PT-symmetric quantum field theory in D dimensions

Abstract

PT-symmetric quantum mechanics began with a study of the Hamiltonian $H=p^2+x^2(ix{)}^{\epsilon }$. A surprising feature of this non-Hermitian Hamiltonian is that its eigenvalues are discrete, real, and positive when $\epsilon \ge 0$. This paper examines the corresponding quantum-field-theoretic Hamiltonian $H=\frac{1}{2}(\nabla \varphi {)}^2+\frac{1}{2}{\varphi }^2(i\varphi {)}^{\epsilon }$ in $D$-dimensional spacetime, where $\varphi$ is a pseudoscalar field. It is shown how to calculate the Green's functions as series in powers of $\epsilon$ directly from the Euclidean partition function. Exact finite expressions for the vacuum energy density, all of the connected $n$-point Green's functions, and the renormalized mass to order $\epsilon$ are derived for $0\le D<2$. For $D\ge 2$ the one-point Green's function and the renormalized mass are divergent, but perturbative renormalization can be performed. The remarkable spectral properties of PT-symmetric quantum mechanics appear to persist in PT-symmetric quantum field theory.

arXiv comment8 pages

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• Constructible reality condition of pseudo entropy via pseudo-Hermiticity
• Massive gauge particles versus Goldstone bosons in non-Hermitian non-Abelian gauge theory
• Relativistic PT-symmetric fermionic theories in 1+1 and 3+1 dimensions
• Dynamical Majorana Neutrino Masses and Axions II: Inclusion of Axial Background and Anomaly Terms
• Towards perturbative renormalization of ${\varphi }^2(i\varphi {)}^{\epsilon }$ quantum field theory
• PT symmetric fermionic field theories with axions: Renormalization and dynamical mass generation
• Non-Hermitian extension of the Nambu--Jona-Lasinio model in 3+1 and 1+1 dimensions
• Exact WKB analysis for $\mathcal{P}\mathcal{T}$ symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture
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• Exact quantization conditions and full transseries structures for $\mathcal{P}\mathcal{T}$ symmetric anharmonic oscillators
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• Fermion and meson mass generation in non-Hermitian Nambu--Jona-Lasinio models
• Wilsonian approach to the interaction ${\varphi }^2(i\varphi {)}^{\epsilon }$
• Original study of the $g{\varphi }^2(i\varphi {)}^{\epsilon }$ theory. Analysis of all orders in $\epsilon$ and resummations
• Scattering in pseudo-Hermitian quantum field theory and causality violation
• Unitarity through PT symmetry in Quantum Quadratic Gravity
• Oscillators with imaginary coupling: spectral functions in quantum mechanics and quantum field theory
• Conformal bridge transformation, $\mathcal{P}\mathcal{T}$- and super- symmetry
• Piecewise linear potentials for false vacuum decay and negative modes
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• Non-linear and weak-coupling expansion in Quantum Field Theory