I am reading Canonical Quantization of Spherically Symmetric Gravity in Ashtekar’s Self-Dual Representation by Thiemann and Kastrup [a] and also the book Modern canonical quantum general relativity by Thiemann [b].
I have noticed that what [b] calls 'diffeomeorphism constraint' is what [a] calls 'vector costraint', i.e. the quantity $$V_a=E^b_iF_{iab}$$ where $E^a_i$ is the densitized triad and $F_{iab}$ is the curvature associated to the connection $A_{ia}$.
In [a] the diffeomorphism constraint is a linear combination of the Gauss and the vector constraints, where the Gauss constraint is given by: $$\mathcal{G}_i=(\delta_{ij}\partial_a+\epsilon_{ijk}A_{ja})E^a_k$$
The two constraints are very different and have different brackets.
I do not understand which of the two actually generates spatial diffeomorphisms, and why there is such a difference in naming these constraints.
Also, I was wondering, in [a] the diffeomorphisms are locked in the $x$ direction, but if they weren't (if they were free to generate diffeomorphisms in another direction) how would they appear? Would there be a second diffeormophism constraint similar to the first one? Or would there be additional terms to the already existing constraint?
Related to the first question: how do I prove that a constraint generates diffeomorphisms?
