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kakarukeys

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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 [tex](1,q_i,p^i)[/tex]. I generate a Lie algebra using all [tex]f(q)[/tex] and [tex]g_i(q)p^i[/tex]. [tex]f(q)[/tex] is seen as multiplication operator, [tex]g_i(q)p^i[/tex] is seen as derivation on space of [tex]f(q)[/tex] (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?

my understanding:

in quantum mechanics the elementary variables are [tex](1,q_i,p^i)[/tex]. I generate a Lie algebra using all [tex]f(q)[/tex] and [tex]g_i(q)p^i[/tex]. [tex]f(q)[/tex] is seen as multiplication operator, [tex]g_i(q)p^i[/tex] is seen as derivation on space of [tex]f(q)[/tex] (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?

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