Ordinary versus $\mathcal{P}\mathcal{T}$-symmetric ${\varphi }^3$ quantum field theory

A quantum-mechanical theory is $\mathcal{P}\mathcal{T}$-symmetric if it is described by a Hamiltonian that commutes with $\mathcal{P}\mathcal{T}$, where the operator $\mathcal{P}$ performs space reflection and the operator $\mathcal{T}$ performs time reversal. A $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian often has a parametric region of unbroken $\mathcal{P}\mathcal{T}$ symmetry in which the energy eigenvalues are all real. There may also be a region of broken $\mathcal{P}\mathcal{T}$ 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 $\mathcal{P}\mathcal{T}$-symmetric $ig{\varphi }^3$ quantum field theory. This quantum field theory is the analog of the $\mathcal{P}\mathcal{T}$-symmetric quantum-mechanical theory described by the Hamiltonian $H=p^2+i{x}^3$, whose eigenvalues have been rigorously shown to be all real. This paper compares the renormalization group properties of a conventional Hermitian $g{\varphi }^3$ quantum field theory with those of the $\mathcal{P}\mathcal{T}$-symmetric $ig{\varphi }^3$ quantum field theory. It is shown that while the conventional $g{\varphi }^3$ theory in $d=6$ dimensions is asymptotically free, the $ig{\varphi }^3$ theory is like a $g{\varphi }^4$ theory in $d=4$ dimensions; it is energetically stable, perturbatively renormalizable, and trivial.

Figure 1

Four RG trajectories in the $(m^2,g^2)$ plane near the non-Gaussian fixed point $m^{2*}=0$, $g^{2*}=128{\pi }^3ϵ/3$ obtained from (38, 39) for $ϵ=0.5$. The four initial values are $m^2(t=0)=-0.1$, 0.1, $-0.1$, 0.1 and correspondingly $g^2(t=0)=0.2$, 0.4, 0.4, 0.2. The eigendirections are the dashed line and the $g^2$ axis.

Figure 2

Four RG trajectories in the $(M^2,g)$ plane for the scalar $g{\varphi }^4$ theory in $d=3$ dimensions near the Wilson-Fisher fixed point. The initial values are: $M^2(t=0)=-0.25$, 0.1, $-0.4$, 0 and correspondingly $g=0.5$, 0.75, 1.1, 1.5. The eigendirections are indicated by the two dashed lines.