Bianchi Haggard volume spectrum paper puts UC Berkeley on Lqg map

[marcus]

I was glad to see this paper for several reasons. The volume operator in Loop Gravity is the locus of some interesting unresolved questions. The kind that requires and attracts creative mathematicians IMHO.
This first paper from Gene Bianchi and Hal Haggard is just a 4-page letter I guess for PRL They have further work in prep, so we will see how this goes in the follow-up.. It looks like an important paper.

http://arxiv.org/abs/1102.5439
Discreteness of the volume of space from Bohr-Sommerfeld quantization
Eugenio Bianchi, Hal M. Haggard
4 pages, 4 figures
(Submitted on 26 Feb 2011)
"A major challenge for any theory of quantum gravity is to quantize general relativity while retaining some part of its geometrical character. We present new evidence for the idea that this can be achieved by directly quantizing space itself. We compute the Bohr-Sommerfeld volume spectrum of a tetrahedron and show that it reproduces the quantization of a grain of space found in loop gravity."

Both Hal and Eugenio are now at Zakopane QG school--where lectures start tomorrow. The organizers may have to schedule a talk by one or the other about volume. Bianchi is giving a talk, but about spinfoam cosmology, not the volume op. They have mornings free so there is time to organize an unscheduled presentation.

Johannes Brunnemann is already scheduled to talk on the Loop Gravity volume operator, and his work is extensively cited by this Bianchi Haggard letter. There is contradiction---it will be interesting to see how it sorts out.

Oh! The real reason I am inordinately delighted by the appearance today of this paper is that Haggard is a PhD student in the Physics department at UC Berkeley. Completing his thesis this year, I believe. Advisor is Littlejohn. It means that there is some significant Loop Gravity awareness at UCB and beginning signs of activity.

I looked up Littlejohn, because the authors thank Rovelli and Littlejohn for discussions contributing to the paper.
http://physics.berkeley.edu/index.p...gement&act=people&Itemid=312&task=view&id=478
It looks like you can do a Loop Gravity PhD at UC Berkeley now! A way has opened for more motivated smart people to get into the field.

 

[negru]

"significant awareness" is a bit of an overstatement. Littlejohn's research interests barely seem even tangential to LQG, or or QG, or G, and even that only based on his last couple of papers (which are purely focused on some technical details).

 

[marcus]

"significant awareness" is a bit of an overstatement. Littlejohn's research interests barely seem even tangential to LQG, or or QG, or G, and even that only based on his last couple of papers (which are purely focused on some technical details).

Thanks Negru! I'm pleased to have your response. We'll see, won't we?

 

[marcus]

There is no need to argue about this. Littlejohn is just one of many physicists at excellent institutions who have gotten interested in Lqg in the past 3 years. There has been a decline of interest in string (which one sees reflected in top string people getting into other research lines, and in the drop-off in citations to string papers) and an upsurge of activity in Lqg. This is well known and not something to squabble pointlessly about.

It is clear that Littlejohn is significantly aware of Loop or else he would not have gone to visit Rovelli's group at Marseille in November 2008.

Then he brought Haggard along and came back for a second visit February-March 2009.

You don't go visit Rovelli and his postdocs for an extended stay of several weeks if you are not aware of Lqg.

===================

Littlejohn has recently done research on Wigner j-symbols. One paper in 2007, one in 2009, and a third in 2010. He has published one of them in Classical and Quantum Gravity (a common journal for Lqg). Wigner j-symbols are how spinfoam amplitudes are defined and how one calculates Lqg dynamics. He seems to be focusing on the asymptotics of j-symbols, and on methods of approximation.

I live a couple of blocks from campus so it would be easy enough to walk over to Birge Hall and ask him what the spread of his current research interest covers. At this point I am only guessing from online stuff, like his recent papers and the interests listed on his faculty webpage and his visits to the Marseille Lqg research group.

