Dual PT-Symmetric Quantum Field Theories

Some quantum field theories described by non-Hermitian Hamiltonians are investigated. It is shown that for the case of a free fermion field theory with a ${\gamma }_5$ mass term the Hamiltonian is $\mathcal{P}\mathcal{T}$-symmetric. Depending on the mass parameter this symmetry may be either broken or unbroken. When the $\cal PT$ symmetry is unbroken, the spectrum of the quantum field theory is real. For the $\mathcal{P}\mathcal{T}$-symmetric version of the massive Thirring model in two-dimensional space-time, which is dual to the $\mathcal{P}\mathcal{T}$-symmetric scalar Sine-Gordon model, an exact construction of the $\mathcal{C}$ operator is given. It is shown that the $\mathcal{P}\mathcal{T}$-symmetric massive Thirring and Sine-Gordon models are equivalent to the conventional Hermitian massive Thirring and Sine-Gordon models with appropriately shifted masses.