Welcome parents, teachers, community members and participants to the 2015 Hunter Youth Leadership Project blog.
This blog will provide an opportunity for you to gain insight into the Year Nine and Year Ten students' seven day journey of personal development while on their Outward Bound program in the beautiful national parks of northern New South Wales.
The Hunter Youth Leadership Project is made possible thanks to the generous support of the Coal and Allied Community Development Fund, which supports local projects in the Hunter Valley in a range of areas including business development, education and training, health, environment, and arts and culture. The aim of this fund is to increase the long-term sustainability of communities surrounding its three operations in the Hunter Valley and ensure the dependence on mining activities decreases over time.
To develop the young people of the Hunter to be positive and engaged members of society.
To bring together young people in the Hunter region to have a shared experience, meet new friends and increase their social support networks.
To encourage young people of the Hunter to discover other regions of NSW.
To demonstrate how collaborative partnerships between business, not for profit and education sectors can have collective impact on a particular issue and deliver broad community benefits.
To provide opportunities for young people to develop into young leaders, improving their personal development and long-term employability.
Don't forget to bookmark this site so you can check in regularly to read about the students' experiences and adventures while on Outward Bound!
In the dark of the night, the participants left the comforts of their homes nestled in the Upper Hunter area of NSW to be greeted by an early morning in the Northern Rivers.
The day began with a welcome and info about the program and area, followed by a tasty morning tea of handmade goodies from the kitchen. The participants were then split into their camp groups for photos and soon after were off to begin their journey Outward Bound!
The year 9's headed to the camp ground where they were to load their hiking bags for the days ahead.
The first activity to kick off their experience was Raft Building! Getting into the water at the dams was a challenge for some of the students, but once in, the laughter was infectious. A couple of the rafts didn't hold together, which helped spread the laughter throughout the group. At the end of a long, first day both groups enjoyed a tasty dinner of beef stir fry. They each settled into camp life for the night wondering what the days ahead would entail.
The year 10's took part in a Leadership workshop for most of the morning. Initially everyone was quiet, but when the speaker introduced "The Moon" game to the group, it got them moving and giggling. This really helped break the nervous anticipation in the air.
Today the participants learnt some valuable lessons around life choices, the ones we can change and also the one we cannot. They all really enjoyed the workshop.
The afternoon was spent setting up camp, however it was a little difficult when the added the challenge of no talking was implemented!
The afternoon was also spent preparing for the climb up Mt Warning which would begin early the next day. It was a huge day for all, and was finished off by being immersed in community service, pulling out invasive weeds from along the Upper Tweed River.
It was an extra early start for the year 10's this morning who crawled out of their sleeping bags before the sun had risen. Today was the day they would climb Mt Warning (1159 meters). Mt Warning is an iconic land mark in the Northern Rivers, due to its size and history. Many years ago it was formed by an active volcano. It also hold great significance to the local Bundjalung tribe. It was a tough hike with a solid 4km's uphill and a rock scramble to the summit. With team support all participants made it to the summit to be greeted with spectacular views. Everyone was blown away with the view and their efforts as a team.
The participants returned to base after their arduous climb for another activity. The Leap of Faith, which involves climbing 8 metres to the top of a pole and jumping off to touch a hanging buoy. With screams, squeals and laughs heard from a far, everyone had a great time. The day finished off with some packing ready for expedition as well as delicious meal cooked by the kitchen staff, a luxury which the group would soon miss.
The year 9's started their day with some mountain biking! Group 1 headed out to Mebbin National Park, where they started mountain biking. The group had a few funny moments whilst learning mountain bike skills. A sudden squeeze of the brakes too hard and they went over the handle but bounced back up like nothing had happened. Everyone laughed for hours.
Group 1 really enjoyed the mountain biking today! Majority of it was down hill, however they found the 1km hill at the start a challenge but they pushed themselves and got though it.
Group 2 had an early bus ride to the start of their big hike through the Nightcap National Park, where they walked along part of the route that was used as the postal trail between Lismore and Murwillumbah at the turn of the century. As the day wore on, the participants found the end of the 13km hike challenging. The team worked together, listening to the leaders of the day and made it to camp without complaining despite beginning to feel the pressure of carrying full hiking packs. After becoming settled in camp the instructor taught the group a game called 'HA HA' it made the camp site roar with laughter and ended in each group member bonding over the funny moments of the day.
The participants woke up to a clear morning on Day 3. All the groups were heading out on expedition today, so it was an early morning for all. By this stage in the program the groups are getting faster and faster with their camp pack down! This will give them more time throughout the day to achieve their goals and get to camp.
Group 1 had a morning filled with a 7 meter abseil. Most of the group felt uncomfortable with the height of the abseil, however when it came time to tackle the challenge the group really pulled together, encouraging each other. Everyone took on the challenge and pushed out of their comfort zone.
Group 2 enjoyed a lovely day of hiking. The group found the navigating and map reading challenging, and found by sharing ideas and information it made it much more comprehensible. The instructor began telling 'dad style' jokes about a tractor, and somehow this kept the spirits alive in the group! The year 9 instructors mentioned that both groups were really starting to bond and get to know each other really well.
The year 10's have been equipped with new leadership ideas, and given the equipment they will need for the next 3 days out in the bush whilst they explore different parts of the Northern Rivers Circuit. While out there, participants will have the opportunity to put this new knowledge to the test and try out different ways of working together.
The staff at Outward Bound Australia have been watching the weather over the past few days, hoping that the weather system approaching was not going to bring as much rain as predicted in the forecast. Unfortunately, this was not the case. Many conversations were had and the decision was made for the Hunter Youth Leadership Program to finish a couple of days early. While we here at Outward Bound Australia encourage healthy risk taking, there were a number of factors that contributed to this decision. Safety is a paramount focus, especially after seeing the power of nature that had recently hit the Hunter Region.
The students were disappointed that the program was going to finish early, they were keen to make the last day in the bush count.
Group 1 had a 6.2km hike to their designated pick up location. A funny part of the day was to try and avoid the cow paddies while walking through the cow paddocks! While this group were happy to be back at base, they were sad that the program was coming to an end. Pacts were made to come back for year 10.
Group 2 were fast at packing down camp and got to canoeing early. Before their activity the group really enjoyed playing the time bomb game. They really enjoyed the canoeing, and wished they could do more days of it! Once back on base, the group participated in an activity called Lily Pads, which involved a lot of team work for it to be successful.
Group 3 were hiking the historic Nightcap Trail and made great time to have lunch at a place called the Postman's Tree. During lunch they were able to have a good laugh about anything and everything, though they found that once they stopped they would get cold quickly. To avoid this, instead of taking long breaks, they would keep breaks short and make sure that everyone pitched in. The students were proud of themselves and excited to be back at base camp.
Group 4 hiked back from the outer edges of the Outward Bound Australia property after sleeping at Mebbin National Park. The group worked well at packing down camp, with today being their fastest yet. Some of the members found the hiking on this day a challenge and they found it easier by keeping positive.
Group 5 were proud of their pack down time as well, considering the rain showers that were lingering. They kept motivated while hiking, hitting their goals and motivating each other to make it happen. They enjoyed getting back to camp and having a chat in the rain along the way. The group talked about their mixed emotions, some were happy, some were tired while others were excited and sad.
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Founded in 1882, Cuervo Y Sobrinos brings to life the resonant zest and exotic glamour of 19th century Havana, Cuba. As can be expected, most of its timepieces are inspired by the famous Cuban cigar trade such as the Cuervo Y Sobrinos Prominente Dual Time - named after what many considers the best cigar ever made- the Prominente. Crafted in stainless steel or 18k rose gold, the rectangular and ergonomic case measures 30.5mm x 52mm with two winding crowns for its dual time zone function which is in turn powered by two automatic movements. The dial of the Cuervo Y Sobrinos Prominente Dual Time looks vintage with the retro inspired hands and numerals to indicate hours and minutes. Recent models now include day/date and 24h display. Waterproof up to 30m, the Cuervo Y Sobrinos Prominente Dual Time lets one live by the brands motto: "el tiempo lento" - to live moment by moment.


The Five Points of Life Bike Ride will raise awareness for blood, apheresis, bone marrow, organ or cord blood donations. The ride starts in Mobile, Ala., and ends in Cedar Key on Oct. 29.
Organized by LifeSouth Community Blood Centers, which is headquarted in Gainesville, the ride features participants who are the recipient of or have donated blood products. Each applied and were selected from applicants all over the United States.
First-time rider Paul "Skip" Choate, 59, is a lecturer in the entomology and nematology department at the University of Florida.
"People are reluctant because of a natural fear of medical procedures," he said.
San Francisco to Salt Lake City with America by Bicycle, when he met Larry Frederick, a policeman from California who was severely injured while on duty 10 years earlier.
"About 5 percent of eligible people give," McGriff said. "We've got a huge job to do."
"People need these things every day," he said.
"I think family support is huge when it comes to donation," she said.
"We couldn't get by without it," he said.


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Love Story at the Jack Studio Theatre
The show is certainly flawed and far from glossy but it's hard to leave at the end without feeling like you've been part of something, whether that's the trials of the characters or the efforts of the cast and crew.
Macbeth at the Little Angel Theatre
A Puppetry interpretation of "The Scottish Play" – The Kingdom of Fife has become a sick menagerie, with the fowl fowls plotting to move up the pecking order to obtain the thrown. Ingenious!
Sunstroke at the Platform Theatre
Official Theatre went to the brand new Platform Theatre by King's Cross to see a new Russian play by Belka Productions that brought together Bunin and Chekhov in a brand new way. Read our review and let us know your comments!
Little Me at Ye Olde Rose and Crown
Official Theatre headed up to Walthamstow to see the first London revival of CY Coleman and Neil Simon's musical 'Little Me' by All Star Productions at the Ye Olde Rose and Crown Theatre. Read our 3 star review and let us know what you thought by adding your comments below!
Title of Show at the Landor Theatre
Official Theatre reviewed the new production of Title of Show at the Landor Theatre in Clapham, which marks the musical's first London production.
Fringe on the Fringe at Upstairs at the Southern Cross
With the exciting new Fringe venue Upstairs at the Southern Cross opening in the heart of Fulham, we decided to check out their first ever Fringe on the Fringe season.
Titanic at the Southwark Playhouse
We saw the Southwark Playhouse's brand new production of Maury Yeston's musical 'Titanic'. Read our 4 star review and catch up with our thoughts on this impressive show.
Our Town at the King's Head Theatre
The Kings Head Theatre is one of the most active fringe venues in London. This time I went down to check out the official 75th anniversary production of Thornton Wilder's Pulitzer Prize winning play, Our Town
Life In A Sketchbook at the Top Secret Comedy Club
There is no denying Fringe Theatre can be a bit of a mixed bag, however it is often where creativity is born and grown and therefore I felt it was only right to include it Official Theatre. Check out this section as I attempt to discover London's less commercial theatrical gems.
The new West End show, Get Up Stand Up! The Bob Marley Musical, is so much more than just a celebration of regg...
George RR Martin's fantasy epic Game of Thrones is being adapted for the stage. Here's what to expect!
Oklahoma! Wyndham's Theatre
Shirley Valentine Duke of York's Theatre
The Great British Bake Off Musical Noel Coward Theatre
Bonnie & Clyde Garrick Theatre
For Black Boys Who Have Considered Suicide When The Hue Gets Too Heavy Apollo Theatre
Ain't Too Proud Prince Edward Theatre
Vardy V Rooney: The Wagatha Christie Trial Ambassadors Theatre
Aspects of Love Lyric Theatre
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In Germany, family law was one of the first areas to establish the thought of mediation, and still is the main field of application of mediation. Conflicts in Family Law are characterized by a high rate of personal conflicts.
A possible definition of conflict might be:

Conflict means … the perceived differences of interests or the conviction that the actual aspirations of the parties can no be received at the same time.

Basis for personal disputes is the fact that conflicting parties don't come to their own in their personal relationships. After various intents of solving the problems on their own or setting an end to the dispute, the last chance is to get help by a third person. Some hope for an improvement of their  marriage by marriage counseling, but in most cases without success. Others try to solve their conflicts by law and hand in a lawsuit. After reviewing this lawsuit, the judge can suggest to consult a mediator, if reasonable.

There are various constellations of conducting the mediation process. Divorces can be regulated by court. Related issues, like child support, visitation rights or household goods can be resolved by mediation. It is also possible to try to solve all controversies in the  mediation process. In this case, the parties can plead for abating legal proceedings. The employed lawyers can recommend a mediator and at the same time can be legal aid during the mediation process.

The mediator, as against the consulting lawyer, deals with concerns and interests of both parties.
The mediator is not suggested or determined by one party alone. This could lead to unwanted disputes or affect the mediation procedure, because one party could feel disadvantaged.
Whether choosing mediation or not is the parties decision. If they decide to conduct the mediation process, but can't agree as intended, it is not of their disadvantage. The still can go back to the legal process and continue the lawsuit, or file a new one.

