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Boardwalk CDJR has been in business for 35 years. We have adapted and moved the dealership to its current location in 2016. We added the Dodge and Ram lines after opening with only the Jeep and Chrysler lines. We are always looking to the future and what that brings. Our team members are our most cherished assets.
Boardwalk CDJR is always looking to grow with a passionate team of people who care about what they do and want to help get to the next level. If you have a good attitude and willingness to learn, then you are the perfect candidate.
New hires have the opportunity to grow as fast as they are willing to learn and master their current roles. Success is promoted and recognized within each individual. There is no limit to how fast you can tackle the next position.


For any other band, the album would be a colossal achievement. But the creative arc established by the White Stripes makes the album something of a letdown. The title track would be wonderful if it had not been so long available. "Conquest"? Yes, if previous covers of Marlene Dietrich and Dolly Parton did not make such a battle-of-the-sexes song expected. "300 MPH Torrential Outpour Blues"? Yes, if we did not already know of Jack's love for Bob Dylan.
The album is as good as anything else they've done, undeniably. That level, however, no longer feels like enough.
Without hearing last year's eponymous release, listeners to the latest release from British Columbia superstars Shapes and Sizes may be left scratching their collective heads. Again, the band's angular guitars rip apart the varied time signatures of individual songs; again, the vocals seem competitive rather than contributive; again, it sounds like one big mess. Tracks like "The Long Indifference" might sound like a more apt description of the band's musical aesthetic: chaotic, jarring and alienating. The title might hint, with a wink, at the personal hazards of listening.
Fear not. Experimentation is the norm here. The discordant musical and vocal sounds do coalesce into a kind of dangerous magical din. Ultimately the joy comes from two unavoidable facts: 14 tracks sound like 40; 50 minutes feels twice that length. Start with "Highlife" and "Alone/Alive" before venturing into the stupefying waters of "The Horse's Mouthy Mouth." A triumph of ambition.


Pamplin Media Group - Time to start thinking about taxes!
Why should my tax advisor be the same person as my financial advisor? These two professions are closer than you might imagine. When you think about a tax return and look down the list of data that is required, you quickly see that much of the information comes from your financial advisor.
When you talk with your advisor about planning your future, they are going to need to know how much money you make, do you have a pension, when are you thinking about starting Social Security, do you have children or grandchildren, and are you planning on saving for their education or to help with future expenses. I hope you can see that the information from on professional is the data the other one needs to do a complete job for you. If those two professionals are but one person, all the planning becomes not only easier but timelier as information is always available. When looking at decision points such as; is this a good time to sell stock and remodel a kitchen, it is not just a question of is the market up or down. It is also a question of how this will affect my taxes. Considering that tax on all of the following are affected by your income level: capital gains, Social Security, Medicare costs, your ACA tax credit, and may other items both from the old and new tax code, it becomes almost imperative that you have the same advisor looking at this data. We have married the two by offering tax services through Diana LH Harrison to our investment clients at Financial Investment Team. Is this of interest to you, please call or visit our website to schedule an appointment.


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Betnüsse oder Gebet(s)nüsse sind nuss- oder schotenförmige Kapseln, die sich in zwei Hälften aufklappen lassen und vorwiegend vom ausgehenden 15. Jahrhundert bis 1530 als Anhänger am Rosenkranz oder an einer Schmuckkette getragen wurden. Das Aufkommen von Betnüssen ging mit der Praxis des häufigeren Rosenkranzgebets im ausgehenden Mittelalter einher.

Der Ursprung der Betnüsse ist ungeklärt. Aus stilkritischen und anderen Gründen ist aber eine flandrische Herkunft am ehesten denkbar. Die deutsche Bezeichnung "Betnuss" stammt erst aus dem späten 19. Jahrhundert und ist wahrscheinlich eine Übersetzung des französischen Begriffs noix de prière.

Aussehen

Betnüsse hatten einen Durchmesser von höchstens 6 cm und bestanden aus Buchsbaumholz, später auch aus Elfenbein, Metall und anderen Materialien. Die ältesten erhaltenen Exemplare sind zumeist mit Maßwerkmustern ornamental durchbrochen oder besitzen eine teils medaillonartig abgeflachte Oberfläche.

Im Innern befinden sich mehr oder weniger figurenreiche Reliefschnitzereien, die Szenen aus dem Leben von Heiligen oder der Passion Christi darstellen. Einige Exemplare beherbergten zusätzliche, aufklappbare Flügelchen. Als Sonderformen erhalten haben sich ein Objekt in Gestalt eines Totenkopfs sowie eine Betnuss, die kleine, mit winzigen Reliefs versehene "Erbschen" enthält.

Nur wohlhabende Personen konnten sich Betnüsse leisten. Als beliebtes Sammelobjekt wurden sie gerne mit einer Metallfassung versehen.

Literatur
 Kurt Dingelstedt: Betnuß. In: Otto Schmitt (Hrsg.): Reallexikon zur Deutschen Kunstgeschichte. Band 2. Druckenmüller, Stuttgart/Waldsee 1948, Sp. 371–377

Weblinks

Sakralbild
Rosenkranzgebet

redpajama_set_name: RedPajamaWikipedia


Modern Tuscany was the heartland of Etruria. The goldsmiths were also mobile, moving from country to country, either to the court of a potential patron or to the workshop of a master goldsmith. Gold itself was central to that fertile period, not so much as a metal although jewellery and gold leaf were everywhere but for its colour.
Many other individual examples of Egyptian gold jewellery or sculpture from earlier and later periods survive in the British Museum, the Louvre or the Metropolitan Museum in New York. Over individual grains of gold may carpet an ornament.
The art of the goldsmith working for a wealthy patron, however, did have a final crescendo in the creations of Carl Faberge for the Russian Czars. But the Sumerians showed the way ahead. On Crete, King Minos built a magnificent palace, with a labyrinth beneath in which lived a ferocious monster, the Minotaur, part bull and part man. What unites Pre-Columbian ornaments is the distinctive verve and style with which they were made by the ancient goldsmiths of the Americas.
Roman goldsmiths applied it as a setting and framing for brilliant and valuable stones - emeralds, sapphires, aquamarines, peridots, diamonds though uncut - and pearls started to become more popular. Just four years later Columbus discovered the Americas. Europe's goldsmiths were busy too. Two churches there, Sant'Apollinaro Nuovo and San Vitale, are unequalled in western Europe as the most precious examples of Byzantine architecture, sculpture and magnificently coloured mosaics. That inheritance, however, is reflected today in Arezzo, once a city-state of ancient Etruria, which is a centre of the modern Italian gold jewellery industry.
Goldsmiths and other craftsmen from the mainland may also have migrated there. Soon, the Portuguese, then the Dutch and the English, were taking gold dust and ornaments directly back to Europe, bypassing the historic Sahara overland route.
Between and the Renaissance transformed European culture and society. Tutankhamun himself was buried in a sarcophagus within which were fitted, one inside the other, three coffins.
Jewellery became incorporated into the symbolism of the dominion of the Christian church and the growing power of kings. This jewellery, created when the Minoan civilisation was at its peak, coincides with the triumph of Egyptian gold work that gave us the treasures of King Tutankhamun. However, by the talent was increasingly to be found north of the Alps, working at the French court in Paris, or in Antwerp, Nurnberg and Augsberg. The goldsmiths had free rein. This tradition, however, was shattered by the Spanish invasion of South America and Pizarro's capture and ransom of the Inca.
Its colour and sheen naturally equated it with the sun, while its incorruptibility which makes the dating of early gold jewellery difficult made it a symbol of permanence. Within the Akan culture was the small kingdom of the Asante, liquidating dividend examples somewhat inland but much closer to the richest gold mines controlled by the Denkyira kingdom. The icons were largely painted on screens separating the sanctuary from the nave of Byzantine churches.


