# Non-abelian Local Gauge Invariance in Field Theories

• samalkhaiat
In summary, The speaker shares that these are notes they made 20 years ago while studying Yang-Mills and related material. They believe the notes can be useful for current students. The speaker also provides some corrections for equations referenced in the notes.
samalkhaiat
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

#### Attachments

• Gauge Invariance.pdf
394.8 KB · Views: 612
Thanks, Sam!

nice,but looks some tough as it starts with non abelian lie group directly!

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

Some corrections:
The notes were originally made using Math-Type, then converted to LaTex. This caused some problems with the references to equations numbers. I corrected most of them but missed the followings:

1) on page 7, the sentence before Eq(3.20) should say "using $Eq(3.1)$ and $Eq(3.19)$"
2) on page 9 the sentence after Eq(3.35) should read "Adding $Eq(3.34)$ to $Eq(3.35)$".
3) on page 10 again you see a reference to $Eq(10)$, this should changed to $Eq(3.1)$.
4) on page 11 reference to $Eq(59)$ is made. The correct equation number is $Eq(4.13)$.
I think that is all. Please do tell me if you find some more of these.

Sam

andrien said:
nice,but looks some tough as it starts with non abelian lie group directly!

but how its start?

## 1. What is non-abelian local gauge invariance?

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.

## 2. What is the significance of non-abelian local gauge invariance in field theories?

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.

## 3. How does non-abelian local gauge invariance differ from abelian gauge invariance?

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.

## 4. Can you give an example of a non-abelian local gauge theory?

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.

## 5. How does the concept of non-abelian local gauge invariance apply to other areas of physics?

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.

• Beyond the Standard Models
Replies
3
Views
2K
• Quantum Physics
Replies
5
Views
1K
• Other Physics Topics
Replies
1
Views
4K
• Electromagnetism
Replies
4
Views
1K
• Quantum Physics
Replies
47
Views
4K
• Other Physics Topics
Replies
1
Views
2K
• Other Physics Topics
Replies
1
Views
2K
• Special and General Relativity
Replies
7
Views
1K
• Science and Math Textbooks
Replies
1
Views
994
• High Energy, Nuclear, Particle Physics
Replies
1
Views
2K