Principles of family mediation

Extrajudicial proceeding

At the beginning, the mediation process was known as an extrajudicial proceeding and was introduced as an alternative to the classical lawsuit. In the meantime, more and more mediations are offered in court. They are conducted by a so-called "judge-mediator" after the commencement of proceedings. The number of judges with this additional qualification is not very high, but it is increasing. This development leads to a change in the point of view of mediations. They are no longer seen as alternative to lawsuits, but are regarded as possible element of the litigation. There is no longer competition between the legal process and the mediation process, but addition.

Voluntariness

It is the marriage partners' decision whether to conduct a mediation process or not, even if it is recommended by court. There is no constraint to take part in the mediation. All parties, also the mediator, are having the right to end the mediation process. So there is no obligation to bring the process to an agreement, exiting the process is possible at any time.

Neutrality

It is the mediator's obligation to be a neutral facilitator. He arbitrates and serves the parties equally under the precondition that he is absolutely neutral and impartial. He does not favorite one site and needs to treat both parties equally. It would be useful for the mediator not knowing the disputing parties, to prevent preferences, discrimination or personal influences on the decision. The confidence of both parties towards to mediator of great importance for a successful cooperation. If a party is not sure whether the mediator is neutral or not, it is best to continue the mediation process with another mediator.

Lack of decision-making power

The above-mentioned criteria of neutrality and impartiality are also concerning judges. So where is the difference to a mediator? The mediator's essential characteristic is not having any decision-making power in a legal sense. He can support the parties finding a solution and guide them to the right way. At the end of the mediation process, he can make a written agreement, to which the parties must uphold. He needs to fulfill the formalities given for such an agreement. To become legally valid, the agreement needs to be recorded by Family Court or notarially certified.

Confidentiality

To protect the parties, the mediation process is confidential. When both parties are sure that their information is not made public and is protected against malpractice, they are able to solve the conflict more open. They can be sure that problems that were discussed stay within the mediation group. If the mediation fails, the principle of confidentiality is of high importance, because parties can be sure that, if it comes to a lawsuit, information revealed in the mediation process can not be used against them. This applies for the participating parties as well as for the mediator. Information given by the parties in one-to-one conversations with the mediator also come under the non-disclosure obligation. He is not allowed to reveal it to the other party, except he obtains the permission of the party.

The process of family mediation

The mediator passes through a process with the parties that is divided into five phases:

Phase 1: Initiation of mediation process

At the beginning the mediator has to comment his mission and function, so that the parties cannot be disappointed during the process. After this discussion, it must be settled if the existent problem of the parties can be answered by a mediation process. At the end of this phase a mediation contract is signed, which contains among other things rights and obligations of the mediator and the parties (modal contract ). The meaning of the contract is to lay a code of behavior, on which the involved must be keeping with.

Phase 2: Theme collection
With the second phase starts the real conflict management. The extent of the conflict will be made clear and the points which are treating over the process are listed in detail. This way the mediator receives an overview about the dimension and is able to create a plan.

Phase 3: Resolve the interests
This phase is the most important chapter of the mediation process. It is basic to deal explicitly with the conflicts. In this phase the parties must reveal their problems and voice their interests, to find the optimal problem-solving approach. If wishes or demands are not communicated at this, it will be difficult to involve those afterwards in the next phase.

Phase 4: Options and solutions
After collecting the interests they can be search for common solution options. The different ideas will be collect and afterwards evaluated together. The ideas can be collected in different ways, for example by brainstorming. A solution concept will be designed in this phase and that will be developed in the last phase.

Phase 5: The finish
At the end of the mediation, the solution concept from phase 4 will be a phrased into binding agreement. The agreement can be discussed with a  consultant lawyer. This way the parties can be sure that they have made a decision which is legally correct. The contract is an obligatory agreement. It needs to be validated, for example by notarization.

Advantages of family mediation in comparison to litigation

There is no categorization in wrong or false. No party will be inculpated to be wrong or to have made mistakes in the past. The conflicts are reviewed, but with a forward-looking view to find the best solution the two parties. An advantage of mediation is, that the problems are not delegated, the parties try to solve the conflict themselves. This course might take longer, but after all the parties came as opponents to the mediation process and not as friends. In most cases, the idea of solving the problem wins after some time. Due to the voluntariness of the mediation process, a high commitment of the two parties is to be estimated. A positive attitude is thereby part of the course because the parties can agree in peace. Both parties must agree to the arrangement and need to implement the changes in the future. Furthermore, the mediation should encourage friendly contact between the parties after the mediation process.  Mediation shall be an impulse to advance the attitude in the future. So  the involved can easily stick to the bargain.

The mediation isn't a judicial action, but for all that the law will be included. Admittedly not with the application of laws but only with the use of them. That means that mediation should be seen as law clarification. The purpose is to represent the three following points, without applying the law:
 Consultation about the legal position: The legal situation will be explained, and how a possible lawsuit would end.
 Decision-making: The legal options will be explained to the parties and they will be encouraged to represent their own interests.
 Form of contract: The parties are shown how the bilateral agreement is transferred into a contract, what legal consequences there are, and which formal requirements are essential.

An advantage of the family mediation in comparison to a lawsuit is the command of openness. The disputing parties are obliged to reveal all necessary facts and information. If any documents are revealed after the end of the mediation process, the agreement can be invalid. In some cases the statutory offense of fraud is given.
Another advantage of family mediation is the time. A lawsuit often takes several years, depending on the number of instances. The mediation process takes only several months depending on the type of conflict and the commitment of the disputing parties.
Another advantage results from the time saving. The mediation process is less expensive. The mediator claims of course a charge, but this doesn't overrun the court costs.

References

External links
   Guidelines BAFM
  Modal Mediation Contract

Mediation
Family law
Law of Germany

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Infestation: Annecy Animation Festival 2019: The New Looney Tunes Cartoons
ANNECY ANIMATION FESTIVAL 2019: THE NEW LOONEY TUNES CARTOONS
During The Annecy Film Festival we were presented with 10 of the new Looney Tunes shorts, as well as some work-in-progress sneak-peaks from the new creators Audrey Diehl, Peter Browngardt and Alex Kirwan. The new Bugs Bunny cartoons may not sound like something to be wowed about, as these characters have never exactly been gone from the public eye, but it's been awhile since they were prominently featured in new content. For the last few decades Warner Bros. were experimenting with different ways to resurrect the brand, from TV shows like "Duck Dodgers", "The Looney Tunes Show" and "Wabbit", to full movies like "Space Jam" and "Looney Tunes Back in Action", or even desperate spin-offs like "Baby Looney Tunes" and the horrid superhero reincarnation "Loonatic Unleashed". While there were certainly some fans of these, no self-respecting Looney Tunes lover would dare to put them on par with the original classics. These shorts were created for the cinema and there was real theatrical quality to the animations. But where Disney decidedly focused on making their theatrical animations feel artistic, Looney Tunes directors like Tex Avery, Bob Clampett, and Chuck Jones put their artistry in slapstick, comedic timing and the zany energy of the characters. If you watch the shorts from the 30's it's clear that the cartoonists had the freedom to go as far as possible with the comedy resulting in some of the most memorable and beloved animated shorts ever made. Finally, after over half a century, Warner Bros. decided to create some new Looney Tunes that could be shown next to the classic cartoons. I'm happy to report they didn't disappoint.
It's not clear if Warner Bros. is planning to release these cinematically or simply on a streaming service. No discussion on this topic was breached. However, if the option were available to watch these on a big screen, I'd jump at the chance. Much like the originals, the animation simply jumps out of the screen, with plenty of clever details as well as terrific backgrounds and at points, truly cinematic storytelling. In fact, the artists stayed so true to the originals that they made a rule of never reusing the same backgrounds, creating each new short entirely from scratch. Porky and Daffy's homes are different in each short, character designs had small alterations each time and even the color palettes are different, all in the name of treating each one as its own mini-movie. Much like with the originals there are no scripts. The process of making each new short starts with the cartoonists making a bunch of drawings until the idea is funny enough and after figuring out the most insane jokes they can, they storyboard the plot. There's also not a set running time to the shorts. Some will take seven minutes, others three or four. These are cartoons made by artists who want to produce timeless entrainment and not a bunch of studio executives who are only interested in what's 'hot' at the moment.
The million-dollar question is, are the shorts funny? I can't remember laughing this hard at anything Bugs Bunny-related in a very long time. In one of the first presented shorts, Bugs Bunny gets captured by Elmer. In order to save his skin, the wrascally wabbit pretends to be sick, forcing the hunter to take care of him and nursing his future meal to get back into shape. The premise is simple enough but everything from the over the top expressions of "dying" Bugs, to the tempo of the slapstick, to Elmer slowly losing his sanity all felt like something Avery or Clampett would have made back in the day. The most pleasant surprise was that the cartoon violence didn't felt toned down in any manner. Characters would chase each other with axes, get hit on the head or get their bones broken in imaginatively painful manners, quite literally at points. In one scene something hits Elmer's spine and there is a quick X-ray showcasing how the pain affects his body.
All the other Bugs cartoons continue this manic energy. In another short, Bugs Bunny is going to have a vacation in Las Vegas but by mistake goes to Egypt (that stupid left turn at Albuquerque!) where he mistakes an ancient pyramid for a luxury hotel-casino and the angry, just-woken mummy for the service. The slapstick set up once again is simple, perhaps reminiscent of Bugs Bunny's visit to Transylvania in one of the old Chuck Jones episodes, and the humor goes to surprisingly dark places. In one scene Bugs mistakes a bunch of organ-stuffed funeral urns for a buffet and starts making a plate. Later, he sees some horrifically brutal ceremonies depicted on hieroglyphs to which he comments, "Meh, these comic strips aren't what they used to be". A different short features Bugs trying to outwit amusement park employee Yosemite Sam who is preventing rabbits from riding the rollercoaster. At one point Sam rescues a baby and after that he turns to the audience, "What? I may like to harm rabbits but I would NEVER hurt a baby… Oh, yeah? I'll prove it to you!" and then runs to a random woman in the street and asks if he can he kiss her baby, much to the woman's confusion. It's such a random non-sequitur but that's the level of trademark insanity I haven't seen from these characters since the '40s.
Bugs here is not the wisecracking heroic character that he evolved into, but his original trickster persona. He isn't there to save the day but to screw with people for the sheer pleasure of it. This is especially evident in one short that almost feels like a middle finger to Space Jam. One of Chuck Jones' key complaints about that movie was that Bugs Bunny would NEVER ask an NBA player for help, he would outsmart his opponents on his own. Here, Bugs challenges a gigantic, muscular, and down-right monstrous bully to a basketball game and tricks the guy so much you can almost hear Chuck yelling from the grave "YES! THANK YOU! THAT'S HOW BUGS WOULD DO IT!" Bugs even destroys his new Jordans, which may have been a tad too on the nose.
The Bugs cartoons are fantastic but my favorite shorts featured Daffy. There have always been two different versions of the character. The later Daffy is the greedy loser whose entire existence seems to be centered around being humiliated by Bugs, Speedy Gonzales, or whichever other of the Looney Tunes happens to be there. For some reason, this is the version they've stuck to for the last few decades. All but forgotten is early Daffy, the zany ball of looniness who exists to spread madcap chaos, usually with Porky as his unfortunate victim. Once again, the new Looney Tunes decided to revert the character to the earlier version and it's very refreshing. A fantastic bit features Porky trying to prevent Daffy from making footsteps on wet cement which goes on for a good four minutes and is all done in one shot. There is also a fun and action-packed story about the two taking a trip to the jungle looking for "The Treasure of The Monkey Bird".
The best of the shorts involved the two playing baseball. Not only is it a hilarious set-up, with Porky breaking his arm and Daffy hiding under his t-shirt to pitch for him, but the Daffy in the short, while still wacky, is also incredibly helpful and nice. There's one brilliant joke after another and even an appearance by the Gas-House Gorillas, the baseball team Bugs played against in the Friz Freleng classic "Baseball Bugs". I certainly hope that in a future short trickster Bugs and zany Daffy clash. For once the two would be equally matched in their bags of tricks and unpredictability.
Road Runner also got a cartoon, which was, of course, a collection of Willie Coyote failures, and a Sylvester and Tweety episode set during the winter. While I enjoyed them and there were some clever bits, they didn't feel as creative as the Bugs and Daffy efforts. The main complaint some have always had about the Road Runner and the Tweety cartoons is they never stray very far from the formula, so in that regard, I suppose this felt true to the predecessors. These two not being as memorable as the rest is the closest thing I have to a complaint. I guess some purists might be bothered that the humor at times is too Ren and Stimpy. There are the strange close-up shots or jokes that have a character's skeleton jumping out of their bodies, but I didn't find it discordant. Comedy has evolved quite a bit since the original Looney Tunes were created; new cartoons had to up the zaniness. John Kricfalusi, the creator of Ren and Stimpy, never hid how much he was strongly influenced by Bob Clampett so this felt like a natural progression. However, it was impossible to miss that the creators didn't feel the need to make the settings or the humor overly modern. Bugs making a jab about Elmer "body shaming him" is the closest moment I can think of that counts as modern commentary.
The best news is that these are only 10 shorts out of 200 that are set to be produced! If this is just the first sample, I can't imagine how much more goodness the future holds. The creators already promised the return of such classic characters like the Tasmanian Devil, Marvin The Martian, the somewhat obscure Pete Puma, Beaky Buzzard, Gossamer, Hubie and Bertie, and even from way back in the black and white shorts, Petunia Pig and Porky's troublemaking nephew Cicero Pig. The character designs often took inspiration from their oldest appearances and at the Annecy presentation, we were shown animatics and storyboards promising some fun future scenarios like Elmer trying to finally grow hair, Bugs fighting the Tasmanian Devil at the Roman Coliseum, or trying to teach Beaky Buzzard how to hunt for rabbits.
Warner Bros should be proud of themselves. While only time will tell if these shorts will become classics, they decidedly will be seen as a noble attempt to bring the "Looney" back into Looney tunes. The cartoons are manic, beautifully animated, and feature amazing voice acting. Even the music tries to recapture the spirit of the originals. However, I do hope that the creators won't be afraid to try new things rather than being nothing but slavish to the old shorts. From what we've seen so far, this may well be the case. In that spirit, unlike a lot of modern reboots of old shows and cartoons, there's no bombardment of Easter-egg references; if they were any at all, they were very subtle. As long as there is imagination there will always be wacky scenarios for these characters to have adventures in. I hope the first 200 shorts are only the beginning.
THAT'S ALL FOLKS!
~Maciej Kur
Check out the first of the new Looney Tunes, Dynamite Dance!
Tags:Annecy Animation FestivalBugs BunnycartoonsDaffy DuckInfestationreviewThe New Looney Tuneswarner brothersWB


