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A quantum-mechanical theory is PT-symmetric if it is described by aHamiltonian that commutes with PT, where the operator P performs spacereflection and the operator T performs time reversal. A PT-symmetricHamiltonian often has a parametric region of unbroken PT symmetry in which theenergy eigenvalues are all real. There may also be a region of broken PTsymmetry in which some of the eigenvalues are complex. These regions areseparated by a phase transition that has been repeatedly observed in laboratoryexperiments. This paper focuses on the properties of a PT-symmetric $ig\phi^3$quantum field theory. This quantum field theory is the analog of thePT-symmetric quantum-mechanical theory described by the Hamiltonian$H=p^2+ix^3$, whose eigenvalues have been rigorously shown to be all real. Thispaper compares the renormalization-group properties of a conventional Hermitian$g\phi^3$ quantum field theory with those of the PT-symmetric $ig\phi^3$quantum field theory. It is shown that while the conventional $g\phi^3$ theoryin $d=6$ dimensions is asymptotically free, the $ig\phi^3$ theory is like a$g\phi^4$ theory in $d=4$ dimensions; it is energetically stable,perturbatively renormalizable, and trivial.

Ordinary versus PT-symmetric $\phi^3$ quantum field theory