Merry Christmas And Happy Holidays From LetterStream!
This holiday season we want to wish you lots of love, time with family, hot chocolate, and of course… all your mail delivered on time. We are thankful for your support in 2014, and we look forward to serving you well in the coming year as well.
Please note that according to the USPS, most post offices will be closing at 12:00 noon on Christmas Eve (Wed. Dec. 24th) and New Year's Eve (Wed. Dec. 31st). Be sure to drop mail in blue collection boxes by noon on those days as well, as the USPS will be picking up mail early.
See LetterStream's holiday hours below, and click here for a link to the 2015 USPS holiday calendar.


Sitting in the port with a café con leche, we are recovering from yet another super early start and mulling over the journey ahead and the journey we've had already that brings us to this point.
The concept of Shifting Sands began 17 months ago and what a ride it has been so far.
After months of planning we set off from Stoke by Nayland at 5.20am on Wednesday 12 March with blurry eyes but in high spirits. 2683 kilometres later and minus one diff, one prop shaft, two half shafts, and a handbrake (now reinstated) we find ourselves in much the same frame of mind as we anticipate our arrival in Morocco.
In the last few days we've done hours of solid driving with the background noise of Landy humming gently to us through our earplugs and ear defenders…interspersed with the odd bone-shattering clunk to keep us on our toes.
Avoiding the monotony of the motorways we have travelled through some spectacular scenery in France and Spain, the highlight being the beauty of driving over the lunar lit Pyrenees just before midnight feeling like fugitives with the mountains to ourselves. Landy was kind enough to save the broken handbrake until just after our snowy descent from the Tunnel Routier d'Aragnouet-Bielsa. Surprisingly it was not the Pyrenees that marked the coldest point of our journey but Teruel where we stopped for a couple of hours kip in the Landy and later discovered to be one of the coldest towns in Spain – we can certainly corroborate that. Our first glimpses of the Mediterranean overlooking Malaga came this morning, and sitting in the sun right now the extreme cold of our night drive is a distant memory.
There are those who might say that we have got to this point through sheer obstinacy and bloody mindedness and – whilst we won't try and dispute that these are factors – we have also benefited hugely from the advice, support and gifts of many many people and without them we would probably still be sitting at our desks in London. So there are lots of people we would really like to thank.
Starting with Andrew and Jill Rigamonti: thank you for donating your beloved Land Rover (and we hope you are enjoying the space we have created in your barn). We promise to look after her.
Thanks Henry Morshead for spending a whole weekend sharing your encyclopedic Land Rover knowledge with us, for getting under the vehicle with a hammer, and for lending us useful tools and parts. David Pledger, thank you for putting hours of work into getting her shipshape in the final couple of days. Thank you to Duncan, Jim and Richard at Britcar for helping us out with so many parts and good advice. And a huge thanks to all of the above for your help in resolving the matter of the differential that broke in France on the way down…our GILB Construction hi-vis vests came in handy at this point too so thank you for those Harry.
Big thanks to the gorgeous Mark Rigamonti for building our bed and then returning at the last minute to make us some curtains. Thanks also to Ian Murray who surprised us with two parcels full of maps covering our entire route kindly donated from his private collection – it's thanks to you we haven't got lost yet. Thank you Dan Rigamonti for your socket set, Sheba for your towel, Kevin Knott for the varnish and paint brush, and Mrs Jefferies for lending us your trowel – we are not looking forward to needing to use it but loos are few and far between in the desert so this is an important piece of kit.
Closer to home, thank you Sophie and Andy Vaughan, John and Mary Engleheart and Rowena Mackay for your massive moral support and dedicated re-posting of our tweets and liking of our Facebook posts. A special mention is due to John and Rowena for making so much marmalade that we were able to steal 10 pots – they are going down very well with the people we have stayed with en route to North Africa. Thanks to the Carruth parents for sharing your invaluable logistics knowledge. Huge thanks also to the Engleheart parents for sponsoring our much needed soundproofing, two new tyres and various other items that you probably haven't yet noticed have disappeared…thanks for coping with the Shifting Sands team en masse over the last month or so and all associated mess and disruption.
A big big thank you to Monique and Chantal in Sable-sur-Sarthe, Alice, Charles and Eleanor Loftie in Cazals, and Paco at the Don Gonzalo Hotel in Montilla for putting us up and feeding us on our journey down through Europe. You all went way beyond the call of duty: food parcels, calls to local Land Rover garages, curtain wire, 70 year old fino etc – in fact, we have just realised that apart from the odd baguette we haven't paid for a single meal since we left home. So we are extremely grateful to all of you.
Thank you to everyone on our support web for all your links, advice and contacts as well as the fantastic and knowledgeable people who have responded to our email, Facebook and Twitter requests for advice. We also want to thank the people who have helped but asked not to be mentioned – you know who you are.
And finally to the thousands of people who have listened enthusiastically to us going on and on and on about Shifting Sands at every available opportunity – your good wishes, tact and patience is hugely appreciated.
More posts and pictures to follow over the next few days and months so please keep checking in.
Tracing a map of NAfrica & plotting our route…please shout with ideas of places to go or people to meet on the way!
The Landy has catastrophically failed the team today…it'll be £600 to get her through her MOT. Haven't dared call A-L yet!
Lunchtime in the park in London: sitting in the sun eating falafel and updating our risk assessment.
Bloody hell…has life suddenly become a string of blog posts?!!


Traveling around the world is quite difficult in many ways, but how about if there is one place where you can find all the flavors, traditions, cultures, colors, textures and customs from all the corners of the world? A little of Japan, a fringe of Ukraine, a piece of the US, a part of the Middle East and even a fragment of Romania? Well, this place really exists and is, of course, in the city of records, Dubai. Where everything becomes possible in the shortest time.
Although I spent a whole day in this mini Terra, as I like to call it, I have to admit that it didn't make me feel bored at all, on the contrary. I could easily "travel" to all the 75 countries represented at the Global Village for at least 2 days, especially since I don't have to spend money on expensive airplane tickets, I don't have to waste time on check-in, and it doesn't even require a visa.
With Uber, taxi or rented car. The first two options are similar in price, but I have often noticed that it was more convenient to choose a taxi in Dubai. In the case of the rented car, I can't give you many details because I did not rent myself any car there. However, I found out that the prices are reasonable, and you can rent a decent car at a price starting at $ 30 / day. As for gas, it is well-known that in the Emirates you can make a fairly inexpensive deal. By the way, if your time in Dubai is limited and you want to do as many things as possible in a short time, I recommend you to visit Miracle Garden in the same day, being very close to the Global Village.
By bus. There are several routes to the Global Village, from several subway stations in the city. These are 103, 104, 102 and 106, and the stations from where you can get there are Union Station, Al Ghubaiba Station, Rashidiya and Mall of the Emirates.
Global Village closes in the spring and opens in autumn, meaning that over 150 days a year you will have the chance to cross its gates. Obviously, the main reason is given by the extremely hot temperatures in the hot season. It opens in November and closes in April, but when it is open, it is open every single day, with the mention that Mondays are reserved exclusively for families and women.
Besides this little detail, you also need to know that although in many places in Dubai you can dress exactly as you want, keeping the limit of decentness of course, at Global Village you have to make sure you have all your body covered, if you're a woman, of course. Otherwise, you will not be allowed get inside.
When it comes to tickets, they can be purchased both online and physically and cost 15 AED in both cases. I recommend the first option, so you can avoid the very long queues from the ticket office. Once you have paid the 15 AED, you can enjoy all the pavilions without having to pay for another ticket.
According to statistics, about 5 million tourists come to the Global Village each year, which means an average of over 30,000 people a day. Well, the Global Village is getting very crowded with the evening, honestly.
With more than 23 restaurants, 120 stalls, nearly 30 pavilions with over 70 countries, amusement park, 7D cinema and all kinds of shows, you have unlimited possibilities of spending time at the Global Village. Every week brings new surprises, from spectacular fireworks to fabulous concerts with international artists. This is in addition to what each country has to offer.
At Global Village, I enjoyed Turkish ice cream, watched an Iranian dance with swords, admired a Bollywood mini puppet theater, had fun in the amusement park, gazed at the Persian carpets, entered the mystical world of Saudi Arabia, and tasted dates from there, was happy seeing the flag of Romania, took a look at the Colosseum of Italy, the Eiffel Tower of France and the Sydney Opera House, witnessed a fabulous concert and I made a small incursion into the past of the United Arab Emirates, when Dubai was not even on the map. I admit, Global Village reminded me of Las Vegas.
Therefore, the fun is guaranteed. The shows are generally the Bollywood brand and include acrobatics, dances, and music. All are very colorful and free of charge. Once you have paid the entrance fee, you have free access to all these shows, except for the attractions on Fantasy Island. At the same time, Global Village is the perfect place to buy souvenirs from all over the world, which is great.
Among the countries at Global Village are: Japan, South Korea, Indonesia, Philippines, Algeria, Romania, Serbia, Ukraine, UAE, Turkey, Jordan, Russia, Pakistan, Yemen, China, Spain, Afghanistan, Qatar, Bahrain, Germany, France, UK, Saudi Arabia, India, Egypt, Syria and many more.
Would you like to visit Global Village when you arrive in Dubai?
The Global Village seems like a place I would definitely love to explore, Bella! It has some special charm and it is so lovely. The pictures you provided are astonishing. When's the best time of the year to explore it?
Thank you, Lydia! It's good to explore it any day when it's open, from November to April 🙂 During the summer, due to the very high temperatures, it's closed.


