The Immirzi Rebellion in Loop quantum gravity

[Martin]

Non-string quantum gravity is already a minority research line compared with string----although it is a growing one.

 

But even that minority is now split over the issue of the Immirzi parameter.

 

Some of the big names in LQG, people who head institutes and guide research, where the grad students and postdocs work on LQG. Have signed on to a value of the Immirzi parameter 1/4.21 (that is, 0.23753...) calculated by Christoph Meissner in Warsaw, with the help of Jerzy Lewandowski.

 

but Lee Smolin and a young researcher at Perimeter Institute named Olaf Dreyer say that this key number is actually 1/8.088

 

about 8.088... about one over that.

 

Lee Smolin has voiced the opinion that the value calculated by Meissner and Lewandowski and others represents deep trouble for LQG. His reasons for preferring 1/8.088 are clear enough that I think I can reproduce the gist---they have to do with the vibration frequencies of black holes and with their entropy. And they are explained in this paper

http://arxiv.org/hep-th/0409056

 

In this paper the splinter group, Fotini Markopoulou, Lee Smolin, Olaf Dreyer, is flipping a challenge to the majority (Ashtekar, Lewandowski...)

 

the strongest statement of it is on page 5, third paragraph from the bottom

 

I should try to say why this number is important.

Another post maybe. It basically tells how big the AREA of surfaces turn out to be measured in the natural area unit of Planck area, if you compare it with the humongous network which is a quantum state of the shape of the universe.

 

since from our standpoint we know roughly how many planck units of area are on any surface---I can tell you, e.g., for a football if you tell me diameter in centimeters

 

and because the area is essentially proportional to how many times the universe-network pokes thru it.

 

then this number is telling us how fine the typical network is.

 

it is so fine that on average every patch of area which is 3 planck units has one link of the network passing thru it. (Just a rough estimate) here's how this 3 comes from the 8.088 :

 

[math]3 = \frac{8\pi}{8.088}\text{planck area}[/math]

 

the warsaw people say the network that describes space is slightly coarser with less than one network link passing thru per 3 plank units of area.

 

the contest is between 4.21 and 8.088, or their reciprocals

seems like a dumb thing to quarrel about but it is connected to some interesting things like the ringing frequencies of black holes, and stuff about the entropy which Hawking discovered, so they are prob ably right to get excited about it

[Martin]

I am trying to see how to put this business about Immirzi in a clear simple way and it dawns on me that in Gen Rel and quantum gravity the Planck area always appears with 8 pi in front. In fact the newton grav. const. G most often appears with 8 pi in front----the way nature likes it I guess.

 

So nature is trying to tell me about a little bit of area which is 8 pi times the conventional planck area. I need a symbol for it so I will call it with the double-hook Greek cap Upsilon and it comes out, in square meters, to be

 

65.65 x 10^-70 m²

 

in other words

[math]\Upsilon = 8\pi\text{ planck area}[/math]

 

or in terms of fundamental constants c, G, hbar, it is

 

[math]\Upsilon = 8\pi\frac{G\hbar}{c^3}[/math]

 

the contest between 4.21 and 8.088, or their reciprocals

is over who gets to divide this natural unit of area down, because

in quantum gravity even this very small bit of area is too large and

it has to be shrunk by a factor of like 4 or 8 and THEN it corresponds to

a spin-network needling thru the surface

 

in quantum gravity the quantum state of the geometry of the universe is a humongous colored network and the network gives area to a surface by how many of its segments poke thru the surface

each poke-thru gives it one bit of area and the area is this

65.65E-70 sq. meters but scaled down by an immirzi number

[Martin]

In Quantum Gravity, area is measured in "pinpricks" and to oversimplify and put it very crudely each pinprick is worth

 

65.65 x 10^-70 m²

 

And it would be sooooooo nice if it were that simple but there is this damned parameter called immirzi so that roughly speaking each pinprick is worth either

 

 

 

[math]\frac{1}{8.088}65.65 \times 10^{-70} m^2[/math]

 

or

 

[math]\frac{1}{4.21}65.65 \times 10^{-70} m^2[/math]

 

AND WE DONT KNOW WHICH IS RIGHT

 

and Abhay Ashtekar says divide by 4.21 and Lee Smolin says divide by 8.088, both are bigleague quantum gravity, so it's a tossup

[Martin]

It does give a notion of how intense the spin-network quantum state of the universe is because it is made of jillions of nodes connected with links so tiny that a square meter area where you are standing has

on the order of 10⁷⁰ links passing thru it.

