Non-abelian Gauge invariance (chapter 15.1 in Peskin/Schroeder)

In summary, the covariant derivative in Peskin/Schroeder (chapter 15.1, page 483) is derived using the limit definition of differentiation. The scalar quantity U(y,x) is expanded in powers of ε nμ, with the coefficient of the linear term defined as Aμ. The pure phase condition is used to simplify the expansion, and the rest is derived from the Taylor expansion and the definition of A_\mu(x).
  • #1
Stalafin
21
0
I am trying to understand the derivation of the covariant derivative in Peskin/Schroeder (chapter 15.1, page 483).

This is the important stuff:
[tex]n^\mu\partial_\mu\psi=\lim_{\epsilon \rightarrow 0} \frac{1}{\epsilon}\left[\psi(x+\epsilon n)-\psi(x)\right][/tex]

Scalar quantity: U(y,x):
[tex]U(y,x) \rightarrow e^{i\alpha(y)} U(y,x) e^{-i\alpha(x)}[/tex]

Conditions for zero separation and pure phase:
[tex]U(y,y)=1[/tex]
[tex]U(y,x)=exp[i\phi(y,x)][/tex]

Covariant derivative:
[tex]n^\mu D_\mu \psi = \lim_{\epsilon\rightarrow 0} \frac{1}{\epsilon}\left[\psi(x+\epsilon n)-U(x+\epsilon n,x)\psi(x)\right][/tex]

Expansion of U(y,x) in the separation of the two points:
[tex]U(x+\epsilon n,x) = 1 - i e\epsilon n^\mu A_\mu(x) + \mathcal{O}(\epsilon^2)[/tex]

A_\mu(x) is a new vector field and is the coefficient of the displacement \epsilon n^\mu. Why?! I don't see how the author gets there.
 
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  • #2
Isn't this just by definition. You expand U in powers of ε nμ, and the coefficient in the linear term, you define that to be Aμ.
 
  • #3
I thought about making an expansion... This is what I came up with (Peskin expands for epsilon):
[tex]\left. U(x+\epsilon n,x) \right|_{\epsilon=0} = U(x,x) + U'(x,x) n \epsilon + \mathcal{O}(\epsilon^2)[/tex]

So, U(x,x) becomes one from the zero separation condition; what I have trouble understanding is the next part. How does that work out?

[tex] U'(x,x) n \epsilon = - ie\epsilon n^\mu A_\mu(x) [/tex]

Do I use the pure phase condition for that? And how am I supposed to read the n? As a four-vector itself? Or is that:
[tex]n=n^\mu A_\mu(x)[/tex]

Where does the rest come from?
 
  • #4
The Taylor expansion is
[tex]U(x+\epsilon n,x)=U(x,x)+\epsilon n^\mu\partial_\mu U(x,x) + O(\epsilon^2)[/tex]
Then we define
[tex]A_\mu(x)\equiv (i/e) \partial_\mu U(x,x)[/tex]
Note that, with the factor of i, unitarity of U implies that A is hermitian. And to be precise, the partial derivative actually acts only on the first x.
 
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1. What is non-abelian gauge invariance?

Non-abelian gauge invariance is a fundamental principle in quantum field theory that states that the physical laws of a system should not change under a change in the gauge or mathematical representation of the system. In other words, the physical predictions of a theory should be independent of the mathematical tools used to describe it.

2. How does non-abelian gauge invariance differ from abelian gauge invariance?

In abelian gauge invariance, the gauge transformations commute, meaning the order in which they are performed does not matter. In non-abelian gauge invariance, the gauge transformations do not commute, and their order can affect the physical predictions of the theory. This is due to the presence of non-commuting group generators in non-abelian gauge theories.

3. What is the role of non-abelian gauge invariance in the Standard Model of particle physics?

Non-abelian gauge invariance plays a crucial role in the Standard Model, which is the most widely accepted theory of particle physics. The Standard Model describes the interactions between elementary particles and the fundamental forces of nature, and non-abelian gauge invariance is necessary for the theory to be consistent with experimental observations.

4. What are some examples of non-abelian gauge theories?

Some examples of non-abelian gauge theories include the Yang-Mills theory, which describes the strong nuclear force, and the electroweak theory, which describes the electromagnetic and weak nuclear forces. These theories are both components of the Standard Model.

5. How is non-abelian gauge invariance tested in experiments?

Non-abelian gauge invariance is tested by comparing experimental results to theoretical predictions based on the principles of gauge invariance. If the predictions match the experimental results, it provides evidence for the validity of non-abelian gauge invariance. Additionally, experiments such as particle colliders can directly probe the behavior of particles and their interactions, providing further evidence for the predictions of non-abelian gauge theories.

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