| This thesis concerns the study of the new kinds of physical theories that are described by PT-symmetric non-Hermitian Hamiltonians. When PT-symmetry is not broken, a PT-symmetric Hamiltonian has a real positive spectrum and the theory is unitary. By applying the methods of PT-symmetric quantum mechanics to the quantum-field-theoretic model called the Lee model and calculating the C operator, we reveal the nature of the ghost states in the Lee model. We find that the C operator transforms as a scalar under Lorentz transformation. This fact suggests that the C operator is the complex continuation of the intrinsic parity PI . We study the quantum anomaly in the negative quartic potential with methods of differential equations and functional integrals. We also study classical PT-symmetric theories. Using numerical analysis, we study in detail the classical trajectories of classical particles which are driven by non-Hermitian PT-symmetric Hamiltonians. Finally, we analyze the KdV equations in the complex domain. |
