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Properties of non-Hermitian quantum field theories. (English) Zbl 1069.81030
Summary: I discuss quantum systems whose Hamiltonians are non-Hermitian but whose energy levels are all real and positive. Such theories are required to be symmetric under \({\mathcal C}{\mathcal P}{\mathcal T}\), but not symmetric under \({\mathcal P}\) and \({\mathcal T}\) separately. Recently, quantum mechanical systems having such properties have been investigated in detail. In this paper I extend the results to quantum field theories. Among the systems that I discuss are \(-\phi^4\) and \(i\phi^3\) theories. These theories all have unexpected and remarkable properties. I discuss the Green’s functions for these theories and present new results regarding bound states, renormalization, and nonperturbative calculations.

81T05 Axiomatic quantum field theory; operator algebras
34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)
34B24 Sturm-Liouville theory
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
81U40 Inverse scattering problems in quantum theory
81S40 Path integrals in quantum mechanics
CPT; non-hermitian
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