Mike (kneemo) paper on hol-flux algebra

[marcus]

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-Schrödinger 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.

 

[kneemo]

Bravo, Mike! that was quick work on a timely topic, and will undoubtably be noticed.

Thanks Marcus!

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 $$\mathfrak{h}_N(\mathbb{C})$$. The relevant automorphism group is then $$SU(N)$$. For the case of $$\mathfrak{h}_2(\mathbb{C})$$, we thus recover the automorphism group $$SU(2)$$, which is relevant for $$SU(2)$$ 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

 

[R0man]

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!