
Their research article is available on arXiv at http://arxiv.org/abs/1108.6083, was published in Physical Review A (http://pra.aps.org/abstract/PRA/v84/i2/e024103), was reported on the PT-Symmeter (http://ptsymmetry.net/?p=552), and can now be seen as part of a documentary on the CBS 60 minutes website here:
http://www.cbsnews.com/video/watch/?id=7395214n&tag=re1.galleries
U. D. Jentschura
In the matter wave equations describing spin one-half particles, one can either enforce superluminal propagation by an explicit substitution of the real mass term for an imaginary mass, or one can use a matrix representation of the imaginary unit that multiplies the mass term. The latter leads to thetachyonic Dirac equation, while the equation obtained by the substitution m -> i*m in the Dirac equation is naturally referred to as the imaginary-mass Dirac equation. Both the tachyonic as well as the imaginary-mass Dirac Hamiltonians commute with the helicity operator. Both Hamiltonians are pseudo-Hermitian and also possess additional modified pseudo-Hermitian properties, leading to constraints on the resonance eigenvalues. The spectrum is found to consist of well-defined real energy eigenvalues and complex resonance and anti-resonance energies. The quantization of the tachyonic Dirac field has recently been discussed, and we here supplement a discussion of the quantized imaginary-mass Dirac field. Just as for the tachyonic Dirac Hamiltonian, we find that one-particle states of right-handed helicity acquire a negative norm and can be excluded from the physical spectrum by a Gupta–Bleuler type condition. This observation may indicate a deeper, general connection of superluminal propagation and helicity-dependent interactions.
http://arxiv.org/abs/1201.6300
High Energy Physics – Phenomenology (hep-ph); High Energy Physics – Theory (hep-th)
Andrea Cavaglia, Andreas Fring
We investigate for a large class of nonlinear wave equations, which allow for shock wave formations, how these solutions behave when they are PT-symmetrically deformed. For real solutions we find that they are transformed into peaked solutions with a discontinuity in the first derivative instead. The systems we investigate include the PT-symmetrically deformed inviscid Burgers equation recently studied by Bender and Feinberg, for which we show that it does not develop any shocks, but peaks instead. In this case we exploit the rare fact that the PT-deformation can be provided by an explicit map found by Curtright and Fairlie together with the property that the undeformed equation can be solved by the method of characteristics. We generalise the map and observe this type of behaviour for all integer values of the deformation parameter epsilon. The peaks are formed as a result of mapping the multi-valued self-avoiding shock profile to a multi-valued self-crossing function by means of the PT-deformation. For some deformation parameters we also investigate the deformation of complex solutions and demonstrate that in this case the deformation mechanism leads to discontinuties.
http://arxiv.org/abs/1201.5809
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)
Miloslav Znojil, Hendrik B. Geyer
A new version of PT-symmetric quantum theory is proposed and illustrated by an N-site-lattice Legendre oscillator. The essence of the innovation lies in the replacement of parity P (serving as an indefinite metric in an auxiliary Krein space) by its non-involutory alternative P(positive)=Q>0 playing the role of a positive-definite nontrivial metric in an auxiliary, redundant, unphysical Hilbert space. It is shown that the QT-symmetry of this form remains appealing and technically useful.
http://arxiv.org/abs/1201.5058
Quantum Physics (quant-ph)
Francisco Correa, Mikhail S. Plyushchay
We study a reflectionless PT-symmetric quantum system described by the pair of complexified Scarf II potentials mutually displaced in the half of their pure imaginary period. Analyzing the rich set of intertwining discrete symmetries of the pair, we find an exotic supersymmetric structure based on three matrix differential operators that encode all the properties of the system, including its reflectionless (finite-gap) nature. The structure we revealed particularly sheds new light on the splitting of the discrete states into two families, related to the bound and resonance states in Hermitian Scarf II counterpart systems, on which two different series of irreducible representations of sl(2,C) are realized.
http://arxiv.org/abs/1201.2750
High Energy Physics – Theory (hep-th)
Miloslav Znojil
A given Hamiltonian matrix H with real spectrum is assumed tridiagonal and non-Hermitian. Its possible Hermitizations via an amended, ad hoc inner-product metric are studied. Under certain reasonable assumptions, all of these metrics are shown obtainable as recurrent solutions of the hidden Hermiticity constraint called Dieudonne equation. In this framework even the two-parametric Jacobi-polynomial real- and asymmetric-matrix N-site lattice Hamiltonian is found tractable non-numerically at all N.
http://arxiv.org/abs/1201.2263
Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Carl M. Bender, V. Branchina, Emanuele Messina
A quantum-mechanical theory is PT-symmetric if it is described by a Hamiltonian that commutes with PT, where the operator P performs space reflection and the operator T performs time reversal. A PT-symmetric Hamiltonian often has a parametric region of unbroken PT symmetry in which the energy eigenvalues are all real. There may also be a region of broken PT symmetry in which some of the eigenvalues are complex. These regions are separated by a phase transition that has been repeatedly observed in laboratory experiments. This paper focuses on the properties of a PT-symmetric ig\phi^3 quantum field theory. This quantum field theory is the analog of the PT-symmetric quantum-mechanical theory described by the Hamiltonian H=p^2+ix^3, whose eigenvalues have been rigorously shown to be all real. This paper compares the renormalization-group properties of a conventional Hermitian g\phi^3 quantum field theory with those of the PT-symmetric ig\phi^3 quantum field theory. It is shown that while the conventional g\phi^3 theory in d=6 dimensions is asymptotically free, the ig\phi^3 theory is like a g\phi^4 theory in d=4 dimensions; it is energetically stable, perturbatively renormalizable, and trivial.
http://arxiv.org/abs/1201.1244
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)