Blog Entries: 4

## Intuitive content of Loop Gravity--Rovelli's program

Well, given that my knowledge in differential geometry is rather poor, I've ordered the book "Differential geometry" of Schaum to Amazon. Hope that will be a good book, like all the other Schaum books that I've read
A question: Then the SO(2) connection used like a variable in Ashtekar's general relativity is a real connection or a complex connection? There are papers that say that is real and others that is complex. I'm dying in the doubt
I've just read that loop quantum gravity violates the "weak energy condition" at short distances, when the granularity of spacetime becomes significant. I've don't have the foggiest idea of what is the weak energy condition, so I'm going to read about it right now

 Recognitions: Gold Member Science Advisor several people have expressed interest in the SciAm January 2004 article by Lee Smolin, "Atoms of Space and Time" The complete article is probably worth a visit to your local public library. It is written for general audience but manages to give a fairly clear picture of the field and how it developed. Here is an exerpt, as a sample, from the section where Smolin is describing how he and some others got started: -------quote page 68---- ...In the mid-1980s a few of us...Ashtekar...Jacobson...Rovelli...decided to reexamine the question of whether quantum mechanics could be combined consistently with general relativity using the standard techniques. We knew that the negative results from the 1970s had an important loophole. Those calculations assumed that the geometry of space is continuous and smooth, no matter how minutely we examine it, just as people had expected matter to be before the discovery of atoms. Some of our teachers and mentors had pointed out that if this assumption was wrong, the old calculation would not be reliable. So we began searching for a way to do calculations without assuming that space is smooth and continuous. We insisted on not making any assumptions beyond the experimentally well tested principles of general relativity and quantum theory. In particular we kept two key principles of general relativity at the heart of our calculations. The first is known as background independence. This principle says that the geometry of spacetime is not fixed. Instead the geometry is an evolving, dynamical quantity. To find the geometry, one has to solve certain equations that include all the effects of matter and energy. Incidentally, string theory, as currently formulated, is not background independent; the equations describing the strings are set up in a predetermined classical (that is, nonquantum) spacetime. The second principle, known by the imposing name of diffeomorphism invariance, is closely related to background independence. This principle implies that, unlike theories prior to general relativity, one is free to choose any set of coordinates to map spacetime and express the equations. A point in spacetime is defined only by what physically happens at it, not by its location according to some special set of coordinates... ...By carefully combining these two principles with the standard techniques of quantum mechanics, we developed....[the means]...to do a calculation... That calculation revealed, to our delight, that space is quantized. We had laid the foundations of...loop quantum gravity... ------end of exerpt-----

Recognitions:
Gold Member
 Originally posted by ranyart Amazing.. http://uk.arxiv.org/PS_cache/gr-qc/pdf/0312/0312103.pdf
thanks for the lead, ranyart! a Loop Quantum Cosmology
article. I will have a look. In case anyone wants the abstract:

http://uk.arxiv.org./abs/gr-qc/0312103

Martin Bojowald and Kevin Vandersloot
"Loop Quantum Cosmology and Boundary Proposals"
invited talk at the 10th Marcel Grossman meeting July 2003
18 pages, 5 figures

edit: this turned out to be more than the title suggests.
there are 45 references (it's a mini-survey article)
a thumbnail sketch of LQG and quick review of current work
in the general theory, not limited to cosmology
followed by another concise review of current Loop cosmology
developments
then, on pages 10-12, they present their results
relating to cosmological boundary conditions
(citing and comparing work of Hartle/Hawking and of Vilenkin)
finally, pages 12-15, they discuss open questions having to do with behavior around the cosmological singularity or bounce, and graph some results of calculation around the bounce.

it is an interesting paper from several standpoints---
for instance what they choose to emphasize in the overview of the general field: on page 4 at the top, they cite two papers by Sahlmann, another two by Sahlmann/Thiemann, and one by Lewandowski/Okolow.
the view of Loop gravity is on the abstract side, through the window of "representations of the classical algebra"

the paper connects to history by citing Hartle and Hawking "Wave Function of the Universe" (1983) and Vilenkin "Quantum Creation of Universes" (1984) and pointing out the central long-standing concern with cosmological boundary conditions in the Wheeler-DeWitt quantum cosmology model.

