Ordinary versus PT-symmetric φ3 quantum field theory

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 ig phi(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 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) theory in 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.