1. Nov 29, 2006

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?

Last edited by a moderator: Apr 22, 2017
2. Dec 2, 2006

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?