Great collage from the 1950s.
The picture, which consists of many different parts of straw, was produced in painstaking craftsmanship.
It is framed behind glass and very well preserved. A small piece is loose and slipped on the left edge of the image.


When irradiated with a high-intensity ultrasonic field, living tissue might undergo phenomena ranging from heat deposition to bubble formation. Such effects have the potential of inducing structural and functional changes in soft tissue, and have been traditionally avoided in diagnostic ultrasound.
In recent years, however, efforts have been shifted towards enhancing and controlling the occurrence of these phenomena so they can be used in other biomedical applications such as non-invasive surgery with high-intensity focused ultrasound (HIFU).
Our research focuses on understanding ultrasound-induced bubble formation and activity in order to model and design appropriate protocols for specific surgical needs.
Matheus graduated in Mechanical Engineering in 2015 from Rio de Janeiro State University (UERJ) in Brazil and undertook a year abroad at the University of Sheffield, where he has worked as a Research Assistant at the Laboratory of Acoustics. Since the second year of his studies Matheus was involved in projects at the Laboratory of Thermodynamics and Optimisation in UERJ.
For his undergraduate research, he was awarded a Scientific Initiation scholarship by the Carlos Chagas Research Foundation in Brazil in 2012 and has also worked for industry-sponsored projects in the Department of Mechanical Engineering at Sheffield University. His PhD at UCL is funded by the Brazilian National Council of Technological and Scientific Development, through the Science without Borders scheme.


My name is Salva, and I'm from Mallorca (Spain).I need some cc from Slovenia and I have many spanish and other countries cc coins, if you are interested in swap my mail is smarcusxamena@gmail.com.
I saw you name in the good swappers list.
I live in Holland but have quite a lot of coins from other places too.
I have no Slovenian 2E Comm coins.
I need some coin from Slovenia.
hi agree to swap currencies with me??


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Омиш (или Almissa) је град у Хрватској, у Сплитско-далматинској жупанији.

Географски положај
Налази се у Далмацији на ушћу ријеке Цетине у Јадранско море.

Историја
До територијалне реорганизације у Хрватској налазио се у саставу старе општине Омиш.

Подручје Јадранске обале, прво насељавају Илири око 1000. п. н. е., да би после њих дошли Грци око 400. п. н. е. који оснивају колоније на обали и оближњим острвима. Након њих Римско царство ступа на власт 34. п. н. е. Око 600. године Хрвати се насељавају на подручју данашње Далмације.

Омиш се први пут спомиње у историји под именом Онеум (), који замире почетком 7. века. С обзиром на позицију Омиша, као важне везе унутрашњости с копном, то није било дугог века. Историјско средиште тадашњег Омиша је било смештено на источној обали ријеке Цетине (име ријеке Цетине потиче од фригијске речи Зетна — врата, и тиме описујући ушће Цетине у море).

У античком периоду насеље Омиш се вероватно налазило у засеоку Баучићима гдје је пронађено много античких проналазака (рељефа, натписа, надгробних споменика, свјетиљки, новца итд.). Најважнији споменици, који се данас чувају у Градском музеју, су налази камених римских натписа. Из текста се закључује да су уклесани за време владавине римских царева Тиберија и Клаудија, а највјероватније су били намијењени некој јавној згради, споменику или путу, што говори о важности насеља у којем су грађени.

Омиш је 1838. године мали градић опасан зидом, у којем живи 786 душа у 154 дома.

Становништво
На попису становништва 2011. године, град Омиш је имао 14.936 становника, од чега у самом Омишу 6.462.

Град Омиш

Напомена: Настао из старе општине Омиш. У 1991. садржи део података за општину Дуги Рат.

Омиш (насељено место)

Напомена: У 1991. повећано припајањем насеља Равнице које је престало да постоји. За то бивше насеље садржи податке од 1857. до 1981. У 1991. такође повећано за део подручја насеља Дуће (општина Дуги Рат). У 1857, 1869, 1921. и 1931. садржи податке за насеља Борак и Станићи.

Попис 1991.
На попису становништва 1991. године, насељено место Омиш је имало 6.079 становника, следећег националног састава:

Партнерски градови
  Рјазањ
  Диксмуиде
  Загорје на Сави
  Хавиржов
  Сан Феличе дел Молизе
  Бол
  Непомук

Извори
  Савезни завод за статистику и евиденцију ФНРЈ и СФРЈ, попис становништва 1948, 1953, 1961, 1971, 1981. и 1991. године

Референце

Литература

Спољашње везе

 Омиш на интернету
 Омиш
 Туристички портал града Омиша

Градови у Хрватској
Википројект географија/Насеља у Хрватској
Омиш
Насељена места у Хрватској
Насељена места у Сплитско-далматинској жупанији


Choose from a variety of Ultimate Software Events to attend around Ohio this year. Cheers!
"UltiPro has changed the way we work in all aspects. Everything is done within one platform, and it has truly changed our lives. Our operations are much more efficient and streamlined thanks to Ultimate Software."


text:
Ticketmaster Lays Off 1/4 Of All Employees, Cut Costs by $500 Million
The entertainment industry has ground to a halt, so it's surprising that Ticketmaster has made it this long without furloughing anyone due to COVID-19. But with little to no projected revenue for the remainder of 2020, the ticketing company has just furloughed a quarter of its workers.
Employees were apparently caught off guard as they were assured they would be taken care of, but as the situation continues to change, early promises would have to be re-examined.
Ticketmaster president Jared Smith said in the letter obtained by Variety, "For the very first time in the history of live entertainment, the industry is completely shut down," describing that a few months ago, the situation would've "seemed incomprehensible."
Furloughs are set to take effect May 1, but workers will still be given full health benefits as well as 1 to 2 weeks of payout from their earned time off. Ticketmaster has also assured those furloughed that they will be brought back "as soon as live performances start again."
The decision comes on the heels of a SEC filing from Live Nation on April 13 that stated that the company was looking toward a $500 million cost reduction plan for this year, including "reducing contractors, renegotiating rent, and paring back other discretionary spending."
Executives were not spared from the culling either, as Live Nation President/ CEO Michael Rapino has already relinquished his remaining salary for the year in a statement earlier this month.
Rapino said that "to date there are 50,000 events that have been canceled, postponed or rescheduled globally on the TM Platform." He further said that "60% of the over 50,000 events that have been canceled, postponed and rescheduled globally on the TM Platform have been authorized for refunds — and we have already returned over $400 million to fans."
With concerts and festivals unlikely to return in 2020 (sports events will likely return sooner, but when remains up in the air), the entire music industry faces a hard year ahead of it.
via Music Times
BREAKING: All 2020 Bassnectar Events Cancelled
Marshmello, Diplo, Martin Garrix & More Playing The Most Epic Prom Ever

