Abstract
The non-Hermitian but $\mathcal{PT}$-symmetric quantum field theories are known to have a pseudo-Hermitian interpretation. However the corresponding intertwining operator happens to be nonlocal that raises the question to what extent this nonlocality affects observable quantities. We consider the case when the intrinsic parity of the interaction terms is determined by degree of coupling constant. We show that the perturbative S-matrix of the equivalent Hermitian description can be easily obtained from the perturbative S-matrix of the non-Hermitian model. Namely, the first order vanishes whereas the second order is given by the real part of the second order T-matrix of the non-Hermitian model. We compute directly the 2-point and 4-point correlation functions in the equivalent Hermitian model for the $i\phi^3$ model and find the results consistent with this relation. The 1-loop correction to the mass happens to be real reflecting the disappearance of 2-body decays. However the 2 to 2 scattering amplitude obtained using LSZ formula has poles taken in principal value which implies the violation of the causality.
Highlights
The non-Hermitian PT -symmetric quantum theories have attracted significant attention due to their unusual properties [1,2,3]
We have considered the perturbative scattering in the local non-Hermitian PT -symmetric quantum field theory interpreted in a pseudo-Hermitian fashion
We explicitly showed that the intertwining operator remains to be nontrivial even when the interactions asymptotically vanishes
Summary
The non-Hermitian PT -symmetric quantum theories have attracted significant attention due to their unusual properties [1,2,3]. This leads to the extremely simple relation for the S-matrices of two equivalent descriptions; we demonstrate the violation of the Bogolyubov microcausality. This allows us to interpret them as a pseudo-Hermitian ones, i.e., related to some Hermitian Hamiltonian with the intertwining operator [4], h ≡ ηHη−1; H†η†η 1⁄4 η†ηH: ð8Þ The latter equation guarantees the Hermiticity of h with respect to the initial product or equivalently the Hermiticity of H with respect to the new inner product, ðΨ; ΦÞ ≡ hΨjη†ηjΦi: ð9Þ. E.g., the vacuum state jΩi is PT symmetric CjΩi 1⁄4 1 that significantly simplifies the computations on the vacuum state [35]
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