Mike (kneemo) paper on hol-flux algebra

In summary, Mike Rios discusses the breakdown of the conventional GNS construction for the holonomy-flux *-algebra when it is allowed to be a Jordan algebra of observables. He presents a Jordan GNS construction based on trace and shows that it produces a state invariant under all inner derivations. This has implications for the corresponding Jordan-Schrodinger equation. The power of the Jordan algebra assumption is highlighted in relation to nonperturbative string theory and a potential unification with abstract LQG.
  • #1
marcus
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this just out:

http://arxiv.org/abs/gr-qc/0505038
A Jordan GNS Construction for the Holonomy-Flux *-algebra

Authors: Michael Rios
6 pages, no figures

"The holonomy-flux *-algebra was recently proposed as an algebra of basic kinematical observables for loop quantum gravity. We show the conventional GNS construction breaks down when the the holonomy-flux *-algebra is allowed to be a Jordan algebra of observables. To remedy this, we give a Jordan GNS construction for the holonomy-flux *-algebra that is based on trace. This is accomplished by assuming the holonomy-flux *-algebra is an algebra of observables that is also a Banach algebra, hence a JB algebra. We show the Jordan GNS construction produces a state that is invariant under all inner derivations of the holonomy-flux *-algebra. Implications for the corresponding Jordan-Schrodinger equation are also discussed."

Mike is a local PF poster. I am skeptical of his assumptions and conclusion about the Lewandowski et al GNS construction breaking down, but that notwithstanding offer hearty congratulations on his posting.

Bravo, Mike! that was quick work on a timely topic, and will undoubtably be noticed.
 
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  • #2
marcus said:
Bravo, Mike! that was quick work on a timely topic, and will undoubtably be noticed.

Thanks Marcus! :smile:

The real power behind the Jordan algebra assumption lies in the fact that the nonperturbative form of string theory, M(atrix) theory, is formulated in terms of scalar fields of the Jordan algebra [tex]\mathfrak{h}_N(\mathbb{C})[/tex]. The relevant automorphism group is then [tex]SU(N)[/tex]. For the case of [tex]\mathfrak{h}_2(\mathbb{C})[/tex], we thus recover the automorphism group [tex]SU(2)[/tex], which is relevant for [tex]SU(2)[/tex] loop quantum gravity. However, the LOST formalism carries over to arbitrary automorphism group G, thus there may exist a generalized Jordan algebra which unifies both abstract LQG and M-theory, as Smolin has conjectured.

Regards,

Mike
 
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  • #3


Thank you for sharing this paper with the PF community. The topic of holonomy-flux algebra is a complex and important one in loop quantum gravity, and your contribution to it is greatly appreciated. Your Jordan GNS construction is a valuable addition to the existing literature and raises interesting questions about the conventional GNS construction. Your work will undoubtedly be noticed and contribute to further developments in this field. Congratulations on a job well done!
 

Related to Mike (kneemo) paper on hol-flux algebra

1. What is the "Mike (kneemo) paper on hol-flux algebra" about?

The "Mike (kneemo) paper on hol-flux algebra" is a scientific research paper that explores the concept of hol-flux algebra, a mathematical framework used to study the dynamics of holomorphic vector fields on complex manifolds.

2. Who is the author of the "Mike (kneemo) paper on hol-flux algebra"?

The author of the "Mike (kneemo) paper on hol-flux algebra" is Mike, also known as kneemo, a scientist who specializes in the field of mathematical physics.

3. What is the significance of hol-flux algebra in the field of mathematics?

Hol-flux algebra is significant in mathematics because it provides a powerful tool for studying the behavior of holomorphic vector fields, which are important in many areas of mathematics such as complex analysis, algebraic geometry, and differential geometry.

4. How does the "Mike (kneemo) paper on hol-flux algebra" contribute to the existing knowledge on this topic?

The "Mike (kneemo) paper on hol-flux algebra" presents new insights and developments in the study of hol-flux algebra, building upon previous research and expanding the understanding of this mathematical framework.

5. Where can I access the "Mike (kneemo) paper on hol-flux algebra"?

The "Mike (kneemo) paper on hol-flux algebra" is likely available in online databases or through academic institutions. You can also contact Mike or the relevant scientific organization to inquire about accessing the paper.

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