redpajama_set_name: RedPajamaCommonCrawl


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Sapa Mid-Autumn Festival set to open
Photo: Aprotravel.vn
Week-long festival to open on September 2 in Sapa.
by Le Diem
Sapa, Mt. Fansipan among Top 10 Southeast Asian hikes
Da Lat & Sapa among Asia's rising destinations
Sapa to be national tourism area by 2030
Sapa among Top 10 Most Magical Places in Asia
The Ministry of Culture, Sports, and Tourism has announced a week-long Mid-Autumn Festival starting on September 2 in the northern hill town of Sapa.
A wide variety of activities are planned, including art, handicraft and photo exhibitions, food and beverage events, concerts, dance and theater performances, and sporting competitions, among others.
Capping off the sporting competitions is a mountain marathon and cycling race. Other featured events include a lantern parade, Mong ethnic minority dancing, and artistic performances.
Sapa was voted as one of the Top 10 most attractive destinations in Asia in 2017 by popular travel website TripAdvisor, based on millions of travelers' reviews. The hill town is known for its terraced rice fields and trekking trails. The harvest season in October is amazing for both sightseeing and trekking, with beautiful yellow rice paddy fields, according to the site. Tourists can also visit the Saturday night "Love Market", the Gothic stone church in the town center that is a reminder of the French missionary influence, and many Vietnamese and European restaurants downtown.
The Prime Minister has approved the general master plan for the development of the Sapa National Tourism Area to 2030, with a goal of developing unique and high-quality tourism products and attracting 5.2 million visitors annually.
Sapa welcomed over 2.7 million tourists last year, up 37 per cent against 2015 and including more than 750,000 international visitors.
Paradise Vietnam debuts new restaurant in Nha...
Grant Thornton: Digital technologies changing...
Alba Wellness Valley by Fusion hosting wellness...
JW Marriott Phu Quoc Emerald Bay recognized in...
Hoi An tops Travel + Leisure's world's best...
Hôtel des Arts Saigon introduces new business luncheon concept
CNN lists best things to do on Phu Quoc
Forbes reveals Vietnam's 10 most beautiful beach destinations
Laguna raises the bar for 19th holes
International visitors in June the lowest this year
InterContinental Danang Sun Peninsula Resort among world's best
Vietnam Airlines baggage allowance to be based on pieces
Central Vietnam among Lonely Planet's Top 10 Asia-Pacific...
Malaysian Culinary and Product Fair on horizon
LuxYacht to distribute luxury Ferretti Group yachts


text:
Bloody Funny
August 16, 2012 by CJ
Jo Nesbø's Jackpot ***1/2 (out of five)
Contrary to its title, Jo Nesbø's Jackpot is not about the Norwegian bestselling author winning the lottery. Rather, it's an original film (not an adaptation of one of his novels) that's based on a story by Nesbø, and a cracking tale it is. A bloody shoot-out in a porn palace near the Swedish / Norweigan border initiates a police investigation into what Oscar (Kyrre Hellum, who's got something of the Sam Rockwells about him), the only survivor, was doing there. We delve into the events leading up to that massacre, beginning with a syndication of ex-cons winning a football pool. Things go wrong, and get funnier, from there.
Ever since John Travolta shot that kid in the back seat of the car in the head in Pulp Fiction and gazillions of people around the world found themselves laughing hysterically at the site of Travolta's face covered in the kid's brains, many films have tried to find the right balance by which heavy violence can be funny. Jackpot, a true "black comedy" if ever there was one, deftly finds that balance, and is entertaining throughout. Constantly engaging, and highly recommended.
One thought on "Bloody Funny"
Ben Carew says:
Hey why such a mediocre score for such a highly recommended flick?


At Big Garlic, we prepare diverse formats for our garlic, with full availability and flexibility to adapt to the needs of each of our clients.
If you can't find a combination that meets your needs, don't hesitate to contact us.
A mesh bag of 100 grams can be prepared with any variety of garlic and, depending on the size, this can be done with 1, 2 or 3 bulbs of garlic.
A mesh bag of 150 gramscan be prepared with any variety of garlic and, depending on the size, this can be done with 2, 3 or 4 bulbs of garlic.
A mesh bag of 170 grams can be prepared with any variety of garlic and, depending on the size, this can be done with 2, 3 or 4 bulbs of garlic.
A mesh bag of 200 grams can be prepared with any variety of garlic and, depending on the size, this can be done with 3, 4 or 5 bulbs of garlic.
A mesh bag of 250 grams can be prepared with any variety of garlic and, depending on the size, this can be done with 3, 4 or 5 bulbs of garlic.
The 1 kilogram mesh bag can be prepared with any variety of garlic and, depending on its size, can be done with between 7 and 12 bulbs. .


Projected starters: NEVADA — F Jordan Caroline (17.6, 8.6) and Caleb Martin (19.6, 5.4); G Lindsey Drew (7.4, 2.1), Cody Martin (14.6, 6.4) and Kendall Stephens (7.9, 1.3); UC IRVINE — F Tommy Rutherford (10.6, 5.2) and Brandon Smith (4.4, 1.8) or Jonathan Galloway (3.7, 7.1); G Max Hazzard (9.0, 4.8), Evan Leonard (10.9, 3.4) and John Edgar Jr. (9.7, 3.6).
The Anteaters are 2-0 at home. They have lost four of their last six after back-to-back wins over Chapman and Denver ... Nevada has won six straight true road games dating back to last season. The Pack is 3-0 away from home this year (UOP, Santa Clara and Hawaii). Nevada is 16-14 in true road games under Musselman, but 10 of those losses came in his first season (5-10) ...Jordan Caroline had his 40-game starting streak end against Illinois. He left the team for a couple of days to be with family after his grandfather, ex-Chicago Bear J.C. Caroline, passed away on Nov. 17. He still managed to score 20 points against ISU, mostly from the key area ... ESPN has the Pack's RPI at 5 and strength of schedule is 33 ... Nevada has scored 81 or more in all but one game this year (67 vs. Hawaii) ... Nevada is the only undefeated team in the Mountain West ... This is the third non-conference foe that Nevada will face in its non-conference schedule, joining Idaho and Rhode Island.
With Texas Tech losing a neutral-site game against Seton Hall, it's possible Nevada could move into the Top 25 with a win at UC Irvine on Saturday at 7 (BigWest.TV and 630AM).


This study was conducted in order to determine if the source of contaminants at formerly used defense sites (FUDS) in Alaska were deposited as a result of military occupancy or from long-distance transport. This determination largely influences whether remediation will occur, and, if so, to what extent. For this reason, plant samples (rinsed and unrinsed) and sediment cores were collected at military and remote sites on St. Lawrence Island (SLI) and Norton Sound, Alaska during the summers of 2002, 2006, and 2007 and analyzed for persistent organic pollutants. On St. Lawrence Island sediment core samples were collected at the Northeast Cape FUDS, also a traditional fishing/hunting camp, and were sectioned and analyzed for concentrations of Polychlorinated biphenyl (PCB) congeners, Mirex, Dichlorodiphenyldichloroethylene (DDE), Hexachlorobenzene (HCB), Mercury and Cesium-137 (137Cs). Differences in the total concentrations and distributions of PCB congeners, Mirex, DDE and Mercury in sediment cores and in plants collected from the two SLI and three Norton Sound mainland formerly used defense sites indicate the majority of the contaminants found can be temporally related to releases during military occupancy and subsequent redistribution of contaminants. Contaminants in plant samples at the SLI sites are elevated relative to the two remote sites located on St. Lawrence Island and the three mainland Norton Sound FUDS at Elim, Unalakleet, and Wales. The concentrations, lateral and vertical distribution of the total PCBs, and congener-specific differences in sediments and plants readily differentiate locally derived from globally transported contaminants. The relative contaminant concentrations in sediment cores and between rinsed and unrinsed plants collected from the NEC FUDS indicate contaminants were remobilized and redistributed during recent site remediation activities.
Copyright 2012 Journal of Local and Global Health Science.