 

the geometry of the universe, which is this network, gives area to things by how much it punctures them, and a square meter of area is something it punctures a lot.

it has to puncture it about 10⁷⁰ times because

each puncture only contributes about 10^-70 sq. meter of area

 

I picture this network as like the fine suds in the sink when we used to wash dishes (instead of putting them in the dishwasher)

it was this fine foam of millions of bubbles

 

and I picture putting a dot or node in the middle of each bubble

and connecting two nodes if their two bubbles touch

and in this way I replace the foam by a kind of network a bit like

a 'ball-and-stick' model of a complex polymer molecule

but with millions of atoms corresponding to the original bubbles

 

and this I picture describing the geometry of the universe and extending everywhere-----the nodes carry volume info and the "legs" or links of the net carry area info.

[Martin]

Reminder: the basic 65.65E-70 square meter pinprick area is

[math]\Upsilon = 8\pi\text{ planck area}=8\pi\frac{G\hbar}{c^3}[/math]

 

What we are doing in this thread is mainly just reading

http://arxiv.org/hep-th/0409056

by Olaf Dreyer, Fotini Markopoulou, and Lee Smolin.

the essential idea is by Olaf Dreyer and he wrote about it in 2002 (see reference in this paper) but Lee and Fotini seem to have taken it up.

(this is good because Olaf is just a postdoc at Perimeter Inst.

and they add weight)

 

Here is their equation for the Immirzi gamma number, which if you solve it with your calculator will give 1/8.088. the equation is

 

[math]\sqrt {2} \gamma = \frac{\ln 3}{2 \pi}[/math]

 

 

Their paper says that when the area of a Schwarz. horizon changes by gaining or losing one puncture by the network then (by the all-righteous LQG area formula) it gains or loses this amount of area

 

[math]\sqrt {2} \gamma \times 65.65 \times 10^{-70} m^2[/math]

 

But by Hod and others' classical analysis of making the Schwarz. solution go boing, and the Bohr corresp. principle, it gains or loses this amount of area

 

[math]\frac {\ln 3}{2\pi} \times 65.65 \times 10^{-70} m^2[/math]

 

You can see where the equation comes from, just putting the two gained-or-lost areas equal. they are two different expressions for the "delta A"

 

Now the squareroot 2 has a hidden message here, about the coloring of a spin network (at least inside a black hole).

the spin network---humongous ball and stick molecule model with very tiny nodes joined by very tiny links---has its links COLORED with "spins" which are halfintegers: 1/2, 1, 3/2, 2, 5/2,....and so on.

 

If a surface is allowed to be intersected by links colored 1/2 then the area of the surface has a squareroot 3/4 term, but if it can only be crossed by links colored 1 and higher (no links colored 1/2) then the area formula has a squareroot 2 term instead.

 

the algebra for this is

[math]\sqrt{p(p+1)}[/math]

where p is the least spin allowed, and you can see that putting

p=1/2 you get squareroot 3/4, while putting p=1 give squareroot 2.

this is an obscure technicality at the heart of how areas of surfaces are

calculated using the spin networks that roger penrose invented and

the Loop people made into quantum states of gravity.

 

Olaf and the others are saying something about the spin-color of a network inside a black hole, or at least at the event horizon. he is saying that the spins there can only be ONE they cannot be one-half.

 

now our job is to connect what has already been said to the Bekenstein Hawking black hole entropy formula

[[Tycho?]]

I love it when Martin respondes to his own posts, and nobody else does.

 

All of this is well above my current level of education.

[Martin]

Reminder: the basic 65.65E-70 square meter Upsilon area is

[math]\Upsilon = 8\pi\text{ planck area}=8\pi\frac{G\hbar}{c^3}[/math]

 

What we are doing in this thread is mainly just reading

http://arxiv.org/hep-th/0409056

by Olaf Dreyer, Fotini Markopoulou, and Lee Smolin.

 

 

Here is their equation for the Immirzi gamma number,

 

[math] \gamma = \frac{\ln 3}{2 \pi \sqrt 2} = \frac{1}{8.088...}[/math]

 

 

Their paper says that when the area of a Schwarz. horizon changes by gaining or losing one puncture by the network then it gains or loses this amount of area

 

[math]\sqrt {2} \gamma \Upsilon = \sqrt {2} \gamma \times 65.65 \times 10^{-70} m^2[/math]

 

NOW FOR THE HAWKING ENTROPY FORMULA

 

 

well, Hawking formula relates entropy S to horizon area A, and the question is WHAT AREA DO YOU HAVE TO DIVIDE A BY TO GET S?

S is a number so to get S you have to divide horizon area A by some area, nothing else will work.

