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Essentially, gauge fields are an important area of research in physics (quantum
field theory, elementary particle theory) (six physicists won Nobel prizes in 1979, 1999 and 2004
for their work directly or indirectly related to gauge fields), and fiber bundles are an important
area of research in mathematics (differential geometry, group theory, Lie algebras) (Donaldson
won the Fields Medal in 1986 for his work on fiber bundle). In recent years, the research of fiber
bundle and gauge field based on Yang-Mills equation is being carried out deeply. Therefore,
this paper focuses on the formation and development of the related concepts of gauge field in
quantum field theory and fiber bundle in differential geometry, and the relationship between
gauge field and Yang-Mills equation, especially from the unification of electromagnetic force,
weak force and strong force. It shows the importance and far-reaching significance of gauge field.
In order to enable more relevant professional readers to quickly acquire necessary professional
knowledge and generate enthusiasm for exploration and innovation in this important field, the
basic concepts and processing methods from Abelian gauge field to non-Abelian gauge field
and spontaneous symmetry breaking are discussed in detail in this review. In particular, the
spatial properties of the Yang-Mills equation are discussed in order to better connect the gauge
field and fiber bundle with the Yang-Mills equation, so as to deepen the understanding of the
connection of the fiber bundles in a deeper level. Since this kind of problem is a meaningful
research direction, it is worth to be explored by interested people.
