When is not necesary to do surface ordering in the non-Abelian Stokes theorem?

[andresB]

Acording to the non-Abelian stokes thoerem

http://arxiv.org/abs/math-ph/0012035

I can transform a path ordered exponential to a surface ordered one.

P e^{$\oint$$\tilde{A}$}= P e^∫F

where F is some twisted curvature;F=U^-1FU, and U is a path dependet operator.So, I have a system where every element of the curvature 2-form F commute with each other, i just have the feeling that it is true that

P e^{$\oint$$\tilde{A}$}=e^∫F (*)

where the RHS is just a ordinary surface exponential.So the questions are

1)is my suposition right?
2) if afirmative how cah i prove that fact?
3) if negative, why? and what properties should have the conection A and/or the curvature F to ensure (*)

 

[azntoon]

is true?The answer to your question is yes, your supposition is right. You can prove this fact by showing that the non-Abelian Stokes theorem holds for the system you are considering. This theorem states that given a smooth connection A and a corresponding curvature F, one can express the path-ordered exponential P e\oint\tilde{A} as a surface ordered exponential e∫F. In other words, the theorem states that the two expressions are equivalent. To prove this, one can begin by introducing a coordinate system and expressing the connection and curvature in terms of the coordinates. Then, one can use the definition of the exponential and the properties of the Lie derivative to calculate the path-ordered exponential. Finally, once the expression is obtained, it can be shown that it is equivalent to the surface ordered exponential. In order for the non-Abelian Stokes theorem to hold, the connection A and the curvature F must satisfy certain conditions. Firstly, the connection must be smooth, meaning that its components must be continuous and differentiable. Secondly, the curvature must be closed, meaning that the components of the curvature must satisfy the Bianchi identities. These conditions ensure that the path-ordered exponential can be expressed as a surface ordered exponential.