 

hawking says divide A by four times the conventional planck area

 

[math]S = \frac{A}{4\text{ planck area}}[/math]

 

[math]S = \frac{2\pi A}{8\pi \text{ planck area}}[/math]

 

[math]S = \frac{2\pi A}{\Upsilon}[/math]

 

this is just another version of the Hawking entropy formula where we divided by the pinprick area and then compensated by multiplying by 2 pi.

 

Now we need to check that the Olaf Dreyer version of the Immirzi gets the correct Hawking entropy formula. Their assumption is that at each puncture the spin p =1 so that the dimension of the little microstate hilbertspace there is 2p+1 = 3

The entropy is the logarithm of the dimension of all the hilbertspaces collectively so it is the number of punctures N multiplied by ln 3.

 

Here is Olaf's gamma

[math] \gamma = \frac{\ln 3}{2 \pi\sqrt {2}}[/math]

each puncture contributes this much area

[math]\sqrt{2} \gamma \Upsilon = \frac{\ln 3}{2 \pi}\Upsilon[/math]

 

We can use that to learn the number of punctures in

a horizon of given area A.

 

[math] N = \frac{A}{\sqrt {2}\gamma \Upsilon}[/math]

[math] N = \frac{A}{\frac{\ln 3}{2 \pi} \Upsilon}[/math]

 

[math]S= N\ln3 = \frac{2\pi A}{\Upsilon}[/math]

 

and that's the right entropy formula

[Dave]

']I love it when Martin respondes to his own posts' date=' and nobody else does.

 

All of this is well above my current level of education.[/quote']

 

Same here. But it makes for very interesting reading anyway. Since I may be doing physics modules in the next few years, I think I may take note of some of this.

[bloodhound]

i am doing some quantum mech modules. dont know if i am doing any serious physics. oh well they should at least be an interesting exercise in solving partial differential equations.

[Martin]

...to get S you have to divide horizon area A by some area' date=' nothing else will work.

 

hawking says divide A by [b']four times the conventional planck area[/b]

 

[math]S = \frac{A}{4\text{ planck area}}[/math]

 

[math]S = \frac{2\pi A}{8\pi \text{ planck area}}[/math]

 

[math]S = \frac{2\pi A}{\Upsilon}[/math]

 

this is just another version of the Hawking entropy formula...

 

Hello Dave and Bloodhound, If you are simultaneously doing physics modules and reading this then please be careful and avoid being misled by notation I am using for my own convenience. I know you will but want to explicitly say this anyway: The real Hawking formula is:

 

[math]S = \frac{A}{4\text{ planck area}}[/math]

 

That is all well and good. You can see I've tinkered trivially with it and introduced a private symbol Upsilon for the Planck area multiplied by 8 pi.

If you want to connect this with a standard physics module you should clean out any idiosyncratic notation.

 

I have noticed more and more people using a slightly unconventional set of planck units where instead of making G the unit they make 8 pi G the unit.

this is essentially because in the main equation of GR you have

an 8 pi G instead of a simple G.

 

If you set hbar=c=8piG=1 then there is no constant in the Einstein equation at all! more and more people I see doing this. It is potentially confusing.

Same kind of confusion as when you start out in lowerdivision courses and people use h, and then somewhere along the line they change over and always use hbar.

[5614]

'']I love it when Martin respondes to his own posts, and nobody else does.

 

he seems to like to do it a lot, i cant really add to what he has said, also it seems to be facts rather than questions so there's nothing to answer

[Martin]

he seems to like to do it a lot, i cant really add to what he has said, also it seems to be facts rather than questions so there's nothing to answer

 

hi Tycho and 5614, I also read your posts (esp. 5614 who occasionally does longer more explanation-type) with interest. thanks for checking out these of mine.

 

I think there is an information-scarcity of sorts about non-string QG.

Quantum Gravity is obscure and a bit on the new side.

 

the biggest results are getting rid of the BigBang singularity (2001) and the BlackHole singularity (2004)

everybody should now know that if you take Einstein 1915 Gen Rel and quantize it then two major singularities go away, but almost nobody seems aware of this!

 

it is something of a side issue that this quantization of GR is done in a way that MAY reproduce the famous Hawking 1976 entropy formula and this is still plagued by uncertainty over the exact value of the Immirzi parameter.

 

getting rid of singularities by quantizing, and getting discrete spectra for (quantum) area and volume observables is in some sense a qualitative result that is not bothered by the numerical uncertainty surrounding the Immirzi number. It seems to work whatever that number finally turns out to be.

 

but the Hawking formula is an exact numerical thing, so the value of the Immirzi number becomes an issue

 

in any case QG is making progress and approaching testability and is still pretty much invisible to the public (partly because of the blinding glare of string popularization involving major communication talents like Brian Greene, and partly because it is a latecomer as well, field hasnt been active as long)----so if you want QG popularization you almost have to do it yourself