Focusing on Spontaneous Symmetry Breaking (SSB), this collection of lectures aims to bridge concepts in theoretical physics with mathematical understanding. It covers a range of topics from statistical mechanics to elementary particle theory, highlighting the significance of SSB in recent scientific advancements. The discussions not only emphasize the practical success of SSB but also explore its philosophical implications, making it essential for students of mathematics and theoretical physics who are interested in foundational principles and mathematical frameworks.
Focusing on Quantum Field Theory and Gauge theories, this book emphasizes mathematical consistency while exploring key elements of the standard model of elementary particles. It delves into the Higgs mechanism and chiral symmetry breaking in quantum Chromodynamics, presenting these concepts without the use of perturbative expansion or instanton calculus, offering a fresh perspective on fundamental physical theories.
Focusing on the advantages of Analytical Mechanics over Newtonian Mechanics, the book emphasizes its efficiency in describing motion through a minimal set of coordinates, eliminating constraint forces. It highlights the form invariance of evolution equations, which addresses fictitious forces, and presents dynamics through a single scalar function instead of multiple vectors. The Hamiltonian formulation is explored, showcasing first-order time evolution governed by the Hamiltonian function. Additionally, it discusses Hamiltonian canonical algebra's role in simplifying dynamical problems and linking symmetries to conservation laws.
The book provides a non-perturbative approach to the symmetry breaking in the
standard model, in this way avoiding the critical issues which affect the
standard presentations.
The intriguing mechanism of spontaneous symmetry breaking is a powerful innovative idea at the basis of most of the recent developments in theoretical physics, from statistical mechanics to many-body theory to elementary particles theory; for infinitely extended systems a symmetric Hamiltonian can account for non symmetric behaviours, giving rise to non symmetric realizations of a physical system. In the first part of this book, devoted to classical field theory, such a mechanism is explained in terms of the occurrence of disjoint sectors and their stability properties and of an improved version of the Noether theorem. For infinitely extended quantum systems, discussed in the second part, the mechanism is related to the occurrence of disjoint pure phases and characterized by a symmetry breaking order parameter, for which non perturbative criteria are discussed, following Wightman, and contrasted with the standard Goldstone perturbative strategy. The Goldstone theorem is discussed with a critical look at the hypotheses that emphasizes the crucial role of the dynamical delocalization induced by the interaction range. The Higgs mechanism in local gauges is explained in terms of the Gauss law constraint on the physical states. The mathematical details are kept to the minimum required to make the book accessible to students with basic knowledge of Hilbert space structures. Much of the material has not appeared in other textbooks.