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\section{Introduction}

Three-particle channels are important in spectroscopy of  excited mesons. For example, $G$ parity forbids a two pion decay of the $a_1(1260)$ meson so that the width of this axial vector resonance is determined by its coupling to three pions. The three pion decay channel is allowed for the exotic, $J^{PC}=1^{-+}$, $\pi_1$ meson, while the exotic $h_2$ with $J^{PC}=2^{+-}$ could decay to $b_1[\omega\pi]\pi$ and $\rho\pi$.
 Because of their possible hybrid nature and relation to
confinement~\cite{gluex} these mesons are a subject of intense theoretical and experimental investigations~\cite{Alekseev:2009aa}.
In the baryon sector, the situation is even more complex because two and three-particle systems often mix. Phenomenologically it is observed that the $\pi\pi N$ channels have significant influence and cannot be neglected when determining resonance parameters
of nucleon excitations.  The number of possible quantum numbers ($\pi\Delta^{(*)}$, $\pi N^*$, $\sigma N$, $\rho N$,\dots) for a given spin and parity $J^P$ is large  so that one has to truncate. Yet, phenomenology can be a guideline to select relevant channels, e.g., by inspecting the involved centrifugal barriers.
For example, it is reasonable to assume that the Roper resonance $N(1440)1/2^+$ is dominated by the $\sigma N$ channel (all three particles in relative $S$-wave) and maybe the $\pi\Delta$ channel, while the $\rho N$ channel could be less important at smaller energies where only pions from the low-energy tail of the $\rho$ can be on-shell.

Pioneering calculations in lattice QCD of the spectra of light excited mesons~\cite{Dudek:2009qf} and baryons~\cite{Edwards:2011jj, Engel:2013ig} deliver a semi-quantitative picture in which pion masses are large and finite-volume effects are usually neglected.
Energy eigenvalues have been calculated for the $a_1(1260)$~\cite{Lang:2014tia} using $q\bar q$ operators but also a $\rho\pi$ meson-meson operator, though taken at zero momentum, $\pi({\bf 0})\,\rho({\bf 0})$. Recently, the HadronSpectrum collaboration calculated isospin $I=2$ $\pi\rho$ scattering although the $\rho$ meson is stable at the pertinent pion mass~\cite{Woss:2018irj}.

Few-body systems above threshold represent the next milestone for the ab-initio understanding of the strong interaction through lattice QCD calculations. Therefore, the infinite-volume extrapolation of three-body systems has attracted much interest recently~\cite{ Klos:2018sen, Konig:2017krd,Briceno:2018mlh, Briceno:2017tce, Hansen:2016fzj, Hansen:2015zga, Hansen:2014eka, Romero-Lopez:2018rcb, Hammer:2017kms, Hammer:2017uqm,  Meissner:2014dea,  Polejaeva:2012ut, Mai:2018djl, Doring:2018xxx, Mai:2017bge,  Briceno:2012rv, Kreuzer:2012sr,Guo:2016fgl,Guo:2017ism,Guo:2017crd,Guo:2018xbv,Agadjanov:2016mao,Hansen:2017mnd,Orasch:2018xxz}, including an extension to coupled $2$ to $3$-body channels~\cite{Briceno:2017tce} and an study of connection between low temperature condensation and scattering in lattice $\phi^4$ theory \cite{Orasch:2018xxz}.
For the three-particle system in the finite volume, the energy eigenvalues are expected to change not only quantitatively but also qualitatively above  threshold, compared to the two-body case.
One can easily understand this by considering the non-interacting energies. For two particles of equal mass $m$ in a cube of side lengths $L$ with periodic boundary conditions, they are given by $E=2E_{\bf n}$ where $E_{\bf n}=\sqrt{m^2+{\bf p_{\bf n}}^2}$ and ${\bf p_n}=(2\pi/L)\,{\bf n}$, ${\bf n}\in \mathbb{Z}^3$.
In contrast, noninteracting three-body energies are given by $E=E_{\bf n}+E_{{\bf n}'}+E_{{\bf n+n}'}$ where ${\bf n,n}'\in \mathbb{Z}^3$. This pattern is entirely different from the two-body case which is expected to reflect in the interacting spectrum, as well.
Apart from conceptual challenges, there is also the problem of underdetermination, {\it i.e.} there being a plethora of possible channels  in contrast to scarce data, and, for the selection of relevant channels, similar considerations as in the infinite volume will have to be made.

Four-body systems above threshold have so many possible combinations of quantum numbers that one can only hope to address the simplest cases. A prime example is the energy region above the four-pion threshold in $\pi\pi$ scattering. In analyses of experiment, usually the four-pion channel below $1\mbox{ GeV}$ is neglected given that the experimentally measured partial wave amplitudes are almost elastic in this energy region (see  Refs.~\cite{Niecknig:2012sj,
GarciaMartin:2011cn} for a calculation of the inelasticity). Similarly, the
 existing lattice studies of  $\pi\pi$ partial waves
   exclude the  data above the four pion intermediate states for the infinite-volume extrapolation (see, e.g., Refs.~\cite{Guo:2018zss, Hu:2017wli, Wilson:2015dqa, Bolton:2015psa}).

   So far, in the three-body calculations only two meson-meson or meson-baryon operators are used, and not yet three of type $\pi\pi\pi$ or $\pi\pi N$. An exception is the $N$-body threshold calculation of positively charged pions by the NPLQCD collaboration~\cite{Beane:2007es, Detmold:2008fn}. Energy eigenvalues for the Roper~\cite{Lang:2016hnn} resonance above the $\pi\pi N$ threshold have been calculated recently including non-local $\pi N$ and $\sigma N$ operators. This calculation was performed close to the physical pion mass such that the extracted energy levels lie above the $\pi N$ but also above the $\pi\pi N$ threshold. An unusual pattern was observed that could originate from the afore-mentioned three-body dynamics in coupled $\pi N$, $\sigma N$, $\dots$ channel, where the $\sigma$ has to be understood as an interacting two-body subsystem and not a stable particle.

In a rough and incomplete classification, there are three major approaches to solving the three-body problem in the finite volume, in the momentum space representation:  An relativistic, all-orders perturbation theory
pursued by Brice\~no, Hansen, Sharpe~\cite{Briceno:2018mlh, Briceno:2017tce, Hansen:2016fzj, Hansen:2015zga, Hansen:2014eka}, a non-relativistic dimer formalism by Hammer, Pang, Rusetsky et al.~\cite{Romero-Lopez:2018rcb, Hammer:2017kms, Hammer:2017uqm,  Meissner:2014dea,  Polejaeva:2012ut} and a method based on three-body unitarity to identify on-shell configurations and, therefore, power-law finite volume-effects by Döring and Mai~\cite{Mai:2018djl, Doring:2018xxx, Mai:2017bge}. For the latter two approaches, the partial diagonalization of the amplitude according to cubic symmetry was discussed in Ref.~\cite{Doring:2018xxx}.

All these approaches aim at fully mapping out the few-body dynamics but there are also attempts to obtain essential information from these system without the need to take explicitly all degrees of freedom into account~\cite{Agadjanov:2016mao, Hansen:2017mnd}. The first-ever prediction of excited three-body energy eigenvalues of a physical system  ($\pi^+\pi^+\pi^+$), from  two-body scattering information and lattice threshold eigenvalues~\cite{Beane:2007es, Detmold:2008fn} was achieved recently~\cite{Mai:2018djl}.

\subsection{Variational approach}
Most of the approaches to the few-body problem, including the examples discussed above, rely on  momentum-representation of the reaction amplitude or correlation function in the finite volume. There is, however, an alternative approach based on Faddeev equations and two- and three-body wave functions in configuration space \cite{Guo:2012hv,Guo:2013vsa,Guo:2016fgl,Guo:2017ism,Guo:2017crd,Guo:2018xbv}. The  finite volume wave function is related to the infinite volume wave function by a  linear superposition over infinite sets of periodic cubic boxes, the quantization conditions are subsequently obtained from  matching conditions   \cite{Guo:2012hv,Guo:2013vsa,Guo:2016fgl,Guo:2017ism,Guo:2017crd,Guo:2018xbv}. One of the advantages of this approach is that the connection between long-distance correlation over boxes and short-distance interaction within each cubic box is made explicit by construction of finite volume wave function.

In the present work, we   set up   and explore a foundation to a potentially convenient  numerical approach to   the $N$-body interaction in finite volume. The formalism presented in this work is based on  the variational method~\cite{Kohn:1948col,Kohn:1954KKR} combined with  the
Faddeev approach \cite{Faddeev:1960su,Faddeev:1965,Gloeckle:1983,Faddeev:1993}.   A brief summary of the variational principle and Faddeev approach are presented in Appendix \ref{variational} and  \ref{faddeeveq} as a short reference for a reader who is not familiar with above mentioned methods.

Based on the traditional variational principle, the secular equation may be obtained   by considering: \mbox{$\delta \Lambda =0$} with     \mbox{$\Lambda =  \langle \Phi | E- \hat{H}^{(L)} | \Phi\rangle $},  where  both the $N$-body  Hamiltonian $\hat{H}^{(L)} = \hat T + \sum_{i<j}\hat{V}_{(ij)}^{(L)} $ and the trial wave function $\Phi$ display periodicity in the cubic lattice, $\hat T$ is the kinetic energy operator and  $\hat{V}_{(ij)}^{(L)}$ stands for the periodic pair-wise interaction between the $i$-th and $j$-th particle. Instead of the traditional approach, in this work we
write the total wave function as a sum of multiple terms \cite{Faddeev:1960su,Faddeev:1965,Gloeckle:1983,Faddeev:1993}: \mbox{$ \Phi  = \sum_{(i<j)=1}^{N} \Phi^{(ij)}$}. Each component of the total wave function,  $\Phi^{(ij)}$, is required to satisfy the equation  \mbox{$(  E- \hat{T} ) | \Phi^{(ij)} \rangle =\hat{V}_{(ij)}^{(L)} | \Phi\rangle $}.  Therefore, a single Schr\"odinger equation, \mbox{$(E- \hat{H}^{(L)} ) | \Phi\rangle =0$},  is turned into $N(N-1)/2$   coupled equations. The advantage of Faddeev approach is to allow one to incorporate the dominant subsystem structures in an adequate way and to use two-body scattering amplitudes as input for the $N$-body dynamical equations.  By splitting the complete $N$-body wave function,  ultimately    $N(N-1)/2$   secular equations may be obtained by considering
\begin{equation}
 \delta \left[ \langle \Phi |   (  E- \hat{T} ) | \Phi^{(ij)} \rangle -  \langle \Phi |  \hat{V}_{(ij)}^{(L)} | \Phi\rangle \right ] =0. \label{introsecular}
\end{equation}
 As will be shown in Section \ref{discussion}, the secular equations obtained from Eq.~(\ref{introsecular})  resemble two-body secular equations.

