Loop quantum gravity

Note that string theory says little about the space in which strings move and vibrate. A relatively new mathematical model known as loop quantum gravity represents an alternate approach in which the rules of quantum mechanics are applied directly to Einstein s description of space and time. In this model, space itself comes packaged in... 

Wikipedia, org, an on-line encyclopedia, offers a very detailed six-part entry on loop quantum gravity written by a loop quantum gravity advocate. A seventh part, written by a string theory advocate, discusses numerous objections to the validity of the loop quantum gravity theory. 

To see how the link polynomials appear in the context of loop quantum gravity we use the following equations. 

There are two pictures in loop quantum gravity [25]. The first one is called the Ahstekar connection representation and the second one is called the loop representation. A functional ( A ) in the connection representation, being A an Ahstekar connection and a functional P( L ) in the loop representation, being L a link, are related by the loop transform whose kernel is the trace of the holonomy of the Ahstekar connection, it is to say the following Wilson loop for the circulation of A on L [25]... 

It is well known that (17) is justly the Witten partition function for the Chern-Simons theory with gauge group SU(2) and that the final result is T (L) = (A,B,J), where (A,B,J) is the Kauffman bracket defined by (1) it is to say that the Kodama state is the loop representation is the Kauffman bracket being the skein variable A certain function of A, where A is the cosmological constant [25]. All these show how the link polynomial naturally arise as non-perturbative physical states in loop quantum gravity. 

In loop quantum gravity the relation between the Jones polynomial and the Kauffman bracket given by (3), is rewritten in the form [6]... 

With all that is very interesting consider the possibility to have a computational model for which the gravitational field is the computer. This possibility can be explored both from the point of view of the general relativity as from the perspective of quantum gravity or more concretely from the optics of the loop quantum gravity. Here we introduce a computational model which is named gravitational topological quantum computation and we expect that our model results relevant for computer science both from the side of complexity theory as from the side of computability theory. 

It is based on the equations (13)-(21) which are showing the role of the Kauffman bracket in loop quantum gravity. 

This gravitational topological adiabatic quantum algorithm, constitutes an example of a quantum adiabatic speedup that relies on the topological order that it is inherent to the space-time in loop quantum gravity. 

Peterson, I. (1998) Loops of gravity calculating a foamy quantum space-time. Science News.]nne 13, 153(24) 376-77. 

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