Nonlinear Su (2) ayar kuramında klasik çözümler

Classical solutions to nonlinear Su (2) gauge theory

Tez No: 46399
Yazar: HALUK ÖZBEK
Danışmanlar: PROF.DR. JAN KALAYCI
Tez Türü: Doktora
Konular: Fizik ve Fizik Mühendisliği, Physics and Physics Engineering
Anahtar Kelimeler: Belirtilmemiş.
Yıl: 1995
Dil: Türkçe
Üniversite: İstanbul Teknik Üniversitesi
Enstitü: Fen Bilimleri Enstitüsü
Ana Bilim Dalı: Belirtilmemiş.
Bilim Dalı: Belirtilmemiş.
Sayfa Sayısı: 25

Özet

The typical examples of nonlinear algebras are W- algebras and quantum groups» The representation theories of quadratic extension of nonlinear SU(2) algebra and some interesting nonlinear algebras are investigated by several authors. Recently, a general formulation of nonlinear Lie algebras is given by Ikeda. Quadratic nonlinear Lie algebras in the context of quantum field theory were first introduced by Schoutens, Sevrin and van Nieuwenhuizen. The authors built a gauge theory with no reference to a Lagrangian by begining from the classical W3 algebra. Ikeda and Izawa proposed a gauge theory Lagrangian of quadratic Lie algebras by utilizing the formulation given by Schoutens et al. Its gauge algebra is closed on mass-shell. We will try to find classical solutions to this Lagrangian. The nonlinear Lie algebra is defined with the commutator, where WAB(T) is some polynomial function of generator products. For quadratically nonlinear algebras, AfcflCT) =f£Tc+vgTcTD+kABI where f£, v%% and k^ denote structure constants which are antisymmetric in their lower indices and I is a central term added to {TA}. This term may be regarded as a zeroth- order term in the genertors TA. These algebras satisfy the Jacobi identities. However, in the foregoing discussion we will omit the central term. The above algebra takes a Lie algebra like form [TA, Tn\ -Zjra-L, AB^C if one defines f a a- -£aH+ VXb Tfs LAB-J-ABT VAB-LD Note that here structure constants are generator dependent. The Lagrangian proposed by Ikeda and Izawa is given by, which comes out to be gauge invariant. Here hf's are gauge VI

Özet (Çeviri)

