The Holonomy Expansion for Hamiltonian in LQG

[jfy4]

In Rovelli's book, in chapter 7 it talks about the Hamiltonian operator for LQG. In manipulating the form for the Hamiltonian operator Rovelli makes the following expansions
$$U(A,\gamma_{x,u})=1+\varepsilon u^a A_a(x)+\mathcal{O}(\varepsilon^2)$$
where by fixing a point $x$ and a tangent vector $u$ at $x$, and a path $\gamma_{x,u}$ of corrdinate length $\varepsilon$ that starts at $x$ and is tangent to $u$. Next he takes a point $x$ and puts two tangent vectors on it $u$ and $v$ and considers a small triagular loop $\alpha_{x,u\,v}$ with one vertex at $x$, and two sides tangent to $u$ and $v$ each of length $\varepsilon$. He expands again
$$U(A,\alpha_{x,u\, v})=1+\frac{1}{2}\varepsilon^2 u^a v^b F_{ab}(x)+\mathcal{O}(\varepsilon^3)$$

My question is, how do these expansions follow from the description above. In one expansions he seems to have series expanded to linear order, and it seems to have to do with $\varepsilon$, but in the other he expanded to second order, but the linear order term seems to vanish for some reason, and it also seems to depend on $\varepsilon$. Please, some enlightenment would be great, thanks.

 

[marcus]

Hi J, several others here can give a more satisfactory answer but I'll take as stab at it. I haven't looked in the chapter, just at your post, but at first sight it strikes me that this is almost *by definition* of the curvature F of a connection A.

BTW you did a nice job of transcribing in LaTex! I assume it is an exact copy of what is in the book. Rovelli has a page near the beginning where he lists his standard notation and if I remember right A is standard for a connection and F for the curvature.

A connection A tells you how things change as you go along a path, and the curvature F tells you what happens if you use that connection A to go around a loop. So the equivalence seems almost automatic to me. I can't think of anything really helpful that I can say about this, but hopefully one of the others (Tom, Demy?) can make it clearer.

Actually the curvature F of a connection A is a "differential form"--a *two-form* that tells you about going around a *very small* loop, and that appears to be what is happening in the second equation. It is the evaluation of a 2-form as if around an infinitesimal loop based at the point x.