• kakarukeys
In summary, the Holonomy-flux algebra (HF) is constructed from the quantum operators of holonomies and fluxes, which are generated by cylindrical functions and derivatives of fluxes. However, the constraints do not belong to HF as they are not polynomial in the flux operators. This creates a problem in seeking a representation of the Holonomy-flux algebra. In other reviews, HF is defined as the algebra generated by holonomies and fluxes, but it is unclear how holonomies, which are valued in a Lie group, can generate an algebra. Some suggest using Wilson loops, which are traces of holonomies, as an alternative, but it is not clear how this is equivalent to the original definition of HF.
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?

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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?

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.

## 1. What is holonomy-flux algebra?

Holonomy-flux algebra is a mathematical framework used in theoretical physics to describe the behavior of gauge fields, which are fundamental forces such as electromagnetism and the strong and weak nuclear forces. It combines concepts from differential geometry, topology, and algebra to study the relationship between the curvature of a space and the flux of a field through that space.

## 2. How is holonomy-flux algebra related to string theory?

Holonomy-flux algebra is an important tool in string theory, which is a theoretical framework that attempts to reconcile quantum mechanics and general relativity. In string theory, holonomy-flux algebra is used to describe the behavior of branes, which are extended objects that play a crucial role in the theory.

## 3. What are some applications of holonomy-flux algebra?

Holonomy-flux algebra has a wide range of applications in theoretical physics. It has been used to study the physics of black holes, to understand the dynamics of quantum fields in curved spacetime, and to explore the behavior of topological phases of matter.

## 4. How does holonomy-flux algebra relate to other mathematical concepts?

Holonomy-flux algebra is closely related to other mathematical concepts such as Lie algebras, differential forms, and differential cohomology. It also has connections to other areas of physics such as quantum field theory and general relativity.

## 5. Are there any current research developments in holonomy-flux algebra?

Yes, there is ongoing research in holonomy-flux algebra, particularly in its applications to string theory and quantum gravity. Some current topics of interest include the use of holonomy-flux algebra to study topological phases of matter in condensed matter systems, and its role in understanding the holographic principle in black hole thermodynamics.

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