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text:
The genetics of the Book of Mormon
by Orson Scott Card
I keep hearing from ex-Mormons or wavering Saints that their testimony problems come from "science." Yet when I hear which particular scientific point is causing them problems, I realize they're suffering from a serious misdiagnosis.
It isn't "science" that's causing their doubts. Rather, their doubts are causing them to seize on science -- or what they think is science -- as an excuse.
I'll begin with an old example -- John C. Kunich's "Multiply Exceedingly: Book of Mormon Population Sizes" from the June 1990 Sunstone. With a great show of scholarliness, Kunich compares the population figures deducible from the Book of Mormon text and compares them with the estimated global population growth for the period of about .04 percent per year.
Simple arithmetic then shows that the Book of Mormon population could not possibly have derived from the original settlers -- Nephite or Mulekite.
Let's just ignore the obvious fact that an estimated global population growth rate reflecting general trends across centuries is hardly applicable to any local population.
The global rate has to take into account plagues, famines, and devastating wars; the actual growth rate when such things are not happening is always higher.
We hardly needed his elaborate figures to see the biggest problem. Within a single generation of arriving in the new world, the Nephites have enough population to build a temple and to wage war -- and enough women that there is a problem with men practicing polygamy.
Instead of Kunich's misused science, we can derive a much simpler answer from the text itself. Obviously, there were indigenous people with whom both the Lamanites and Nephites mixed. They are not mentioned because this is the story of the people God led to the Promised Land -- not of the locals who joined them.
But such a joining would be almost inevitable. If a high-tech, metal-tool-making, alphabet-writing group entered a population that lacked these skills, the less-advanced group will either accord high prestige to the newcomers -- or slaughter them.
For whatever reason, apparently the Nephites and Lamanites weren't slaughtered. Instead, the populations joined, each one taking on the name of the high-prestige group that led them.
Others have elaborated on other Book of Mormon examples of groups changing their names and forming themselves into new tribes at will. I will, however, point out that seen in this context, Nephi's racism begins to take on a different meaning.
"After they had dwindled in unbelief they became a dark, and loathsome, and a filthy people, full of idleness and all manner of abominations," said Nephi, reporting on his vision of the future of his people (1 Ne 12).
Jacob and Enos echo his attitude (Jacob 3:5, Enos 1:20), and even Mormon, writing in his abridgment of the book of Alma, says, "And the skins of the Lamanites were dark, according to the mark which was set upon their fathers, which was a curse upon them because of their transgression and their rebellion against their brethren" (Alma 3:5-6).
This is Nephi's story, and Mormon is sticking with it. But what was actually going on when, near the end of his life, Nephi wrote his account on the small plates?
Suppose that when the Nephites and Lamanites divided, the Lamanites immediately intermarried (or took concubines) from among the indigenous population. The Lamanites might also have adapted to the indigenous lifestyle -- which makes perfect sense, since the locals already knew how to live in this place.
From what we see of Nephi's character, however, it would be nearly impossible for him to compromise on any of the customs or practices of the Jews he had lived among in Jerusalem. He would have insisted that his people adopt none of the "lazy" and "filthy" practices of the locals. He might even have tried to keep them from intermarrying.
What looks like racism to us might well be Nephi trying to preserve the purity of Jewish culture in a strange land. It is no surprise that he seized upon the darkness of skin as well as the cultural shortcomings (as he saw them) of the locals in order to condemn the Lamanites, if they were the first to bring forth a mestizo generation.
By Jacob's time, though, the need for exogamy must surely have come into play -- which may be part of what was going on with the polygamy. Presumably, the population they intermarried with would have converted to their religion as well -- complete with the law of Moses and a temple like Solomon's.
Ultimately, the racial distinction is completely erased -- probably long before the time of the first King Mosiah -- but the formulaic language denigrating the Lamanites remained. (If there were racial distinctions, Amulek probably would not have to have told anybody he was a descendant of Nephi.)
My point is not that we can prove this, though this interpretation (especially as buttressed by better scholars than I) does a good job of accounting for the text. Nor is this an "all things are possible with God" explanation -- it is culturally plausible in every way -- including Nephi's nonmention of indigenous people.
Once you allow the possibility that implied populations in the Book of Mormon include recruitment of and intermarriage with indigenous populations, which are then viewed forever after as "Nephite" and "Lamanite" on political, cultural, or religious grounds, rather than genetic or racial, Kunich's problem simply disappears.
If the text fought against this interpretation, there would still be a problem. But it does not. Instead, this interpretation illuminates other portions of the text, unrelated to population.
But let's step back and look at what Kunich's article actually shows about Kunich and those who thought his article was publishable.
It should have been rejected by a serious editor because of its absurd misapplication of an estimated global population growth rate. Why wasn't it?
The science is not only unconvincing, it's obviously so, because of the false premise Kunich's entire argument is based on -- i.e., that local populations cannot grow markedly faster than the (estimated) global growth rate.
It is also textually absurd, since Nephi's account has them almost immediately engaging in wars and building temples, an obvious impossibility regardless of what population growth rate you use. The text clearly must be asserting something other than birth rate of Nephi's people to account for the population.
Kunich and those who took him seriously were not following science to its logical conclusions. They were looking for reasons to disbelieve the Book of Mormon, and seized upon a misunderstood scientific "fact" as a way to convince others to join them in their disbelief.
Their enthusiasm for the outcome ("Behold! The Book of Mormon is disproved!") keeps them from even noticing (or caring about) the borderline fraudulent misuse of science. They get just enough science to claim it as the authority for the conclusion they have already reached.
It's identical to the way people always manage to find proof-texts in the scriptures for whatever wacko doctrine they want to teach. ("If you're supposed to love your neighbor as yourself, that must mean you should love yourself first!")
The Book of Mormon asserts itself as a genuine document from another culture, translated from another language. Like most of history and science, it can't be proven true -- but we can fail to prove it false, after subjecting it to the most rigorous tests.
The July 2008 Scientific American has a wonderful article about the way mitochondrial DNA, Y-chromosomes, and haplotypes are being used to track the gross movements of populations throughout history.
Naturally, the eager-to-prove-the-Book-of-Mormon-false group will triumphantly proclaim that since the haplotypes, mitochondrial DNA, and y-chromosomes all agree that the origin of native Americans is traceable only to Siberia and/or southeast Asia, the Book of Mormon is thus proven false.
Scientists and students of science will hardly be taken in by such a claim. But many naive people who lack the knowledge or experience to recognize a pseudo-science scam when they see one may well face a completely needless crisis of faith.
I especially worry about our most vulnerable population: Young Mormons who are in an age when doubt comes naturally. They are most susceptible to such fakery, not because they "lack faith," but because they are hungry for truth, and are likely to take "facts" over testimonies.
So let me explain why perfectly good and useful science -- which the tracing of DNA in large populations certainly is -- turns into junk science when those who are committed to unbelief find a spin that serves their purpose.
1. Haplotype, mitochondria, and Y-chromosome tracking is done using tiny samples from the populations in any given area. This is necessary and perfectly acceptable, because the scientists are not trying to eliminate the possibility of intermixing of populations, but rather trying to trace the general ancestry of large groups.
2. Any variation from the predominant DNA strains will be interpreted, correctly, as "contamination" and either disregarded or removed from the study as long as it exists in only trivial amounts. The only question that would be hard for them to answer is when the contamination took place.
3. Since the dominant strain that populated the Americas shares a common ancestry with Fertile Crescent ancestors, some haplotypes that might have pointed to the Middle East are already in the entire population and therefore invisible.
4. Many of their findings deal with populations that have been tracked through history. Yet the genetic record does not account for "trivial" population movements like the conquest of India, Persia, all of Europe, and much of Asia Minor by Indo-European tribes, or repeated conquests of China by borderland nomads.
If you can't track major conquests genetically, even though we know from the historical and archaeological record that they occurred, and if small contaminations are ignored, why would any educated person expect that these methods would reveal even a hint of a group of only a few dozen culturally elite people who arrived in America 2600 years ago and (probably) almost immediately intermarried with the local population?
Also, the Book of Mormon says that 1600 years ago there was an attempt to eradicate precisely those people most likely to have maintained some genetic distinction. It is obvious that the real science on this subject simply has nothing to say about the claims of the Book of Mormon.
When unbelievers claim to have found scientific disproof of the Book of Mormon, they usually understand neither the science nor the book.
Instead, they revealed their hunger to believe in whatever will buttress their account of the Mormon religion.
That is a natural human tendency -- all religious believers do exactly the same thing. The difference is that these folks claim to be above religion; to be post-religious.
They're not. Their "discoveries" are as driven by their preexisting faith as anyone else's. They may claim to be intellectuals, but they lack the rigor to doubt their own doubts, to question their own questions.
Here is the remarkable thing: Within the Church, among believers, there are many genuine skeptics who question everything, who test everything.
We are not afraid to look at any evidence, and when we are convinced, we change our frame-of-view to accommodate the new information. At the same time, we recognize that none of our knowledge is final and might be revised -- by new evidence, by new revelation.
This is where intellectual rigor leads you -- in religion as well as science. You believe and act upon all that you now understand to be true, but you know that there will yet be revealed or discovered many great and wonderful things of which you are currently ignorant.
You don't let each new wind blow away all that you knew before. Instead, you hold all things in abeyance and take time to work them out and see what the implications might be. Eventually the religion and the science both get clearer.
Neither has anything to fear from the other -- as long as there is complete candor and rigor on both sides. They can coexist in the same mind. The only war between science and religion occurs among those who fail to understand one or the other -- or both.
Copyright © 2019 by Orson Scott Card
• A New Thanksgiving Hymn
• First Class
- - August 20, 2015
• The Gifts of Conference
- - March 23, 2015
• Christmas Is About A Baby
• What Tithing Means
- - October 2, 2014
• Earning Leisure
- - April 25, 2014
• Mormon Materialism
• Noah the Movie
- - April 3, 2014
• Literature, Logic, and the Scientific Method
• On Terminology
- - May 2, 2013
More by Orson Scott Card
About Orson Scott Card
Orson Scott Card is the author of the novels Ender's Game, Ender's Shadow, and Speaker for the Dead, which are widely read by adults and younger readers, and are increasingly used in schools.
Besides these and other science fiction novels, Card writes contemporary fantasy (Magic Street, Enchantment, Lost Boys), biblical novels (Stone Tables, Rachel and Leah), the American frontier fantasy series The Tales of Alvin Maker (beginning with Seventh Son), poetry (An Open Book), and many plays and scripts.
Card was born in Washington and grew up in California, Arizona, and Utah. He served a mission for the LDS Church in Brazil in the early 1970s. Besides his writing, he teaches occasional classes and workshops and directs plays. He also teaches writing and literature at Southern Virginia University.
Card currently lives in Greensboro, North Carolina, with his wife, Kristine Allen Card, and their youngest child, Zina Margaret.
More about Orson Scott Card
Orson Scott Card currently serves as second counselor in the bishopric.

redpajama_set_name: RedPajamaCommonCrawl


text:
Boeing says software fix for grounded 737 MAX complete
©Reuters / Lindsey Wasson / Free
Boeing said it has finished a software update for its 737 MAX airliner, part of the company's effort to address critical safety issues with the plane after two fatal crashes this year.
The company said it would soon work to schedule a certification flight with the Federal Aviation Administration. The agency has come under fire for its previous certification of the aircraft despite ongoing problems with the automatic flight control system.
"With safety as our clear priority, we have completed all of the engineering test flights for the software update," Boeing chairman, CEO and president Dennis Muilenburg said in a statement Thursday.


Hello all my dear Galavant Girl Followers….
After a much needed hiatus, and now with some advice from Mr. David Iwanow Mr. David Iwanow at The Lost Agency–who has written books and counseled big wigs about their sites and if Google likes them or not–The Galavant Girl will be undergoing some nice changes!
This is taking a wee bit of time to get set up and put together with all the ûber-technical BS writers must wade through these days to make things happen… but it will be up very very soon.
Thank you everyone for following me and keeping up! Stay tuned!!!


text:
\section*{Introduction}

Spinfoam models provide a background independent framework for a regularized path integral for (loop) quantum gravity. They define transition amplitudes between quantum states of geometry through state-sum models which can be understood as discrete space-time geometries. In this context, a recent development has been a proposal for reconstructing the graviton propagator in this discrete setting from correlations between geometrical observables such as the areas of elementary surfaces \cite{graviton}. This allows to extract some semi-classical correlations at large scales, which we hope to compare with the perturbative calculations performed in quantum general relativity treated as a quantum field theory.

Since the original proposal, they has been a lot of work developing this line of research, mainly focusing on creating the new mathematical tools needed for these semi-classical calculations and on using these computations as a criteria to select spinfoam models with a correct semi-classical behavior and discriminate between them (see e.g \cite{alesci}). All these developments hint towards the fact that the graviton propagator in spinfoam models lead back to Newton's law for gravity at large scales while being regularized at the Planck scale. This behavior has been confirmed by numerical simulations in the simplest cases \cite{numeric1,numeric2}.

The main ingredients of these ``spinfoam graviton propagator" calculations are: a suitable boundary state peaked on a classical 3-geometry, the spinfoam amplitudes for the bulk geometry and relevant observables probing the space-time geometry.
The main spinfoam models used to define the bulk amplitudes are the Barrett-Crane model, which exists in both its Euclidean version \cite{bc1,bubblediv} and its Lorentzian counterpart \cite{bc2,bc2bis}, and the more recent EPRL-FK models \cite{epr,eprl,fk,ls1,ls2}
and their generalizations \cite{eprl,polish}.

Considering boundary states, while most of the recent work has focused on building quantum coherent states with good semi-classical properties, one important issue is the requirement that the states are physical i.e solve the Hamiltonian constraint on the 3d boundary.
As we will review and discuss later, this ``physical state" criteria can be entirely formulated in term of a compatibility equation between the boundary state and the spinfoam bulk amplitude, which translates the requirement of stability of the boundary state under the evolution defined by the spinfoam model.
This question was investigated in the framework of the Ponzano-Regge spinfoam model for 3d quantum gravity, more particularly in a toy model where the three-dimensional space-time is triangulated by a single tetrahedron \cite{toymodel3d}. In this case, the correlations between length observables can be thoroughly studied both analytically and numerically \cite{graviton3d}. In this context, the requirement of working with a physical boundary state was shown to determine explicitly and uniquely the width of the gaussian wave-packet defining the state \cite{physical3d}. This width is relevant in the context of the geometrical correlations because it enters the exact numerical factor in front of these correlations. Therefore, if we would like to have the exact correlations and not only their scaling properties, we need a definite prediction of that width.