they use the algebraic representation-theory raised earlier, carried over and specialized to cosmology, to say how and why Loop quantum cosmology differs from vintage 1980s (Wheeler-DeWitt) quantum cosmology-----different Hilbertspace, different operators, discrete spectra instead of continuous---references to the Bohr compactification and the Stone-von Neumann theorem at bottom of page 6.

then starting on page 7 they focus on the dynamics of loop quantum cosmology---the Hamiltonian constraint and difference equation that determines evolution around the bounce---and
show how the loop model matches up with Wheeler-DeWitt: in effect has the right limiting behavior (see for example Figure 1).
They also discuss ways the modern theory differs from the vintage model
(eliminates the singularity, provides for varying degrees of inflation depending on assumptions, and gives rise to somewhat different boundary conditions, or to similar ones in a different way)

this paper ties a number of threads together.
the other research currents it draws on and connects to
are as significant as the research results
it relates current research in loop cosmology with the historical antecedents (connects it to Wheeler/DeWitt/Hawking/Hartle/Vilenkin) by addressing issues that were traditionally central to earlier work

and it points up linkage between the specialized field of loop cosmology and the algebraic approach to the broader field of LQG associated with Ashtekar/Lewandowski/Thiemann/Sahlmann

the bibliography is extensive and up to date, as you might want from a "mini-survey". For example, the Husain/Winkler "On Singularity Resolution" paper that ranyart just found posted a couple of days ago on arxiv (gr-qc/0312094) is their reference 25.

an earlier brief overview of loop cosmology this year
http://arxiv.org./abs/astro-ph/0309478
is only 6 pages and less abstract
that is Bojowald's
"Quantum Gravity and the Big Bang"
it is less hilbert spacey but gives a quick idea of what
the field is about and how the calculations are done
both papers are good, just different introductions to the same thing

 Marcus I know someone asked for a good link for loop Quantum Gravity? I presume you have this link somewhere?..if so I can always delete it: http://arxiv.org./abs/gr-qc/0306008 I will post just to the Abstract as I automatically link directly to pre-print papers, but maybe the abstract is more usual practice.

Recognitions:
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 Originally posted by ranyart I presume you have this link somewhere?..if so I can always delete it: http://arxiv.org./abs/gr-qc/0306008
"Cosmological applications of loop quantum gravity"

I know the paper but I didnt have the link handy, not handy to this thread anyway. Thanks for mentioning it. Bojowald co-authored that with Hugo Morales-Tecotl, in Mexico City. It's good because it is introductory, part of a loop gravity seminar taught for undergrad and grad students. I read somewhere that Rovelli was Morales-Tecotl's thesis advisor, which makes me think that Morales-Tecotl is also a young person too, like Bojowald----recent PhD or recent postdoc.

 I will post just to the Abstract as I automatically link directly to pre-print papers, but maybe the abstract is more usual practice.
Whichever you prefer! We can provide links to either abstract and full text. I don't know that one is more useful or usual than the other. I always look at the abstract first because a long PDF download ties up my computer and the abstract tells me how many pages.

Right now I feel a bit to lazy to bother but probably all these links should be gathered in a list-----or two lists: one for the full theory and one for the specific application to cosmology. It gets tedious playing librarian but it is actually easier than having to go through piles of paper on my desk. Thanks again for contributing these good links!

