LOGO Researches’ Center : Sudan University of Science and Technology College : Sciences Department : Mathematics Research Title : The Geometrical Formulation of Gauge Theories Researcher Name : Dr. Omima Elnour Saeed LOGO Abstract Conclusions Vector bundles are utilized in the study of physical fields . The connection of a fiber bundle corresponds to the gauge potential , and the physical field corresponds to the curvature of the fiber bundle . In this study we consider the geometrical interpretation of fields interaction using the language of fiber bundle . We illustrate our geometrical picture by : Dirac fields , Nucleon field and Einstein field . We found from Dirac assumptions for the spin curvature λ(p A) = λ(p) (A) that this curvature is same as Maxwell inhomogeneous equation [ . j = d p / d t ] that changing charge in a closed cylindrical tube gives us the induced current .This leads to unification of weak interactions and electromagnetic ones . This encouraged us to splice them to find the global inhomogeneous Maxwell equation which is unified with Einstien field equation under specific situation (changing the bundle metric) in chapter five . Introduction In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations . Many powerful theories in physics are described by Lagrangian which are invariant under some symmetry transformation groups . When they are invariant under a transformation identically performed at every point in the space in which the physical processes occur , they are said to have a local symmetry. The requirement of global symmetry, the cornerstone of gauge theories, is a stricter constraint . In fact , a global symmetry is just a local symmetry whose group's parameters are fixed in space-time . Gauge theories are important as the successful field theories explaining the dynamics of symmetry particles . Quantum electrodynamics is an abelian gauge theory with the symmetry group U(1 ) and has one gauge field , the electromagnetic four potential , with the photon being the gauge boson . The Standard Modal is a nonabelian gauge theory with the symmetry group U(1)× SU(2 ))× SU(3 ) and has a total of twelve gauge bosons: the photon , three weak bosons and eight gluons . Methods and Materials We defined differentiable manifolds and their kinds , homogeneous physical spaces. We defined the Principal Fiber Bundle (PFB) , the connection . We defined the – valued differential forms and the curvature of vector space . We made a one to one correspondence between the special linear group of transformations and Lorentz group to form a principal fiber bundle above which we studied the spin curvature (spinor) and deduced the global Dirac equation for the free electron . We set up the bundle splicing between Electroweak interaction and the electromagnetic field equation and of the nucleons with Yang-Mills field and deduced their global Dirac equation and global Lagrange equation respectively. We made a global tensor analysis and finding the Lie derivative of the Lagrangian , We proved its Lagrange equation and deduced the inhomogeneous field equation . Recommendations 1.The researcher recommends to establish a mathematical model for the spin curvature to be suitable for gravitation field . 2.The researcher recommends those people who call for unification of the four forces in nature to generalize the spin curvature to be suitable for gravitation field equation . 3.The researcher recommends physicians to treat particles inside atoms as a vibration not distinct points , this leads to improve the idea of super string theory . References 1. Adler , Ronald , (1975) . Introduction To General Relativity Second Edition . McGraw-HILL Book Company , New York . 2. Artzy , Rafael , (1965) . Linear geometry , Addison – Wesley Publishing Company , Inc . New York . 3. Beiser , Arthur , (1995) .Theory and Problems of Applied Physics , third edition, McGraw-Hill , Inc . New York . 4. Bleecker , David , (1981) . Gauge Theory and Variational Principle , Addison - Wesley , London . 5. Boothby , William M ., (1975 ) . An Introduction to Differentiable Manifolds and Riemannian Geometry , Acadami Press , New York . 6. Fadeev , L . D . , Slavnov , A . A . , (1991) . Gauge Fields : Introduction to Quantum Theory , Westwiew Press , USA . 7. Felice , F. De . , Clarke , C.J.S. , (1990) . Relativity on curved manifolds , Cambridge University Press , New York . 8. F. Gesztesy, Univ. of Missouri , Foundations of Physics, Vol. 23, No. 7 . 24 Mar 2011 .The Dirac Equation ,SpringerVerlag , New York . 9 . Lee , H . C . , (1984) . An introduction to Kaluza ــKlein theories , world scientific ,Canada . ، الطبعة األولى، مقدمة في ميكانيا الكم. )1998 ( ، أسامة زيد ابراهيم. ناجي.10 . القاهرة، الدار الدولية للنشر والتوزيع 1 ط، ميكانيكا الكم. )2005( ، صالح الدين البكري. ويس، محمد نبيل. يس. 11 . القاهرة، دارالفكر العربي، دار، 1 ط، النظرية النسبية الخاصة والعامة. )2010 (، عادل طه. يونس.12 . القاهرة، الفكر العربي
© Copyright 2025 Paperzz