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samalkhaiat said:These are notes I made when I was studying the subject 20 years ago. They seem fine considering that I was student then. I believe they can be useful for those who are studying Yang-Mills and other related material.
Sam
andrien said:nice,but looks some tough as it starts with non abelian lie group directly!
Non-abelian local gauge invariance is a fundamental principle in field theory that describes the symmetry of a system under local transformations. It means that the equations of motion for a system remain unchanged when the fields are transformed by a non-abelian group of continuous transformations at each point in space.
Non-abelian local gauge invariance plays a crucial role in the formulation of gauge theories, such as the Standard Model of particle physics. It allows for the inclusion of interactions between particles and the description of their interactions in terms of gauge fields. This leads to a more complete and elegant understanding of the fundamental forces of nature.
Abelian gauge invariance is a special case of non-abelian gauge invariance where the gauge transformations commute with each other. This means that the order in which the transformations are applied does not matter. In contrast, non-abelian gauge invariance allows for the gauge transformations to not commute, leading to a more complex and rich structure.
The most well-known example of a non-abelian local gauge theory is the Standard Model of particle physics. It describes the interactions between the fundamental particles and their associated gauge bosons - the photon, W and Z bosons, and gluons. The theory is based on the non-abelian gauge group SU(3) × SU(2) × U(1) and has been extensively tested and confirmed by experiments.
Non-abelian local gauge invariance is not limited to particle physics and has applications in other areas of physics, such as condensed matter physics and cosmology. In condensed matter systems, it can describe the behavior of interacting electrons in materials. In cosmology, it plays a crucial role in the theory of inflation, which explains the rapid expansion of the universe in its early stages.