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Should Your Client Come in with Clean Hair for Your Color Service?
Dana Simnovec is a 4th generation hairstylist. After two decades of doing hair, her clients have become family. Her biggest passion (when not behind the chair) is her role as the Artistic Educator for Matrix Canada.
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Pinterest's Top Three Beauty Trend Predictions for 2023
From micro-manis to barely-there bobs, this year's beauty looks will be a cut above the rest. Millennials and Gen Z are trading in their long, coffin-shaped nails for mini French manis. They'll opt for short braids and chopped bangs instead of long, luscious locks. In 2023, small beauty gets its big break.
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Festive Fingertips from NAILGOD
Canadian beauty brand NAILGOD shares its favourite holiday-inspired nail looks to rock this season. Go full-on Christmas crazy with a melody of different festive nail designs. Show everyone you're in the holiday spirit by wearing all the wonderful symbols of the seasons on your nails.
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Charlotte Tilbury Reveals Her Magic Gifting Universe
Charlotte's magic gifting universe of sparkling new innovations and best-selling, globally loved beauty favorites are inspired by the magic and euphoria of Studio 54. Charlotte Tilbury Beauty sets the mood with an iconic lineup of beauty looks inspired by the sparkling energy of the disco era, set against a backdrop of glittering mirrorballs and disco lighting.
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Trevor Sorbie new collection CHROMA
The International award-winning hair salon Trevor Sorbie has announced the release of their latest collection, 'CHROMA'. Created by newly appointed Creative Director, Giuseppe Stelitano, this is the first collection in over a decade from the brand that focuses on color at the forefront, rather than styling.
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Interview with British Beauty Queen Trinny Woodall of Trinny London
The famous UK fashionista from What Not to Wear, Trinny Woodall, experienced makeovers on thousands of women around the world. Her vast understanding of what women want and need
was the catalyst that started her Trinny London beauty brand online in 2017.
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MYNC joins Blo Blow Dry Bar Franchise
Vanessa Yakobson, Blo Blow Dry Bar Partner, and CEO, and/or Nathalie Smith, MYNC Founder and Brand President partner to create a powerhouse beauty franchise.
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Fall is the Best Season for Laser Hair Removal
With the fall season in effect and winter not far, the industry buzz suggests that this season is the best time for those thinking of having laser hair removal services. However, there has been a rise in misinformation about laser services/benefits, causing clients to feel uncertain about the viability of laser hair removal services.
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Sola Salons Suites The World's Largest and Fastest-Growing Salon Studio Franchise
As the workforce continues to shift towards a self-employment model, individuals from many different industries from tech moguls to farmers are becoming Sola Salons franchise owners. Helping over 19,000 beauty professionals in 628 locations in North America, salon suites have become the business trend for those seeking to become their own bosses.
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LANCÔME partners with iconic [email protected]
America's premier Livestream shopping platform, NTWRK, announces a new partnership with LANCÔME. The two brands are teaming up for the exclusive release of 10 LANCÔME x [email protected] customized bears, the wildly popular art collectible toys, which GQ dubbed "streetwear's most enduring icon." Made by Japanese company MEDICOM TOY, the bears have been behind some of the most groundbreaking collaborations with the world's best brands, fashion houses, and artists.
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EVO; Honest Products that Respect People and the Planet
in Blog, News
The creative concept for 'higher ground' was a true collaboration between our Creative Team. Each team member built their own individual brief which we fused together, resulting in a collection with no restraints, boundaries, or limits.
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Phorest Re-Ignites the Annual Salon Owners Summit
Phorest Salon Software invites salon & spa owners to travel to Dublin, Ireland, in 2023 for the highly anticipated Salon Owners Summit.
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Shawntel \sha(wn)-tel\ as a girl's name is a variant of Shawna (English). See for meaning of Shawntel. The baby name Shawntel sounds like Shantel, Shontel, Shawntile, Shauntell, Shantell, Shantal, Chauntel, Chantel and Shandel. Other similar baby names are Shawnte, Shauntee, Shantee, Shannel, Shanel, Shawnta, Shante, Shantey, Shawntay, Shawntae and Shaunte.


HO Scale Craftsman KIT - A. Monnette & Son....featuring laser-cut walls and roof panels. Custom laser-cut store front, window details and corbel trim. Tichy windows and door. Footprint approximately 2-3/4"x2-1/2". ASSEMBLY REQUIRED. Scenery, figures, vehicles and details not included.


text:
Q: Строковые WinAPI функции Допустим, мне надо использовать строку в какой-л. апишной функции, принимающую один из параметров (и более) переменную типа LPWSTR/LPCWSTR/LPCSTR. Все это псевдонимы одного типа. Вот как я поступаю в этом случае:
   StringCbCat(szTitle,MAX_LOADSTRING,L"123123");

StringSbCat Function
Как видите, последний параметр передан "в чистом виде" без использования переменных. Я использую предикат "L". А можно ли как-то обойтись без предиката, сделав какие-либо предварительные преобразования?
Спасибо!
A: Если вы будете передавать строку без использования макроса L или _T (который более универсальный и работает для TCHAR), то ваша строка будет автоматически трактоваться как ANSI-строка.

Соответственно если в настройках проекта стоит Use Wide Character Encoding, то код не скомпилируется. Единственное, что можно сделать в таком случае - воспользоваться какой-либо функцией для перекодирования LPCSTR -> LPCWSTR, однако это намного сложнее, нежели просто использовать L или _T.

redpajama_set_name: RedPajamaStackExchange


Report Card and Parent/Teacher Conference changes.
I hope you are all aware of the change we are making to provide access to your child's report card via Infinite Campus instead of sending home a copy of the report card in the mail or with your child. Bryce DeRoos sent a letter to each family last week outlining this change and with the procedures necessary to locate the report cards on Infinite Campus. If you need any assistance, please do not hesitate to contact us at the high school and we will assist you as necessary. If you need a hard copy of your child's report card, we will provide that as well. The first quarter report cards will be posted to the Infinite Campus parent portal by the end of the day on Monday, November 14th.
I also want to share another change we are implementing this year that I am really excited about personally. We would have normally held Parent/Teacher Conferences shortly after the first quarter ended, but for some time we have contemplate how we could make these meetings more valuable for students and parents, and thus, dramatically increase attendance. This year we plan to have the Parent/Teacher Conferences for Freshmen, Sophomores and Seniors (and their parents) on February 6th and 7th, shortly after the semester. Our approach will also be dramatically different in that we plan to implement student-led conferences. These conferences will have a "script" that students will follow to sharing with their parents their career goals, researching and presenting the requirements they will have to meet to be ready for the next phase of their lives, and discussing their current performance at school, and their ACT/Aspire scores, in the context of their career goals. Our purpose in moving to student-led conferences is for students to take even greater ownership in their present level of work, goals, and future plans in the hopes of insuring they are well prepared for their futures by the time they graduate OHS.
Our general plan is to have each of our faculty members working with 10-12 students to prepare for the student-led conferences with a week of homeroom periods just after the semester. The student will gather information and answer the questions on the guide form in preparation to share with their parents about their career goals and current progress toward those goals. Each teacher will also schedule a 15-20 minute meetings with each of those 10-12 students and their parents in the evenings of Feb. 6th and 7th, and they will also facilitate the meeting as necessary and support each student as they present to their parents. It is our hope that the information from this meeting will be a great springboard for high school students and their parents to have more discussions about their futures and their current preparations toward their goals. We also hope this process is a very intentional means of helping students take ownership of their current work as students and to think more deeply about their futures. Additionally, we hope this process really helps students to see that their high school classes represent a critical phase of their preparation for the future, and that what they are learning right now is important and necessary for future success.
We sincerely hope that this new approach of student-led conferences will be more meaningful for our students, and you as their parents. We hope the conferences are simply a starting point or take off point for some really important discussions between all students and their parents.
If you have any questions or suggestions related to either of these changes, please give me a call at 920-564-2346 x4001.
I sure hope everyone is staying dry on this wet and rainy day! I wanted to inform you that a mailing should be arriving to students' homes today or tomorrow. The mailing includes important information related to parent-teacher conferences and report cards.
Parent-Teacher Conferences: In an effort to streamline the scheduling of parent-teacher conferences for both parents and staff, we are going to use Sign Up Genius this Fall. Therefore, on Friday afternoon of this week, you should receive an email from info@signupgenius.com informing you that you have been invited by Sherri Stengel to sign up for "2016 Parent Teacher Conferences". Click on the link and sign up for a conference time that works for you on either 11/15 or 11/17/2016. To view the letter sent home that explains this information in more detail, please visit Letter regarding Parent-Teacher Conferences.
Report Cards: Another letter explaining our move to online report cards will also be in the envelope you receive in the mail. Report cards will become live in the parent portal of Infinite Campus on the afternoon of 11/14/16. In order to access your child's report card, you will need to have an account with Infinite Campus (IC). If you don't already have an account to sign in to the parent portal of Infinite Campus, the letter explains how to go about creating an account. If you already log into IC to see your child's grades, attendance, and other information, you already have what you need to access your child's report card. The letter explaining our move to paperless report cards as a district that was sent home can be found at Report Card Information.
We understand that there are some families that do not have internet access at home or otherwise may not prefer online methods. Therefore, if you'd prefer a paper copy of your child's report card or want Deb DenBoer, OMS Secretary, to sign you up for parent-teacher conferences, please call the office at 920-564-2383.
There will be a dance on Thursday from 2:30 – 4:00 with a costume contest. Cost is $3.
The OMS ROV Team will be hosting tryouts after school on Wednesday, October 26th from 2:40 – 3:30 in high school room 129. Any middle school students who are interested in robotics and are able to practice after school on Wednesdays should attend this tryout on Wednesday. Tryout students guides are available and can be picked up anytime in room 129. Please see Mr. Hendrikse in the high school for more details.


