Solution of Schwinger-Dyson Equations for $\mathcal{P}\mathcal{T}$-Symmetric Quantum Field Theory

In recent papers it has been observed that non-Hermitian Hamiltonians, such as those describing $ig{\varphi }^3$ and $-g{\varphi }^4$ field theories, still possess real positive spectra so long as the weaker condition of $\mathcal{P}\mathcal{T}$ symmetry holds. This allows for the possibility of new kinds of quantum field theories that have strange and quite unexpected properties. In this paper a technique based on truncating the Schwinger-Dyson equations is presented for renormalizing and solving such field theories. Using this technique it is argued that a $-g{\varphi }^4$ scalar quantum field theory in four-dimensional space-time is renormalizable, is asymptotically free, has a nonzero value of $<0\mathrm{\mid }\varphi \mathrm{\mid }0>$, and has a positive definite spectrum. Such a theory might be useful in describing the Higgs boson.