 In solid state and condensed matter physics, the Linear Combination of Atomic Orbital (LCAO) method for the calculations of the electronic structure of periodic systems \cite{Ziman:1964} provides a elegant way to construct a wave function that satisfies periodic boundary conditions. The trial wave function that describes electrons traveling in a periodic crystal is constructed by linear superposition of all atomic orbital solutions of  an isolated atom centered at each cell of the crystal. In this way, the periodic boundary conditions of the wave function is automatically guaranteed, and the energy spectra are given by   secular equations from the variational principle.
 Similarly   as also suggested in Refs.\cite{Guo:2013vsa,Guo:2016fgl,Guo:2017ism,Guo:2017crd},  the trial finite-volume $N$-body wave function    may be constructed from the $N$-body    infinite-volume wave function,  \mbox{$ \Phi (\{ \mathbf{ x}  \})  = \sum_{ \{ \mathbf{ n}_{\mathbf{ x}} \} \in \mathbb{Z}^{3}} \Psi( \{ \mathbf{ x} + \mathbf{ n}_{\mathbf{ x}} L \} )$},
 where $\Psi$ is the solution of the corresponding infinite-volume Schr\"odinger equation, \mbox{$(E- \hat{H} ) | \Psi\rangle =0$}, and the set $\{ \mathbf{ x}  \}$ stands for the complete set of particle positions.
Hence,  the trial finite-volume $N$-body wave function satisfies periodic boundary conditions by construction. In principle, the infinite-volume wave function may be solved by standard methods, such as Faddeev's approach \cite{Faddeev:1960su, Faddeev:1965, Gloeckle:1983, Faddeev:1993}.

The immediate gain of the variational approach for finite volume systems is  evident.
(1) The construction of the finite-volume wave function from the infinite-volume wave function presents a clear connection  between short-range $N$-body dynamics within each image of the cubic box and long-range correlations in the entire periodic lattice structure. The
$N$-body dynamics  within each cubic box is  determined  by $N$-body Faddeev equations. The long-range correlation effect accumulated from all cubic boxes is implemented by the linear superposition of all     wave functions   centered at each image of cubic boxes. (2) The quantization conditions (secular equations) due to the periodic structure of the lattice are imposed by a variational approach, and eventually yield the discrete energy spectra of the system in the finite volume.  (3) The  variational formalism is   mathematically  transparent and may be suitable for the numerical evaluation of the $N$-body finite volume problem. However, it comes at the price of sacrificing the explicit analytical expressions of quantization conditions as present in the Lüscher formula \cite{Luscher:1990ux} for the two-body problem and  at the high cost of computation of the $N$-body phase space integration.

As  we present here the first attempt to calculate the $N$-body interaction in finite volume with the proposed methods, and also for the sake of simplification of the discussion, in this work, we only consider a simple model with $N$ non-relativistic spinless particles interacting through pair-wise short-range potentials. Given the   infinite-volume wave function, $\Psi  $, that may be solved by the Faddeev approach and  is used as input to the finite-volume problem, the quantization conditions for the finite volume $N$-body interaction are  obtained and presented in Section \ref{Nbform}.
In principle, three-body forces and coupled-channel effects may be included in the formalism as well. However,
 such type of effects are not considered in the present work. In addition, bound states below threshold may be  described  by the analytic continuation of scattering amplitudes. In the present work, the focus is only given on the presentation of the $N$-body problem with pair-wise interactions. The discussion of three-body force, coupled-channel effect, and bound states below threshold will be given in the  future publications.

An effective two-body formalism may be accomplished by integrating out the rest of the degrees of freedoms of the $N$-body system.
The resemblance of the $N$-body quantization condition   to the two-body quantization condition will be discussed in Section \ref{Nbeff}. At last, in the Appendix we also present  some main results   for a  pair-wise short-range $\delta$-function potential. The renormalization issue due to the singular nature of the $\delta$-function interaction in $3D$ is also discussed.

The paper is organized as follows. In Section \ref{Nbform} we present the derivation and main results of the variational approach to the $N$-body interaction  in finite volume. The discussion and summary are given in Section \ref{discussion}.

\section{Variational $N$-body interaction formalism  in finite volume}\label{Nbform}

\subsection{Secular equations of $N$-body interaction in finite volume}
 The  dynamics of $N$ non-relativistic particles interacting via  pair-wise  interactions in the finite volume with periodic boundary conditions is determined by the Schr\"odinger  equation,
 \begin{equation}
 \left [ E+  \sum_{k=1}^{N} \frac{\nabla^{2}_{\mathbf{ x}_{k}} }{2 m}   -  \sum_{(i < j)=1}^{N} V^{(L)}_{(ij)} (\mathbf{ x}_{i } - \mathbf{ x}_{j})    \right ]   \Phi( \{ \mathbf{ x}  \}    )=0. \label{fvschr}
\end{equation}
where we take all masses to be equal to $m$,
 $\mathbf{ x}_{i}$ denotes
  particle position and   $\{\mathbf{ x}\} \equiv \{\mathbf{ x}_{1},\mathbf{ x}_{2}, \cdots, \mathbf{ x}_{N} \}$.     The     potential between  the  $i$-th and $j$-th particles is described by   \mbox{$ V^{(L)}_{(ij)}$}. In finite volume,   \mbox{$ V^{(L)}_{(ij)}$}   displays periodicity when the  distance of  the $i$-th and $j$-th particles is larger than the size of the cubic box.
The    \mbox{$ V^{(L)}_{(ij)}$} may be written as the superposition of all the  potentials centered at each image of the periodic cubic box,
\begin{equation}
V^{(L)}_{(ij)} (\mathbf{ r}  ) = \sum_{\mathbf{ n} \in \mathbb{Z}^{3}} V_{(ij)} (\mathbf{ r} + \mathbf{ n} L) ,
\end{equation}
 where     \mbox{$ V_{(ij)}$} is the   potential between  the  $i$-th and the $j$-th particle in  the same box, and $L$ is the size of  the three-dimensional cube.
 Because of the periodicity of the finite-volume potential,
 the finite-volume  $N$-particle wave function, $\Phi  (\{ \mathbf{ x}  \})$, must also satisfy the periodic boundary condition,
 \begin{equation}
 \Phi   (\{ \mathbf{ x} + \mathbf{ n}_{\mathbf{ x}} L \} ) = \Phi  (\{ \mathbf{ x} \} ),
 \end{equation}
 where $\{ \mathbf{ n }_{\mathbf{ x}} \} = \{ \mathbf{ n }_{\mathbf{ x}_1},\mathbf{ n }_{\mathbf{ x}_2}, \cdots, \mathbf{ n }_{\mathbf{ x}_N} \} $ and $\mathbf{ n }_{\mathbf{ x}_i} \in \mathbb{Z}^{3}$.

Following the Faddeev approach  \cite{Faddeev:1960su,Faddeev:1965,Gloeckle:1983,Faddeev:1993}, that is briefly summarized in Appendix \ref{faddeeveq}, the $N$-body finite-volume wave function may be expressed   as the sum of  $N(N-1)/2$ terms,
\begin{equation}
\Phi  (\{ \mathbf{ x} \} )  = \sum_{(i<j)=1}^{N} \Phi^{(ij)}  (\{ \mathbf{ x} \} ) , \label{fvwavsum}
\end{equation}
 with wave function
  $\Phi^{(ij)}$ required to satisfy the equation
  \begin{equation}
 \left ( E - \hat{T}       \right )  \Phi^{(ij)}( \{ \mathbf{ x}  \}  )  =  V^{(L)}_{(ij)} (\mathbf{ x}_{i } - \mathbf{ x}_{j})     \Phi ( \{ \mathbf{ x}  \}  ) , \label{phiij}
\end{equation}
where \mbox{$\hat{T} = -\sum_{k=1}^{N} \frac{\nabla^{2}_{\mathbf{ x}_{k}} }{2 m} $}.
  Hence,   the Schr\"odinger equation, Eq.~(\ref{fvschr}) is converted to $N(N-1)/2$
coupled equations.
The $N$-body finite volume wave function is normally expanded in terms of a set of periodic basis functions,
 \begin{equation}
  \Phi  (\{ \mathbf{ x} \} ) =  \sum_{[J]} c_{[J]}  \Phi_{[J]}  (\{ \mathbf{ x} \} ), \label{exp}
 \end{equation}
 where $[J]$ refers to a complete set of quantum numbers for the $N$-particle basis  wave functions, and $c_{[J]}$ stands for the expansion coefficients that may be determined by the variational principle. Similar to Eq.~(\ref{fvwavsum}), the basis function $\Phi_{[J]}$ is also given by the sum of $N(N-1)/2$ terms: \mbox{$\Phi_{[J]}  =\sum_{(i<j)=1}^{N} \Phi^{(ij)}_{[J]}  $}. We remark that the choice of basis functions may be arbitrary, depending on the symmetry of the specific physical system and the convenience of numerical computation. However, the different choices  of basis functions should   lead to consistent results. The reasonable choice of basis functions should preserve the symmetry of physical system, such as periodic boundary conditions in finite volume, and numerical results should   be stable and converge.

In the finite volume, to fulfill the periodic boundary condition, the basis functions, $\Phi_{[J]}$, are constructed from   infinite volume solutions of the Schr\"odinger equation. The complete details are given in following Section \ref{wavandsecular}. In the following we use the two-body problem as a specific example to explain  our choice of  basis functions $\Phi_{[J]}$ in finite volume. We denote the infinite volume wave functions by $\Psi_{[J]}$.
 In the case of two-particles in the center-of-mass frame, the solutions of the Schr\"odinger equation in the infinite volume may be determined by the partial wave expansion of the free incoming wave: $\Psi^{(0)}_{[J]} (\mathbf{ r}) = (4\pi) i^J Y_{[J]} (\mathbf{ r}) j_{J} (q r)$, where $[J]=JM$ are partial wave quantum numbers, $\mathbf{ r}$ stands for the relative position of  the two particles, and \mbox{$q=\sqrt{m E}$} is the relative momentum of the particles. Thus, the asymptotic solution of two-body scattering in infinite volume is given by
\begin{equation}
\Psi_{[J]} (\mathbf{ r}) \rightarrow (4\pi) i^J Y_{[J]} (\mathbf{ r}) \left [ j_J(qr) + i t_J (q) h_J^+ (qr) \right ],
\end{equation}
where $t_J(q)$ stands for the two-body scattering amplitude.
Hence, $\Psi_{[J]}$ can be
used as basis functions in the infinite volume to construct the corresponding finite-volume basis functions, $\Phi_{[J]}$ in order
 to  fulfill the requirement of periodic boundary conditions,
 \begin{equation}
 \Phi_{[J]} (\mathbf{ r})  = \sum_{\mathbf{n} \in \mathbb{Z}^{3}  } \Psi_{[J]} (\mathbf{ r} + \mathbf{n}  L).
 \end{equation}
 Such finite volume basis function  reflect the periodic lattice  structure. It will be shown
   in Section \ref{reltoLuscher},  that the two-body basis functions,  $\Phi_{[J]}$, will be related to L\"uscher's zeta function \cite{Luscher:1990ux}. As a general remark, the basis functions should be constructed respecting the cubic symmetry of the problem; yet, we are not explicitly addressing this topic in here. Such basis functions will be given by linear superpositions of infinite-volume basis functions; see, for example, Ref.~\cite{Doring:2018xxx} in which basis functions for ``shells'' are derived as linear combinations of cubic harmonics which by themselves are superpositions of spherical harmonics.

With the finite volume wave function expanded in terms of the basis function, Eq.~(\ref{exp}), the  variational principle,
 $\partial \Lambda^{(ij)}/\partial c^*_{[J']} =0$,
with $\Lambda^{(ij)} $ given by
\begin{align}
 \Lambda^{(ij)} & =  \sum_{[J], [J']} c^*_{[J']} \langle \Phi_{[J']} | \left [  ( E- \hat{T}) | \Phi^{(ij)}_{[J]} \rangle -    \hat{V}^{(L)}_{(ij)} |\Phi_{[J]}  \rangle \right ]c_{[J]}  .
\end{align}
yields a set of coupled secular equations,
\begin{equation}
 \sum_{[J]} \left [  \langle \Phi_{[J']} \left  | E- \hat{T}  \right    | \Phi^{(ij)}_{[J]} \rangle   -    \langle \Phi_{[J']} \left   |  \hat{V}^{(L)}_{(ij)}  \right  |  \Phi_{[J]} \rangle   \right ]  c_{[J]}  =0,\label{seceq}
\end{equation}
and summing the above $N(N-1)/2$  equations leads a secular equation of familiar form (see also Eq.~(\ref{eq:seculareasy})),
   \begin{equation}
 \sum_{[J]} \left [  \langle \Phi_{[J']} \left  | E- \hat{H}^{(L)}    \right  |  \Phi_{[J]} \rangle   \right ]  c_{[J]}  =0  ,\label{secvar}
\end{equation}
where $\hat{H}^{(L)} = \hat{T}  + \sum_{(i<j)=1}^{N} \hat{V}^{(L)}_{(ij)} $.
A non-trivial solution of Eq.~(\ref{seceq})  exists,  provided all $N(N-1)/2$    determinant conditions,
   \begin{equation}
 \det  \left [  \langle \Phi_{[J']}  \left  | E-  \hat{T} \right    | \Phi^{(ij)}_{[J]} \rangle -   \langle \Phi_{[J']} \left  |  \hat{V}^{(L)}_{(ij)}  \right   |  \Phi_{[J]} \rangle \right ] =0, \label{detfaddeev}
\end{equation}
are satisfied simultaneously.
To summarize, given a set of
 basis functions, $\Phi^{(ij)}_{[J]}$ that satisfy periodic boundary conditions,
the variational solution for the spectrum of
the $N$-body finite volume problem is reduced to a solution of the determinant conditions given by Eq.~(\ref{detfaddeev}).

