Covariant Loop Gravity and Livine's Thesis

[marcus]

next step: positive Lambda

It seems likely that the next step in developing Covariant Loop Gravity will be a continuation of Livine's thesis that introduces positive Lambda----positive dark energy, or cosmological constant---into the theory.

The key new idea we need to grasp is that the "quantum hyperboloid becomes a stack of fuzzy spheres"

see first paragraph page 2 of Girelli/Livine "Quantizing Speeds with the Cosmological Constant" gr-qc/0311032
see also their equation (1)

The basis for "Quantizing Speeds..." is what now seems to be a very important paper by Karim Noui and Philippe Roche "Cosmological Deformation of Lorentzian Spin Foam Models" gr-qc/0211109
and in particular page 13 where the hyperboloid of possible speeds is put in terms of some 2x2 matrices---a right coset homog. space SU(2)\SL(2,C) and the Iwasawa decomposition is used to get it as A x N (diagonal and nilpotent) How bad can it be? It is just a few 2x2 matrices of especially simple form

lambda 0
0 lambda^-1

plus

0 0
n 0

the first is the real number diagonal det=1 type
and the second is the complex number n for nilpotent
lower triangular type, the language here is heavier than the matrices

let's see if I can type the sum of those two matrices

[font=symbol]l[/font]  0
n  [font=symbol]l[/font][sup]-1[/sup][/size]

so the hyperboloid of possible speeds, or moving observers, or boosts or whatever, is happily pictured algebraically as some 2x2 matrices

NOW Noui/Roche will tell us how to q-deform them by introducing a cosmological constant.

See page 16 and 17
You will see elegant french style. first the quantum hyperboloid is presented in a fearsomely succinct and categorical way, then in equation (55) one sees that it is simply a stack of spheres made of essemtially the same matrices except the lower left entry, the complex number n, has been multiplied by something EXTREMELY NEAR ONE.

and then presto on the next page there is equation (58) that Girelli and Livine used to see the spectrum of quantized speeds.

The fearsome and succinct definition they give first is something else. They refer to the algebra of compactly supported functions on the quantum hyperboloid as
Fun_c(H_+q) = Fun_c(AN_q)

this is just the Iwasawa decomposition into diagonal (we saw before) and q-deformed nilpotent (here just means lower left nonzero entry)

And they say "therefore as an algebra it has the structure

Fun_c(H_+q)= +_IMat_2I+1(C)

And they say "this description is the deformation of the foliation of H₊ by quantum fuzzy spheres. Quantum fuzzy spheres have been introduced and studied in hep-th/0005273 (Grosse, Madore, Steinacker "Field Theory on the q-deformed Fuzzy Sphere")

At this point my outrage knows no bounds. But what can one do. The speeds that things were traveling at the instant the universe began to expand has according to good authority somewhat to do with q-deformed Fuzzy Spheres. Speeds were quantized. Oh damn the matrix looks the same but the entries are "non-commutative numbers". Oh hell it is awful. It looks like

[font=symbol]l[/font]  0
n  [font=symbol]l[/font][sup]-1[/sup][/size]

except the n has been multiplied by something extremely close to one, namely
√((q²+1)/2)
you can see that since the deformation parameter is very close to one namely like
q = exp(-10^-123) as it is today, then
this square-root thingee is very close to √(2/2) = 1

So the matrix Karim and Philippe (Noui/Roche) give us is

[font=symbol]l[/font]             0
√((q[sup]2[/sup]+1)/2)n  [font=symbol]l[/font][sup]-1[/sup][/size]

 

[marcus]

[font=symbol]l[/font]             0
√((q[sup]2[/sup]+1)/2)n  [font=symbol]l[/font][sup]-1[/sup][/size]

My outrage at the term "Fuzzy Sphere" has subsided and I can think more clearly. Actually this matrix is kind of intriguing. It is
a familiar sensible 2x2 matrix except EXCEPT the numbers in it, the lambda and the n, just barely DO NOT COMMUTE.

right under the matrix on page 16 of Noui/Roche it gives the equations of their non-commutativity and its rather nice. There is this parameter q very very near one. If q were exactly one then the arithmetic would not depend on the order of mults at all. But to the tiny tiny extent that q is not exactly one, the arithmetic depends on the order in which you multiply factors. I'm beginning to think its real cool.

Remember that q is the number e (the base of logs, 2.7...) raised to an unprecedentedly small number 10^-123

one over (one followed by 123 zeros)

this is how we are incorporating dark energy into our local everyday business. the cosmo constant is 10^-123 a number closer to zero than science has ever dealt with so far
and we take that number and we raise e = 2.7... to that power

well e raised to the zero is exactly one

so e raised to something extremely close to zero is extremely close to one-----unprecedentely close to one---I can't think of any number in science that differs from one by that little

and that number is by how much the numbers in the matrix do not commute. and that matrix affects local business---it tells how the speeds around us are quantized.

so suddenly the radius of the cosmological horizon----some 60 billion lightyears---which is basically the square root of that 10¹²³ number---has been flipped over to be a very small 10^-123---and Livine and Girelli are explaining to us that it enters into how speeds around us are quantized in little steps of speed.

and it hinges on the fact that some numbers we compute coordinate framechanges with, numbers which for all practical purposes are ordinary commutative numbers where AxB is the same as BxA, actually do not quite commute. What can I say. It is awesome.

true or not, it is awesome, and it could even be true.

 

[marcus]

Originally posted by Ambitwistor
Good, now you can survive the giant fuzzy moose.

http://arXiv.org/abs/hep-th/0111079

Aiiieee! Oh no! Not the giant Fuzzy Moose!