Question about holonomy-flux algebra

[kakarukeys]

1. the holonomy-flux algebra (HF) is generated by cylindrical functions and operators linear in momenta, so both the Gauss constraint and Hamiltonian constraint do not belong to HF, is this not a problem?

my understanding:
in quantum mechanics the elementary variables are $$(1,q_i,p^i)$$. I generate a Lie algebra using all $$f(q)$$ and $$g_i(q)p^i$$. $$f(q)$$ is seen as multiplication operator, $$g_i(q)p^i$$ is seen as derivation on space of $$f(q)$$ (hamiltonian vector fields). http://sps.nus.edu.sg/~wongjian/lqg.html for details. Then I promote them to quantum operators. Then one generate a free associative algebra from the quantum operators. The elementary variables used in LQG are the holonomies and fluxes. The algebra generated in this way is called Holonomy-Flux algebra (HF). Now since an element in HF is at most polynomial in E's. The Gauss constraint and Hamiltonian constraint do not belong to them. Both are not polynomial in E's. If I seek a representation of Holonomy-Flux algebra, the two constraints will not be represented. Is this not a problem?

2. In other reviews (Perez, Ashtekar), HF is simply defined to be the algebra generated by operators of holonomies and fluxes. Holonomies are valued in a Lie group, how do they generate an algebra?

3. some claim that, we can use Wilson loops (trace of holonomies) instead, how is this equivalent to the above?

 

[kakarukeys]

Question Clarified

Nobody helps me?

Holonomy-flux algebra (HF) is the kinematical algebra of operators in LQG and is supposed to be the quantum analogue of the algebra of functions on phase space. But Holonomy-flux algebra (HF) is constructed from the free associative algebra generated by the so called elementary variables: cylindrical functions and fluxes (as well as poisson brackets of fluxes, etc).

See
http://arxiv.org/abs/gr-qc/0302059
http://arxiv.org/abs/gr-qc/0504147

in the second paper, the set of generators is slightly different: it contains variables in this form: $$f(q)$$ and $$g_i(q)p^i$$

The consequence of this construction is that any element of HF is only "polynomial" in flux operators or E's.

The constraints are functions on phase space. So their operators should be in HF, but they are not "polynomial" in E's, both of them contain factors of the square root of determinant of E. So constraint operators do not belong to HF. Is this not a problem?

 

[Entropia]

1. The fact that the Gauss constraint and Hamiltonian constraint do not belong to the Holonomy-Flux algebra (HF) is not a problem. This is because the HF algebra is generated by cylindrical functions and operators linear in momenta, which are not sufficient to represent the constraints. However, this does not mean that the constraints cannot be represented at all. In fact, they can be represented using other methods, such as the Dirac bracket or the Ashtekar formalism. So while they may not belong to the HF algebra, they can still be taken into account in the overall framework of loop quantum gravity.

2. The Holonomy-Flux algebra is generated by operators of holonomies and fluxes, but these operators are not simply defined as elements of a Lie group. Instead, they are constructed from the holonomies and fluxes using the techniques of loop quantum gravity. This means that while the holonomies themselves are elements of a Lie group, the operators generated from them are not the same as the original holonomies. They are modified in a way that allows them to generate the HF algebra.

3. Some researchers use Wilson loops (trace of holonomies) instead of the holonomies themselves to generate the HF algebra. This is because the Wilson loops are easier to work with and can be used to construct the HF algebra in a more straightforward manner. However, this does not mean that the Wilson loops are equivalent to the holonomies. They are simply another way of generating the same algebra. Both the holonomies and Wilson loops can be used to construct the HF algebra and are equivalent in this sense.