\subsection{Construction of finite volume wave function and reduction of the secular equations}\label{wavandsecular}
To proceed,  a proper choice of basis functions, with
periodic symmetry,
$\Phi^{(ij)}_{[J]}$, has to be made.
There exist numerous examples
 of periodic basis functions,
{\it e.g.} devised for  calculations of electronic structure in a periodic lattice  and other condensed matter or solid state systems.
Specifically, in the LCAO method   \cite{Ziman:1964}, the periodic variational basis functions for describing electronic states tunneling in  a crystal are constructed as a  linear superposition of  all    atomic orbital solutions for an isolated atom located
at each unit cell of the crystal.
In application to hadron scattering, instead of using bound state solutions, as done previously in ~\cite{Guo:2013vsa,Guo:2016fgl,Guo:2017ism,Guo:2017crd},  the basis functions may be constructed from  the  linear superposition of infinite-volume scattering wave functions centered at each image of the cubic box,
  \begin{equation}
  \Phi^{(ij)}_{[J]}  (\{ \mathbf{ x} \} ) = \frac{1}{\mathcal{N}} \sum_{ \{ \mathbf{ n}_{\mathbf{ x}} \} \in \mathbb{Z}^{3}}   \Psi^{(ij)}_{[J]}  (\{ \mathbf{ x} + \mathbf{ n}_{\mathbf{ x}}  L \} )  , \label{fvwavijconstr}
 \end{equation}
 where $\{ \mathbf{ n}_{\mathbf{ x}} \}  = \{ \mathbf{ n}_{\mathbf{ x}_{1}},\mathbf{ n}_{\mathbf{ x}_{2}}, \cdots , \mathbf{ n}_{\mathbf{ x}_{N}} \} $,  \mbox{$\mathcal{N} = \sum_{\mathbf{ n} \in \mathbb{Z}^{3}}$} is a normalization factor, and $ \Psi^{(ij)}_{[J]} $ satisfies the  Schr\"odinger  equation in  the infinite volume with  potentials $V_{(ij)}$,
   \begin{equation}
   ( E -\hat{T}        ) |   \Psi^{(ij)}_{[J]} \rangle =  \hat{V}_{(ij)}  |   \Psi_{[J]}  \rangle . \label{Psiij}
\end{equation}
The  total $N$-body wave function in  the infinite volume,
$\Psi_{[J]}$, is the scattering solution of the Schr\"odinger  equation, \mbox{$(E-\hat{T}-\hat{V} ) |   \Psi_{[J]}  \rangle=0$}, and  may be expressed as
\begin{equation}
\Psi_{[J]} =\Psi^{(0)}_{[J]}   +\sum_{(i<j)=1}^{N} \Psi^{(ij)}_{[J]},
\end{equation}
where $\Psi^{(0)}_{[J]} $ stands for the free incoming wave and satisfies the free Schr\"odinger equation: $( E -\hat{T}        ) |   \Psi^{(0)}_{[J]} \rangle=0$. For example, the two-body
 partial wave   free incoming wave are given by   $\Psi^{(0)}_{[J]}( \mathbf{r}) = (4\pi) i^J Y_{[J]} (  \mathbf{r} )j_{J} (q r) $, while the corresponding wave function in the three-body case in given in
  ~\cite{Guo:2016fgl}.
   The periodic symmetry of the finite-volume wave function $\Phi$ is satisfied automatically by the construction in Eq.~(\ref{fvwavijconstr}). The  infinite-volume $N$-body wave function, $\Psi$, may be solved by a standard Faddeev approach.  Locally, at each image of the cubic box, assuming the size of the box is large enough,  the short distance  $N$-body dynamics is thus described by the solution of  the infinite-volume Schr\"odinger equation, $\Psi $. The long distance correlations  at scales larger than the range of the potential is taken in to account through the linear superposition of the   infinite-volume wave functions centered at each image of the periodic cubic box, see Eq.~(\ref{fvwavijconstr}).

Using Eqs.~(\ref{fvwavijconstr}),~(\ref{Psiij}) and taking into account periodicity of the finite-volume potential,  the secular equations given by
Eq.~(\ref{seceq}) reduce to
  \begin{equation}
 \sum_{[J]}  \langle \Phi_{[J']}    |  \hat{V}_{(ij)}    \left [   |    X^{(ij)}_{[J]}  \rangle - | \Phi_{[J]} \rangle   \right ]  c_{[J]}  =0   , \label{rdseceq}
\end{equation}
where
  \begin{equation}
 X^{(ij)}_{[J]}   (\{ \mathbf{ x} \} ) = \sum_{ \overline{\{ \mathbf{ n}_{\mathbf{ x}} \}}_{(i,j)} \in \mathbb{Z}^{3}}     \Psi_{[J]}  (\{ \mathbf{ x} + \mathbf{ n}_{\mathbf{ x}}  L \} )  ,\label{xijwavconstr}
 \end{equation}
 and $ \overline{\{ \mathbf{ n}_{\mathbf{ x}} \}}_{(i,j)}   $ stands for the set $ \{ \mathbf{ n}_{\mathbf{ x}} \} $   excluding two elements: $ \mathbf{ n}_{\mathbf{ x}_{i} }$ and $ \mathbf{ n}_{\mathbf{ x}_{j} }$.
  Furthermore, the $N(N-1)/2$    determinant conditions given by  Eq.(\ref{detfaddeev})  become
   \begin{equation}
 \det  \left [  \langle \Phi_{[J']}    |  \hat{V}_{(ij)}    \left (     |      X^{(ij)}_{[J]}       \rangle - | \Phi_{[J]} \rangle   \right )  \right ] =0. \label{rddet}
\end{equation}
Given the scattering solution of the $N$-body problem in infinite volume, $\Psi$, as input, the  finite volume wave functions, $\Phi$ and $X^{(ij)}$, can be constructed by using
Eq.~(\ref{fvwavijconstr}) and Eq.~(\ref{xijwavconstr}) respectively.
  We remark that  the energy dependence of the infinite volume wave function, $\Psi_{[J]}$, has been suppressed so far in our presentation.
  For the scattering solutions, $\Psi_{[J]}$ does indeed depend on the incoming momenta of particles, because the finite volume wave functions, $\Phi_{[J]}$ and $X^{(ij)}_{[J]}$, are constructed from $\Psi_{[J]}$,   momenta dependence remains in finite volume wave functions as well.
     Therefore, using the infinite volume scattering solutions, $\Psi_{[J]}$, as inputs,    the finite volume quantization conditions, Eq.(\ref{rddet}),  yield discrete energy spectra as the consequence of periodic cubic lattice structure. In other words, the discrete energy spectra in finite volume is the result of long distance correlation effects of particles in a periodic lattice structure. Meanwhile, the specific patterns of discrete spectra rely on the short distance interaction that is described by scattering amplitudes or "amplitudes carrier" wave functions, $\Psi_{[J]}$, in infinite volume. Ultimately, the quantization conditions play the role of imposing constraints on energy spectra due to periodic boundary conditions. The scattering information in infinite volume and periodic boundary conditions are  combined together by  finite volume wave functions,  $\Phi_{[J]}$ and $X^{(ij)}_{[J]}$.

The secular equations, Eq.~(\ref{rdseceq}), and corresponding quantization conditions, Eq.~(\ref{rddet}), can be further reduced by removing the center-of-mass motion. However, the choice of relative coordinates for the $N$-body system normally can be made quite arbitrary. In Section \ref{relmotion},  the specific choice  \cite{Faddeev:1993}  is described by forming the succession of subsystems of $N$ particles in such a way that the subsystems are obtained by successive joining of particle-$2$, particle-$3$, $\cdots$, particle-$N$  to particle-$1$, {\it i.e.} $(12)$, $((12)3)$, $(((12)3)4)$, $\cdots$.

\subsection{Removal of center of mass motion}\label{relmotion}

Since in this work, we are mainly interested in scattering solutions, the set of incoming  particle's momenta is also introduced to label the $N$-body wave functions:  $\{ \mathbf{p}\} =\{ \mathbf{p}_1, \mathbf{p}_2, \cdots, \mathbf{p}_N\}$, where $\mathbf{p}_i$ stands for the  momentum of the $i$-th incoming particle. The total energy of  $N$ particles is given by $E=  \sum_{i=1}^{N} p^{2}_{i}/2 m$.
The center-of-mass motion and internal motion of the  $N$-body system can be separated out by changing variables of coordinates system. One particular choice of  the relative coordinates and momenta \cite{Faddeev:1993} may be given respectively by
\begin{align}
&\bm{\rho}_{(12), n} = \sqrt{\frac{2n}{n+1}} \left ( \frac{1}{n} \sum_{k=1}^{n}  \mathbf{ x}_{k} - \mathbf{ x}_{n+1} \right ),   \nonumber \\
& \mathbf{ q}_{(12), n} = \sqrt{\frac{n}{2(n+1)}} \left ( \frac{1}{n} \sum_{k=1}^{n}  \mathbf{ p}_{k} - \mathbf{ p}_{n+1} \right ),   \label{relrqcord}
\end{align}
where \mbox{$n<N$}.  The quantities  $ \bm{ \rho}_{(12), n}$ and $ \mathbf{ q}_{(12), n}$ may be interpreted as the relative coordinate and momentum between the \mbox{$(n+1)$}-th particle and the center of  mass coordinate or momentum of the cluster of particles  $(1,2,\cdots, n)$, respectively.
 The index $(12)$ is used to label the special choice we made on  relative coordinates and momenta, so that  for \mbox{$n=1$}, the relative coordinate and momenta are defined between  particle-1 and particle-2,
\begin{equation}
 \bm{ \rho}_{(12), 1} =  \mathbf{ x}_{1} - \mathbf{ x}_{2}  , \ \ \mathbf{ q}_{(12), 1} =  \frac{1}{2} \left ( \mathbf{ p}_{1} - \mathbf{ p}_{2} \right ).
\end{equation}
 The    complete sets of relative coordinates and momenta  are given by  \mbox{$ \{ \bm{ \rho}_{(12)}  \} = \{\bm{ \rho}_{(12),1}  , \bm{ \rho}_{(12),2}, \cdots , \bm{ \rho}_{(12),N-1}    \}$} and  \mbox{$ \{ \mathbf{ q}_{(12)}  \} = \{\mathbf{ q}_{(12),1}  , \mathbf{ q}_{(12),2}, \cdots , \mathbf{ q}_{(12),N-1}    \}$}   respectively.
The center of mass coordinate and momentum of the $N$-body system are
\begin{equation}
\mathbf{ R} = \frac{1}{N}  \sum_{i=1}^{N}  \mathbf{ x}_{i}, \ \ \ \  \mathbf{ P} = \sum_{i=1}^{N}  \mathbf{ p}_{i}.
\end{equation}
The total energy of the  $N$-particle  system  is given by \mbox{$E=  \sum_{n=1}^{N-1} \mathbf{ q}^{2}_{(12),n}/m + \mathbf{ P}^{2}/2mN $}, and the kinetic energy operator of the $N$-body system is
\begin{equation}
\hat{T} =-  \sum_{n=1}^{N-1}  \frac{ \nabla^{2}_{\bm{ \rho}_{(12),n}}  }{m} - \frac{1}{N}  \frac{\nabla^{2}_{\mathbf{ R}} }{2m}.
 \end{equation}

