ADVANCES IN QUANTUM FIELD THEORY Edited by Sergey Ketov Advances in Quantum Field Theory Edited by Sergey Ketov Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2012 InTech All chapters are Open Access distributed under the Creative Commons Attribution 3.0 license, which allows users to download, copy and build upon published articles even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. As for readers, this license allows users to download, copy and build upon published chapters even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. Notice Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published chapters. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Romana Vukelic Technical Editor Teodora Smiljanic Cover Designer InTech Design Team First published February, 2012 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from [email protected] Advances in Quantum Field Theory, Edited by Sergey Ketov p. cm. ISBN 978-953-51-0035-5 Contents Preface IX Part 1 Quantization 1 Chapter 1 The Method of Unitary Clothing Transformations in Quantum Field Theory: Applications in the Theory of Nuclear Forces and Reactions 3 A. V. Shebeko Chapter 2 Ab Initio Hamiltonian Approach to Light-Front Quantum Field Theory 31 James P. Vary Chapter 3 Quantizing with a Higher Time Derivative 51 Sergei V. Ketov, Genta Michiaki and Tsukasa Yumibayashi Part 2 Applications to Low-Energy Physics 73 Chapter 4 Electroweak Interactions in a Chiral Effective Lagrangian for Nuclei 75 Brian D. Serot and Xilin Zhang Chapter 5 Ginzburg–Landau Theory of Phase Transitions in Compactified Spaces 103 C.A. Linhares, A.P.C. Malbouisson and I. Roditi Chapter 6 Quantum Field Theory of Exciton Correlations and Entanglement in Semiconductor Structures 125 M. Erementchouk, V. Turkowski and Michael N. Leuenberger Part 3 Applications in Mathematics 153 Chapter 7 Quantum Field Theory and Knot Invariants 155 Chun-Chung Hsieh VI Contents Chapter 8 Solution of a Linearized Model of Heisenberg’s Fundamental Equation 185 E. Brüning and S. Nagamachi Chapter 9 Topological Singularity of Fermion Current in Abelian Gauge Theory 215 Ai-Dong Bao and Shi-Shu Wu Preface Quantum Field Theory was developed as the theory unifying Quantum Mechanics and Relativistic Field Theory for the purpose of describing physics of elementary particles. Quantum Electrodynamics, electro-weak theory, Quantum Chromodyna- mics and the Standard Model of elementary particles are all particular examples of Quantum Field Theory, which had great success in high-energy physics. At the same time, it was realized that a straightforward application of Quantum Field Theory to Einstein gravity does not give a physically sensible quantum gravity theory because of its nonrenormalizability. More recent applications of Quantum Field Theory are no longer limited to physics of elementary particles. They also include many successful applications to nuclear physics, condensed matter physics and pure mathematics. At the same time, the formalism of Quantum Field Theory has to be further developed because of the new challenges, such as quantum gravity and quantization with higher derivatives, strong coupling and bound states, computational techniques in quantum perturbation theory, and more rigorous mathematical foundations. In the first part of the book, some recent progress in describing clothed particles by unitary transformations together with their physical applications to nucleon scattering and deuteron form-factors are discussed. A light-front quantization in the Hamiltonian approach and a quantization of Pais-Uhlenbeck oscillator, both in the Hamiltonian approach and in the path integral approach, are given. In the second part of the book, some applications of Quantum Field Theory to low- energy physics are considered, namely, (i) electro-weak interactions in a chiral effective Lagrangian for nuclei, (ii) Landau-Ginsburg theory of phase transitions, and (iii) exciton correlators and quantum entanglement in semiconductors. In the third part of the book, various relations between Quantum Field Theory and mathematics are presented, including knot invariants and Chern- Simons-Witten field theory, solutions to a linearized model of Heisenberg equation, quantum anomalies and Atiyah-Singer index theorems. Though the book does not include all actual aspects of Quantum Field Theory and all its recent advances, it does include some of the relevant ones, with a broad spectrum of physical and mathematical applications. X Preface The book was made possible by the generous and professional assistance of Publishing Process Manager, Ms. Romana Vukelic, during all stages of editing. I am very grateful to her for pleasant collaboration. Prof. Dr. Sergey Ketov Tokyo Metropolitan University Japan Part 1 Quantization 1 The Method of Unitary Clothing Transformations in Quantum Field Theory: Applications in the Theory of Nuclear Forces and Reactions A. V. Shebeko Institute for Theoretical Physics National Research Center “Kharkov Institute of Physics & Technology” Ukraine 1. Introduction In what follows we will show how one can realize the notion of "clothed " particles (Greenberg & Schweber, 1958) for field theoretical treatments based upon the so-called instant form of relativistic dynamics formulated by Dirac (Dirac, 1949). In the context, let us recall that the notion points out a transparent way for including the so-called cloud or persistent effects in a system of interacting fields (to be definite, mesons and nucleons). A constructive step (see surveys (Shebeko & Shirokov, 2000; 2001) and refs. therein) is to express the total field Hamiltonian H and other operators of great physical meaning, e.g., the Lorentz boost generators and current density operators, which depend initially on the creation and destruction operators for the "bare" particles, through a set of their "clothed" counterparts. It is achieved via unitary clothing transformations (UCTs) (see article (Korda et al., 2007)) in the Hilbert space H of meson-nucleon states and we stress, as before, that each of such transformations remains the Hamiltonian unchanged unlike other unitary transformation methods (Glöckle & Müller, 1981; Kobayashi, 1997; Okubo, 1954; Stefanovich, 2001))1 for Hamiltonian-based models. In the course of the clothing procedure a large amount of virtual processes associated in our case with the meson absorption/emission, the NN¯ -pair annihilation/production and other cloud effects turns out to be accumulated in the creation (destruction) operators for the clothed particles. The latter, being the quasiparticles of the method of UCTs, must have the properties (charges, masses, etc.) of physical (observable) particles. Such a bootstrap reflects the most significant distinction between the concepts of clothed and bare particles. At the same time, after Dirac, any relativistic quantum theory may be so defined that the generator of time translations (Hamiltonian), the generators of space translations (linear momentum), space rotations (angular momentum) and Lorentz transformations (boost operator) satisfy the well-known commutations. Basic ideas, put forward by Dirac with his "front", "instant" and "point" forms of the relativistic dynamics, have been realized in many relativistic quantum mechanical models. In this context, the survey (Keister & Polyzou, 1991), being a remarkable introduction to a subfield called the relativistic Hamiltonian dynamics, 1 Some specific features of these methods are discussed in (Shebeko & Shirokov, 2001) and (Korda et al., 2007) 42 Advances in QuantumWill-be-set-by-IN-TECH Field Theory represents various aspects and achievements of relativistic direct interaction theories. Among the vast literature on this subject we would like to note an exhaustive exposition in lectures (Bakker, 2001; Heinzl, 2001) of the appealing features of the relativistic Hamiltonian dynamics with an emphasis on "light-cone quantization". Following a pioneering work (Foldy, 1961), the term "direct" is related to a system with a fixed number of interacting particles, where interactions are rather direct than mediated through a field. In the approach it is customary to consider such interactions expressed in terms of the particle coordinates, momenta and spins. Along the guideline the so-called separable interactions and relativistic center-of-mass variables for composite systems were built up by assuming that the generators of the Poincaré group (Π) can be represented as expansions on powers of 1/c2 or, more exactly, (v/c)2, where v is a typical nuclear velocity (cf. the (p/m) expansion, introduced in (Friar, 1975), where