 Recognitions: Gold Member Science Advisor Loop gravity is a planck-scale theory and planck units have a special place in it. This month the National Institute of Standards and Technology (NIST) posted new values for the basic planck units http://physics.nist.gov/cuu/Constants/ choose "universal" from the menu to find (among other things) the 2002 CODATA recommended values of planck mass planck length planck time planck temperature the uncertainties have been reduced by an order of magnitude since the values of planck units were posted in the 1998 CODATA set. Maybe this is no big deal but it is nice that the natural units for Loop Gravity are gradually beginning to look more like a recognized system of units A good article on timekeeping, discussing GR effects on the GPS http://www.allanstime.com/Publicatio...imekeeping.pdf
 Recognitions: Gold Member Science Advisor Giovanni Amelino-Camelia, Jerzy Kowalski-Glikman, and two others "Phenomenology of Doubly Special Relativity" dated 30 December 2003 (recent) about 22 pages http://arxiv.org/gr-qc/0312124 Giovanni A-C is the most eminent person in quantum gravity phenomenology and I believe the fastest riser is Jerzy K-G. QG Phenomenology is a hot field with a lot of recent papers---both theoretical and observational. The theoretical part says what are the various quantum gravity models and what (in the case of those models that actually predict and can be tested by possible observation) do they predict and how---with planned space observatories etc---can they be tested. Like pruning a tree, the observationalists can do the loop gravity/spin foam theorists a favor by chopping off the bad branches (that actually make testable predictions but the predictions are wrong). So there is growing interest and visibility for this QG Phenomenology business. And so when Giovanni A-C and Jerzy K-G get together on a paper and give the latest word on the subject it is apt to be worth paying some attention. So I posted it. I thought it was. It is probably time to gather the links in this "surrogate sticky" thread into a single post----there are enough links now so they are too spread out
 Recognitions: Gold Member Science Advisor Loop Gravity is a theory under construction, so Rovelli's Chapter 7 "Dynamics and Matter", pages 199-212, is describing work in progress. Section 7.1 discusses the hamiltonian and 7.2 the inclusion of matter. Table 7,1 on page 208 gives "Quantum numbers of the spin network states for gravity and matter." The graph $$\Gamma$$ with N nodes and L links, is like a big quantum number describing adjacency. Nodes correspond to regions or chunks of space and links to the surfaces between those volumes $$\Gamma$$ adjacency $$i_n$$ volume of node n $$j_l$$ area of surface l $$F_n$$ number of fermions at node n $$S_n$$ number of scalars at node n $$w_n$$ field strength at node n $$k_l$$ electric flux across surface l In section 7.2.4 "The quantum states of space and matter", notation is given for |s> a quantum state of space and matter. As one has come to expect, quantities like volume/area, fieldstrength/flux appear as irreducible representations/intertwiners. More details about this on pages 208 and 209. "thus we can write $$|s> = |\Gamma,i_n,j_l,F_n,S_n,w_n,k_l>$$ This state describes a quantum excitation of the system that has a simple interpretation as follows. There are N regions n, that have volume and where fermions and Higgs scalars can be located. These are separated by L surfaces l, that have area and are crossed by flux of the (electric) gauge field. The quantum numbers are related to observable quantities as in Table 7.1. This completes the definition of the kinematics of the coupled gravity+matter system." the next section, 7.3 "Matter: dynamics and finiteness" writes the hamiltonian compounded of four pieces. $$H = H_{Einstein} + H_{YangMills} + H_{Dirac} + H_{Higgs}$$ One brief exerpt from Section 7.3, "...The fact that the total hamiltonian turns out to be finite is extremely remarkable. It is perhaps the major payoff of the background independent quantization strategy on which LQG is based..." For the finiteness result Rovelli cites "Lectures on Quantum Gravity" http://arxiv.org/gr-qc/0210094 these are notes at the grad student level prepared by Thomas Thiemann, which are to appear in a textbook series called "Lecture Notes in Physics" (Springer, Berlin) and also "Quantum Gravity as the Natural Regulator of the Hamiltonian Constraint of Matter Quantum Field Theories" http://arxiv.org/gr-qc/9705019 It is a bit of luck that quantizing space makes the ordinary infinities of QFT go away:"...the ultraviolet divergences of ordinary quantum field theory can be directly interpreted as a consequence of the approximation that disregards the quantized, discrete, nature of quantum geometry..."