Popularity of a person means his being well known for his special qualities and achievements among the people of his city, state or country, and even beyond the frontiers of his country. A person becomes popular when the Lagna and Lagna lord are very strong and there are other supportive powerful yogas in his horoscope. The dasa-bhukti of strong Lagna lord and yoga forming planets should run during his youth and middle age, when the native can put in maximum effort and work hard to reap the reward. Success and popularity is the result of strong destiny and sincere and dedicated efforts to unfold it.
The general public is indicated by the 4th house of the horoscope and 10th house refers to one's profession. If these are strong and their lords are also strong along with strong Lagna and its lord, the person becomes popular and gets recognition for his work. In addition, when the 5th house (Poorvapunya), the 9 house (God's Grace) and their lords are also strong and well placed, the person is destined for getting recognition and fame.
Saturn is the significator of masses. When it is strong and favorably placed, the person achieves mass following. Rahu when placed in 10th house of a horoscope also helps in native's rise in politics. Venus in Sagittarius is reputed to give much name and fame. Mercury in 8th house makes an individual capable, rich and famous. Similarly, malefics placed in evil (6, 8, 12) house destroy the negative effect of those house and confer recognition and success on the native.
The Moon stands for personality of a person and Jupiter indicates expansion. Their harmonious location in birth chart makes the native successful and popular. The Kendra location of Moon and Jupiter is known as Gajakesari Yoga, which is reputed for conferring name, fame, wealth and reputation on the individual. The presence of Pancha Mahapurusha Yoga and Karak Yoga makes a person famous and popular in life. Another yoga which helps a person born in ordinary circumstances to rise in life due to destiny and hard work is Neecha Bhanga Raja Yoga. In addition, Vipreet Raja Yoga gives sudden rise, but if there is no supportive Raja Yoga, the down-fall follows.
Conversely, downfall takes place during the dasa-bhukti of planets forming Arishta (Inauspicious) Yogas. The dasa-bhukti of functional malefics (Lords of 6, 8, 12 house) and badhaka (obstructive) planets also cause downfall of the native. It is also observed that some persons rise to a very high position, but also suffer sudden downfall and their reputation and public image gets smeared. This occurs due to operation of good youga forming planet's dasa followed by that of bad yogas present in the horoscope.
The strength of Tagore's horoscope is the Parivartana Yoga (interchange of singns) between Lagna and 10th lord Jupiter and the 5th lord Moon. The Karka of mind Moon is also 5th lord and highly fortified by the aspect of exalted Jupiter, which conferred intuitive and receptive mind with versatile imagination. There is strong Budha-Aditya Yoga formed by Mercury and exalted Sun in 2nd house. The conjunction of Mercury with poetic Venus in 2nd house made him a successful poet, painter and philosopher.
The exaltation of Lagna lord Jupiter is 5th house, and presence of Saraswati, Chamar , Chatra and Vimal Yogas Made him world famous.
Tagore was conferred Nobel Prize for literature in 1913 during Moon dasa-Mercury bhukti. He founded Vishwa Bharati University at Shantiniketan in 1921 (Mars-Venus).
Rahu posited in 10th house in Jupiter's sign gave the excellent result of exalted Lagna and 10th lord Jupiter posited in 5th house. During Rahu dasa he toured Europe, America and Japan and met local authors, thinkers and statesmen.
He passed away on 7th august, 1941 in Jupiter dasa-Jupiter Bhukti. Generally, strong Lagna Lord Jupiter should not cause death, but it is in the Nakshatra of Mercury which is a powerful marka both by ownership of 7th house and its location in 2nd house.
Libra Lagna with its lord there forms 'Malavya' Pancha Mahapurusha Yoga. Jupiter in Kendra to Moon forms Gajakesari Yoga. The aspect of Jupiter on Lagna and presence of Mercury in Lagna Gives keenness of mind and humanitarian approach. The Lagna in Papkartari Yoga gave him frail body, but presence of Mars in Lagna made him energetic and active. Swakshetre Moon is with Rahu due to which his mind was never at rest for one reason or the other. However, Libra Lagna gave him calm and balanced approach in the face of greatest provocation. Jupiter is in 10th from Moon and aspected by Lagna Lord Venus and 9th lord Mercury, which gave him mass following for a noble cause.
The presence of Rahu in 10th from Lagna is always indicative of some notable work by the native. The commencement of Rahu dasa in 1921 brought Gandhiji into active independence struggle and he made a call for Swarajya (self rule). In Rahu dasa-Rahu bhukti Gandhiji was arrested for civil disobedience campaign. He was released in Jupiter Bhukti. He started mass civil disobedience and was again arrested in 1942. His non-violent struggle made him world famous.
The independence of India was finally realized in August, 1947 but Gandhiji was unhappy because of the partition of the country and suffering caused to millions of people. He advocated for restraint and humanitarian approach and even went on hungerstrike for Hindu-Muslim harmony. As a result he was shot dead by a Hindu fanatic on 30.01.1948 during Jupiter dasa-Venus bhukti. Jupiter is in maraksthan (7th) with Pluto and aspected by maraka Mars, and Venus is in Jupiter's Nakshatra with maraka Mars and aspected by Hershel and Pluto.
Lagna lord Saturn is in Kendra in friendly sign owned by Yogakarka Venus. Lagna lord Saturn aspecting Lagna and 10th house gave him success through hard work, name, fame and wealth. Yogakarka Venus in 8th gets Neecha Bhanga in the company of exalted Mercury. Airy sign in Lagna and Moon in 9th has given him excellent thinking mind. The aspect of the Sun, Mercury, Venus, Mars and Jupiter on the 2nd house gave him an impressive voice, expressive face and expertise for portrayal of different roles and excellent success in film line. Jupiter in 10th from Moon forms Gajakesari Yoga.
He tasted success in films with the start of Lagna lord Saturn dasa (1971-90) which is located in 4th house (public) and aspects 10th house (profession). He made his mark in 'Anand' which was released in 1972. In 1973 'Zanzeer' became hit and gave him recognition as 'Angry Young Man of Indian Cinema'. His excellent voice and dialogue delivery scored over contemporary heroes and he continued to give hit films during Mercury dasa (5th lord exalted in 8th) as well. Thus he ruled Bollywood for more than three decades and became common man's hero. However, as Uranus is with Saturn, and aspects a bunch of planets in 8th house and also 12th house, Amitabh also faced sudden and serious health and financial set-backs during adverse bhuktis and transit. Because of his popularity countrywide prayers were offered by his fans for early recovery from serious health problems.
With the beginning of Ketu dasa from January, 2007, which is posited in Lagna in Rahu Nakshatra and unfavourable transit, he is in the midst of controversies.
Lagna and Larna lord Mercury are vargottam. Mercury is in 5th, with 5th lord Saturn in Shubha kartari Yoga formed by Venus and Jupiter. The debilitation of Mars in 11th house gets cancelled because of its being with sign lord Moon. Presence of Venus, Mercury and Jupiter in 6th, 7th and 8th house respectively form Moon forms prosperous Chandradhi Yoga. The 2nd and 9th lord Venus is in Sagittarius in 4th house ensuring all comforts of life, name and fame. He was a popular cricketer and successful Captain of Indian Cricket Team.
As 5th lord Saturn is in own sign in 5th house, he got immense popularity and recognition during the dasa of favorable planet for his Lagna. Venus, 2nd and 9th lord, is the most benefic planet for Virgo Lagna and Mercury is Lagna Lord. He made his debut in Test Cricket in 1984-85 during Venus-Saturn-Mercury with a century against England. In 1987 (Venus-Saturn) he received Padma Shree and Arjuna Award. In 1990 (Venus-Mercury) he was made Captain of the Indian team.
With the Start of 12th lord Sun dasa posited in inimical sign in 5th house his performance started deteriorating and in 1996 (Sun-Jupiter) he was removed from Captaincy. Jupiter is in 6th house and placed in 2/12 position with dasa lord Sun.
In April, 2000 (Moon-Moon) he was banned from playing cricket for life on the charge of match-fixing. Moon is inimical to lagna lord Mercury and is afflicted by Mars, the greatest malefic for Virgo lagna, and Rahu.
This is another example of spectacular rise in life and ignominious end.
The 5th lord Sun is exalted in Lagna with Uranus and shows a revolutionary personality with desire to rule. Venus lord of 2nd and 7th, is exalted in 12th with 10th and 11th lord Saturn. Venus is also Navamasa Lagna lord and exalted. It is also vargottam and assures best result during its dasa. Lagna lord Mars is in own sign in 8th house but afflicted by Rahu. 4th lord Moon and 9th lord Jupiter are debilitated. Jupiter is aspected by Pluto and Ketu. Moon is afflicted by Rahu and Mars in 8th house which indicates troubled mind and tendency of bold and rash actions. Mercury with Ketu in 2nd house made him a revolutionary speaker.
After facing troubled life in childhood and youth during Mercury and ketu dasas, Saddam Hussain participated in a successful military coup in 1969 (Venus-Venus) and became Deputy to the new President. He slowly consolidated his position and became a 4-Star General. After forcing the then President to resign Saddam Hussain became President in 1979 (Venus-Jupiter), and ruled as socialist revolutionary with a firm hand, and modernized Iraq first with the help of Soviet Union and the of USA. His popularity with masses was a mixture of admiration and fear.
Due to his attack on Kwait, an ally of USA, in 1991 (Sun-Rahu, located in Lagna and 8th house) his relations with USA deteriorated. During Moon dasa (in 8th) afflicted by Rahu and Mars, tension increased further, and after USA-led invasion of Iraq he was deposed on 9th April, 2003 (Moon-Ketu-Saturn) and went into hiding.
He was captured from a hideout on 12.12.2003 (Moon-Venues-Mars) and tried for crimes against humanity by US backed Iraqi Tribunal. He was sentenced to death on 05.11.2006 (Mars-Rahu-Moon) and was hanged to death on 30.12.2006 (Mars-Rahu-Mars).
In both the above examples, the natives were very popular, prosperous and basked in fame during favorable dasa and suffered downfall during unfavorable dasa, which proves the sublime truth of Astrology.