 

[atyy]

Interesting. Here are papers I found by looking at Littlejohn's collaborators and those who've cited his work.

http://arxiv.org/abs/quant-ph/0410105
Computing Spin Networks
Annalisa Marzuoli, Mario Rasetti
(Submitted on 13 Oct 2004)
Abstract: We expand a set of notions recently introduced providing the general setting for a universal representation of the quantum structure on which quantum information stands. The dynamical evolution process associated with generic quantum information manipulation is based on the (re)coupling theory of SU(2) angular momenta. Such scheme automatically incorporates all the essential features that make quantum information encoding much more efficient than classical: it is fully discrete; it deals with inherently entangled states, naturally endowed with a tensor product structure; it allows for generic encoding patterns. The model proposed can be thought of as the non-Boolean generalization of the quantum circuit model, with unitary gates expressed in terms of 3nj coefficients connecting inequivalent binary coupling schemes of n+1 angular momentum variables, as well as Wigner rotations in the eigenspace of the total angular momentum. A crucial role is played by elementary j-gates (6j symbols) which satisfy algebraic identities that make the structure of the model similar to "state sum models", employed in discretizing Topological Quantum Field Theories and quantum gravity. The spin network simulator can thus be viewed also as a Combinatorial QFT model for computation. The semiclassical limit (large j's) is discussed.

http://arxiv.org/abs/0901.1074
Quantum and semiclassical spin networks: from atomic and molecular physics to quantum computing and gravity
V. Aquilanti, A.C.P. Bitencourt, C. da S. Ferreira, A. Marzuoli, M. Ragni
(Submitted on 8 Jan 2009)
The mathematical apparatus of quantum--mechanical angular momentum (re)coupling, developed originally to describe spectroscopic phenomena in atomic, molecular, optical and nuclear physics, is embedded in modern algebraic settings which emphasize the underlying combinational aspects. SU(2) recoupling theory, involving Wigner's 3nj symbols, as well as the related problems of their calculations, general properties, asymptotic limits for large entries, play nowadays a prominent role also in quantum gravity and quantum computing applications. We refer to the ingredients of this theory -and of its extension to other Lie and quantum group- by using the collective term of `spin networks'. Recent progress is recorded about the already established connections with the mathematical theory of discrete orthogonal polynomials (the so-called Askey Scheme), providing powerful tools based on asymptotic expansions, which correspond on the physical side to various levels of semi-classical limits. These results are useful not only in theoretical molecular physics but also in motivating algorithms for the computationally demanding problems of molecular dynamics and chemical reaction theory, where large angular momenta are typically involved. As for quantum chemistry, applications of these techniques include selection and classification of complete orthogonal basis sets in atomic and molecular problems, either in configuration space (Sturmian orbitals) or in momentum space. In this paper we list and discuss some aspects of these developments -such as for instance the hyperquantization algorithm- as well as a few applications to quantum gravity and topology, thus providing evidence of a unifying background structure.

http://arxiv.org/abs/0907.3724
Microscopic description of 2d topological phases, duality and 3d state sums
Zoltan Kadar, Annalisa Marzuoli, Mario Rasetti
(Submitted on 21 Jul 2009 (v1), last revised 7 Sep 2009 (this version, v2))
Abstract: Doubled topological phases introduced by Kitaev, Levin and Wen supported on two dimensional lattices are Hamiltonian versions of three dimensional topological quantum field theories described by the Turaev-Viro state sum models. We introduce the latter with an emphasis on obtaining them from theories in the continuum. Equivalence of the previous models in the ground state are shown in case of the honeycomb lattice and the gauge group being a finite group by means of the well-known duality transformation between the group algebra and the spin network basis of lattice gauge theory. An analysis of the ribbon operators describing excitations in both types of models and the three dimensional geometrical interpretation are given.

http://arxiv.org/abs/1004.1737
Counterexamples in Levin-Wen string-net models, group categories, and Turaev unimodality
Spencer D. Stirling
(Submitted on 10 Apr 2010)
We address the claim that the string-net models of Levin and Wen provide a microscopic Hamiltonian formulation of Turaev-Viro invariants. Using simple counterexamples (group categories) we demonstrate that this correspondence is not true (without modifications to the work of Levin and Wen). The language of the paper is meant to informative and useful to both the condensed matter community and the mathematical physics community. We minimize the use of category theory and explain by analogy with many-body bosonic and fermionic physics and ordinary group representation theory. The main feature of group categories under consideration is Turaev's unimodality. We pinpoint where unimodality should fit into the Levin-Wen construction, and show that the simplest example computed by Levin-Wen fails to be unimodal. Unimodality is straightforward to compute for group categories, and we provide a complete classification at the end of the paper.

