Non-Hermitian Quantum Field Theory

The general aim of this thesis is to investigate how some well-known features of Hermitian quantum field theory extend to a non-Hermitian setting. We analyse many different versions of bosonic field theories, where some of them have application to particle physics (standard model) and nuclear physics (Skyrme model). We establish the validity of the Goldstone theorem [2] and Englert-Brout-Higgs-Guralnik-Hagen- Kibble mechanism [3, 4, 5, 6] for the complex scalar field theory with global U(1), global SU(2) and local U(1) symmetry with anti-linear CPT symmetry [7, 8, 9], in the bounded region of the parameter space. Both are shown to hold in the CPT symmetric regime, but need to be treated dfferently or even break down at the boundaries of these regions in parameter space, that is at different types of exceptional points corresponding to the algebraic singularity of the particle masses. Some particular type of these singularities were not previously found in the literature. We also analyse particular non-trivial solutions of the equations of motion, including t'Hooft-Polaykov monopoles [10], kink and BPS solutions [11] and BPS Skyrmions [12]. We show that some of the solutions are complex and yet, possess finite real energy. Drawing an analogy from the non-Hermitian quantum mechanics, we develop a reality constraint on the solutions and show that the Hamiltonian and pair of solutions needs to satisfy symmetry relation simultaneously to realise the real energy. We also show for the first time, the complex t'Hooft-Polyakov monopole solution with real energy which vanishes at the exceptional point of the Higgs particles