PT-symmetric -gφ4 theory

The scalar field theory with potential V(phi) = 12 m2 phi 2 -14 g phi 4 (g > 0) is ill defined as a Hermitian theory but in a non-Hermitian PT-symmetric framework it is well defined, and it has a positive real energy spectrum for the case of spacetime dimension D = 1. While the methods used in the literature do not easily generalize to quantum field theory, in this paper the path-integral representation of a PT-symmetric -g phi 4 theory is shown to provide a unified formulation for general D. A new conjectural relation between the Euclidean partition functions ZPT(g) of the non-Hermitian PT-symmetric theory and ZHerm(lambda) of the lambda phi 4 (lambda > 0) Hermitian theory is proposed: log ZPT (g) = 12 log ZHerm(-g + i0+) + 12 log ZHerm(-g - i0+). This relation ensures a real energy spectrum for the non-Hermitian PT-symmetric -g phi 4 field theory. A closely related relation is rigorously valid in D = 0. For D = 1, using a semiclassical evaluation of ZPT(g), this relation is verified by comparing the imaginary parts of the ground-state energy EPT 0 (g) (before cancellation) and E0,Herm(-g +/- i0+).