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(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

U(A,\gamma_{x,u})=1+\varepsilon u^a A_a(x)+\mathcal{O}(\varepsilon^2)

[/tex]

where by fixing a point [itex]x[/itex] and a tangent vector [itex]u[/itex] at [itex]x[/itex], and a path [itex]\gamma_{x,u}[/itex] of corrdinate length [itex]\varepsilon[/itex] that starts at [itex]x[/itex] and is tangent to [itex]u[/itex]. Next he takes a point [itex]x[/itex] and puts two tangent vectors on it [itex]u[/itex] and [itex]v[/itex] and considers a small triagular loop [itex]\alpha_{x,u\,v}[/itex] with one vertex at [itex]x[/itex], and two sides tangent to [itex]u[/itex] and [itex]v[/itex] each of length [itex]\varepsilon[/itex]. He expands again

[tex]

U(A,\alpha_{x,u\, v})=1+\frac{1}{2}\varepsilon^2 u^a v^b F_{ab}(x)+\mathcal{O}(\varepsilon^3)

[/tex]

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 [itex]\varepsilon[/itex], 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 [itex]\varepsilon[/itex]. Please, some enlightenment would be great, thanks.

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# The Holonomy Expansion for Hamiltonian in LQG

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