Abstract MISHCHENKO, YURIY. Applications of Canonical Transformations and Nontrivial Vacuum Solutions to flavor mixing and critical phenomena in Quantum Field Theory. (Under the direction of Chueng-Ryong Ji.) In this dissertation we consider two recent applications of Bogoliubov Transforma- tion 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 treat- ment. 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 Repre- sentation 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 Decom- position Problem. Here, we attempt to substitute variational problem in terms of complicated Fock space with a constrained minimization problem in terms of expec- tation 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. APPLICATIONS OF CANONICAL TRANSFORMATIONS AND NONTRIVIAL VACUUM SOLUTIONS TO FLAVOR MIXING AND CRITICAL PHENOMENA IN QUANTUM FIELD THEORY by Mishchenko, Yuriy a thesis submitted to the graduate faculty of north carolina state university in partial fulfillment of the requirements for the degree of doctor of philosophy department of physics raleigh November 2004 approved by: Prof. Chueng-Ryong Ji Prof. Dean Lee chair of advisory committee Prof. Ronald Fulp Prof. David Brown Biography I was born on 5th of June, 1978 in Boiarka, which is near the capital city of Ukraine Kiev. When I was about one year old my family moved to town Tynda, which is in Russian Siberia. In Tynda I grew up till age of about 10. My primary school and half of my childhood passed there. In 1988, after almost 10 years, my mother decided to return to Ukraine. That was a big change for me - change of place, change of class, change of life. In Ukraine we returned to a house in Boiarka that my family owned since 1970th. There I also continued on in school until age of 14. Since then first signs of my interest to physics and mathematics became apparent. I still remember myself sitting for hours in front of TV and watching educational astronomical programs. My special interests included space mechanics and introductory physics and chemistry school texts. I also learned differential and integral calculus at that time. While the tale of those times is worth telling on its own, this isn’t the proper place for it. In 1995 I took examination and was admitted to specialized board school of ”Kiev Physical-Mathematical Lyceum”, which is a division of Kiev Taras Shevchenko Uni- versity. In the Lyceum my interest in natural sciences took sharper focus and shape of theoretical physics so that when the time came in 1995 to choose the path of life between math, cybernetics or physics, my choice was quick. That is to say, I passed examinations to Physics Department of Kiev Taras Shevchenko University and was enrolled in the Quantum Field Theory division. In Taras Shevchenko University passed next 5 years of my life. That was fun time of learning and getting to know the world around us. I should say little about ii this time as I would have to say too much otherwise. There I began to work with Prof. I. Simenog on the applications of variational approach in Quantum Mechanics and Field Theory. My first research project, completed for Bachelor of Science degree, was concerned with high-precision variational calculations of the ground state of 3- body meso-molecules. The beauty of this problem is in that it is completely defined by Coulomb interaction and still the task of achieving record precision is by far a nontrivial one. We used so called Galyorkin expansion in Gaussian basis with special emphasis on the basis optimization which allowed us to reduce 10-fold the number of Gaussians needed to reach precision comparable with the best groups in the world using merely a home PC. Still, pounding the digits of numerically generated answer for higher and higher precision for the years to come didn’t make me very excited. In 2000, for my Master Degree project, I concentrated on applications of variational principle for mass renormalization and nonperturbative relativistic 2-body Shrodinger equation in scalar Yukawa quantum field theory. Using a special form of coherent trial state we were able to obtain a full numerical solution of this problem for any values of coupling and also developed analytical approximations useful in weak and strong couplings. Also, at that time I continued to work on 3-body Coulomb problem. Nonetheless, by the time I graduated from Kiev Taras Shevchenko University with a M.Sc. worsening economical situation in the country left me little choice if the physical research would stay in my primary focus. In 2000 I applied and was admitted to Physics Department of North Carolina State University in Raleigh where I eventually started in August of 2000. Here I concentrated on the phenomenology of flavor mixing and thereafter some topics in the field theory of phase transitions and critical phenomena that eventually lead to this dissertation. iii Acknowledgments I owe a debt of gratitude to my academic adviser Dr. Chueng-Ryong Ji for his guidance and support throughout the term of my PhD study not only via many discussions but also by encouraging me at all times during this dissertation research. I am greatly thankful to Dr. Dean Lee for many useful comments, advises and his constant enthusiasm to help with any question. I would like to thank my other committee members Dr. David Brown and Dr. Ronald Fulp for their time, help and advises on topics related to my dissertation as well as many others. I am also thankful to SURA for two years of partial financial support through SURA/JLAB fellowship which played a significant role in making research, in part incorporated in this dissertation, possible. I would like to express my appreciation to Graduate Program Director Dr. Michael Paesler for continuing care about the Graduate Program and Graduate Students. Also, without our Graduate Program Secretary Jenny Allen my PhD student life would definitely be much more complicated. Finally, I want to thank all professors at North Carolina State University as well as at Taras Shevchenko Kiev University whose courses I have taken and who have made, undoubtedly, significant contribution into my growth and maturing as a scientist. iv Table of Contents List of Figures vii Introduction 1 1 Canonical Transformations in Quantum Physics 6 2 Quantum Field Theory of Flavor Mixing 17 2.1 Introduction . 17 2.2 Flavor Oscillations in Quantum Mechanics . 20 2.3 Quantum Field Theory of Mixing of Blasone and Vitiello . 27 2.4 Blasone and Vitiello Theory in Mixing of Two Bosons . 31 2.5 General Quantum Field Theory of Flavor Mixing . 48 2.6 Applications of General Quantum Field Theory of Flavor Mixing . 68 2.6.1 Vector meson mixing (S=1) for two flavors . 68 2.6.2 Fermion mixing (S=1/2) for two flavors . 70 2.6.3 Space-oscillations in two flavors mixing . 72 2.7 Phenomenological Aspects of the Oscillation Formula . 74 2.8 Summary . 77 3 Critical Phenomena and Oscillator Representation Method 85 3.1 Introduction . 85 3.2 Formulating Oscillator Representation Method . 93 3.3 Nonlinear Sigma Model . 103 3.3.1 Physical instances of nonlinear σ-model . 103 3.3.2 Properties of nonlinear σ-model . 107 3.4 The ORM in Nonlinear Sigma Model . 112 3.4.1 The ORM in nonlinear σ-model with Lagrange multipliers . 112 3.4.2 The ORM in nonlinear σ-model in perturbative form . 116 3.4.3 The RGORM in the nonlinear σ-model . 123 v 3.5 The ORM in Scalar φ4 Theory with Magnetic Field . 125 3.6 ORM as a General Field-Theoretic Technique . 132 3.6.1 Nonperturbative interpretation of ORM . 132 3.6.2 ORM as variational method . 140 3.6.3 The ORM as Quantum Effective Potential . 150 3.7 Summary . 156 4 Quantum Gaussian Packets and Symmetric Decomposition Problem166 4.1 Symmetric Decomposition Problem . 167 4.2 Quantum Gaussian Packets . 172 4.3 Symmetric Decomposition Problem: Solution . 177 4.4 Application of Symmetric Decomposition Problem in a Variant of Non- linear Sigma Model . 185 4.5 Summary . 189 5 Conclusions 192 List of References 199 A Essential Cases of Mixing Parameters 212 B The ORM functions reference formulas 214 C Unitary nonequivalence of the ORM states 215 vi List of Figures 2.1 Mixing illustration: flavor eigenstates, produced in weak interaction, propagate as superposition of energy eigenstates which then decay as different flavor eigenstates . 21 2.2 Population density in SU(2) scalar mixing at k = 0.1GeV in QFT and QM....................................