Doubt with differential forms (YM)

In summary, the conversation discusses a calculation of the field strength for a given connection and the use of the commutation relation for the half Pauli matrices. The solution is found to not have a factor of 2 as initially thought.
  • #1
Dox
26
1
Hi everyone,

Homework Statement



I've been studying a paper in which there is a connection given by,

[tex]A = f(r)\sigma_1 dx+g(r)\sigma_2 dy,[/tex]​

where [tex]\sigma[/tex]'s are half the Pauli matrices. I need to calculate the field strength,

[tex]F = dA +[A,A].[/tex]​

Homework Equations



[tex]A = f(r)\sigma_1 dx+g(r)\sigma_2 dy,[/tex]

[tex]F = dA +[A,A][/tex]​


The Attempt at a Solution



I have computed it, but a factor is given me problems. I would say,

[tex]dA = f' \sigma_1 dr\wedge dx + g'\sigma_2 dr\wedge dy[/tex]​

and

[tex][A,A] = 2 f g \sigma_3 dx\wedge dy,[/tex]​

with a factor 2 coming from the fact that there are two contributions... like a binomial.

Is it OK or there is a half factor hidden in the definition of [tex][A,A][/tex]?

Thank you so much.

DOX​
 
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  • #2
If the [tex]\sigma_i[/tex] are equal to 1/2 the Pauli matrices, then their commutation relation is

[tex][\sigma_i,\sigma_j] = i\epsilon_{ijk}\sigma_k.[/tex]

If you recheck your calculation, you'll find that there's no factor of 2, [tex]
[A,A] = f g \sigma_3 dx\wedge dy.
[/tex]
 
  • #3
fzero said:
If the [tex]\sigma_i[/tex] are equal to 1/2 the Pauli matrices, then their commutation relation is

[tex][\sigma_i,\sigma_j] = i\epsilon_{ijk}\sigma_k.[/tex]

If you recheck your calculation, you'll find that there's no factor of 2,


[tex]
[A,A] = f g \sigma_3 dx\wedge dy.
[/tex]

Thank you fzero. I've found out that people use to write a commutator for this factor but it is just a wedge product... that's why I was getting a different factor of 2.

THX.
 

1. What is differential forms in the context of YM?

Differential forms in the context of YM (Yang-Mills) theory are mathematical objects used to describe the dynamics of gauge fields. They are differential geometric concepts that represent the local behavior of a physical quantity, such as the strength of a force field, at each point in space.

2. How does differential forms relate to gauge symmetry?

In YM theory, differential forms are used to describe the gauge symmetry of the theory. Gauge symmetry is the invariance of physical laws under certain transformations. Differential forms provide a mathematical framework for understanding and working with gauge symmetry in YM theory.

3. Can differential forms be used in other areas of physics?

Yes, differential forms are a general mathematical concept that has applications in many areas of physics, including classical mechanics, electromagnetism, and general relativity. In each of these areas, differential forms are used to describe physical quantities and their symmetries.

4. How do differential forms help solve problems in YM theory?

Differential forms provide a powerful mathematical tool for solving problems in YM theory. They allow for the manipulation and transformation of complex equations, which can simplify calculations and reveal important relationships between physical quantities and symmetries.

5. Are there any limitations to using differential forms in YM theory?

While differential forms are a useful tool in YM theory, they are not always the most efficient or intuitive way to solve problems. Some calculations can become very complicated when using differential forms, and other mathematical methods may be better suited for certain problems. It is important to understand when and how to use differential forms in order to effectively apply them in YM theory.

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