  In terms of the set $\{ \bm{\rho}_{(12)}\}$, the infinite-volume $N$-body wave function is given by \mbox{$ \Psi( \{ \mathbf{ x}  \} ,  \{ \mathbf{ p}  \}) = e^{i \mathbf{ P} \cdot \mathbf{ R}}  \psi( \{ \bm{ \rho}_{(12)}  \} ,  \{ \mathbf{ q}_{(12)}  \})$}, where the center of mass motion is represented by a plane wave, $ e^{i \mathbf{ P} \cdot \mathbf{ R}}$, and the wave function $\psi$ describes the internal motions of the $N$-body system. The same applies to the finite-volume wave function.
   The $N$-body wave function representing the relative motion in  finite volume is thus given  by
  \begin{align}
  &  \phi( \{ \bm{ \rho}_{(12)}  \} ,  \{ \mathbf{ q}_{(12)}  \})    \nonumber \\
  &   = \frac{1}{\mathcal{N}} \sum_{ \{ \mathbf{ n}_{\mathbf{ x}} \} \in \mathbb{Z}^{3}} e^{i \frac{ \mathbf{ P} }{N} \cdot \sum_{i=1}^{N} \mathbf{ n}_{\mathbf{ x}_{i}}   }       \psi  (\{ \ddot{\bm{ \rho}}_{(12)}   \}, \{ \mathbf{ q}_{(12)} \})  ,  \label{phiconstr}
 \end{align}
 where the $n$-th element of the set $\{ \ddot{\bm{ \rho}}_{(12)}  \}$ is
 \begin{equation}
\ddot{ \bm{ \rho}}_{(12),n}  = \bm{ \rho}_{(12),n}  + \sqrt{\frac{2n}{n+1}} \left ( \frac{1}{n} \sum_{k=1}^{n}  \mathbf{ n}_{ \mathbf{ x}_{k}  }-  \mathbf{ n}_{\mathbf{ x}_{n+1} } \right ) L.
 \end{equation}
In order to remove one redundant element of the set $\{ \mathbf{ n}_{\mathbf{ x}} \} $, a  subset $ \{ \mathbf{ n}_{(12)} \} $  is introduced,
 \begin{equation}
 \{ \mathbf{ n}_{(12)} \} = \{ \mathbf{ n}_{(12),1},  \mathbf{ n}_{(12),2},   \cdots,  \mathbf{ n}_{(12),N-1} \}   \in \mathbb{Z}^{3} ,
\end{equation}
 where    the $k$-th element is given by \mbox{$\mathbf{ n}_{(12),k} = \mathbf{ n}_{\mathbf{ x}_{k }} -\mathbf{ n}_{\mathbf{ x}_{k+1}}$}. With the introduction of the set $ \{ \mathbf{ n}_{(12)} \} $ in this particular way, all the elements of set $\{ \mathbf{ n}_{(12)} \} $  still belong  in  $\mathbb{Z}^{3}$. We also  find
   \begin{equation}
\ddot{ \bm{ \rho}}_{(12),n}  = \bm{ \rho}_{(12),n}  + \sqrt{\frac{2}{n(n+1)}}   \sum_{k=1}^{n} k  \mathbf{ n}_{ (12),k   } L,
 \end{equation}
 and
 \begin{equation}
 \sum_{i=1}^{N}  \mathbf{ n}_{ \mathbf{ x}_{i}  } = \sum_{k=1}^{N-1} k  \mathbf{ n}_{ (12),k   } L  + N  \mathbf{ n}_{ \mathbf{ x}_{N}  }.
 \end{equation}
 Hence, after removal  of the center-of-mass motion,  the finite-volume relative wave function, $\phi$, is related to $\psi$ by
\begin{align}
& \phi    ( \{ \bm{ \rho }_{(12) }   \},  \{ \mathbf{ q}_{(12)} \})  \nonumber \\
&  =    \sum_{ \{ \mathbf{ n}_{(12)} \}  \in \mathbb{Z}^{3} }   e^{i \frac{\mathbf{ P}}{N} \cdot    \left ( \sum_{k=1}^{N-1 } k \mathbf{ n}_{(12),k}     \right ) L  }   \psi   ( \{\ddot{ \bm{ \rho}}_{(12)}  \}, \{ \mathbf{ q}_{(12)} \}).
\end{align}
 and $\phi  $ satisfies periodic boundary condition,
\begin{align}
& \phi   (\{ \ddot{\bm{ \rho}}_{(12)}  \}, \{ \mathbf{ q}_{(12)} \})  \nonumber \\
& \quad \quad =      e^{-i \frac{\mathbf{ P}}{N} \cdot    \left ( \sum_{k=1}^{N-1 } k \mathbf{ n}_{(12),k}     \right ) L  }   \phi  (\{ \bm{ \rho}_{(12) }   \}, \{ \mathbf{ q}_{(12)} \}) .
\end{align}

 Clearly, the   set   $\{ \bm{ \rho}_{(12)} \}$  cannot be the only choice of independent relative coordinates. By exchanging the labels of particle, such as $1 \leftrightarrow i$ and $2 \leftrightarrow j$  in  set $\{ \bm{ \rho}_{(12)} \}$,     another  independent coordinate set,   $\{ \bm{ \rho}_{(ij)} \}$, may be obtained. For example,  relabeling $2 \leftrightarrow 3$ in $\{ \bm{ \rho}_{(12)} \}$,  the elements of set $\{ \bm{\rho}_{(13)} \}$  are given   by
 \begin{align}
 & \bm{ \rho}_{(13), 1} =  \mathbf{ x}_{1}-  \mathbf{ x}_{3} = \frac{\bm{ \rho}_{(12), 1}   + \sqrt{3}  \bm{ \rho}_{(12), 2}  }{2} , \nonumber \\
 &   \bm{ \rho}_{(13), 2}  =\frac{2}{\sqrt{3}} \left ( \frac{ \mathbf{ x}_{1}+  \mathbf{ x}_{3} }{2} - \mathbf{ x}_{2}   \right ) = \frac{  \sqrt{3}  \bm{ \rho}_{(12), 1}   - \bm{ \rho}_{(12), 2}  }{2} , \nonumber \\
 &\bm{ \rho}_{(13), n} = \bm{ \rho}_{(12), n}   , \ \ n >2.
 \end{align}
 The different sets $\{ \bm{ \rho}_{(ij)} \}$ and $\{ \bm{ \rho}_{(i'j')} \}$  are linked by a linear transformation,
\begin{equation}
\{ \bm{ \rho}_{(ij)} \} = \Gamma^{(ij),(i'j')}    \{ \bm{ \rho}_{(i'j')} \},
\end{equation}
for example,
\begin{align}
\Gamma^{(13),(12)} = \begin{bmatrix}  \frac{1}{2} & \frac{\sqrt{3}}{2} & 0 \\ \frac{\sqrt{3}}{2} & - \frac{1}{2} &  0 \\  0 & 0 & \mathbb{I} \end{bmatrix}  .
\end{align}
Because of the way of construction, the transfer matrix $\Gamma$ between two different sets is an orthogonal matrix, $  \Gamma^{T}    \Gamma   = \mathbb{I}$. Hence,
both the sum, $\sum_{n=1}^{N-1} \bm{ \rho}_{(ij),n}^{2} $, and the $N$-body volume element, $\int \prod_{n=1}^{N-1} d \bm{ \rho}_{(ij),n}$  are  invariant under
 this transformation of sets. Similarly, both total energy and kinetic energy operator are also  invariant under transformation between  sets of relative coordinates and momenta.

 With introduction of these sets of relative coordinates,  the finite volume $N$-body wave function, $\phi_{[J]}$,   may be written as,
\begin{equation}
  \phi_{[J]}( \{ \bm{ \rho}_{(12)}  \} ,  \{ q_{(12)}  \}) =  \sum^{N}_{(i  < j)=1}  \phi^{(ij)}_{[J]}( \{ \bm{ \rho}_{(ij)}  \} ,  \{ q_{(12)}  \}), \label{phiJ}
\end{equation}
where $\phi^{(ij)}_{[J]}$ satisfies
\begin{align}
& \left ( \sigma^{2} - \hat{T}_{\bm{\rho}}       \right )   \phi^{(ij)}_{[J]}( \{ \bm{ \rho}_{(ij)}  \} ,  \{ q_{(12)}  \}) \nonumber \\
& \quad \quad  \quad \quad     =  m V^{(L)}_{(ij)} (\bm{ \rho}_{(ij),1})   \phi_{[J]} ( \{ \bm{ \rho}_{(12)}  \} ,  \{ q_{(12)}  \})  ,\label{fvrelwavschr}
\end{align}
 and we introduced rescaled total energy and kinetic energy operators, $\sigma^{2} =  m E - \mathbf{ P}^{2}/2 N$ and $\hat{T}_{\bm{\rho}} = -\sum_{n=1}^{N-1}  \nabla^{2}_{\bm{ \rho}_{(ij), n}}  $. The construction of  $\phi^{(ij)}_{[J]}$ is given by
 \begin{align}
\phi^{(ij)}_{[J]}    ( \{ \bm{ \rho}_{(ij) }   \}, & \{ q_{(12)} \})  =    \sum_{ \{ \mathbf{ n}_{(ij)} \}  \in \mathbb{Z}^{3} }   e^{i \frac{\mathbf{ P}}{N} \cdot    \left ( \sum_{n=1}^{N-1 } n \mathbf{ n}_{(ij),n}     \right ) L  }  \nonumber \\
& \quad   \times  \psi^{(ij)}_{[J]}   ( \{ \ddot{\bm{ \rho}}_{(ij)}   \}, \{ q_{(12)} \}), \label{fvrefwavconstr}
\end{align}
where     $ \{ \mathbf{ n}_{(ij)} \} =  \{ \mathbf{ n}_{(ij),1} , \mathbf{ n}_{(ij),2} , \cdots , \mathbf{ n}_{(ij),N-1} \}  $, and  the set  $ \{ \mathbf{ n}_{(ij)} \}  $  may be related to  $ \{ \mathbf{ n}_{(12)} \} $ by relabeling $1 \leftrightarrow i$ and $2 \leftrightarrow j$.  The set $ \{ \ddot{\bm{ \rho}}_{(ij)}   \}$ is defined by
\begin{equation}
 \ddot{ \bm{ \rho}}_{(ij),n}  = \bm{ \rho}_{(ij),n}  + \sqrt{\frac{2}{n(n+1)}}   \sum_{k=1}^{n} k  \mathbf{ n}_{ (ij),k   } L, \  \ (  \mathbf{ n}_{ (ij),k   } \in \mathbb{Z}^{3}).
\end{equation}
 The wave functions, $ \psi^{(ij)}_{[J]}$, are the solutions of  infinite-volume Schr\"odinger equations,
 \begin{align}
&\left ( \sigma^{2} - \hat{T}_{\bm{\rho}}       \right )    \psi^{(ij)}_{[J]}( \{ \bm{ \rho}_{(ij)}  \} ,  \{ q_{(12)}  \}) \nonumber \\
& \quad \quad  \quad \quad      =  m V_{(ij)} (\bm{ \rho}_{(ij),1})   \psi_{[J]} ( \{ \bm{ \rho }_{(12)}  \} ,  \{ q_{(12)}  \})  , \label{freerelwavschr}
\end{align}
and the total  infinite-volume $N$-body wave function is given by, $\psi_{[J]} = \psi^{(0)}_{[J]} +\sum^{N}_{(i  < j)=1}  \psi^{(ij)}_{[J]}$,  where $ \psi^{(0)}_{[J]}$ refers to the incoming free wave.

  After removal of the center-of-mass motion, the determinant conditions    are now given by
 \begin{equation}
 \det \left [  \langle \phi_{[J']}    |  \hat{V}_{(ij)}    \left (   |   \chi^{(ij)}_{[J]}   \rangle  - | \phi_{[J]} \rangle   \right ) \right ]   =0   , \label{rdreldetcond}
\end{equation}
where
   \begin{align}
  & \chi^{(ij)}_{[J]}  (\{ \bm{ \rho}_{(12)} \}, \{ q_{(12)} \})  = \sum_{ \overline{\{ \mathbf{ n}_{ (ij)} \}} \in \mathbb{Z}^{3}}     e^{i \frac{\mathbf{ P}}{N} \cdot    \left ( \sum_{n=2}^{N-1 } n \mathbf{ n}_{(ij),n}     \right ) L  }   \nonumber \\
 & \quad \quad \times  \psi_{[J]}  ( \Gamma^{(12),(ij)}  \{  \overline{ \ddot{\bm{\rho}}_{(ij)} } \} , \{ q_{(12)} \})  .   \label{overlinephiijpwa}
 \end{align}
The set $\overline{\{ \mathbf{ n}_{(ij)} \}}  $ refers to   $  \{ \mathbf{ n}_{(ij),2} , \mathbf{ n}_{(ij),3} , \cdots , \mathbf{ n}_{(ij),N-1} \}  $ which is a subset of $\{ \mathbf{ n}_{(ij)} \} $ by excluding element $ \mathbf{ n}_{(ij),1}$.    The  set $ \{  \overline{ \ddot{\bm{\rho}}_{(ij)} }\} $ is defined by
  \begin{equation}
\overline{\ddot{ \bm{ \rho}}_{(ij),n}  } = \bm{ \rho}_{(ij),n}  + \sqrt{\frac{2}{n(n+1)}}   \sum_{k=2}^{n} k  \mathbf{ n}_{ (ij),k   } L, \   \  (\mathbf{ n}_{ (ij),k   } \in \mathbb{Z}^{3}).
 \end{equation}

Noting   the  invariance of set $\{ n_{(ij)} \}$ we are able to write $ \phi_{[J]} $ in a similar way  as Eq.~(\ref{overlinephiijpwa}),
    \begin{align}
  & \phi_{[J]}  (\{ \bm{ \rho}_{(12)} \}, \{ q_{(12)} \})  = \sum_{  \{ \mathbf{ n}_{ (ij)} \} \in \mathbb{Z}^{3}}     e^{i \frac{\mathbf{ P}}{N} \cdot    \left ( \sum_{n=1}^{N-1 } n \mathbf{ n}_{(ij),n}     \right ) L  }   \nonumber \\
 & \quad \quad \times  \psi_{[J]}  ( \Gamma^{(12),(ij)}  \{    \ddot{\bm{\rho}}_{(ij)}  \} , \{ q_{(12)} \})  ,
 \end{align}
 which is equivalent to Eq.~(\ref{phiconstr}). Hence, the relation between $\phi_{[J]}$ and $\chi^{(ij)}_{[J]} $ can be made more clear,
   \begin{align}
  & \phi_{[J]}  (\{ \bm{ \rho}_{(12)} \}, \{ q_{(12)} \})  = \sum_{  \mathbf{ n}_{(ij),1} \in \mathbb{Z}^{3}}     e^{i \frac{\mathbf{ P}}{N} \cdot      \mathbf{ n}_{(ij),1}       L  }   \nonumber \\
 & \quad \quad \times  \chi^{(ij)}_{[J]}  ( \Gamma^{(12),(ij)}  \{  \widetilde{ \bm{ \rho}}_{(ij)}  \} , \{ q_{(12)} \})    , \label{phichiij}
 \end{align}
where the $n$-th element of  $\{  \widetilde{ \bm{ \rho}}_{(ij)}  \}$  is given by
  \begin{equation}
 \widetilde{ \bm{ \rho}}_{(ij),n}  = \bm{ \rho}_{(ij),n}  +   \mathbf{ n}_{ (ij),1   } L, \  \  \   (  \mathbf{ n}_{ (ij),1   } \in \mathbb{Z}^{3} ),
 \end{equation}
 and $ \{ \ddot{ \bm{ \rho}}_{(ij)} \}  = \{ \overline{\ddot{ \bm{ \rho}}_{(ij)}  } \} + \{  \widetilde{ \bm{ \rho}}_{(ij)}  \}  $.