Financiers LOVE their acronyms more than $$. The more not obvi(ous) the better!
ETFs are giving Mutual Funds (MFs) a run for their money in the popularity contest that is the stock market.
Best of both worlds: ETFs allow for diversification (like a fund) and frequent trading (like a stock).
Easy access: ETFs don't have minimum investment requirements like MFs. Good news for newbie investors.
Tips: Owners have to pay extra $$ per trade. If dollar cost averaging is your investment strategy, then commissions are your enemy.
Passive Action: Index investing is really the only option for these babies. You have to believe the market is efficient (and be anti-Warren). Do you believe??
When you make the MF/ETF decision…ALWAYS read the prospectus.


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Between May 30 and June 2, CPI held two workshops to train 44 CHVs in basic rescue techniques. These CHVs provide the first line of basic health care services to fellow refugees in four refugee camps in Cox's Bazar - Kutupalong (Camp 1W), Hakimpara (Camp 14), Jamtoli Camp (15), and Potibonia Camp (16). During the workshops, participants learned how to lift patients safely, take precautions to ensure spinal protection, correctly use stretchers, and practice different types of drags and carries to safely move patients during emergencies.
CPI also equipped the CHVs with rescue kits that include life jackets and throwlines for flood rescue, stretchers, head torches, and basic first aid supplies so that they can provide rescue and first response services, day or night, to their fellow refugees across Cox's Bazar.


text:
\section{Introduction}
\label{sec:level1}

The self-dual (SD) model, put forward by Townsend, Pilch and Van
Nieuwenhuizen\cite{TPN1}, has been the object of several investigations.
On a semiclassical level, it has been shown to be equivalent to the
Maxwell-Chern-Simons (MCS) theory\cite{Jackiw1,Frad}. That this equivalence
holds on the level of the Green functions was proved in Ref.\cite{Rothe1}.
Lately, the second-class constraints of the SD model were successfully
converted into first-class by means of the embedding procedure proposed by
Batalin, Fradkin and Tyutin\cite{BFT}. It was then found that the SD model and
the MCS theory are just different gauge-fixed versions of a parent
theory\cite{Kim,Rothe2,Rothe3}.

The statements above refer to the free SD and MCS models. The central purpose
of this paper is to present the canonical quantization of the SD theory
minimally coupled to fermions, while unraveling possible links with the MCS
theory minimally coupled to fermions.