 

[marcus]

Wow! much more than I can assimilate at one sitting. A lot of connections with your former physics prof X-G Wen and his collaborator Levin. I note numerous references to Levin-Wen.

btw so pleased to see you've been awarded SA medal. Congratulations!

 

[atyy]

btw so please to see you've been awarded SA medal. Congratulations.

Oh thank you - but do you know how to get rid of it - you know, I want to be able to spout rubbish freely

 

[marcus]

Oh thank you - but do you know how to get rid of it - you know, I want to be able to spout rubbish freely

Go ahead and spout. There may be canny insight and words-of-wisdom hidden in the spouting. You can count on us to sort it out.

 

[marcus]

It seems that UC Berkeley is really on the LQG map. Robert Littlejohn (Nonlinear Dynamics) has THREE PhD students whose work relates to Loop. I mentioned Hal Haggard (and Nadir Jeevanjee, in another thread) earlier and today I learned of Liang Yu, thanks to Jason11 a new PF member! Hal has already one or more Loop articles, one with a member of Rovelli's group. About Liang Yu, he is doing a simultaneous Masters in Math, with Jon Wilkening, and a PhD with Littlejohn. Here are four recent papers. Incidentally two of the authors of the first paper will be attending the Loops 2011 conference next month in Madrid.

http://arxiv.org/abs/1009.2811
Semiclassical Mechanics of the Wigner 6j-Symbol
Vincenzo Aquilanti, Hal M. Haggard, Austin Hedeman, Nadir Jeevanjee, Robert G. Littlejohn, Liang Yu
48 pages, 22 figures
(Submitted on 14 Sep 2010)
"The semiclassical mechanics of the Wigner 6j-symbol is examined from the standpoint of WKB theory for multidimensional, integrable systems, to explore the geometrical issues surrounding the Ponzano-Regge formula. The relations among the methods of Roberts and others for deriving the Ponzano-Regge formula are discussed, and a new approach, based on the recoupling of four angular momenta, is presented. Special attention is devoted to symplectic reduction, the reduced phase space of the 6j-symbol (the 2-sphere of Kapovich and Millson), and the reduction of Poisson bracket expressions for semiclassical amplitudes. General principles for the semiclassical study of arbitrary spin networks are laid down; some of these were used in our recent derivation of the asymptotic formula for the Wigner 9j-symbol."
From the introductory section:
"The 6j-symbol is the simplest, nontrivial, closed spin network (one that represents a rotational invariant). Spin networks are important in lattice QCD and in loop quantum gravity where they provide a gauge-invariant basis for the field.
Applications in quantum gravity are described by Rovelli and Smolin (1995), Baez (1996), Carlip (1998), Barrett and Crane (1998), Regge and Williams (2000), Rovelli (2004) and Thiemann (2007), among others...
...In addition, the 6j-symbol has been taken as a test case for asymptotic studies of amplitudes that occur in quantum gravity (Barrett and Steele 2003, Freidel and Louapre 2003), in which the authors developed integral representations for the 6j-symbol as integrals over products of the group manifold. There have also been quite a few other studies of asymptotics of particular spin networks, including Barrett and Williams (1999), Baez et al (2002), Rovelli and Speziale (2006), Hackett and Speziale (2007), Conrady and Freidel (2008), Alesci et al (2008), Barrett et al (2009), among others. We also mention the works of Gurau (2008),..."
These are all familiar names to anyone who follows Loop gravity research.

As I mentioned earlier, both Littlejohn and Haggard have spent several weeks as visitors at Marseille with Rovelli's LQG group. The Wigner 15j symbol is of vital importance to LQG, in particular for proving that the theory has the right semiclassical limit. The pattern I see in Liang Yu's research is that he is building up to an understanding of the asymptotics of the 15j symbol, but starting with simpler cases: 6j, 9j, 12j, ...

