Applications of canonical transformations and nontrivial vacuum solutions to flavor mixing and critical phenomena in quantum field theory

In this dissertation we consider two recent applications of Bogoliubov Transformation to the phenomenology of quantum mixing and the theory of critical phenomena. In recent years quantum mixing got in the focus of the searches for New Physics due to its unparalleled sensitivity to SM parameters and indications of neutrino mixing. It was recently suggested that Bogoliubov Transformation may be important in proper definition of the flavor states that otherwise results in problems in perturbative treatment. As first part of this dissertation we investigate this conjecture and develop a complete formulation of such a mixing field theory involving introduction of general formalism, analysis of space-time conversion and phenomenological implications.; As second part of this dissertation we focus our attention on Oscillator Representation Method relevant to the study of degrees-of-freedom rearrangement during phase transitions in which vacuum condensation and mass change are analyzed using Bogoliubov Transformation. Given parallels with the duality between quarks and hadrons as well as constituent and current quarks, this method presents attractive and interesting idea. We review this method and consider its applications to nonlinear sigma model and other models. We also discuss possible schemes for its improvement.; We further introduce a novel approach in QFT---method of Symmetric Decomposition Problem. Here, we attempt to substitute variational problem in terms of complicated Fock space with a constrained minimization problem in terms of expectation values of only the relevant operators. Application of this principle to quadratic operators and the exact constraints on their expectation values are discussed along with an example of study of the ground state in a variant of nonlinear sigma model.