Lineer olmayan Su(3) cebrinin instanton tipi cözümleri

Abstract

Üç bölümden oluşan bu çalışmada, lineer olmayan SU(3) cebrinin instanton tipi çözümleri incelenmiştir. İlk bölümde, yarı-basit bir Lie cebrinin standart formu hakkında geniş bir özet verilip, bu formun bir özdeğer probleminin çözülmesiyle elde edilebileceği gösterilmiştir. îkinci bölüm, dört alt kısımdan oluşmaktadır. Birinci kısımda, lineer olmayan Lie cebri, ikinci kısımda lineer olmayan ayar alanları, üçüncü kısımda invaryant eylem ve son kısımda da lineer olmayan SU(3) cebrinin kanonik komütasyon ilişkileri hakkında bilgiler verilmiştir. Son bölümde ise, lineer olmayan SU(3) cebrinin instanton tipi çözümleri, bir önceki bölümde verilen dönüşüm kurallarına göre kovaryant olan hareket denklemlerinin yardımıyla bulunmuştur.The standart form of a Lie algebra is obtained by solving an eigenvalue problem [A, X] =sX The operator A and the eigenvector X are linear combinations of the Lie algebra generators. The eigenvalue s is, in general, a complex number, solution of an algebraic equation of degree r, where r is the dimension of the Lie algebra. To each root of this equation corresponds an eigenvector. A semi-simple group has no abelian invariant subgroup besides itself, the identity and possible discrete subgroups. A semi-simple Lie algebra has no invariant abelian subalgebra. Of course the Lie algebra of a semi-simple group is a semi- simple Lie algebra. A nice characterization of semi-simple algebras is obtaind by defining, from the structure constants C"p / the symmetrical Cartan tensor Cartan has shown that a necessary and sufficient condition for a Lie algebra to be semi-simple is det [gpa\ *0. Nonlinear Lie algebras are a generalization of ordinary Lie algebras which contain squares and possibly higher order products of the generators on the right-hand side of the defining brackets without violating the Jacobi identites. Quantum groups are an example of nonlinear algebras with an infinite set of products. Algebras with at most squares are considered here. Quadratic nonlinear Lie algebras in the context of quantum field theory were first introduced by K. Schoutens, A. Sevrin and P. van Nieuwenhuizen. The Jacobi identites restrict the possible quadratically nonlinear algebras severely, and reveal that they are always an extension of ordinary (linear) Lie algebras if the brackets are Poisson brackets. Only, Poisson brackets are considered here.