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?

  • $\begingroup$ Have you tried computing Poisson brackets of smeared constraints (diffeomorphism constraint is smeared by a vector field that generates the diffeomorphism) with field variables? They should be equal to the Lie derivative of the variables w.r.t. the vector field, that's what it means for the constraint to generate diffeomorphisms $\endgroup$ – Prof. Legolasov Jan 31 at 13:05

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