Parity-time (PT ) symmetric Klein-Gordon oscillator is presented using PTsymmetric minimal substitution. It is shown that wave equation is exactly solvable, and energy spectrum is the same as that of Hermitian Klein-Gordon oscillator presented by Bruce and Minning. Landau problem of PT -symmetric Klein-Gordon oscillator is discussed.Keywords Klein-Gordon oscillator · PT symmetry · Landau problem Although Hermitian Hamiltonian holds mainstream for quantum mechanics and quantum field theory, the study of non-Hermitian Hamiltonian has attracted a great deal of attention. This interest is triggered by the development of the studies of PT -symmetric Hamiltonian. In a fundamental paper [1] Bender and Boettcher found that it is possible to see that the energy eigenvalues of non-Hermitian Hamiltonians such as H = p 2 + x 2 (ix) ( ≥ 0) are all real as long as they preserve PT symmetry. Then, Dorey et al. provided a rigorous proof of spectral positivity [2,3]. In [4,5], it was shown that the time-evolution operator for the PT -symmetric Hamiltonian is unitary. A large number of PT -symmetric models have been studied, including PT -symmetric quantum mechanics [6], PT -symmetric quantum electrodynamics [7], PT -symmetric quantum field theory [8] as well as optical PT -symmetric structures [9]. For a comprehensive review of the basic ideas and techniques responsible for the recent developments in non-Hermitian Hamiltonian, see Ref. [10].Harmonic oscillator is one of the most useful and well studied system. In the nonrelativistic limit, the positive energy states of relativistic oscillator reduces to the spectrum of non-relativistic harmonic oscillator. The study of relativistic oscillator is of special interest in particle physics. Following the study of Dirac oscillator first proposed by Moshinsky and Szczepaniak [11], its physical applications and extensions to other case have attracted a lot of attention and been studied intensively by various authors [12][13][14][15][16][17][18][19][20][21][22][23][24].

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We introduce an integral expression of Euclidean-space propagators. The Euclidean-space propagators can be expressed as Fourier-Laplace transform of the free-field operator commutator. Based on the integral expression of Euclidean-space propagators, we find a sufficient condition for the ultraviolet (UV) finiteness of Feynman graphs calculated with various modified propagators.

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Unanyan, Otterbach, and Fleischhauer [Phys. Rev. A 79, 044101 (2009)] found that the confinement limit of a one-dimensional Dirac particle in a symmetric potential is half its corresponding Compton length and can be derived from the Dirac equation. Then, Toyama and Nogami [Phys. Rev. A 81, 044106 (2010)] conjectured that a more stringent limit holds for any symmetric potential. I would like to show, in this Brief Report, that the confinement limit conjectured by Toyama and Nogami can be derived from the Dirac equation.