A quantum-mechanical theory is PT-symmetric if it is described by a Hamiltonian that commutes with PT, where the operator P performs space reflection and the operator T performs time reversal. A PT-symmetric Hamiltonian often has a parametric region of unbroken PT symmetry in which the energy eigenvalues are all real. There may also be a region of broken PT symmetry in which some of the eigenvalues are complex. These regions are separated by a phase transition that has been repeatedly observed in laboratory experiments. This paper focuses on the properties of a PT-symmetric $igϕ^3$ quantum field theory. This quantum field theory is the analog of the PT-symmetric quantum-mechanical theory described by the Hamiltonian $H=p^2+ix^3$, whose eigenvalues have been rigorously shown to be all real. This paper compares the renormalization-group properties of a conventional Hermitian $gϕ^3$ quantum field theory with those of the PT-symmetric $igϕ^3$ quantum field theory. It is shown that while the conventional $gϕ^3$ theory in $d=6$ dimensions is asymptotically free, the $igϕ^3$ theory is like a $gϕ^4$ theory in $d=4$ dimensions; it is energetically stable, perturbatively renormalizable, and trivial.
Comment on ``PT-Symmetric versus Hermitian Formulations of Quantum Mechanics [hep-th/0511229]''
我们解释了为什么Bender等人，hep-th/0511229 [J. Phys. A 39（2006）1657]关于PT对称量子系统的非厄米描述相对于它们的厄米描述的实际优越性的主要结论是无效的。回顾厄米描述在描述和解释物理观测量中所起的重要作用，我们认为，就理论的物理方面而言，厄米描述不仅是不可避免的，而且是必不可少的。
Ghosts versus Unstable Particles in Quantum Field Theory
Departamento de Física de Partículas, Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela（圣地亚哥-德孔波斯特拉大学加利西亚高能物理研究所粒子物理系）