\subsection{Relation to two-body L\"uscher formula}\label{reltoLuscher}

 In the simplest case, $N=2$, there is only one  relative coordinate and relative momentum so that the particle index can be
  dropped and in  what follows they are
   denoted by $\mathbf{ r}$ and $\mathbf{ q}$, respectively. Notice that $\chi^{(ij)}_{[J]}$ is now reduced to $\psi_{[J]}$ for the two-body system. Therefore, the
determinant condition in Eq.~(\ref{rdreldetcond}) is  given by
   \begin{equation}
 \det \left [  \langle \phi_{[J']}    |  \hat{V}   \left (     | \psi_{[J]}   \rangle  - | \phi_{[J]} \rangle   \right )   \right ]   =0   ,  \label{2bdetcond}
\end{equation}
where $[J]=JM$ stand for quantum numbers of a specific partial wave, and
 \begin{equation}
\phi_{[J]}    ( \mathbf{ r}   , q  )  =    \sum_{  \mathbf{ n}  \in \mathbb{Z}^{3} }   e^{i \frac{\mathbf{ P}}{2} \cdot     \mathbf{ n}    L  }    \psi_{[J]}   (  \mathbf{ r}  +   \mathbf{ n}     L ,  q  ).
\end{equation}
Assuming the potential is of finite range, \mbox{$V(r)=0$} for \mbox{$R<r<L$},  in  the region outside of the potential the  infinite-volume wave function has the form,
\begin{align}
 & \psi_{[J]} (\mathbf{ r},q) \stackrel{r>R}{ =}  \psi^{(out)}_{[J]} (\mathbf{ r},q) ,\nonumber \\
 & \psi^{(out)}_{[J]} (\mathbf{ r},q)  = (4\pi) i^{J} Y_{[J]}  (   \mathbf{ r}  ) \left [ j_{J} (q r) + i t_{J} (q) h_{J}^{(+)} (q r) \right],
\end{align}
where   $t_{J}$ is the partial-wave two-body scattering amplitude, and $t_{J}$ is normalized by the unitarity relation, Im~$ \left [ 1/ t_{J} \right ]=-1$.  Inside the potential region, the wave function is given by the solution of the Schr\"odinger equation, and is denoted by $\psi^{(in)}_{[J]} (\mathbf{ r},q) $ from now on. The matrix element of the determinant condition, Eq.~(\ref{2bdetcond}), is given by
\begin{align}
\int_{r<R} d \mathbf{ r} \phi^{*}_{[J']} (\mathbf{ r},q) V(r) \left [ \psi^{(in)}_{[J]} (\mathbf{ r},q) - \phi_{[J]} (\mathbf{ r},q)  \right ],
\end{align}
Using the fact that
 \begin{align}
& \phi_{[J]}    ( \mathbf{ r}   ,  q  )  \stackrel{r<R}{ =}\psi^{(in)}_{[J]} (\mathbf{ r},q)  - \psi^{(out)}_{[J]} (\mathbf{ r},q)    \nonumber \\
& +  (4\pi) i^{J} i t_{J} (q) \sum_{  \mathbf{ n}  \in \mathbb{Z}^{3} }   e^{i \frac{\mathbf{ P}}{2} \cdot     \mathbf{ n}    L  }      Y_{[J]}  (  \mathbf{ r}  +   \mathbf{ n}     L )   h_{J}^{(+)} (q |\mathbf{ r}  +   \mathbf{ n}     L |) ,
\end{align}
and the relation
\begin{align}
&  \sum_{  \mathbf{ n}  \in \mathbb{Z}^{3} }   e^{i \frac{\mathbf{ P}}{2} \cdot     \mathbf{ n}    L  }      Y_{[J]}  (  \mathbf{ r}  +   \mathbf{ n}     L )   h_{J}^{(+)} (q |\mathbf{ r}  +   \mathbf{ n}     L |)   \nonumber \\
&= \sum_{[j]} Y_{[j]} (\mathbf{ r}) \left [ \delta_{[J], [j]} i n_{J} (qr) - i \mathcal{M}_{[J] ,[j]}^{(\frac{\mathbf{ P}}{2})}  (q) j_{j} (q r)\right ]  , \label{luscherlatsum}
\end{align}
derived in Eq.~(B1) and Eq.~(B3) in
 Ref.~\cite{Guo:2012hv}, for expansion coefficient function of the finite volume two-body Green's function,  $\mathcal{M}_{[J] ,[j]}^{(\frac{\mathbf{ P}}{2})}$,     we find
\begin{align}
&   \langle \phi_{[J']}    |  \hat{V}   \left (     | \psi_{[J]}   \rangle  - | \phi_{[J]} \rangle   \right )    \nonumber \\
&=  \sum_{[j]} \left [   \int_{r<R} d \mathbf{ r} \phi^{*}_{[J']} (\mathbf{ r},q) V(r)    j_{j} (q r) Y_{[j]} (\mathbf{ r}) \right ]  \nonumber \\
& \times (4\pi) i^{J}  \left [  \delta_{[J],[j]} \left (1+ i t_{J} (q)  \right)  - t_{J} (q)  \mathcal{M}_{[J], [j]}^{(\frac{\mathbf{ P}}{2})}  (q)\right ] .
\end{align}
Therefore, the determinant condition for the two-body problem, Eq.~(\ref{2bdetcond}),  yields  L\"uscher's  formula,
\begin{align}
\det \left [  \delta_{[J],[j]} \left (\frac{1}{  t_{J} (q) } + i \right)  - \mathcal{M}_{[J], [j]}^{(\frac{\mathbf{ P}}{2})}  (q)\right ] =0.
\end{align}
Hence, the variational approach to the finite -volume few-body system  is  consistent with the L\"uscher approach.

\subsection{Effective two-body formalism}\label{Nbeff}
In this section, we would like to show that the $N$-body    quantization conditions may be recast in a similar form as  the two-body quantization condition  in Eq.~(\ref{2bdetcond}).

Before proceeding to $N$-body interaction, first of all, let us rewrite the two-body quantization condition of Eq.~(\ref{2bdetcond}) by using the periodicity of the finite-volume wave function. We find
\begin{equation}
\phi^{*}_{[J']}(\mathbf{ r}, q)\phi_{[J]}(\mathbf{ r},q ) =  \sum_{\mathbf{ n} \in \mathbb{Z}^{3}}  \phi^{*}_{[J']}(\mathbf{ r} + \mathbf{ n} L, q)    \psi_{[J]}(\mathbf{ r} + \mathbf{ n} L, q).
\end{equation}
Therefore, the two-body quantization condition, Eq.~(\ref{2bdetcond}), may be expressed as
\begin{align}
& \det \Bigg [ \int d \mathbf{ r} V(r) \bigg [\Omega_{[J'], [J]}  (\mathbf{ r},q)  \nonumber \\
&\quad \quad \quad \quad\quad \quad \quad -  \sum_{\mathbf{ n} \in \mathbb{Z}^{3}} \Omega_{[J'], [J]}  (\mathbf{ r} + \mathbf{ n} L,q) \bigg ]  \Bigg ]=0, \label{2beff}
\end{align}
where $\Omega_{[J'], [J]}  (\mathbf{ r},q)=\phi^{*}_{[J']}(\mathbf{ r}  , q)    \psi_{[J]}(\mathbf{ r}  , q) $.

Next, for the $N$-body problem,  we revisit the determinant condition in channel $(ij)$ in a   explicit format,
\begin{align}
&  \det \Bigg [ \int \prod_{n=1}^{N-1} d \bm{\rho}_{(ij),n} \phi^{*}_{[J']} ( \{ \bm{\rho}_{(12)} \}, \{q_{(12)} \}) m V_{(ij)}( \bm{\rho}_{(ij),1}) \nonumber \\
& \times \left [\chi^{(ij)}_{[J]} ( \{  \bm{\rho}_{(12)} \}, \{q_{(12)} \}) -\phi_{[J]} ( \{ \bm{\rho}_{(12)} \}, \{q_{(12)} \})  \right ]  \Bigg ] =0, \label{seculareqij2b}
\end{align}
where $ \chi^{(ij)}_{[J]}$ and $\phi_{[J]} $  are given by Eq.~(\ref{overlinephiijpwa}) and Eq.~(\ref{phichiij}), respectively.
 Because of the relation between $ \chi^{(ij)}_{[J]}$ and $\phi_{[J]} $ given in Eq.~(\ref{phichiij}), similarly to Eq.~(\ref{2beff}), it is  advantageous  to introduce the  quantity  $\Omega^{(ij)} (\bm{\rho}_{(ij),1})$
again by integrating out all the relative coordinates  except for $\bm{\rho}_{(ij),1}$,
 \begin{align}
& \Omega^{(ij)}_{[J'],[J]} (\bm{\rho}_{(ij),1} , \{q_{(12)} \} )=\int \prod_{n=2}^{N-1} d \bm{\rho}_{(ij),n} \nonumber \\
 & \quad \times \phi^{*}_{[J']} ( \{ \bm{\rho}_{(12)} \}, \{q_{(12)} \}) \chi^{(ij)}_{[J]} ( \{  \bm{\rho}_{(12)} \}, \{q_{(12)} \}).
 \end{align}
Using the periodicity of the wave function $\phi$,    the quantization conditions, Eq.~(\ref{seculareqij2b}), now have a form  that resembles the two-body quantization condition given in Eq.~(\ref{2beff}),
 \begin{align}
 & \det  \Bigg\{ \int d \bm{\rho}_{(ij),1}  V_{(ij)}( \bm{\rho}_{(ij),1})    \bigg [ \Omega^{(ij)}_{[J'],[J]} (\bm{\rho}_{(ij),1} , \{q_{(12)} \} ) \nonumber \\
 & \quad \quad  - \sum_{  \mathbf{ n}  \in \mathbb{Z}^{3}}    \Omega^{(ij)}_{[J'],[J]} (\bm{\rho}_{(ij),1}+\mathbf{ n}   L  , \{q_{(12)} \} )\bigg ] \Bigg \}=0. \label{Nbeffsecular}
 \end{align}
 When all the interactions $V_{(i'j')}$ except the interaction between the $i$-th and $j$-th particle are turned off, clearly,  the quantization condition in the $(ij)$ channel in Eq.~(\ref{Nbeffsecular}) is thus reduced to the two-body quantization condition given in Eq.~(\ref{2beff}).