Explicit calculations of probability amplitudes for an interacting field
theory in 2+1 dimensions are, on
general grounds, possible only in perturbation theory. Our main goal
will then be to formulate the quantum dynamics of the minimally
coupled SD-fermion system in the interaction picture.
But to make this operational one must {\it a priori}
solve for the quantum dynamics of the free SD and fermion fields. Hence,
we start in section 2 by quantizing the uncoupled SD theory through the Dirac
bracket quantization procedure (DBQP)\cite{Di,FVil,Su,Gi1}. The model is found
to contain only one independent
polarization vector, which is explictly determined and turns out
to be identical to the polarization vector of the MCS in the Landau
gauge\cite{Gi2}. By combining this result
with the field commutators at different times, obtained afterwards, we are led
to conclude that the space of states is positive definite. Ordering ambiguities
arise in the construction of the (symmetric) energy-momentum tensor and we are
forced to adopt an ordering prescription. The components of the energy-momentum
tensor are shown to fulfill the Dirac-Schwinger algebra, which secures the
existence of a set of charges obeying the Poincar\'e algebra\cite{DiSch,Sch}.
The summary of this section is that
the SD model is a relativistically invariant quantum field theory
describing massive quanta whose spin can be $\pm 1$ depending
upon the sign of the mass parameter. Thus, the particle content of the
SD model is identical to that of the MCS theory in the Coulomb
gauge\cite{Gi2,Ha1,Jackiw2}. This establishes the equivalence of these two
models within the operator approach. We emphasize that the developments in this
section do not duplicate those in
Refs.\cite{Jackiw1,Rothe1,Kim,Rothe2,Rothe3}, which
were carried out either within the semiclassical approximation\cite{Jackiw1} or
within the functional approach\cite{Rothe1,Kim,Rothe2,Rothe3}.

The quantization of the free fermion field through the DBQP can be found in
the literature\cite{Su,Fleck} and leads to standard results quoted in
texbooks on quantum field theory. We then turn in section 3 into applying
the DBQP for quantizing the SD model minimally coupled to fermions.
Aside from technicalities which will be discussed in detail, the main point
arising along this process of quantization is that the
equal-time commutators (ETC's) involving the fermions and one of the SD
field components do not vanish,
thus obstructing the standard way for arriving at the formulation of the
quantum dynamics in the interaction picture. The situation resembles that
encountered when quantizing the MCS theory minimally coupled to fermions
in the Coulomb gauge. There, the ETC's involving the fermionic fields and the
longitudinal component of the momenta canonically conjugate to the gauge
potentials do not vanish, also. The way out from the trouble consists in
getting rid off these longitudinal components by means of the Gauss
law constraint. This operation brings, in turns, two new interaction terms to
the Hamiltonian\cite{Gi2,fo1}. In the present case, the elimination of the
corresponding SD field variable also brings two new interaction terms, being
both local in space-time and non-renormalizable by power counting. One of these
terms is the time component
of a Pauli interaction  while the other is the time component of a Thirring
interaction. After having determined all the vertices of the coupled theory
we end this section by presenting the expressions for the free fermionic and
SD-field propagators.

The relativistic invariance of the interacting model is tested in section 4.
As in Ref.\cite{Gi2}, we use, for this purpose, the lowest order
contribution to the elastic fermion-fermion scattering amplitude. We
demonstrate that the combined action of the
non-covariant pieces that make up the interaction Hamiltonian can, after all,
be replaced by the minimal covariant field-current interaction. The high
energy behavior of the self-dual field
propagator corroborates that the SD field when minimally coupled to fermions
defines a non-renormalizable theory.

Some final remarks and the conclusions are contained in section 5.

\section{Quantization of the free SD model}
\label{sec:level2}

The dynamics of the SD theory is described by the Lagrangian
density\cite{TPN1,Jackiw1}

\begin{equation}
\label{201}
{\cal L}^{SD}\,=\,- \frac{1}{2m} \epsilon^{\mu\nu\rho}\,
(\partial_{\mu} f_{\nu})\,
f_{\rho}\,+\,\frac{1}{2}\,f^{\mu}f_{\mu}\,\,\,,
\end{equation}

\noindent
where $m$ is a parameter with dimensions of mass. We use natural units
($c=\hbar=1$) and our metric is $g_{00}=-g_{11}=-g_{22}=1$. The fully
antisymmetric tensor is normalized such that $\epsilon^{012}=1$ and we define
$\epsilon^{ij}\equiv\epsilon^{0ij}$. Repeated Greek indices sum from $0$ to $2$
while repeated Latin indices sum from $1$ to $2$.

By computing the momenta ($\pi_{\mu}$) canonically conjugate to the field
variables ($f^{\mu}$) one finds that the theory possesses the primary
constraints

\begin{mathletters}
\label{202}
\begin{eqnarray}
&&P_0\,=\,\pi_0\,\approx\,0\,\,\,, \label{mlett:a202} \\
&&P_i\,=\,\pi_i\,+\,\frac{1}{2m} \epsilon_{ij} f^j\,\approx\,0\,\,,i=1,2\,\,\,,
\label{mlett:b202}
\end{eqnarray}
\end{mathletters}

\noindent
where the sign of weak equality ($\approx$) is used in the sense of
Dirac\cite{Di}. Furthermore, the canonical Hamiltonian ($H^{SD}$) is found to
read

\begin{equation}
\label{203}
H^{SD}\,=\,\int d^2x\,\left(-\frac{1}{2}f^{\mu}f_{\mu}\,+\,\frac{1}{m} \epsilon_{ij}
f^0 \partial^i f^j \right)\,\,\,.
\end{equation}

\noindent
Hence, the total Hamiltonian ($H^{SD}_T$) is given by\cite{Di}
$H^{SD}_T = H^{SD} + \int d^3x \left( u^0 P_0 + u^i P_i \right)$, where the
$u$'s are Lagrange multipliers.

The persistence in time of $P_0\approx0$,

\[
\dot{P_0}\,=\,[P_0\,,\,H^{SD}_T]_P\,\approx\,0\,\,\,,
\]

\noindent
where $[,]_P$ denotes the Poisson bracket, leads to the existence of the
secondary constraint

\begin{equation}
\label{204}
S\,=\,\frac{1}{m}\left(f^0\,-\,\frac{1}{m}\epsilon_{ij}\partial^if^j\right)
\,\approx\,0\,\,\,.
\end{equation}

\noindent
On the other hand, $[P_i(\vec{x}),P_j(\vec{y})]_P \neq 0$ implies that by demanding
persistence in time of $P_i\approx 0, i=1,2$, one determines the Lagrange
multipliers $u^i, i=1,2$.
Similarly, $[P_0(\vec{x}),S(\vec{y})]_P \neq 0$, together with the persistence in time
of $S(\vec{x})\approx 0$, enables one to determine $u^0$.

From the above analysis follows that all constraints are
second-class and, therefore, Dirac brackets  with respect to them can be
introduced in the usual manner\cite{Di}. The phase-space variables are,
afterwards, promoted to operators obeying an equal-time
commutator algebra which is to be abstracted from the corresponding Dirac
bracket algebra, the constraints and gauge conditions thereby translating
into strong operator relations. This is the DBQP, which presently
yields\cite{f2,f3}

\begin{mathletters}
\label{205}
\begin{eqnarray}
&&[f^0(\vec{x}),f^j(\vec{y})]\,=\,i\, \partial^j_x \delta(\vec{x}-\vec{y})\,\,\,,\label{mlett:a205}\\
&&[f^k(\vec{x}),f^j(\vec{y})]\,=\,-im\, \epsilon^{kj}\, \delta(\vec{x}-\vec{y})\,\,\,,\label{mlett:b205}\\
&&[f^0(\vec{x}),\pi_k(\vec{y})]\,=\,-\frac{i}{2m}\,\epsilon_{kj}\,
\partial^j_x \delta(\vec{x}-\vec{y})\,\,\,,\label{mlett:c205}\\
&&[f^j(\vec{x}),\pi_k(\vec{y})]\,=\,\frac{i}{2}\, g^j_k\, \delta(\vec{x}-\vec{y})\,\,\,,
\label{mlett:d205}\\
&&[\pi_j(\vec{x}),\pi_k(\vec{y})]\,=\,-\frac{i}{4m}\,\epsilon_{jk}\,
\delta(\vec{x}-\vec{y})\,\,\,,\label{mlett:e205}
\end{eqnarray}
\end{mathletters}

