Symmetries in physics.pdfVIP

a r X i v : n u c l - t h / 0 5 0 9 0 0 7 v 1 2 S e p 2 0 0 5 Symmetries in physics ? Roelof Bijker ICN-UNAM, AP 70-543, 04510 Me?xico, DF, Me?xico E-mail: bijker@nucleares.unam.mx August 24, 2005 Abstract The concept of symmetries in physics is briefly reviewed. In the first part of these lecture notes, some of the basic mathematical tools needed for the understanding of symmetries in nature are presented, namely group theory, Lie groups and Lie algebras, and Noether’s theorem. In the sec- ond part, some applications of symmetries in physics are discussed, ranging from isospin and flavor symmetry to more recent developments involving the interacting boson model and its extension to supersymmetries in nuclear physics. 1 Introduction Symmetry and its mathematical framework—group theory—play an increasingly important role in physics. Both classical and quantum systems usually display great complexity, but the analysis of their symmetry properties often gives rise to simplifications and new insights which can lead to a deeper understanding. In addition, symmetries themselves can point the way toward the formulation of a correct physical theory by providing constraints and guidelines in an otherwise intractable situation. It is remarkable that, in spite of the wide variety of systems one may consider, all the way from classical ones to molecules, nuclei, and elementary particles, group theory applies the same basic principles and extracts the same kind of useful information from all of them. This universality in the applicability of symmetry considerations is one of the most attractive features of group theory. Most people have an intuitive understanding of symmetry, particularly in its most obvious manifestation in terms of geometric transformations that leave a body or system invariant. This interpretation, however, is not enough to readily grasp its deep connections with physics, and it thus becomes necessary to generalize the notion of symmetry transformations