CLASSICAL SOLUTIONS TO NONLINEAR Sü(2) GAUGE THEORY STJMMARY in the second half of eighties it has become clear that conformal field theories in two dimensions play an important role in string theories and in statistical systems at the critical point. Each conformal field theory is built from a set of representations of the two- dimensional conformal algebra, which is the product of two copies of the Virasoro algebra. However, in actual models there is of ten more symmetry than just conformal invariance. in fact, ali rational conformal field theories correspond to the Virasoro algebra ör some extension of it. in general such extended algebras are generated by a finite set of currents of definite conformal dimension. A systematic study of finitely generated conformal algebras was initiated by Zamolodchikov and has been developed further by many authors. The extended algebras that turn up in d=2 conformal field theory are guantum mechanical, i.e. they describe the commutation relations of operator-valued fields. The classical versions of these algebras, where the bracket is interpreted as a Poisson ör Dirac bracket are relevant in the study of certain hierarchies of completely integrable systems generalizing the KdV-hierarchy. in some examples the Fourier modes of the currents of an extended conformal algebra form an ordinary Lie algebra, ör Lie superalgebra. The more general case falls outside the scope of ordinary Lie (süper) algebras and involves algebras that may be called nonlinear Lie algebras. Nonlinear Lie algebras are a generalization of ordinary Lie algebras which contain sguares, and possibly higher order products, of the generators on the right-hand side of the def ining brackets vrithout violating the Jacobi identities. We shall consider below algebras with at most sguares. The Jacobi identities restrict the possible guadratically nonlinear algebras severely, and reveal that they are always an extension of ordinary (linear) Lie algebras if the brackets are Poisson brackets. We shall only consider Poisson brackets in this thesis. vThe typical examples of nonlinear algebras are W- algebras and quantum groups» The representation theories of quadratic extension of nonlinear SU(2) algebra and some interesting nonlinear algebras are investigated by several authors. Recently, a general formulation of nonlinear Lie algebras is given by Ikeda. Quadratic nonlinear Lie algebras in the context of quantum field theory were first introduced by Schoutens, Sevrin and van Nieuwenhuizen. The authors built a gauge theory with no reference to a Lagrangian by begining from the classical W3 algebra. Ikeda and Izawa proposed a gauge theory Lagrangian of quadratic Lie algebras by utilizing the formulation given by Schoutens et al. Its gauge algebra is closed on mass-shell. We will try to find classical solutions to this Lagrangian. The nonlinear Lie algebra is defined with the commutator, where WAB(T) is some polynomial function of generator products. For quadratically nonlinear algebras, AfcflCT) =f£Tc+vgTcTD+kABI where f£, v%% and k^ denote structure constants which are antisymmetric in their lower indices and I is a central term added to {TA}. This term may be regarded as a zeroth- order term in the genertors TA. These algebras satisfy the Jacobi identities. However, in the foregoing discussion we will omit the central term. The above algebra takes a Lie algebra like form [TA, Tn\ -Zjra-L, AB^C if one defines f a a- -£aH+ VXb Tfs LAB-J-ABT VAB-LD Note that here structure constants are generator dependent. The Lagrangian proposed by Ikeda and Izawa is given by, which comes out to be gauge invariant. Here hf's are gauge VICLASSICAL SOLUTIONS TO NONLINEAR Sü(2) GAUGE THEORY STJMMARY in the second half of eighties it has become clear that conformal field theories in two dimensions play an important role in string theories and in statistical systems at the critical point. Each conformal field theory is built from a set of representations of the two- dimensional conformal algebra, which is the product of two copies of the Virasoro algebra. However, in actual models there is of ten more symmetry than just conformal invariance. in fact, ali rational conformal field theories correspond to the Virasoro algebra ör some extension of it. in general such extended algebras are generated by a finite set of currents of definite conformal dimension. A systematic study of finitely generated conformal algebras was initiated by Zamolodchikov and has been developed further by many authors. The extended algebras that turn up in d=2 conformal field theory are guantum mechanical, i.e. they describe the commutation relations of operator-valued fields. The classical versions of these algebras, where the bracket is interpreted as a Poisson ör Dirac bracket are relevant in the study of certain hierarchies of completely integrable systems generalizing the KdV-hierarchy. in some examples the Fourier modes of the currents of an extended conformal algebra form an ordinary Lie algebra, ör Lie superalgebra. The more general case falls outside the scope of ordinary Lie (süper) algebras and involves algebras that may be called nonlinear Lie algebras. Nonlinear Lie algebras are a generalization of ordinary Lie algebras which contain sguares, and possibly higher order products, of the generators on the right-hand side of the def ining brackets vrithout violating the Jacobi identities. We shall consider below algebras with at most sguares. The Jacobi identities restrict the possible guadratically nonlinear algebras severely, and reveal that they are always an extension of ordinary (linear) Lie algebras if the brackets are Poisson brackets. We shall only consider Poisson brackets in this thesis. vThe typical examples of nonlinear algebras are W- algebras and quantum groups» The representation theories of quadratic extension of nonlinear SU(2) algebra and some interesting nonlinear algebras are investigated by several authors. Recently, a general formulation of nonlinear Lie algebras is given by Ikeda. Quadratic nonlinear Lie algebras in the context of quantum field theory were first introduced by Schoutens, Sevrin and van Nieuwenhuizen. The authors built a gauge theory with no reference to a Lagrangian by begining from the classical W3 algebra. Ikeda and Izawa proposed a gauge theory Lagrangian of quadratic Lie algebras by utilizing the formulation given by Schoutens et al. Its gauge algebra is closed on mass-shell. We will try to find classical solutions to this Lagrangian. The nonlinear Lie algebra is defined