It is possible that his work will be instrumental in answering questions about the long distance (i.e. large j) limit of LQG.

http://arxiv.org/abs/1104.1499
Semiclassical Analysis of the Wigner 9J-Symbol with Small and Large Angular Momenta
Robert G. Littlejohn, Liang Yu
(Submitted on 8 Apr 2011)
"We derive a new asymptotic formula for the Wigner 9j-symbol, in the limit of one small and eight large angular momenta, using a novel gauge-invariant factorization for the asymptotic solution of a set of coupled wave equations. Our factorization eliminates the geometric phases completely, using gauge-invariant non-canonical coordinates, parallel transports of spinors, and quantum rotation matrices. Our derivation generalizes to higher 3nj-symbols. We display without proof some asymptotic formulas for the 12j-symbol and the 15j-symbol in the appendices. This work contributes a new asymptotic formula of the Wigner 9j-symbol to the quantum theory of angular momentum, and serves as an example of a new general method for deriving asymptotic formulas for 3nj-symbols."

http://arxiv.org/abs/1104.3275
Semiclassical Analysis of the Wigner 12J-Symbol with One Small Angular Momentum: Part I
Liang Yu
(Submitted on 17 Apr 2011)
"We derive a new asymptotic formula for the Wigner 12j-symbol, in the limit of one small and eleven large angular momenta. There are two kinds of asymptotic formulas for the 12j-symbol with one small angular momentum. We present the first kind in this paper. Our derivation relies on the techniques developed in the semiclassical analysis of the Wigner 9j-symbol, where we used a gauge-invariant form of the multicomponent WKB wavefunctions to derive new asymptotic formulas for the 9j-symbol with small and large angular momenta. When applying the same technique to the 12j-symbol in this paper, we find that the spinor is diagonalized in the direction of an intermediate angular momentum. In addition, we find that the geometry of the new asymptotic formula for the 12j-symbol is expressed in terms of the vector diagram for a 9j-symbol. This illustrates a general geometric connection between asymptotic limits of the various 3nj-symbols. This work contributes the first known asymptotic formula for the 12j-symbol to the quantum theory of angular momentum, and serves as a basis for finding asymptotic formulas for the Wigner 15j-symbol with two small angular momenta."
From the conclusions, final sentence of paper:
"... Since the Wigner 15j-symbol is used extensively in loop quantum gravity and topological quantum field theory, we suspect that there are deeper, and more geometrical interpretations of these approximate relations of the 3nj-symbol in their various semiclassical limits."

http://arxiv.org/abs/1104.3641
Asymptotic Limits of the Wigner 15J-Symbol with Small Quantum Numbers
Liang Yu
(Submitted on 19 Apr 2011)
"We present new asymptotic formulas for the Wigner 15j-symbol with two, three, or four small quantum numbers, and provide numerical evidence of their validity. These formulas are of the WKB form and are of a similar nature as the Ponzano-Regge formula for the Wigner 6j-symbol. They are expressed in terms of edge lengths and angles of geometrical figures associated with angular momentum vectors. In particular, the formulas for the 15j-symbol with two, three, and four small quantum numbers are based on the geometric figures of the 9j-, 6j-, and 3j-symbols, respectively, The geometric nature of these new asymptotic formulas pave the way for further analysis of the semiclassical limits of vertex amplitudes in loop quantum gravity models."

A third Berkeley PhD student, Nadir Jeevanjee, is involved enough in Loop to be attending the May Loops 2011 conference in Madrid, along with Haggard. I've seen only one paper he has co-authored so far. Here is a facebook page about Nadir. http://www.facebook.com/note.php?note_id=381095338385
I see he is teaching Cosmology and Math Olympiad courses this summer in a Stanford program for gifted students.
In case anyone's curious here is a page or two about Wilkening in the math department:
http://math.berkeley.edu/~wilken/
http://math.berkeley.edu/~wilken/wilkeningCV.pdf