\smallskip

In the present paper, we investigate the consequences of the physical state requirement for the Euclidean Barrett-Crane model for the simplest case of a space-time triangulation constructed from a single 4-simplex. In this context, we show that this requirement fixes uniquely the width of the quantum boundary state (in term of the classical data) similarly to what happens in the 3d toy model. In the next section, we will introduce and quickly review the framework of the spinfoam graviton propagator. Then in section II, we will review the definition of the (Euclidean) Barrett-Crane spinfoam model. Section III presents our results about identifying suitable physical boundary states for the Barrett-Crane model.
We consider both the simplest cases of a decoupled Gaussian ansatz and the more interesting case of a coupled Gaussian ansatz involving the Hessian of the Regge action (for a single 4-simplex).
Finally, we will conclude by a short discussion on the relevance of our physical states under renormalization of the Barrett-Crane model and on the possible extension of our results to other spinfoam models.

\smallskip

Before proceeding to the analysis of the Barrett-Crane spinfoam amplitude for a 4-simplex, we would like to discuss to which extent our notion of boundary states stable under spinfoam evolution can be considered as an actual physical state. From the strict perspective of the canonical framework, a physical quantum state should solve the Hamiltonian constraint (and actually also solve the vector constraints generating space diffeomorphisms). Thinking of this requirement in classical terms, this means that the 3-metric on our boundary should be physical, i.e satisfy the constraints, and that we should furthermore consider equivalence classes of boundary 3-metrics under the gauge transformation(s) generated by the constraint(s). In our spinfoam framework, considering the 3d case with a single tetrahedron in the Ponzano-Regge model, there is a single truly physical state from the canonical point of view, which solves all the constraints. This is the completely flat boundary state, which is given by the 6j-symbol. The procedure followed in \cite{physical3d} is to look for semi-classical states on the same tetrahedron, which are peaked around a certain classical 3-metric satisfying the constraints (here defined as the lengths of the tetrahedron's edges) and which are stable under evolution defined by the Ponzano-Regge spinfoam amplitude. The requirement of being fixing the tetrahedron's edge lengths amounts to a gauge-fixing of the  Hamiltonian (and diffeomorphism) constraint(s). This allows us to identify a single quantum boundary state for each classically admissible edge-length configurations of the tetrahedron \cite{physical3d}. Each of these quantum boundary states describe a flat physical 3d-metric within the tetrahedron. Thus we refer to these boundary states as physical in the spinfoam context, although they are not strictly physical states from the canonical viewpoint.

We follow the same logic in the present study of the quantum 4-simplex in the Barrett-Crane model. The basic spinfoam amplitude for the 4-simplex implies that the bulk 4-metric is flat, although we will discuss in the conclusion the possibility of effective spinfoam amplitudes for the 4-simplex inducing curvature. Thus we expect once again a unique physical boundary state for 4-simplex from the strict canonical theory definition. Nevertheless, we will look for semi-classical states which are peaked on fixed values of the area of the boundary triangles and which are stable with respect to the Barrett-Crane spinfoam amplitude. Fixing the triangle areas amounts again to a partial gauge-fixing of the Hamiltonian and diffeomorphism constraints. And we then expect a unique such stable semi-classical boundary state for each sets of classically admissible areas. These can be considered as particular gauge-fixings of the unique canonical physical flat state. By misuse of language, we shall nevertheless refer, in the  specific spinfoam context, to our quantum boundary states stable under spinfoam evolution as physical states.

\section{Spinfoams and the ``Graviton Propagator" Framework}

The spin foam approach to quantum gravity provides a path integral formulation for the canonical framework of loop quantum gravity (LQG).
It is based on a reformulation of general relativity of an ``almost-topological" gauge field theory and the spinfoam path integral is constructed as a discretization of the continuum path integral. Without focusing especially on the specific case of quantum gravity, we describe below the generic features of the spinfoam framework for an arbitrary gauge group $G$.
As it is well-known, the Hilbert space of quantum geometry states for loop quantum gravity  is spanned by spin network states $|\psi \rangle$ where  in $\psi=(\gamma, j_l, i_n)$: $\gamma$ is a graph, $j_l$ is a ``spin" labeling an irreducible representation of the gauge group $G$ associated to the link $l$ of the graph, and $i_n$ is associated to the node $n$ of $\gamma$ and labels  intertwiners. We consider a 4d space-time region $\mathcal{M}$  with a 3d boundary $\Sigma$. The spin network state $|\psi\rangle$ defines the quantum state of geometry of the boundary $\Sigma$, then the spin foam amplitudes define the dynamical probability amplitude of that state and are supposed to contain the whole dynamical content of quantum gravity. More precisely, the standard ansatz for local spinfoam amplitudes can be presented in the following general form:
\begin{equation} \label{propaSF}
K[\psi]=\sum_{C|\partial C= \gamma_\psi} w(C) \sum_{j_f, i_e} \prod_f A_f(j_f) \prod_e A_e(j_f, i_e) \prod_vA_v(j_f, i_e)
\end{equation}
where the sum is taken over two-complexes $C$ and their ``coloring"  $c\,\equiv\,\{j_f, i_e\}$. The 2-complexes are constrained to fit the graph $\gamma_\psi$ of the spin network $\psi$ at the boundary.
The representations $j_f$ are associated to the faces $f$ of $C$ and the intertwiners $i_e$ to its edges $e$. They are also constrained to fit the representations $j_l$ and intertwiners $i_n$ of the state $\psi$ living on the boundary.
Then a spinfoam is defined as a colored two-complex, namely a couple $(C, c)$.
Finally, the spinfoam amplitude is made of four types of factors. First, $w(C)$ is a statistical weight that depends only on the two-complex $C$ (similar to the symmetry factor for Feynman diagrams in standard quantum field theory). $A_f$ are weight factors  associated to the faces and $A_e$ are amplitudes associated to the edges of the 2-complex. These three types of factors can be interpreted as measure factors of the discrete path integral defined by $K[\psi]$. All the dynamical information is encoded in the vertex amplitude, $A_v(j_f, i_e)$, which is an amplitude associated with each vertex $v$ of the two-complex $C$ and depends on the spins $j_f$ and intertwiners $i_e$ living on the faces and edges around that vertex.

The amplitudes are usually assumed to be local, that is they depend only on the coloring of adjacent simplicial elements. Thus, $A_f$ is a function of the representations located on the face $f$, $A_e$ is a function of the intertwiner assigned to $e$ and the representations on the faces containing $e$, whereas $A_v$  depends on the representations on the faces and on the intertwiners on the edges containing the vertex $v$.
Generally speaking, the choice of the vertex amplitude $A_v$ corresponds to the choice of a specific form of the hamiltonian operator in the canonical theory.

Since a spin network state defines a quantum state for the boundary geometry, spin foams are thus interpreted as representing the states of the bulk defining the quantum space-time interpolating between given boundary data. This  can be clearer when considering an initial spin network $\psi$ and a final spin network $\psi^\prime$. Then a spin foam $\mathcal{F}: \psi \rightarrow \psi^\prime$ defined by the 2-complex $C$ bordered by the supporting graphs of $\psi$ and  $\psi^\prime$, respectively $\gamma$ and $\gamma^\prime$, represents a transition from the spin network state $\psi=(\gamma, j_l, i_n)$ into $\psi^\prime=(\gamma^\prime, j_l^\prime, i_n^\prime)$. Nodes and links in the initial spin network $|\psi\rangle$ evolve into 1-dimensional edges and faces. New links are created and spins are reassigned at vertexes. This defines a foam-like structure whose components inherit the spin representations from the initial spin network $\psi$ and are at the end compatible with the spin representations of the final spin network $\psi^\prime$. The propagator kernel $K[\psi,\psi^\prime]$ is obtained summing over all the spinfoams compatible with the boundary data:
\begin{equation} \label{propaSFcompact}
K[\psi,\psi^\prime]=\sum_{\mathcal{F}|_{\partial \mathcal{F}=\psi \cup \psi^\prime }} \mathcal{A}_\mathcal{F}[\psi,\psi^\prime]
\end{equation}
where we have introduced here a more compact notation than in the previous formula (\ref{propaSF}): the sum over the compatible spin foam $\mathcal{F}$ gathers the sum over the two-complexes $C$ compatible with the graphs $\gamma$ and $\gamma^\prime$ and the sums over the G-representations compatible with the representations $j_l$ and $j_l^\prime$ and over the intertwiners compatible with $i_n$ and $i_n^\prime$.  $\mathcal{A}_{\mathcal{F}}[\psi,\psi^\prime]$ is the spin foam amplitude associated to the spin foam $\mathcal{F}: \psi \rightarrow \psi^\prime$ interpolating between the initial and final states. It is given as above as a product of vertex, edge and face amplitudes: $A_v, \, A_e, \, A_f$. This special case where the boundary is made of two disconnected pieces, which we can interpret as the initial boundary and the final boundary, allows a clearer connection with the canonical framework, whose goal is to define transition amplitudes between initial and final states.

In the present work, we will consider the case where the boundary is connected and the kernel $K$ is then defined as a function of only one boundary spin network $\psi$:
\begin{equation}
K[\psi]=\sum_{\mathcal{F}|_{\partial\mathcal{F}=\psi }} \mathcal{A}_\mathcal{F}[\psi]
\end{equation}
where we used the same compact notation as above.

Before going further, we have to emphasize that the sum over spinfoam $\mathcal{F}=(C,c)$ is not exactly well-defined. Usually, for fixed $C$, the sum over colorings $c$ is well-controlled. On the other hand, controlling the full sum over two-complexes is a much more subtle issue. It is nevertheless non-perturbatively defined as quantum field correlations  in the context of group field theory (see e.g. \cite{GFT}). In this context, $\mathcal{A}_\mathcal{F}$ actually  depends on the GFT coupling constant $\lambda$. Indeed the statistical weight $w(C)$ for a two-complex $C$ is given by the symmetry factor of the two-complex (considered as a Feynman diagram for the GFT) times a factor $\lambda^V$ where $V$ is the number of vertices of the two-complex. Then the sum over 4-geometries in $K[\psi]$ can be written as a power series in $\lambda$; that is:
$$
K[\psi]=\sum_{V=0}^\infty \lambda^{V} K_V[\psi]
$$
where $K_V[\psi]=\sum_{\mathcal{F}^V|_{\partial \mathcal{F}^V=\psi }} \mathcal{A}^V_\mathcal{F}[\psi]$ is the sum over spinfoams $\mathcal{F}^V$ with $V$ vertices. Assuming that the perturbative expansion in $\lambda$ is well-defined and that we are working with a suitable semi-classical boundary state $\psi$, we expect that the dominant contributions come from the simplest spinfoams and we consider the lowest order terms in the expansion of $K[\psi]$ in $\lambda$ as giving the leading order for the path integral and correlations.

\medskip

Let us now describe the spinfoam framework for deriving the graviton propagator from correlations between area observables.
We start by considering a semi-classical spin network functional $\Psi_{q}[\psi]$ peaked on a classical 3d metric $q$ for the boundary $\Sigma$. Let us remind the reader that $\psi=(\gamma,j_l,i_n)$ labels basis states of the Hilbert space of spin networks. We further require that this boundary state $\Psi_{q}[\psi]$ induces a space-time structure in the bulk peaked around the flat Minkowski metric. In particular, this normally fixes the classical boundary $q$ to be the 3-metric induced on $\Sigma$ by the Minkowski metric on ${\cal M}$. Then we construct correlations between the metric fluctuations for the chosen boundary state $\Psi_{q}$:
\begin{equation} \label{twopointSF}
W^{abcd}(x,y;q)=\sum_{\psi}K[\psi] \langle \psi | h^{ab}(x) h^{cd}(y) |\psi \rangle \Psi_{q}[\psi]
\,=\,{\textrm Tr}\, \left[K \,h^{ab}(x) h^{cd}(y)\,\Psi_q\right]\,,
\end{equation}
where the trace is taken over the Hilbert of spin networks. Here $x$ and $y$ are two points localized on the boundary $\Sigma$. In our discrete spin network setting, they are usually determined as nodes of the graph $\gamma$ underlying the spin network state $\Psi_q$. The metric fluctuations $h^{ab}(x) h^{cd}(y)$ are usually constructed as geometrical quantities. They are two basic types of such geometric observables in the discretized geometry setting of spinfoams: the diagonal components of the metric tensor can be interpreted as areas and the off-diagonal components as dihedral angles between simplices.