with the commutator, where WAB(T) is some polynomial function of generator products. For quadratically nonlinear algebras, AfcflCT) =f£Tc+vgTcTD+kABI where f£, v%% and k^ denote structure constants which are antisymmetric in their lower indices and I is a central term added to {TA}. This term may be regarded as a zeroth- order term in the genertors TA. These algebras satisfy the Jacobi identities. However, in the foregoing discussion we will omit the central term. The above algebra takes a Lie algebra like form [TA, Tn\ -Zjra-L, AB^C if one defines f a a- -£aH+ VXb Tfs LAB-J-ABT VAB-LD Note that here structure constants are generator dependent. The Lagrangian proposed by Ikeda and Izawa is given by, which comes out to be gauge invariant. Here hf's are gauge VICLASSICAL SOLUTIONS TO NONLINEAR Sü(2) GAUGE THEORY STJMMARY in the second half of eighties it has become clear that conformal field theories in two dimensions play an important role in string theories and in statistical systems at the critical point. Each conformal field theory is built from a set of representations of the two- dimensional conformal algebra, which is the product of two copies of the Virasoro algebra. However, in actual models there is of ten more symmetry than just conformal invariance. in fact, ali rational conformal field theories correspond to the Virasoro algebra ör some extension of it. in general such extended algebras are generated by a finite set of currents of definite conformal dimension. A systematic study of finitely generated conformal algebras was initiated by Zamolodchikov and has been developed further by many authors. The extended algebras that turn up in d=2 conformal field theory are guantum mechanical, i.e. they describe the commutation relations of operator-valued fields. The classical versions of these algebras, where the bracket is interpreted as a Poisson ör Dirac bracket are relevant in the study of certain hierarchies of completely integrable systems generalizing the KdV-hierarchy. in some examples the Fourier modes of the currents of an extended conformal algebra form an ordinary Lie algebra, ör Lie superalgebra. The more general case falls outside the scope of ordinary Lie (süper) algebras and involves algebras that may be called nonlinear Lie algebras. Nonlinear Lie algebras are a generalization of ordinary Lie algebras which contain sguares, and possibly higher order products, of the generators on the right-hand side of the def ining brackets vrithout violating the Jacobi identities. We shall consider below algebras with at most sguares. The Jacobi identities restrict the possible guadratically nonlinear algebras severely, and reveal that they are always an extension of ordinary (linear) Lie algebras if the brackets are Poisson brackets. We shall only consider Poisson brackets in this thesis. vThe typical examples of nonlinear algebras are W- algebras and quantum groups» The representation theories of quadratic extension of nonlinear SU(2) algebra and some interesting nonlinear algebras are investigated by several authors. Recently, a general formulation of nonlinear Lie algebras is given by Ikeda. Quadratic nonlinear Lie algebras in the context of quantum field theory were first introduced by Schoutens, Sevrin and van Nieuwenhuizen. The authors built a gauge theory with no reference to a Lagrangian by begining from the classical W3 algebra. Ikeda and Izawa proposed a gauge theory Lagrangian of quadratic Lie algebras by utilizing the formulation given by Schoutens et al. Its gauge algebra is closed on mass-shell. We will try to find classical solutions to this Lagrangian. The nonlinear Lie algebra is defined with the commutator, where WAB(T) is some polynomial function of generator products. For quadratically nonlinear algebras, AfcflCT) =f£Tc+vgTcTD+kABI where f£, v%% and k^ denote structure constants which are antisymmetric in their lower indices and I is a central term added to {TA}. This term may be regarded as a zeroth- order term in the genertors TA. These algebras satisfy the Jacobi identities. However, in the foregoing discussion we will omit the central term. The above algebra takes a Lie algebra like form [TA, Tn\ -Zjra-L, AB^C if one defines f a a- -£aH+ VXb Tfs LAB-J-ABT VAB-LD Note that here structure constants are generator dependent. The Lagrangian proposed by Ikeda and Izawa is given by, which comes out to be gauge invariant. Here hf's are gauge VICLASSICAL SOLUTIONS TO NONLINEAR Sü(2) GAUGE THEORY STJMMARY in the second half of eighties it has become clear that conformal field theories in two dimensions play an important role in string theories and in statistical systems at the critical point. Each conformal field theory is built from a set of representations of the two- dimensional conformal algebra, which is the product of two copies of the Virasoro algebra. However, in actual models there is of ten more symmetry than just conformal invariance. in fact, ali rational conformal field theories correspond to the Virasoro algebra ör some extension of it. in general such extended algebras are generated by a finite set of currents of definite conformal dimension. A systematic study of finitely generated conformal algebras was initiated by Zamolodchikov and has been developed further by many authors. The extended algebras that turn up in d=2 conformal field theory are guantum mechanical, i.e. they describe the commutation relations of operator-valued fields. The classical versions of these algebras, where the bracket is interpreted as a Poisson ör Dirac bracket are relevant in the study of certain hierarchies of completely integrable systems generalizing the KdV-hierarchy. in some examples the Fourier modes of the currents of an extended conformal algebra form an ordinary Lie algebra, ör Lie superalgebra. The more general case falls outside the scope of ordinary Lie (süper) algebras and involves algebras that may be called nonlinear Lie algebras. Nonlinear Lie algebras are a generalization of ordinary Lie algebras which contain sguares, and possibly higher order products, of the generators on the right-hand side of the def ining brackets vrithout violating the Jacobi identities. We shall consider below algebras with at most sguares. The Jacobi identities restrict the possible guadratically nonlinear algebras severely, and reveal that they are always an extension of ordinary (linear) Lie algebras if the brackets are Poisson brackets. We shall only consider Poisson brackets in this thesis. v

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