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ABSTRACT
This Project addresses the fundamental problem of determining the conditions under which the Lagrangian remains invariant under local and global transformations in the context of quantum field theory. We investigate the key question of whether the Lagrangian, describing fermionic fields, retains its form when subjected to these transformations.
For global transformations, we examine the effect of a uniform phase change on the fermionic field. In contrast, local transformations necessitate the introduction of covariant derivatives and gauge fields to accommodate variations across spacetime.
Our results reveal that the Lagrangian remains invariant under global transformations but requires the incorporation of covariant derivatives and gauge fields for local invariance. This study underscores the pivotal role of gauge symmetries in modern theoretical physics, highlighting their significance in understanding the behavior of fermionic fields and the emergence of conservation laws in the universe.