Finally, this formula defines the 2-point function for the gravitational field in the spinfoam framework. It can be considered as the equivalent of the standard 2-point function of the conventional quantum field theory  framework, which defines the graviton propagator \cite{graviton}:
\begin{equation} \label{2point}
W_{\mu \nu \rho \sigma} (x, y)= \langle 0 | T\{ h_{\mu \nu} (x) h_{\rho \sigma} (y) \}| 0\rangle.
\end{equation}

In the spinfoam setting,  $\Psi_{q}[\psi]$ is a semi-classical state peaked on both intrinsic and extrinsic geometry \cite{graviton}.
This boundary data will determine the 4-metric induced in the bulk.
Moreover, it should be a physical state, that is a solution a the Hamiltonian constraint. This ``physical state criteria" can be formulated by the two following conditions:
\begin{equation} \label{conditionsSF} \left\{ \tabl{l}{
\sum_\psi |\Psi_q[\psi]|^2=1 \\
\sum_\psi K[\psi] \Psi_q[\psi]=1
} \right.
\end{equation}
The first condition is the normalization of the boundary state, while the second condition translates truthfully the requirement to work with a physical state and can be considered as the ``Wheeler-deWitt" condition. It is the consequences of these conditions on $\Psi_q$ that we are going to study.

\section{The Barrett-Crane Spinfoam Model}
We now focus  on a specific spinfoam model: the Barrett-Crane model of 4d Riemannian Quantum Gravity \cite{bc1}.

\subsection{Simple Spin Networks and the Barrett-Crane Amplitudes}
The spin foam amplitude for the Euclidean Barrett-Crane model  is given by the evaluation of a spin network $\psi$ with group $\mathrm{Spin}(4)$. $\mathrm{Spin}(4)$ is the double cover of $\mathrm{SO}(4)$. Using the homomorphism $\mathrm{Spin}(4)=\mathrm{SU}(2)_L\times\mathrm{SU}(2)_R$ the irreducible representations (irreps) of $\mathrm{Spin}(4)$ are labelled by two halh-integers: ${\mathcal I}=(j_L,j_R)$ corresponding to the irreps of the two $\mathrm{SU}(2)$ sectors and group elements decompose as the product of two left and right rotations $G=g_Lg_R$. In the Barrett-Crane model, each node $n$ of the graph $\gamma$ of $\psi$ is associated to a $\mathrm{Spin}(4)$ group element $G_n$ and each link $l$ is labelled by a simple irreps of $\mathrm{Spin}(4)$ which corresponds to a couple of half-integers ${\mathcal I}_l=(j_l,j_l)$: the SU(2) irreps are the same in the left and right sectors. The evaluation then reads \cite{barrett
\begin{equation}
\mathcal{A}[\psi] \equiv \int_{\mathrm{Spin}(4)} \prod_n dG_n \prod_i
\mathcal{K}_{\mathcal{I}_i} (G^{-1}_{s(i)}G_{t(i)}),
\end{equation}
where we use the Haar measure $dG=dg_L dg_R$.
The kernel $\mathcal{K}_{\mathcal{I}}(G)$ is the matrix element of $G$ on the $\mathrm{SU}(2)$-invariant vector $|{\mathcal I}, 0\rangle$ in the ${\mathcal I}$ representation with ${\mathcal I}=(j, j)$ a simple representation of $\mathrm{Spin}(4)$. Here $\mathrm{SU}(2)$ is the diagonal rotation group, corresponding to the subgroup of 3d rotations.
Expressing the invariant vector in term of left/right components
\begin{equation}
|{\mathcal I}, 0\rangle = \frac{1}{\sqrt{d_j}} \sum_m (-1)^{j-m} |j, m\rangle_L |j, -m\rangle_
\end{equation}
where $d_j=2j+1$ is the dimension of the $\mathrm{SU}(2)$ representation of spin $j$. It is straightforward to show that the kernel $\mathcal{K}_{\mathcal{I}}$ is simply given by  the $\mathrm{SU}(2)$ character $\chi_j(g)$, defined as the trace of the group element in the $j$-representation of $\mathrm{SU}(2)$.
Parameterizing the $\mathrm{SU}(2)$ group elements as
\begin{equation}
g(\theta, \hat{n})= \cos(\theta) \mathbb{I} + i \sin(\theta) \hat{n} \cdot \vec{\sigma}, \quad \theta \in [0, \pi],
\end{equation}
the characters depend entirely on the class angle $\theta$ (half the rotation angle) and are expressed as:
\begin{equation}
\chi_j(g)=\frac{\sin d_j \theta}{\sin \theta}.
\end{equation}
Then using the properties of invariance of the Haar measure, it can be proved that the relativistic spin network evaluation is actually a 3d object involving only integrals over $\mathrm{SU}(2)$ \cite{simone1}:
\begin{equation}
\mathcal{A}= \int_{\mathrm{SU}(2)} \prod_n dg_n \prod_i \frac{1}{d_{j_i}} \chi_{j_i}(g^{-1}_{s(i)} g_{t(i)}).
\label{evaluation}
\end{equation}
This spin network evaluation then defines the vertex amplitude for the Barrett-Crane model.

The Barrett-Crane model was originally derived for  specific two-complexes $C$, which are the dual 2-skeleton of 4-dimensional triangulations $\Delta_4$.
Starting with the triangulation  $\Delta_4$, the dual two-skeleton $C$ is constructed  by associating to each simplex a point in its interior. An edge of the skeleton connects two points, which corresponds to two 4-simplices having a common tetrahedron. Then, a triangle of the triangulation $\Delta_4$ correspond to a face of the two-complex $C$ formed by the edges of the dual tetrahedra having this triangle in common. This duality allows to derive the spin foam amplitudes from a quantization of the geometry of a 4-simplex.
Indeed, the amplitude for a single 4-simplex defines the vertex amplitude of the spinfoam model. Arbitrary 2-complexes are constructed by gluing 4-simplices together and the corresponding spinfoam amplitudes are given by the product of these vertex amplitudes corresponding to each 4-simplex.

Considering a single 4-simplex, the vertex amplitude is constructed as the evaluation of its boundary spin network. The boundary graph is constructed as the dual of the boundary of the 4-simplex: each tetrahedron is mapped on a node of the graph and each triangle shared by two tetrahedra is mapped to a link between these nodes. This gives a graph $s$ with 5 nodes $n \in \{1, \cdots, 5\}$ and 10 links between them, as represented in fig. \ref{4simplexPlusTetra}.
 \begin{figure}[ht]
\begin{center}
\includegraphics[width=5cm]{4simplexPlustetra.eps}
\end{center}
\caption{The 4-simplex (or pentahedral) boundary spin network. We label the nodes $n= 1, . . . 5$. In the dual picture, they are in correspondence with tetrahedra of the boundary triangulation. Two of them are represented. The ten links $nm$, on the other hand, are dual to triangles. Consider for instance the link 45: this is dual to the triangle shared by the tetrahedra 4 and 5. Associated with the link 45 is the dihedral angle $\theta_{45}$ between the tetrahedra 4 and 5.} \label{4simplexPlusTetra}
\end{figure}
Then the vertex amplitude $\mathcal{A}[s]$ is given up to normalization factors by the evaluation of the boundary spin networks and defines the Barrett-Crane $\{10j\}$-symbol:
\begin{equation} \label{10j}
\{10j\}\equiv  \int_{\mathrm{SU}(2)} \prod_{n=1}^{5} dg_n \prod_{i=1}^{10} \chi_{j_i}(g^{-1}_{s(i)} g_{t(i)}),
\end{equation}
following equation (\ref{evaluation}).
The normalization ambiguity leads to an ambiguity in the definition of the Barrett-Crane model. It corresponds to arbitrary gluing factors between 4-simplices, or equivalently to ambiguity in defining the spinfoam edge amplitude ${\cal A}_e$. We will come back to this issue in the next section.

\subsection{The 4-Simplex and the Boundary States}
We now focus on the simplest 4d boundary spin network: a pentahedral that we denote $s=(\Gamma, j_{ab}, i_a)$ which is the dual of a single 4-simplex. We have  introduced  the notation $(a\, b)$ for $i$ linking the nodes $a$ and $b$.  $a, b=1\cdots 5$ label the five nodes of the pentahedral; in the triangulation picture, there are in correspondence with tetrahedra of the boundary  4-simplex and links $i\equiv (a\, b)$ are dual to triangles. Thus, in the triangulation picture,  while the five group elements $g_a \, \in$ $\mathrm{SU}(2)$ are associated to the 5 tetrahedra of the 4-simplex, the ten representations $j_i \equiv j_{ab}$ for $a\ne b$ can be seen as attached to the triangles of the 4-simplex.
The amplitude $\mathcal{A}_\mathcal{F}$ of a spin foam $\mathcal{F}\equiv(C,c)$ where the two-complex $C$ is bordered by the boundary 4-simplex graph $\Gamma$ and its color $c$ is compatible with $\{j_{ab}, i_{a}\}$ is then given (up to normalization factors) by:
\begin{equation} \label{SFamplitude}
\mathcal{A}[s]= \prod_{a<b} (d_{j_{ab}})^2 \prod_{a<b}\left(A_e(d_{j_{ab}})\right)^{1/2}{\{10j\}}
\end{equation}
where  $A_e(j_f)$ is the edge amplitude left unspecified here and the ${\{10j\}}$-symbol, the vertex amplitude of the spin foam, defined by (\ref{10j}). Let us point out that $\mathcal{A}[s]=K[s]$ in the first order in the group field theory coupling constant $\lambda$ in the specific case of the pentahedral boundary spin network $s$. We now focus on the ${\{10j\}}$-symbol which is defined by the following group integral:
\begin{equation} \label{10jsymbol}
{\{10j\}}\,=\,
\int_{\mathrm{SU}(2)} [dg_a]^{\otimes 5}\,
\prod_{a<b}\chi_{j_{ab}}(g_ag_b^{-1})
\end{equation}
The $\{10j\}$ symbol admits a geometrical interpretation associated to the 4-simplex. The key fact is that it can be written as an integral over ten class angles $\theta_{ab} \in [0, \pi]$ of the group elements $g^{-1}_{a}g_{b}$:  $\theta_{ab}$ is the dihedral angle and we use the convention $\theta_{aa}=0$. In the triangulation picture, it is the angle between the outward normals to the tetrahedron $a$ and the tetrahedron $b$ of the 4-simplex. The $\{10j\}$ symbol then reads,
\begin{equation}
\{10j\}=\int d\mu[\theta_{ab}] \prod_{a<b} \chi_{j_{ab}}(\theta_{ab}),
\end{equation}
where the measure on the ten class angles $d\mu[\theta_{ab}]$ takes into account that the group elements $g^{-1}_{a}g_b$ are obviously not independent. The measure is in fact simply given by a constraint \cite{6jsaddle2}
\begin{equation}
d\mu[\theta_{ab}] = \prod_{a<b} d\theta_{ab} \sin \theta_{ab} \delta \left(\det G_{ab} \right)
\end{equation}
$G$ is the Gram matrix, a symmetric $5\times 5$ matrix defined as $G_{ab} = \cos \theta_{ab}$. This constraint $\delta(G)$ contains all the geometric information and allows to relate the $\{10j\}$-symbol to the Regge amplitude (which describes discretized gravity) for a Euclidean 4-simplex in the large $j$ asymptotics \cite{6jsaddle2, simone1}.
The independent variables are the areas of the triangles of the 4-simplex representing the triangulation or equivalently the dimension of the representation $j_{ab}$, $d_{j_{ab}}=2j_{ab}+1$, since $A_{ab}=l_P^2d_{j_{ab}}$. The geometry of the 4-simplex is fixed by given the ten areas $A_{ab}=d_{j_{ab}}$ and the dihedral angles can be considered as functions $\theta_{ab}(d_{j_{cd}})$ of the areas. Actually, we scale the area $d_{j_{ab}}$ instead of the spins $j_{ab}$, then the asymptotic behavior on a single 4-simplex at the leading order\footnote{The fact to scale the amplitude with the irrep dimension $d_{j_{ab}}$ instead of the spin $j_{ab}$ has no effect on the leading order.} is given by:
\begin{equation} \label{asympt}
{\{10j\}} \, \sim \, P(d_{j_{ab}}) \cos \left( \frac{1}{l_P^2}S_R[d_{j_{ab}}]\right) + D(d_{j_{ab}})
\end{equation}
where $S_R[d_{j_{ab}}]= \sum_{a<b}d_{j_{ab}}\theta_{ab}$. The function $P(d_{j_{ab}})$ is a slowly varying factor, that grows as $\zeta^{-9/2}$ when scaling all triangle areas $d_{j_{ab}}$ by $\zeta$.
$D(d_{j_{ab}})$ is a contribution coming from degenerate configurations of the 4-simplex. This is a non-oscillating term which has no geometrical meaning. It was found to scale like $1/\zeta^2$ and thus dominate the large spin limit of the ${\{10j\}}$-symbol. Moreover, it was shown in a recent paper \cite{numeric2} that even after having removing the dominating contribution of order $1/\zeta^2$, there are additional non-oscillating contributions still hiding the Regge term. However, this sick term is negligible in most computations we are interested in, such as in the computation of  (\ref{twopointSF}) (see \cite{graviton}) or as we will see in the computation of (\ref{conditionsSF}), because it will not match the boundary data induced by $\Psi_q$ which peaks the asymptotic around the non-degenerate semi-classical configuration.

