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Day September 13, 2014

PT-symmetric interpretation of the electromagnetic self-force

Carl M. Bender, Mariagiovanna Gianfreda

In 1980 Englert examined the classic problem of the electromagnetic self-force on an oscillating charged particle. His approach, which was based on an earlier idea of Bateman, was to introduce a charge-conjugate particle and to show that the two-particle system is Hamiltonian. Unfortunately, Englert’s model did not solve the problem of runaway modes, and the corresponding quantum theory had ghost states. It is shown here that Englert’s Hamiltonian is PT symmetric, and that the problems with his model arise because the PT symmetry is broken at both the classical and quantum level. However, by allowing the charged particles to interact and by adjusting the coupling parameters to put the model into an unbroken PT-symmetric region, one eliminates the classical runaway modes and obtains a corresponding quantum system that is ghost free.
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Solvable non-Hermitian discrete square well with closed-form physical inner product

Miloslav Znojil

A non-Hermitian N−level quantum model with two free real parameters is proposed in which the bound-state energies are given as roots of an elementary trigonometric expression and in which they are, in a physical domain of parameters, all real. The wave function components are expressed as closed-form superpositions of two Chebyshev polynomials. In any eligible physical Hilbert space of finite dimension \(N<\infty\) our model is constructed as unitary with respect to an underlying Hilbert-space metric \(\Theta\neq I\). The simplest version of the latter metric is finally constructed, at any dimension N=2,3,…, in closed form. This version of the model may be perceived as an exactly solvable N−site lattice analogue of the \(N=\infty\) square well with complex Robin-type boundary conditions. At any \(N<\infty\) our closed-form metric becomes trivial (i.e., equal to the most common Dirac’s metric \(\Theta(Dirac)=I\)) at the special, Hermitian-Hamiltonian-limit parameters.
Mathematical Physics (math-ph); Quantum Physics (quant-ph)