\section{Discussion and conclusion}\label{discussion}

\subsection{Resemblance to isobar model}\label{isobar}
In the past, isobar models \cite{Khuri:1960zz,Bronzan:1963xn,Aitchison:1965kt,Aitchison:1965zz,Aitchison:1966kt,Pasquier:1968zz,Pasquier:1969dt,Guo:2014vya,Guo:2014mpp,Danilkin:2014cra,Guo:2015kla,Guo:2015zqa,Guo:2016wsi, Mai:2017vot}  have been a useful tool to describe few-body interactions, in which the few-body interaction is treated  by taking into account all possible recombinations of two-body subsystems. The two-body subsystems are considered as the dominant contribution compared to three-body force,
and the few-body interaction correction to the two-body subsystem  is generated by rescattering  between all possible pairs.
In order to show the similarity of this
 approach to the isobar  formulation \cite{Khuri:1960zz,Bronzan:1963xn,Aitchison:1965kt,Aitchison:1965zz,Aitchison:1966kt,Pasquier:1968zz,Pasquier:1969dt,Guo:2014vya,Guo:2014mpp,Danilkin:2014cra,Guo:2015kla,Guo:2015zqa,Guo:2016wsi, Mai:2017vot},  we consider a special case, {\it i.e.}, a three-body system with two light spinless particles and one infinitely heavy spinless particle stationed at the origin. The heavy particle is labeled as third particle. The system may be described by
\begin{align}
& \left [ E+ \frac{1}{2m} \sum_{i=1}^{2} \nabla^{2}_{\mathbf{ r}_{i}}  - \sum_{i=1}^{2} V^{(L)} (r_{i}) + U^{(L)}(\mathbf{ r}_{1} - \mathbf{ r}_{2})  \right ]  \nonumber \\
& \quad \quad \quad \quad \times \phi(\mathbf{ r}_{1}, \mathbf{ r}_{2}; \mathbf{ q}_{1}, \mathbf{ q}_{2}) =0,
\end{align}
where $V^{(L)}$ represents the interactions between  the heavy particle and one of the light particles, and $U^{(L)}$ stands for the interaction between the two light particles. Again, we use the superscript $(L)$ to identify the periodic potential in finite volume.
We also assume that $U^{(L)}$ is a weak interaction, so that the interaction between the two light particles  is treated as a perturbation, which serves the purpose of discussion in this work. Therefore, the corresponding infinite-volume wave function must have the form of
\begin{equation}
 \psi(\mathbf{ r}_{1}, \mathbf{ r}_{2}; \mathbf{ q}_{1}, \mathbf{ q}_{2})  = \psi(\mathbf{ r}_{1} ; \mathbf{ q}_{1}) \psi(\mathbf{ r}_{2} ; \mathbf{ q}_{2})   + \delta \psi(\mathbf{ r}_{1}, \mathbf{ r}_{2}; \mathbf{ q}_{1}, \mathbf{ q}_{2}) ,
\end{equation}
where the first term is the solution of the system with zero interaction between two lights particles.
The second term, $\delta \psi$, can be considered as perturbative contribution when the weak $U$-potential is turned on. The two-body  infinite-volume wave function, $\psi(\mathbf{ r}_{i}; \mathbf{ q}_{i})$, is given by the solution of
\begin{equation}
\left [ \frac{q_{i}^{2}+\nabla^{2}_{\mathbf{ r}_{i}} }{2m}    -  V (r_{i})    \right ]  \psi(\mathbf{ r}_{i}; \mathbf{ q}_{i})=0,
\end{equation}
with  $E = q_{1}^{2}/2m + q_{2}^{2}/2m$. Following the argument provided in previous sections, the finite-volume wave function is constructed from the infinite-volume wave function. The two ingredients $\phi$ and $\chi^{(1/2)}$ of the secular equations thus also have the forms,
\begin{equation}
\phi(\mathbf{ r}_{1}, \mathbf{ r}_{2}; \mathbf{ q}_{1}, \mathbf{ q}_{2})  = \phi(\mathbf{ r}_{1} ; \mathbf{ q}_{1}) \phi(\mathbf{ r}_{2} ; \mathbf{ q}_{2})   + \delta \phi(\mathbf{ r}_{1}, \mathbf{ r}_{2}; \mathbf{ q}_{1}, \mathbf{ q}_{2}) ,
\end{equation}
and
\begin{align}
& \chi^{(1)}(\mathbf{ r}_{1}, \mathbf{ r}_{2}; \mathbf{ q}_{1}, \mathbf{ q}_{2}) \nonumber \\
 & \quad \quad  = \psi(\mathbf{ r}_{1} ; \mathbf{ q}_{1}) \phi(\mathbf{ r}_{2} ; \mathbf{ q}_{2})   + \delta \chi^{(1)}(\mathbf{ r}_{1}, \mathbf{ r}_{2}; \mathbf{ q}_{1}, \mathbf{ q}_{2}) ,  \nonumber \\
 & \chi^{(2)}(\mathbf{ r}_{1}, \mathbf{ r}_{2}; \mathbf{ q}_{1}, \mathbf{ q}_{2}) \nonumber \\
 & \quad \quad = \phi(\mathbf{ r}_{1} ; \mathbf{ q}_{1}) \psi(\mathbf{ r}_{2} ; \mathbf{ q}_{2})   + \delta \chi^{(2)}(\mathbf{ r}_{1}, \mathbf{ r}_{2}; \mathbf{ q}_{1}, \mathbf{ q}_{2}) ,
\end{align}
where $\phi(\mathbf{ r}_{i}; \mathbf{ q}_{i}) = \sum_{\mathbf{ n}_{i} \in \mathbb{Z}^{3}} \psi(\mathbf{ r}_{i} + \mathbf{ n}_{i} L; \mathbf{ q}_{i})$.  The construction of $\delta \phi$ and $\delta \chi^{(i)}$ can also be obtained based on $\delta \psi$, accordingly, but the specific expressions are not crucial for our brief   discussion. The  two secular equations may also be treated as a perturbation; for example, for channel $(13)$, we obtain
\begin{align}
& \sum_{[J]} \bigg[ \int d \mathbf{ r}_{1} \left ( \int  d \mathbf{ r}_{2}  \phi_{[J']}^{*}(\mathbf{ r}_{1}, \mathbf{ r}_{2}; q_{1}, q_{2}) \phi_{[L_{2}]} (\mathbf{ r}_{2} ; q_{2})  \right )  V(r_{1})   \nonumber \\
& \quad \quad        \times \left [ \psi_{[L_{1}]} (\mathbf{ r}_{1} ; q_{1})  -  \phi_{[L_{1}]} (\mathbf{ r}_{1} ; q_{1})  \right ]  + \delta U_{[J'],[J]} \bigg ] c_{[J]} =0,
\end{align}
where  $\delta U$ stands for the perturbative contribution to the secular equation from the weak $U$-potential. The three-body quantum numbers set, $[J]$, is constructed from two-body quantum numbers, $[L_{i}]$, by $[J]=[L_{1}] \bigotimes [L_{2}]$. In the limit of $\delta U \rightarrow 0$,   two secular equations yield two independent two-body quantization conditions,
\begin{align}
\det \left [  \delta_{[L_{i}],[l_{i}]} \left (\frac{1}{  t_{L_{i}} (q_{i}) } + i \right)  - \mathcal{M}_{[L_{i}], [l_{i}]}^{(\mathbf{ 0})}  (q_{i})\right ] =0, \ i=1,2.
\end{align}
and both $q_{i}$'s are quantized  independently according to the corresponding L\"uscher  formula. The three-body correction  with weak $U$-potential may be obtained by  perturbation. In the current approach, the physical picture is presented in a way that is similar to the three-body rescattering effect corrected isobar model, \cite{Khuri:1960zz,Bronzan:1963xn,Aitchison:1965kt,Aitchison:1965zz,Aitchison:1966kt,Pasquier:1968zz,Pasquier:1969dt,Guo:2014vya,Guo:2014mpp,Danilkin:2014cra,Guo:2015kla,Guo:2015zqa,Guo:2016wsi}. That is to say that the three-body system considered in this subsection can be treated as two two-body isobar subsystems: $(13)$ and $(23)$, which yield two independent two-body quantization conditions to $q_{1,2}$.   The three-body rescattering correction to isobar subsystems produces an energy shift on quantized two-body energies: $q_1^2/2m +q_2^2/2m + \delta E$.

\subsection{Summary}

In this work, we propose a variational approach to the finite-volume $N$-body problem. In order to fulfill the periodic boundary conditions, the trial wave functions are constructed by linear superposition of all the solutions of the infinite-volume wave functions centered at each image of the periodic cubic boxes, given that the infinite-volume wave functions may be obtained by standard methods. In this
 approach short-range $N$-body dynamics is    local to each box and long-range
  correlations correspond to particles
   travel through the entire periodic structure of the lattice. In  other words, the short-range dynamics are determined by  infinite-volume wave functions, and finite-volume wave functions control the long-range correlations that eventually yield the discrete energy spectra because of the periodic structure of lattice.
No explicit analytic expressions between discrete lattice eigenvalues and the scattering amplitude, such as present in the two-body L\"uscher formula,  can be given by the variational approach for $N>2$. Instead, the discrete energy spectra and $N$-body scattering amplitudes are linked in a  rather complicated  way. Nevertheless
 the method is has potential advantages
  for systems with $N >>2$.
By combining the variational approach with the Faddeev approach, the $N(N-1)/2$  quantization conditions are obtained ultimately. In the end, these quantization conditions can be expressed in a way that   resemble   rescattering in the isobar approach, The overall rescattering corrections can also be written as a collective effect by integrating out the  of degrees of freedom other then as selected  two-body subsystem.

\section{ACKNOWLEDGMENTS}
  We   acknowledge support from the Department of Physics and Engineering, California State University, Bakersfield, CA.  P.G.   would like to thank Vladimir~Gasparian and David~Gross for numerous fruitful discussions.   This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958,  NSF PHY-1415459, and by the U.S. Department of Energy, Office of Science, Office of Nuclear
Physics under contracts
 no. ~DE-AC05-06OR23177 and DE-FG02-87ER40365. M.D. acknowledges support by the NSF Career grant No. PHY-1452055.

redpajama_set_name: RedPajamaArXiv


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The Blogqueer voicesGay RightsQueer Voices
Moving Beyond 'Gay Is the New Black': NOM's Divisive Racial Politics and the LGBT Movement's Need for a Racial Justice Agenda
We're right to get outraged over NOM's racial strategy to divide "gays and blacks," but this division is one we have to take a serious look at within the LGBT movement, which currently lacks a racial justice agenda.
Jessie Daniels, Columnist
There's a lot of buzz about the just-revealed internal memos from the National Organization for Marriage (NOM) that make plain their divisive racial strategies to oppose marriage equality. The key strategy NOM has employed is wedge politics, that is, seeking to drive a wedge between African Americans and Latinos on one side and those in the LGBT movement on the other.
Here is just some of what the NOM memos say about African Americans (emphases are mine):
The strategic goal of this project is to drive a wedge between gays and blacks -- two key Democratic constituencies. Find, equip, energize and connect African American spokespeople for marriage; develop a media campaign around their objections to gay marriage as a civil right; provoke the gay marriage base into responding by denouncing these spokesmen and women as bigots. No politician wants to take up and push an issue that splits the base of the party. Fanning the hostility raised in the wake of Prop 8 is key to raising the costs of pushing gay marriage to its advocates and persuading the movement's allies that advocates are unacceptably overreaching on this issue.
NOM's strategy for Latinos looks like this:
Will the process of assimilation to the dominant Anglo culture lead Hispanics to abandon traditional family values? We can interrupt this process of assimilation by making support for marriage a key badge of Latino identity.
You can read all the documents here, thanks to HRC.
To say that NOM's strategy is racist is stating the obvious. Sometimes it's worth stating the obvious, but I want to make a slightly less obvious point, and that is that the revelations about NOM's racial politics highlight the LGBT movement's need for a racial justice agenda.
The truth is that African Americans and Latinos are just as likely to be gay, lesbian, bisexual, and/or transgender as white people. Moreover, Zack Ford makes the excellent point that "NOM's tactics seek to erase an entire population of people who live at the intersections of these experiences, limiting their ability to fulfill their complete identities."
That's exactly right. The NOM strategies not only are racist but they assume that "gay" and "black" or "lesbian" and "Latina" are somehow mutually exclusive categories, that you can't be both gay and black, or lesbian and Latina. The reality is that the LGBT movement has also ignored the "both/and" identities. How else can we explain the popularity of the "gay is the new black" slogan, popularized during the Prop 8 campaign? We're right to get outraged over NOM's racial strategy to divide "gays and blacks," but this division is one we have to take a serious look at within the LGBT movement, which currently lacks a racial justice agenda.
What would it look like if the LGBT movement had a racial justice agenda? Well, for starters, we'd see our struggle for equality tied to other movements for justice, not just by analogy. So, for example, there's been a noticeable silence about Trayvon Martin on most of the mainstream gay blogs, probably because most (white) gay folks don't see the case as "our issue." But as Zach Stafford recently pointed out here on HuffPost, gay folk should care about Trayvon Martin because all of us who are "outsiders" -- whether because of sexual orientation, gender non-conformity, or race -- can be targets of violence.
When we say that the gay right movement is the new civil rights movement, we're playing into the divisive racial politics of NOM. We have to do better than "gay is the new black." We have to see that the fight for sexual equality hasn't replaced the fight for racial equality, because that's not over. When the LGBT movement moves beyond shallow slogans like "gay is the new black" to embrace a racial justice agenda that sees our struggle as tied to others, then we'll have truly won a victory against opponents like NOM that can only see "gays and blacks" as an easy place to drive a wedge.
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