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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
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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
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4. Bleecker , David , (1981) . Gauge Theory and Variational
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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
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