On the other hand, the unspecified edge amplitude $A_e$ in (\ref{SFamplitude})  reflects the normalization ambiguity of the ${\{10j\}}$-symbol and different choices leads to different versions of the Barrett-Crane model; we will specify this normalization factor latter.

Since in the Barrett-Crane model, the vertex amplitude $A_v$, defined by (\ref{10jsymbol}),
and therefore the propagator kernel $K[s]=\mathcal{A}[d_{j_{ab}}]$, given by (\ref{SFamplitude}), are independent of the intertwiners, we  can  rewrite the two conditions (\ref{conditionsSF}), restricting $\psi_q$ to be a physical state,  as\footnote{It is often useful for spinfoam asymptotic calculations to do a Fourier transform and go back to group variables (see e.g. \cite{simone1} for the application to the spinfoam graviton propagator). In the present case, this is straightforward to do if we choose a factorizable ansatz, i.e. a product state of the form $\psi(j_1,..,j_{10})=\prod_{i=1}^{10} \phi_i(j_i)$. We further simplify it by taking an equilateral ansatz, i.e $\phi_i(j)=\phi(j)$ for all triangles $i=1..10$. Then following \cite{simone1}, we introduce the Fourier transform of the state:
$$
f(g)=\sum_j \phi(j) \chi_j(g).
$$
Then the two conditions for a physical state simply translates to:
$$
\int dg\, |f(g)|^2=1,
$$
$$
\int [dg_m]^5\,\prod_{a<b} f(g_mg_n^{-1})=
\int [d\theta_{mn}]^{10}\delta(\det [\cos\theta_{mn}])
\prod_{m<n}f(\theta_{mn})\,=\,1.
$$
}:
\begin{equation} \label{norm}
\sum_{j_{ab}} |\psi_q(d_{j_{ab}})|^2=1 \quad \textrm{ for the normalized condition,}
\end{equation}
\begin{equation} \label{dyna}
\sum_{j_{ab}} K(d_{j_{ab}})\psi(d_{j_{ab}})=1 \quad \textrm{ for the "Wheeler-deWitt" condition,}
\end{equation}
with the propagator kernel a function of the dimensions $d_{j_{ab}}$ only
\begin{equation} \label{kernel}
K[j_{ab}]= \mu \left(\prod_{a<b}d_{j_{ab}}\right)^{\sigma}{\{10j\}}
\end{equation}
with $\mu$ and $\sigma$ undetermined coefficients. The edge amplitude $A_e$ is taken into account in this factor $\mu \left(\prod_{a<b}d_{j_{ab}}\right)^{\sigma}$

For more generic spinfoam models such as the EPRL-FK models, we need to take into account the intertwiners in the definition of the spinfoam vertex and of the physical states. This is postponed to a further work.

\section{Looking for Physical States}
In the following, we will  work on the Barrett-Crane model in the large spin limit; that is we will consider the asymptotic formula of the ${\{10j\}}$-symbol (\ref{asympt}) in the propagator kernel formula (\ref{kernel}).
We now focus on the issue of the semi-classical boundary state. The function $\Psi_q$ should describe the boundary value of the gravitational field on the boundary 4-simplex. We thus consider a state peaked on the geometry of a regular 4-simplex: $q=(d_{j_0}, \Theta)$. The simplest possibility is to choose a Gaussian peaked on theses values:
\begin{equation} \label{gaussian}
\Psi_{(d_{j_0}, \Theta)}(d_{j_{ab}})= e^{-\sum_{a<b,c<d} \alpha_{cd}^{ab}(d_{j_{ab}}-d_{j_0})(d_{j_{cd}}-d_{j_0})+i\sum_{a<b} \Theta \, d_{j_{ab}}}
\end{equation}
The phase of this semi-classical state determines where the state is peaked in the conjugate variables: $\Theta$ is the variable conjugate to the spin $j_0$ and it codes the extrinsic geometry of the boundary. $\alpha_{cd}^{ab}$ is a given ten by ten matrix. It depends on $d_{j_0}$ in such a way that the relative uncertainties of area and angle on this state become small in the large $d_{j_0}$ limit, namely:
\begin{equation}
\frac{\langle \Psi_q|\Delta d_{j_{ab}}| \Psi_q\rangle}{\langle\Psi_q| d_{j_{ab}}| \Psi_q\rangle} \rightarrow 0, \qquad \qquad \frac{\langle\Psi_q|\Delta \theta_{ab}| \Psi_q\rangle}{\langle\Psi_q| \theta_{ab}| \Psi_q\rangle} \rightarrow 0 \qquad \qquad \forall \,a<b
\end{equation}
Assuming  that the matrix elements $\alpha_{cd}^{ab} \sim \alpha d_{j_0}^{-n}$ in the large spin limit with $\alpha$ which does not scale with $d_{j_0}$, the fluctuation determined by the gaussian state (\ref{gaussian}) are of the order:
\begin{equation} \label{uncertainties}
\frac{\langle\Psi_q|\Delta d_{j_{ab}}| \Psi_q \rangle}{\langle\Psi_q| d_{j_{ab}}| \Psi_q \rangle} \sim \frac{d_{j_0}^{n/2-1}}{\sqrt{\alpha}}, \qquad \qquad \frac{\langle\Psi_q|\Delta \theta_{ab}| \Psi_q \rangle}{\langle \Psi_q| \theta_{ab}| \psi_q \rangle} \sim d_{j_0}^{-n/2}\sqrt{\alpha}
\end{equation}
which restricts $n \in  ] 0, 2[$. \\
In the following, we thus focus on a Gaussian state as boundary state such as the matrix elements of the ten by ten matrix $\alpha^{ab}_{cd}$ are given by
\begin{equation}
\alpha^{ab}_{cd} \sim \frac{\alpha}{d_{j_0}^n}= \frac{a+ib}{d_{j_0}^n} \quad \textrm{ with } n \in ]0,2[ \textrm{ and } a, b \in \mathbb{R}
\end{equation}
in the large spin limit. We now study the consequences of the two conditions (\ref{norm}) and (\ref{dyna}) on this boundary state. That is the aim is now to determine the consequences on $\alpha_{cd}^{ab}$ of the requirement that our boundary state is a physical state.

\subsection{Semi-Classical States: the Decoupled Gaussian Ansatz}
We first start by considering an additional ansatz for the boundary state. We take a factorized boundary state:
\begin{equation}
\Psi_q[j_{ab}]= \prod_{a<b}\phi(j_{ab})
\end{equation}
where each $\phi(j_{ab})$ is peaked around the background value $q=(d_{j_0}, \Theta)$ which corresponds to an equilateral 4-simplex.
In this simplest case, the ten by ten matrix $\alpha_{cd}^{ab}$ reduces to a diagonal matrix $\alpha \mathbb{I}_{10\times 10}$.
Such a factorized boundary state has been used in  \cite{numeric2} since it is up to now the only setting in which numerical simulations can be performed. However, this assumption has not been tested yet. Could such a decoupled gaussian state capture a true physical state? Could it satisfy conditions (\ref{norm}) and (\ref{dyna})?

Two choices for a factorized boundary state have so far appeared in the literature: \begin{itemize}
\item  the Gaussian state \cite{graviton, numeric1}, where each factor is given by,
\begin{equation} \label{gaussianf}
\phi(j)= e^{-\alpha (d_{j}-d_{j_0})^2}e^{i\Theta d_{j}}
\end{equation}
where $\alpha$ is a complex number.
\item The Bessel-based state \cite{simone1, graviton3d}, where each factor is given by:
\begin{equation} \label{bessel}
\phi_B(j)= \frac{I_{|d_j-d_{j_0}|}(\frac{d_{j_0}}{\alpha})-I_{d_j+d_{j_0}}(\frac{d_{j_0}}{\alpha})}{\sqrt{I_0(\frac{2d_{j_0}}{\alpha})-I_{2d_{j_0}+1}(\frac{2d_{j_0}}{\alpha})}} \cos\left(d_j\Theta\right)
\end{equation}
\end{itemize}
In the large spin limit regime, we focus only on the Gaussian ansatz. Indeed, in the large spin limit the Bessel part of (\ref{bessel}) reduces to the Gaussian in (\ref{gaussianf}). Therefore, at the leading order, the only difference between (\ref{gaussianf}) and (\ref{bessel}) is in the phase: the phase in (\ref{gaussianf}) is complex whereas the phase in (\ref{bessel}) is real. A gaussian state with a real phase would lead to the same results as a gaussian state with a complex phase. The interested reader can find details concerning the case of a gaussian state with a real phase in appendix \ref{real} and we now tackle the issue of defining a physical state coming from a decoupled gaussian state with a complex phase.
Therefore, we consider a boundary state of the form,
\begin{equation} \label{Factorizedgaussian}
\Psi_{(d_{j_0}, \Theta)}[j_{ab}]= \frac{1}{\mathcal{N}}\prod_{a<b}e^{-\alpha (d_{j_{ab}}-d_{j_0})^2}e^{-i\Theta d_{j_{ab}}}
\end{equation}
with $\mathcal{N}$ the normalization constant and $\alpha \in \mathbb{C}$.
We now want to test this assumption using conditions (\ref{norm}) and (\ref{dyna}). These conditions lead to the two following equations on $\mathcal{N}$ and $\alpha$:
\begin{equation} \label{normapprox}
1=\sum_{j_{ab} }|\Psi_{(d_{j_0}, \Theta)}(j_{ab})|^2= \frac{1}{(\mathcal{N}^2)^{10}}\sum_{\{j_{ab}\}} e^{-2\Re(\alpha)\sum_{a<b}(d_{j_{ab}}-d_{j_0})^2}
\end{equation}
and,
\begin{equation} \label{WWapprox}
1=\sum_{j_{ab} }K[d_{j_{ab}}] \Psi_{(d_{j_0}, \Theta)}(j_{ab})\simeq \frac{\mu}{\mathcal{N}^{10}}\sum_{\{j_{ab}\}}(\prod_{a<b}d_{j_{ab}})^\sigma P(d_{j_{ab}}) \sum_{\epsilon=\pm1} e^{-\Re(\alpha)\sum_{a<b}(d_{j_{ab}}-d_{j_0})^2+i\sum_{a<b}[d_{j_{ab}}(\epsilon \theta_{ab}-\Theta)-\Im(\alpha)(d_{j_{ab}}-d_{j_0})^2]}
\end{equation}
The first equation corresponds to the normalization condition (\ref{norm}) for $\Psi_{(d_{j_0}, \Theta)}[j_{ab}]$. The second equation is the "Wheeler-deWitt" condition for $\Psi_{(d_{j_0}, \Theta)}[j_{ab}]$ in the simple case $K[j_{ab}]= \mu \left(\prod_{a<b}d_{j_{ab}}\right)^{\sigma}{\{10j\}}$.  Solving them allow to determine uniquely $\mathcal{N}$ and $\alpha$ in term of the coefficients $\mu$ and $\rho$.
The analysis is done in the large spin limit regime.
In equation (\ref{WWapprox}), we have already used the asymptotic formulae of the ${\{10j\}}$-symbol. Moreover, we also replace $\alpha$ in (\ref{normapprox}) and (\ref{WWapprox}) by its asymptotic expression
\begin{equation}
\alpha \sim \frac{a+ib}{d_{j_0}^n}
\end{equation}
with $a,\, b \in \mathbb{R}$ and $n$ restricted to belonging to $ ]0,2[$ in order that the asymptotic behavior of the  relative uncertainties of the area and angle on this state is correct.  Then solving the two obtained equations requires to distinguish three cases with respect to the value of the power $n \in \,]0,2[$. Keeping in mind that the standard choice of scaling for the Gaussian width $\alpha$ is $n=1$, the final result can be stated as:

