Srednicki Ch90: How to Identify Electromagnetic Gauge Field

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In summary, the conversation discusses Srednicki's identification of the electromagnetic gauge field with his l,r,b fields. Srednicki is trying to match covariant derivatives and has set l_{\mu}=l_{\mu}^a T^a+b_{\mu} and r_{\mu}=r_{\mu}^a T^a+b_{\mu}. However, there is confusion in arriving at the desired result of eA_{\mu}=l^3_{\mu}+r_{\mu}^3+1/2b_{\mu}. The solution may involve expanding the T^{a} gens in terms of Pauli and solving simultaneously.
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LAHLH
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Does anyone know exactly how Srednicki identitifies the electromagnetic gauge field with his [itex]l,r,b [/itex] fields. I know he is trying to match covariant derivatives, i.e.

[itex] D_{\mu} p=(\partial_{\mu}-il_{\mu})p[/itex] with [itex]D_{\mu} p=(\partial_{\mu}-ieA_{\mu})p[/itex]

and that he has set [itex]l_{\mu}=l_{\mu}^a T^a+b_{\mu} [/itex]

and also match [itex] D_{\mu} n=(\partial_{\mu}-ir_{\mu})n[/itex] with [itex]D_{\mu} n=(\partial_{\mu})n[/itex]

and that he has set [itex]r_{\mu}=r_{\mu}^a T^a+b_{\mu} [/itex]

But I don't seem to be able to work out the fine print of arriving at (90.20):

[tex] eA_{\mu}=l^3_{\mu}+r_{\mu}^3+1/2b_{\mu} [/tex]

from these.

I guess it must be quite simple, and I thought maybe I should just expand the [itex]T^{a} [/itex] gens in terms of Pauli then solve simultaneously, but this didn't quite seem to work out..
 
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I seem to be finding:

[tex]D_{\mu} p=\partial_{\mu} p-\frac{i}{2}\left[(l_{\mu}^2+r_{\mu}^1)n-i(l_{\mu}^2+r_{\mu}^2)n+(l_{\mu}^3+r_{\mu}^3+2b_{\mu}) p\right] [/tex]

and

[tex]D_{\mu} n=\partial_{\mu} n-\frac{i}{2}\left[(l_{\mu}^2+r_{\mu}^1)p+i(l_{\mu}^2+r_{\mu}^2)p+(l_{\mu}^3+r_{\mu}^3+2b_{\mu}) n\right] [/tex]

when what I want to demand consistency with is

[tex] D_{\mu}p=\partial_{\mu}p-ieA_{\mu}p [/tex]
and
[tex] D_{\mu}n=\partial n [/tex]

and Srednicki says to do this I need to demand [itex] eA_{\mu}=l^3_{\mu}+r_{\mu}^3+1/2b_{\mu} [/itex]

Anyone tell me where I am going wrong?
 

Related to Srednicki Ch90: How to Identify Electromagnetic Gauge Field

1. What is an electromagnetic gauge field?

An electromagnetic gauge field is a mathematical concept used in the study of electromagnetism. It describes the interaction between electric and magnetic fields and how they are affected by changes in the surrounding environment.

2. How do you identify an electromagnetic gauge field?

To identify an electromagnetic gauge field, you can use the equations developed by physicists to describe its behavior. These equations include Maxwell's equations, which describe the relationship between electric and magnetic fields, and gauge transformations, which allow for the manipulation of these fields.

3. What is the significance of identifying an electromagnetic gauge field?

Identifying an electromagnetic gauge field is important for understanding the behavior of electromagnetism and its role in the physical world. It allows scientists to make predictions and calculations about the behavior of electric and magnetic fields in various situations, from everyday electronics to the interactions of particles in high-energy physics experiments.

4. Can you give an example of an electromagnetic gauge field in action?

One example of an electromagnetic gauge field in action is in the transmission of radio waves. These waves are created by oscillating electric and magnetic fields and are guided through space by the properties of the electromagnetic gauge field. Other examples include the behavior of particles in an electromagnetic field, such as in particle accelerators, and the electromagnetic interactions between atoms and molecules.

5. Are there any practical applications for understanding electromagnetic gauge fields?

Yes, understanding electromagnetic gauge fields has many practical applications in technology and engineering. This includes the development of communication systems, such as radios and cell phones, as well as the design of electrical circuits and devices. It also plays a crucial role in fields such as medical imaging, where knowledge of electromagnetic fields is used to create detailed images of the human body.

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