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arXiv · 1810.12479

PT-symmetric quantum field theory in D dimensions

Carl M. Bender·Nima Hassanpour·S. P. Klevansky·Sarben Sarkar
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Abstract

PT-symmetric quantum mechanics began with a study of the Hamiltonian $H=p^2+x^2(ix)^\varepsilon$. A surprising feature of this non-Hermitian Hamiltonian is that its eigenvalues are discrete, real, and positive when $\varepsilon\geq0$. This paper examines the corresponding quantum-field-theoretic Hamiltonian $H=\frac{1}{2}(\nablaϕ)^2+\frac{1}{2}ϕ^2(iϕ)^\varepsilon$ in $D$-dimensional spacetime, where $ϕ$ is a pseudoscalar field. It is shown how to calculate the Green's functions as series in powers of $\varepsilon$ 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 $\varepsilon$ are derived for $0\leq D<2$. For $D\geq2$ 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.

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Carl M. Bender, Nima Hassanpour, S. P. Klevansky, Sarben Sarkar. 2018-10-30. PT-symmetric quantum field theory in D dimensions. https://doi.org/10.1103/physrevd.98.125003

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