\begin{proposition} The requirement on a factorized Gaussian state (\ref{Factorizedgaussian}) to be a physical state fixes the width of the Gaussian $\alpha$ for certain values of the power $n$:
\begin{enumerate}
\item $0<n<1$: the width of the Gaussian $\alpha$ is uniquely determined and the coefficient $\sigma$ is restricted. More precisely, \begin{equation} \label{result1} \left\{ \tabl{l}{
\alpha \in \mathbb{R}_+, \\
a= \frac{(\mu P)^{2/5} \pi}{2}, \; b=0\\
\sigma= \f14\left( \f95-n \right) \textrm{ and } \sigma>\f15,
} \right.
\end{equation}
where $P$ is the numerical factor in front of the asymptotics of the $\{10j\}$-symbol.
\item $1<n<2$: there is no solution in this case.
\item the standard case $n=1$: A solution exists if and only if $$\sigma=\f15$$.
Then the value of $\alpha$ can be determined graphically or numerically in term of the value of $\mu P$. For example,
$$
a\simeq 1 \textrm{ and } b\simeq 1.99 \quad \textrm{ for } \mu P=1.01\,10^5.
$$
\end{enumerate}
\end{proposition}
Before proving this result, let us comment on the third case. This last case should be the most natural since the semi-classical state is then peaked on the same way on $\Theta$ encoding the extrinsic geometry of the boundary and on $d_{j_0}=A_0$ encoding the intrinsic geometry. However, the calculations do not look particularly more natural or transparent in that case. This leads to the conclusion that the choice of a decoupled gaussian state to define a physical state might be too simple and should be modified. A new proposition is given in the next subsection. Let us now give the proof of the results stated above.

\begin{proof}
In the large spin limit, the summation in equations (\ref{normapprox}) and (\ref{WWapprox}) can then be approximated with an integral  and we can write:
\begin{equation} \label{intNorm}
1\simeq \frac{1}{2\mathcal{N}^2} \int d(d_{j})e^{-2\Re(\alpha)(d_{j}-d_{j_0})^2}
\end{equation}
and
\begin{equation} \label{intDyna}
1\simeq \frac{\mu}{(2\mathcal{N})^{10}}\int \prod_{a<b}d[d_{j_{ab}}] (\prod_{a<b}d_{j_{ab}})^\sigma P(d_{j_{ab}}) \sum_{\epsilon=\pm1} e^{-\Re(\alpha)\sum_{a<b}(d_{j_{ab}}-d_{j_0})^2+i\sum_{a<b}[d_{j_{ab}}(\epsilon \theta_{ab}-\Theta)-\Im(\alpha)(d_{j_{ab}}-d_{j_0})^2]}
\end{equation}
The first integral (\ref{intNorm}) is just a Gaussian integral, which can been integrated directly:
\begin{equation} \label{equa1}
\mathcal{N}^2=\frac{1}{2}\sqrt{\frac{\pi}{2\Re(\alpha)}}
\end{equation}
 given a first relation at the leading order between $\mathcal{N}$ and $\alpha$ .
 %
 \\
 %
 To evaluate the second integral (\ref{intDyna}) in the large spin limit, we first notice that the Gaussian implies
 $$\delta d_{j_{ab}}=d_{j_{ab}}-d_{j_0}\ll1\qquad \forall a<b,
 $$
  thus we can expand the Regge action around $d_{j_0}$:
 \begin{equation}
 S_R[d_{j_{ab}}]=\sum_{a<b}d_{j_{ab}} \theta_{ab}(d_{j}) \simeq \sum_{a<b} d_{j_0} \Theta +\sum_{a<b} \frac{\partial S_R}{\partial d_{j_{ab}}}|_{d_{j}=d_{j_0}} \delta d_{j_{ab}} +\f12\sum_{a<b, c<d} \frac{\partial^2 S_R}{\partial d_{j_{ab}}d_{j_{cd}}}|_{d_{j}=d_{j_0}} \delta d_{j_{ab}} \delta d_{j_{cd}}
 \end{equation}
The Schafli identity implies that
$$\frac{\partial S_R}{\partial d_{j_{ab}}}|_{d_{j}=d_{j_0}} =\theta_{ab}(d_j)|_{d_{j}=d_{j_0}}= \Theta$$
since in the equilateral 4-simplex, all the dihedral angles are equal to $\Theta$. And we introduce the Hessian,
\begin{equation} \label{hessian}
N^{ab}_{cd}= \frac{\partial^2 S_R}{\partial d_{j_{ab}}d_{j_{cd}}}|_{d_{j}=d_{j_0}}=\frac{\partial \theta_{ab}}{\partial d_{j_{cd}}}|_{d_{j}=d_{j_0}},
\end{equation}
 then:
\begin{equation}
 S_R[d_{j_{ab}}] \simeq 10d_{j_0} \Theta + \Theta \sum_{a<b} \delta d_{j_{ab}} +\f12 \sum_{a<b, c<d} N^{ab}_{cd}  \delta d_{j_{ab}} \delta d_{j_{cd}}=\Theta \sum_{a<b}  d_{j_{ab}} +\f12 \sum_{a<b, c<d} N^{ab}_{cd}  \delta d_{j_{ab}} \delta d_{j_{cd}}.
 \end{equation}
We replace $ S_R[d_{j_{ab}}]$ by  this expansion in (\ref{intDyna}):
\begin{equation} \tabl{ll}{
1 \simeq \frac{\mu}{(2\mathcal{N})^{10}}\int \prod_{a<b}d[d_{j_{ab}}] (\prod_{a<b}d_{j_{ab}})^\sigma P(d_{j_{ab}}) &[ e^{-2i\Theta\sum_{a<b}d_{j_{ab}}-\frac{i}{2}\sum_{a<b,c<d}N^{ab}_{cd}  \delta d_{j_{ab}} \delta d_{j_{cd}}-\alpha \sum_{a<b}\delta d_{j_{ab}}^2}\\
&+ e^{\frac{i}{2}\sum_{a<b,c<d}N^{ab}_{cd}  \delta d_{j_{ab}} \delta d_{j_{cd}}-\alpha \sum_{a<b}\delta d_{j_{ab}}^2}]
}\end{equation}
The first exponential is a rapidly oscillating term in $d_{j_{ab}}$ which will vanish when we perform the integration over $d_{j_{ab}}$ so we only have to consider the second term. Moreover, at the leading order in $\delta d_{j_{ab}}$ we can replace $(\prod_{a<b}d_{j_{ab}})^\sigma P(d_{j_{ab}}) $  by $d_{j_{0}}^{10\sigma} P(d_{j_{0}})$ in the integral (\ref{intDyna}).

At this point, we recall that  $P(d_{j_{ab}})$ grows as $\zeta^{-9/2}$ when scaling all triangle areas $d_{j_{ab}}$ by $\zeta$. So we can write $P(d_{j_{ab}}=d_{j_0})\sim\frac{P}{d_{j_0}^{9/2}}$ in the large $d_{j_0}$ limit, where $P>0$ is a constant. The precise value of $P$ can be computed analytically or determined numerically, but whose its explicit value does not affect the present discussion since it can be re-absorbed in the parameter $\mu$. Indeed, all the equations in the asymptotical regime will depend on $P$ only through the combination $\mu P$.

Then, once again we have to integrate a Gaussian integral:
\begin{equation} \label{WWapprox2}
1\simeq \frac{\mu(d_{j_{0}})^{(10\sigma-9/2)} P}{(2\mathcal{N})^{10}}\int \prod_{a<b}d[\delta d_{j_{ab}}]  e^{-\delta d_{j_{ab}}M^{ab}_{cd}\delta d_{j_{cd}}}
\end{equation}
where $M$ is a ten by ten matrix defined by
\begin{equation}
M^{ab}_{cd}=\alpha \delta_{cd}^{ab} -iN^{ab}_{cd}\qquad a<b, \; c<d,
\end{equation}
with  $\delta_{cd}^{ab}=1$ if the two couples of indices are the same and it vanishes otherwise;  $N^{ab}_{cd}=\frac{\partial \theta_{ab}}{\partial d_{j_{cd}}}|_{d_{j}=d_{j_0}}$ was explicitly computed in \cite{graviton}\footnote{
$$N^{ab}_{cd}=\frac{\sqrt{3}}{4\sqrt{5}d_{j_0}}\left(\tabl{cccccccccc}{
-4 & 7/2 & 7/2 & 7/2& 7/2 & 7/2 & 7/2 & -9 & -9 & -9\\
7/2 & -4 & 7/2 & 7/2 & 7/2 &-9 &-9 & 7/2 & 7/2& -9 \\
7/2 & 7/2 & -4 & 7/2 & -9 & 7/2 & -9 & 7/2 & -9 & 7/2\\
7/2 & 7/2 & 7/2 & -4 & -9 & -9 & 7/2 & -9 & 7/2 & 7/2\\
7/2 & 7/2 & -9 & -9 & -4 & 7/2 & 7/2 & 7/2 & 7/2 & -9 \\
7/2 & -9 & 7/2 & -9 & 7/2 & -4 & 7/2 & 7/2 & -9 & 7/2\\
7/2 & -9 & -9 & 7/2 & 7/2 & 7/2 & -4 & -9 & 7/2 & 7/2\\
-9 & 7/2 & 7/2 & -9 & 7/2 & 7/2 & -9 & -4 & 7/2 & 7/2 \\
-9 & 7/2 & -9 & 7/2 & 7/2 & -9 & 7/2 &7/2 & -4 & 7/2 \\
-9 & -9 & 7/2 & 7/2 & -9 & 7/2 & 7/2 & 7/2 & 7/2 & -4} \right)
$$}:
\begin{equation}
N^{ab}_{cd}=\frac{\tilde{N}^{ab}_{cd}}{d_{j_0}}
\end{equation}
where $\tilde{N}$ is a ten by ten real symmetric constant matrix with all coefficients independent of $d_{j_0}$. In particular, $\tilde{N}^{ab}_{ab}=-\sqrt{\f35} \, \;\forall a<b$. Therefore, $M$ is a symmetric matrix with all its diagonal coefficients equal to $\alpha+i \sqrt{\f35}\frac{1}{d_{j_0}}$.

At this stage we have to distinguish the three cases mentioned above. Assuming that $\alpha$ is of the form $\alpha=\frac{a+ib}{d_{j_0}^n}$, with $a, b \in \mathbb{R}$, $n\in ]0,2[$. The three cases with respect to the power $n \in ]0,2[$ are: \begin{enumerate}
\item $0<n<1$ corresponds to $\alpha \gg \frac{1}{d_{j_0}}$. In this case, we can neglect the terms of order $\frac{1}{d_{j_0}}$ compared to $\alpha$.
\item $1<n<2$ corresponds to $ \alpha \ll \frac{1}{d_{j_0}}$. In this case, $\alpha$ will be negligible with respect to the terms of the order $\frac{1}{d_{j_0}}$.
\item $n=1$ corresponds to $\alpha \sim \frac{1}{d_{j_0}}$. This case should be the most natural case since it peaks in the same way the triangle areas of the 4-simplex around the background value $A_0=d_{j_0}$ and the dihedral angles around the background value $\Theta$.
\end{enumerate}
Let us now separately study the three cases to solve equation (\ref{WWapprox2}).
\begin{enumerate}
\item {\bf The first case $0<n<1$} is the easiest one.
\\
Indeed in this case, the matrix $M\sim \frac{a+ib}{d_{j_0}^n}+ \frac{\tilde{N}}{d_{j_0}}$  can be approximated by a  ten by ten diagonal matrix: $M \simeq \alpha \mathbb{I}=(a+ib) \mathbb{I}$ since $\alpha \sim  \frac{a+ib}{d_{j_0}^n} \gg \frac{1}{d_{j_0}}$ and the ten integrals are then decoupled. Thus, we just have to compute a one-dimensional Gaussian integral:
\begin{equation} \label{equa2-3}
\mathcal{N}= \frac{(\mu P)^{1/10}}{2} (d_{j_{0}})^{(\sigma-9/20)} \int d[\delta d_j] e^{-\alpha (\delta d_{j})^2}= \frac{(\mu P)^{1/10}}{2} d_{j_{0}}^{(\sigma-9/20)} \sqrt{\frac{\pi}{\alpha}}
\end{equation}
which gives a second equation for $\mathcal{N}$ and $\alpha$. Therefore using equation (\ref{equa1}), we obtain the following equation on $\alpha$:
\begin{equation}
\frac{1}{2}\sqrt{\frac{\pi}{2\Re(\alpha)}}= \frac{(\mu P)^{1/5}}{4} d_{j_{0}}^{(2\sigma-9/10)}\frac{\pi}{\alpha}
\end{equation}
Finally, expressing $\alpha$ under the form $\frac{a+ib}{d_{j_0}^n }$, $0<n<1$), we get that:
\begin{equation}
\left\{ \tabl{l}{
\alpha \in \mathbb{R}_+, \\
\sigma= \f14\left( \f95-n \right) \textrm{ and } \sigma>\f15, \\
a= \frac{(\mu P)^{2/5} \pi}{2}, \quad b=0.
} \right.
\end{equation}
In this case, $\alpha$ is real and positive. Furthermore, we get a condition on the normalization factor of the spinfoam vertex $\sigma > 1/5$.

\item {\bf The second case $1<n<2$} implies that $ \alpha \sim  \frac{a+ib}{d_{j_0}^n} \ll \frac{1}{d_{j_0}}$.
\\
In this case, the matrix $M$ reduces to $M=-iN$. We have to integrate: $\int d[\delta d_{j_{ab}}] \exp( i \sum \delta d_{j_{ab}} N^{ab}_{cd} \delta d_{j_{cd}})$. Recall that for a $m \times m $ symmetric invertible matrix $A$ with signature $\sigma(A)$ we have:
\begin{equation}
\int_{\mathbb{R}^m}[dX_i] \exp \left[ i\left( \sum_{i,j} X_i A_{ij}X_j\right)\right]= e^{i\sigma(A) \frac{\pi}{4}}\sqrt{\frac{\pi^m}{|\det(A)|}}
\end{equation}
In our case, $N$ is real, symmetric and its signature, which is the difference between the number of positive eigenvalues and the number of negative eigenvalues, is equal to $-2$. Therefore, $e^{i\sigma(N)\frac{\pi}{4}}=-i$ and our Gaussian integral is an imaginary number which is not compatible with the first equation on $\mathcal{N}$ (\ref{equa1}). So, we cannot have $ \alpha \sim \frac{a+ib}{d_{j_0}^n }$ with $1<n<2$.

\item {\bf The third case $n=1$} corresponds to $\alpha=\frac{a+ib}{d_{j_0}}$.%
\\
 Then the determinant of $M$ is a complex number: its real part  and its imaginary part are polynomials of degree 10 and of arguments $a$ and $b$. We thus obtain a complex equation for $\mathcal{N}$ and $\alpha$:
\begin{equation}
1\simeq \frac{\mu (d_{j_{0}})^{(10\sigma-9/2)} P}{(2\mathcal{N})^{10}}\int \prod_{a<b}d[\delta d_{j_{ab}}]  e^{-\delta d_{j_{ab}}M^{ab}_{cd}\delta d_{j_{cd}}}=\frac{\mu (d_{j_{0}})^{(10\sigma-9/2)} P}{(2\mathcal{N})^{10}} \sqrt{\frac{\pi^{10}}{ \det(M)}}
\end{equation}
Combining this equation with the first equation (\ref{equa1}) that we already have on $\mathcal{N}$ and $\Re(\alpha)=a$ we get a complex equation on $a$ and $b$:
\begin{equation} \label{equa2-1}
2^5\left(\frac{d_{j_0}\pi}{2a}\right)^{5/2}-\mu (d_{j_{0}})^{(10\sigma-9/2+5)} P\sqrt{\frac{\pi^{10}}{ \det(\tilde{M}(a,b))}}=0
\end{equation}
with $\tilde{M}=\frac{M}{d_{j_0}}$. This equation implies a condition on the normalization factor of the spinfoam vertex: $$
\sigma=\f15.
$$
 We can then solve numerically the previous equation by plotting a 3d graph representing the square of the norm of the complex number given by the left-hand side of the previous equation (\ref{equa2-1}) in terms of $a$ and $b$ using Maple (see graph fig. \ref{graphsurface}). For example, the value of this norm vanishes for $$
 a\simeq 1 \textrm{ and } b\simeq 1.99 \quad \textrm{ for } \mu P=1.01\,10^5.
 $$
  Therefore, there exists a specific value $\alpha$ for which $\Psi_{(d_{j_0}, \Theta)}$ is a physical state. However, whereas this case for which $\Psi_{(d_{j_0}, \Theta)}$ is peaked in the same way around the intrinsic and extrinsic geometry $(d_{j_0}, \Theta)$ should be the most natural, it is quite complicated.

\end{enumerate} \qed
\end{proof}

\begin{figure}[h]
\begin{center}
\includegraphics[height=50mm]{3dgraph_a0.95_1.05.eps}
\hspace{20mm}
\includegraphics[height=40mm]{graph_a1_b1.9_2.1.eps}
\caption{We plot the norm squared of the left hand side of eqn \Ref{equa2-1} for $\mu P=1.01\,10^5$ in term of $a$ and $b$. The second graph is a slice of the 3d graph for fixed $a=1$ and variable $b$. \label{graphsurface}}
\end{center}
\end{figure}

The alternative is then to work with a more complication semi-classical state. Indeed although a factorized gaussian wave-packet  is up to now the only one which has allowed to perform numerical simulations, the previous analysis seems to show that it is too simple to catch all the features of a physical state. We show in the next section that  a tensorial Gaussian state is more adapted to describe a physical state. Indeed we will see that such a state allows to compensate the imaginary part which comes from the second derivative of the Regge action and given by the matrix $iN$ and consequently to simplify the resolution of this third case studied above. This is what's commonly expected from the more general perspective of Regge calculus \cite{bianca}.

\subsection{The Coupled Gaussian Ansatz}

To go beyond the decoupled Gaussian ansatz, our new assumption is to consider a boundary state of the form:
\begin{equation} \label{coupledGaussian}
\psi_q[d_{j_{ab}}]= \frac{1}{\mathcal{N}}e^{-\sum_{a<b,c<d}\alpha_{cd}^{ab} (d_{j_{ab}}-d_{j_0})(d_{j_{cd}}-d_{j_0})}e^{i\Theta \sum_{a<b}d_{j_{ab}}}
\end{equation}
where $\alpha$ is now a ten by ten complex matrix. Furthermore we choose
\begin{equation} \label{coupledWidth}
\alpha_{cd}^{ab}= \beta(d_{j_0}) \delta^{ab}_{cd} + iN_{cd}^{ab}
\end{equation}
 where $\beta \in \mathbb{R}$ and the imaginary part of $\alpha$ is  now  the conjugate variables of the dihedral angles of the tetrahedron in the semi-classical regime introduced in (\ref{hessian}):$$
 N^{ab}_{cd}=\frac{\partial \theta_{ab}}{\partial d_{j_{cd}}}|_{d_{j}=d_{j_0}}.
 $$
$N^{ab}_{cd}$ is the Hessian of $S_R$ and we will see that it is this choice which allows to simplify the construction of a physical semi-classical state peaked in the same way on the extrinsic and extrinsic geometry of the 3d boundary. We recall that $N$ depends on $d_{j_0}$ such that $N=\frac{\tilde{N}}{d_{j_0}}$ with $\tilde{N}$ a matrix with constant coefficients.
\begin{proposition}
For $0<n\leq 1,$ we take as ansatz for the width $\beta =\frac{a}{d_{j_0}^n}$ (at least at leading order for large $j_0$) of the Gaussian state (\ref{coupledGaussian}). This defines the real part of the matrix $\alpha$ through \eqref{coupledWidth}. Then the width is uniquely defined and the coefficient $\sigma$ is restricted. More precisely,
\begin{equation} \label{resultcoupled1} \left\{ \tabl{l}{
\beta \in \mathbb{R}_+, \\
a= \frac{(\mu P)^{2/5} \pi}{2}\\
\sigma= \f14\left( \f95-n \right) \textrm{ and } \sigma\geq\f15,
} \right.
\end{equation}
\end{proposition}

Therefore, the case $n=1$ appears now in the continuity of the case $0<n<1$. This can be considered as an improvement compared to the previous case of a factorized Gaussian semi-classical state. However, we have not solved the case $1<n<2$, which admitted no solution in the factorized Gaussian state. The resolution of this case needs the knowledge of the next-to-leading order of the asymptotic expansion of the $\{10j\}$-symbol, which hasn't yet been computed.

\begin{proof}
Regarding the asymptotic behavior of $\beta$, the real part of the matrix $\alpha$, we distinguish the three same cases as in the decoupled gaussian state case.

\begin{enumerate}
\item  The result of the {\bf first case}, $\beta \sim \frac{a}{d_{j_0}^n}$  with $0<n<1$, is not modified. therefore, {\bf $\beta \gg \frac{1}{d_{j_0}}$} and the matrix $M$ can be approximated by a ten by ten diagonal matrix $M\simeq \beta \, \mathbb{I}$ and the state is again factorized and we obtain the same result as in (\ref{result1}):
\begin{equation} \label{result2} \left\{ \tabl{l}{
\beta \in \mathbb{R}_+ \\
a= \frac{(\mu P)^{2/5} \pi}{2}\\
\sigma= \f14\left( \f95-n \right) \textrm{ and } \sigma> \f15
} \right.
\end{equation}
\item The {\bf second case} is when $1<n<2$ in $\beta \sim \frac{a}{d_{j_0}^n}$ ($a\in \mathbb{R}$). Since {\bf $\beta \ll \frac{1}{d_{j_0}}$} in this case, the real part of $\alpha$ is negligible compare to its imaginary part.  Consequently since this imaginary part compensates the second derivative of the Regge action in the exponential of the second condition (\ref{dyna}), we should go to the next order in $\delta d_j$. To compute this expansion in $\delta d_j$ we need to know the next-to-leading order of the asymptotic expansion of the ${\{10j\}}$-symbol. We can thus say nothing now in this case for the moment.

\item The {\bf third case}, {\bf $\beta \sim \frac{a}{d_{j_0}}$} ($a \in \mathbb{R}$), is greatly simplified compared to the equivalent case for the decoupled gaussian wave-packet. Indeed, applying condition (\ref{norm}) and condition (\ref{dyna}) to this state, we obtain the two following equations for $\mathcal{N}$ and $a$:
\begin{equation} \left\{ \tabl{l}{
1\simeq \frac{1}{2^{10}\mathcal{N}^2}\left(\int d[\delta d_{j}] e^{-2\frac{a}{d_{j_0}}(\delta d_{j})^2} \right)^{10}= \frac{1}{2^{10}\mathcal{N}^2} \left(\frac{\pi d_{j_0}}{2a} \right)^5\\
1 \simeq \frac{d_{j_0}^{10\sigma-9/2}\mu P}{2^{10}\mathcal{N}}\left(\int d[\delta d_j] e^{-\frac{a}{d_{j_0}}(\delta d_j)^2}\right)^{10}=  \frac{d_{j_0}^{10\sigma-9/2}\mu P}{2^{10}\mathcal{N}}\left(\frac{\pi d_{j_0}}{a} \right)^5
} \right.
\end{equation}
where we use to compute the second equation the same analysis as in the factorized gaussian state case.  The key element which simplifies everything is that the second derivative of the Regge action added to the imaginary part of $\alpha$ is null since $\alpha$ imaginary part is the Hessian. This system of equations can then be simplified in a single equation for $a$:
\begin{equation}
\frac{1}{2^{10}}\left( \frac{\pi d_{j_0}}{2a} \right)^5 =\frac{d_{j_0}^{(20\sigma-9)}(\mu P)^2}{2^{20}} \left( \frac{\pi d_{j_0}}{a}\right)^{10}
\end{equation}
which implies a condition on the normalization factor of the ${\{10j\}}$-symbol in the spinfoam vertex:
\begin{equation}
\sigma= \frac{1}{5}
\end{equation}
 and the unique solution for $a$:
 \begin{equation}
 a= \frac{\pi (\mu P)^{2/5}}{2}.
 \end{equation}
 Therefore, we now get the same value of $a$ in this case as in the first case. Moreover, $\sigma$ is now uniquely determined since  $n=1$.
\end{enumerate}
\end{proof}
This analysis on the consequences of conditions (\ref{norm}) and (\ref{dyna}) on a coupled gaussian wave-packet (\ref{coupledGaussian})
is therefore more convicting on the ability of such a state to capture a true physical rather than a too simple state defined  as a factorized gaussian state. Indeed, the width $\alpha$ of the coupled gaussian state is now uniquely determined by the requirement of a normalized physical state, as expected from the 3d case \cite{physical3d}.

\section*{Conclusion and Outlook}
The analysis done shows that it is possible to determine a physical semi-classical state for the Barrett-Crane model in the case of the simplest triangulation given by a 4-simplex. The requirement of a normalized physical state determines uniquely the Gaussian width (in term of the precise normalization of the $\{10j\}$ vertex amplitude). This analysis shows also that we cannot take a too simple semi-classical state. For e.g. we have seen that a decoupled gaussian state which is peaked on the geometry of an equilateral 4-simplex does not seem to be able to capture a true physical state. A coupled gaussian state appears as defining more naturally a physical semi-classical state.
\smallskip
We recall that in all our calculations, we have used $K_1[s]$ the lowest order term in the expansion of $K[s]$ in the group field theory coupling constant $\lambda$  and that  $s$ symbolizes the 4-simplex boundary graph. The total propagator kernel $K[s]$ is given by  $K[s]=\sum_{V=1}^{\infty} \lambda^{V}K_V[s]$. This operator should describe a unitary "evolution". In the spin foam framework the notion of evolution is not well-defined,
however we can require the following normalization condition on $K[s]$
\begin{equation}
K[s]\bar{K}[s]=(K[s])^2=1
\end{equation}
that describes the process of creation of a  4-simplex starting from a null 4-volume following by the  annihilation of this 4-simplex into a null 4-volume again. This condition also says that $K$ is by definition a physical state. Using the $\lambda$ expansion of  $K[s]=\sum_{V=1}^{\infty} \lambda^{V}K_V[s]$, we can expand this normalization condition in power of $\lambda$
\begin{equation}
\sum_{V=1}^\infty \lambda^{V}\sum_{V_1, V_2/V_1+V_2=V+1}K_{V_1}K_{V_2}=1
\end{equation}
which simplifies  at the leading order in $\lambda$ in
\begin{equation}
\lambda^2K_1[s]^2=1
\end{equation}
meaning that approximatively $\lambda K_1[s]$ is a physical state and  introducing a new constraint on $\lambda K_1[s]=  \lambda\mu \prod_{a<b}d_{j_{ab}}^{\sigma}{\{10j\}}$. It is this new constraint which allows to determine $\mu$:
\begin{equation}
(\lambda\mu)^2=\frac{1}{\left(\prod_{a<b}d_{j_{ab}}\right)^{2\sigma}{\{10j\}}^2}
\end{equation}
where $\sigma \geq 1/5$ in order that $\mu$ is finite. This restriction on the domain of validity of $\sigma$ is consistent with the results obtained previously.
\smallskip
Since up to now we have only considered the lowest order term in the expansion of $K[s]$ in  $\lambda$, we have in fact fixed  the simplest bulk triangulation and taken into account only a finite number of degrees of freedom.
The next step would be, keeping the same boundary spin network $s$, to explicitly write the next terms: $K_2$, $K_3$,... of the $\lambda$-expansion of $K[s]=\sum_{V=1}^{\infty} \lambda^{V}K_V[s]$. This is not obvious.  A priori,  $K_4$ could be determined from $K_1$ by doing a 5-1 move. Otherwise, from an effective field theory point of view we can  write $K[s]=K_1[s]+ k[s]$ where $k$ takes into account all the possible counter-terms and therefore all the possible bulk geometries for a given boundary spin network. This is equivalent to look at a coarse-grained lattice in which all vertices can be considered as shrunk to a single effective vertex. $k[s]$ is then the weight associated to this effective vertex. In this weight $k[s]$, we expect to get a term proportional to the original Barrett-Crane spinfoam vertex amplitude, $\epsilon {\{10j\}}$, which would come from the contribution of the flat 4-simplex. Moreover, we expect additional contributions representing curved 4-simplex. These extra terms can be written in a condensed way as $\rho  \int dG\, P(G) \,\Gamma_{10j}(G)$, where $\Gamma_{10j}(G)$ is the  spin network functional, on the 4-simplex boundary, evaluated on the non-trivial holonomies $G$. The flat 4-simplex is then simply the term
$\Gamma_{10j}(\mathbb{I})={\{10j\}}$, and $P(G)$ is the measure factor giving the relative weight of every curved configuration.
Other terms will certainly appear in a more precise analysis.
The issue is then to determine which terms of $k[s]$ contribute in the requirement of a physical state for $\psi_q$; that is, we should understand how the condition $\sum_{j_{ab}} K[d_{j_{ab}}] \psi(d_{j_{ab}})=1$ is explicitly modified by the term $k[d_{j_{ab}}]$ of $K[d_{j_{ab}}] $. We would expect that the requirement for the boundary state to be a physical state selects the bulk triangulation. From this perspective, the choice of the phase seems crucial: indeed it is the choice of the flat dihedral angles in the phase factor of the boundary state that is used to select the flat 4-simplex at leading order. The choice of  a different phase for $\Psi_q$ should select another bulk triangulation $\Gamma$, which might correspond a non-trivial space-time curvature. A detailed investigation is required to confirm this rough analysis and to really understand the dominant terms in the computation of the two-point
 function of the gravitational field.
\smallskip
Up to now, we have discussed the case of the simplest 4d triangulation defined by a single 4-simplex. We should go beyond and study more general backgrounds defined on more complex triangulations. We would then expect a more complicated dependence on the boundary spins of both the Gaussian matrix and the phase factor, as happens in Regge calculus (see e.g. \cite{bianca}). To deal with such more complex backgrounds, we would however need to develop more powerful analytical tools to probe the spinfoam amplitudes and their asymptotics. We nevertheless hope to tackle this issue soon.
\smallskip
Finally, another step would be to deal with the EPRL-FK models instead of the Barrett-Crane model to define the propagator kernel $K[s]$. Since the EPRL-FK vertex has also for asymptotic the exponential of the Regge action, we are expected to get similar results in the restricted case of a single 4-simplex. However, the behavior under renormalization should be quite different since the space of intertwiners in the EPRL-FK model introduces new degrees of freedom in the 3d space geometry compared to the Barrett-Crane geometry which is describe by a unique intertwiner and  seems to have too many frozen degrees of freedom. These degrees of freedom will have to be taken into account when gluing 4-simplices together.

redpajama_set_name: RedPajamaArXiv


text:
How digital financial services can support economic opportunity in Central America
With Latin America facing its worst economic downturn in two hundred years and vaccine distribution lagging, there is more urgency than ever for a collective response to drive an equitable, sustained recovery. The necessary acceleration in adoption of digital tools by governments and their residents presents a renewed opportunity to rebuild the economy in a way that builds pathways for families and small business owners to a more financially secure future.
The implications of living in a cash economy and being left out of the formal financial system are vast. Providing more services online is more efficient, safe and creates opportunities for growth. Access to a simple, secure digital account is far more than a convenience. It can mean transparency in getting the market price for your crop, an ability to build savings and gain access to credit, or safely receive a paycheck.
That's why Mastercard is committing to bring 5 million people in El Salvador, Guatemala and Honduras into the formal financial economy and to digitizing 1 million micro and small businesses following a call from U.S. Vice President Kamala Harris to accelerate economic opportunities to the Northern Triangle region of Central America.
This is not new territory for us. Last year we reached our five-year goal of bringing 500 million previously unbanked people around the world into the financial system. We then doubled down and committed to bring 1 billion people and 50 million small businesses into the digital economy by 2025 with support for 25 million women entrepreneurs. Our new commitment in the Northern Triangle is an important part of this initiative.
Our partnerships with banks, fintechs, NGOs and governments in Latin America are already bearing fruit. A digital payments program for farmers in southern Mexico shifted coffee growers' payments to an electronic account, with simple electronic point-of-sale systems set up in more than 200 small shops where the coffee growers purchased their food and supplies. That means no more long trips to the nearest city to cash checks. Training offered to farmers helps them manage and save their money, reducing earnings lost to intermediaries and increasing transparency between farmers and their buyers.
Our work in Latin America continues to be a focal point for our financial inclusion efforts, including forging partnerships with governments to accelerate digitization. We are now working with Panama to digitize its infrastructure and economy with an aim to accelerate inclusive economic growth. That includes digitizing public transit, providing access to affordable tools and services for small businesses to grow, and building a digital infrastructure to reach remote communities, helping them secure access to not only financial tools but vital services such as healthcare and education.
For example, we have been been supporting the digital transformation of micro, small and medium-sized companies to help them survive and thrive. Through the launch of Digital Allies in Latin America and the Caribbean, we're working with technology partners to help them make the shift to digital business strategies and online platforms.
In partnership with Mercado Libre, the largest e-commerce site in Latin America, Common Cents Lab, and the Inter-American Development Bank, we're piloting work in Mexico to design strategies based on behavioral science to support financial resilience for small businesses and gig workers.
Latin America also has one of the highest rates of early-stage entrepreneurship by women in the world, so alongside this, we have been unlocking new models of microcredit for businesses with Unilever, working with USAID under the Women's Global Development and Prosperity Initiative to support the growth of women entrepreneurs in Latin America. In January, Mastercard and USAID launched Start Path Empodera, a business accelerator focused on promoting women's entrepreneurship and financial inclusion in Colombia.
LEADS Mujer, a partnership between Mastercard and the INCAE Business School in Costa Rica, helps women business owners across Central America strengthen their professional skills by providing targeted training. Thelma Martinez de Anda of Honduras was able to pivot her wholesale prepared food business to produce delivery to families in lockdown, crediting the program for helping her business surviving the pandemic: "I applied everything I learned – above all, to be flexible when it comes to change."
The past year has shown how critical digital technology is for people and businesses across the world to be able to access opportunity and have pathways to economic mobility. And we remain committed to harnessing innovation and technology to create these opportunities to ensure the economic recovery includes everyone.
Helping farmers bank on a robust future for coffee
This leadership program helped an entrepreneur in Honduras weather the pandemic
By Sophie Hares
Taking it to go: A coffee entrepreneur's pandemic lessons
By Rebecca Abraham


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