\noindent
whereas all other ETC's vanish. As for the quantum mechanical Hamiltonian
($H^{SD}$) it can be read off directly from (\ref{203}), in view of the
absence of ordering ambiguities in the classical-quantum transition. This is
true although $f^0$  and $f^j$ do not commute at equal
times (see(\ref{mlett:a205})). Alternative expressions for the operator
$H^{SD}$ can be obtained after recalling that, within the algebra (\ref{205}),
the constraints act as strong identities. Thus,

\begin{equation}
\label{206}
H^{SD}\,=\,\int d^2x\,\left(-\frac{1}{2}f^{\mu}f_{\mu}\,+\,\frac{1}{m} \epsilon_{ij}
f^0 \partial^i f^j \right)\,=\,
\int d^2x\,\left(-\frac{1}{2}f^{\mu}f_{\mu}\,+\,f^0f_0  \right)\,\,\,.
\end{equation}

What we need next is to solve the equations of motion

\begin{equation}
\label{207}
\partial_0 f^0(\vec{x})\,=\,i\,[H^{SD} , f^0(\vec{x})]\,=\,-\partial_jf^j(\vec{x})
\Longrightarrow \,\partial_{\mu}f^{\mu}\,=\,0\,\,\,,
\end{equation}

\noindent
\begin{equation}
\label{208}
\partial_0f^j(\vec{x})\,=\,i\,[H^{SD} , f^j(\vec{x})]\,=\,\partial^jf_0(\vec{x})\,+\,m\,
\epsilon^{jk}\,f_k(\vec{x})\,\,\,,
\end{equation}

\noindent
obeyed by the field operators $f^{\mu}$. Notice that this last equation and
(\ref{204}) can be unified into the single covariant expression

\begin{equation}
\label{209}
\epsilon^{\mu\sigma\alpha}\,\partial_{\sigma} f_{\alpha}\,-\,m\,f^{\mu}\,
=\,0\,\,\,,
\end{equation}

\noindent
which is formally identical to the Lagrange equation of motion deriving from
(\ref{201}). Of course, (\ref{207}) follows from (\ref{209}). The solving of
(\ref{209}) is greatly facilitated if one observes that its solutions
also fulfill the Klein-Gordon equation

\begin{equation}
\label{210}
\left(\Box\,+\,m^2\right)f^{\mu}\,=\,0\,\,\,,
\end{equation}

\noindent
the converse not being necessarily true. One can then write

\begin{equation}
\label{211}
f^{\mu}(x)\,=\,\int d^2z\,\Delta(x-z)\,\partial_0^z f^{\mu}(z)\,+\,
\int d^2z\,\partial_0^x \Delta(x-z)\,f^{\mu}(z)\,\,\,,
\end{equation}

\noindent
where the 2+1 dimensional Pauli-Jordan delta function ($\Delta$) of mass $m$
verifies, as known, $\left(\Box_x+m^2\right)\Delta(x-z)=0$ together with the
initial conditions $\Delta(x-z)|_{x^0=z^0}=0$ and
$\partial_0^x \Delta(x-z)|_{x^0=z^0}=\delta(\vec{x}-\vec{z})$. The velocities, in the right hand
side of (\ref{211}), can be eliminated in favor of the phase-space variables
by using Eqs.(\ref{207}) and (\ref{208}). Once this has been done, the
commutators at different times can be computed by using, as input, the equal
time algebra (\ref{205}). In this way one obtains

\begin{equation}
\label{212}
[f^{\mu}(x)\,,\,f^{\nu}(y)]\,=\,i\,\left(m^2\,g^{\mu\nu}\,+\,\partial^{\mu}_x
\partial^{\nu}_x\,-\,m\,\epsilon^{\mu\nu\rho}\,\partial^x_{\rho}\right)
\Delta(x-y)\,\,\,,
\end{equation}

\noindent
which is easily seen to satisfy the first order differential equation

\begin{equation}
\label{213}
\epsilon_{\alpha\beta\mu}\,\partial^{\beta}_x\,[f^{\mu}(x)\,,\,f^{\nu}(y)] \,
-\,m\,[f_{\alpha}(x)\,,\,f^{\nu}(y)]\,=\,0\,\,\,.
\end{equation}

\noindent
Thus, the field configurations entering the commutator (\ref{212}) are, as
required, solutions of Eq.(\ref{209}). Since the Pauli-Jordan delta function
admits a decomposition into positive and negative frequency parts\cite{f4}, we
learn from (\ref{212}) that the same is true for the field operators
$f^{\mu}$, namely,

\begin{equation}
\label{214}
f^{\mu}(x)\,=\,f^{\mu(+)}(x)\,+\,f^{\mu(-)}(x)\,\,\,,
\end{equation}

\noindent
with

\begin{equation}
\label{215}
f^{\mu(\pm)}(x)\,=\,\frac{1}{2\pi} \int \frac{d^2k}{\sqrt{2\omega_{\vec{k}}}}
\exp \left[\pm i(\omega_{\vec{k}} x^0 - \vec{k} \cdot
\vec{x})\right]\,f^{\mu(\pm)}(\vec{k})\,\,\,,
\end{equation}

\noindent
$\omega_{\vec{k}} \equiv +\sqrt{|\vec{k}|^2+m^2}$, and

\begin{mathletters}
\label{216}
\begin{eqnarray}
&&[f^{\mu (+)}(\vec{k}),f^{\nu (+)}({\vec{k}}\prime)]\,= \,
[f^{\mu (-)}(\vec{k}),f^{\nu (-)}({\vec{k}}\prime)]=0\,\,\,,\label{mlett:a216}\\
&&[f^{\mu(-)}(\vec{k})\,,\,f^{\nu(+)}({\vec{k}}\prime)]\,=\,-\left( m^2\,g^{\mu\nu}\,
-\,k^{\mu}k^{\nu}\,+\,im\,\epsilon^{\mu\nu\rho}\,k_{\rho} \right)
\delta(\vec{k}\,-\,{\vec{k}}\prime)\,\,\,.\label{mlett:b216}
\end{eqnarray}
\end{mathletters}

To go further on we must recognize that not all variables spanning
the phase are independent variables. Indeed, the system under analysis
possesses three coordinates
($f^{\mu},\mu=0,1,2$), three momenta ($\pi_{\mu}, \mu=0,1,2$) and four
constraints ($P_0, P_1, P_2 ,S$). Thus, we are left with only one
independent degree of freedom which implies that there is only one
polarization vector in the theory, to be designated by
$\varepsilon^{\mu}(\vec{k})$. We can then write

\begin{mathletters}
\label{217}
\begin{eqnarray}
&&f^{\mu (+)}(\vec{k})\,=\,\varepsilon^{\mu}(\vec{k})\,a^{(+)}(\vec{k})\,\,\,,\label{mlett:217a}\\
&&f^{\mu (-)}(\vec{k})\,=\,\varepsilon^{\ast\mu}(\vec{k})\,a^{(-)}(\vec{k})\,\,\,,
\label{mlett:217b}
\end{eqnarray}
\end{mathletters}

\noindent
where $a^{(\pm)}(\vec{k})$ are operators whose commutator algebra will be
determined later on. By going back with (\ref{217}) into (\ref{215})
and with this into (\ref{209}) one finds that $\varepsilon^{\mu}$ verifies the
homogeneous equation

\begin{equation}
\label{218}
\Sigma^{\rho\alpha}(\vec{k})\,\varepsilon_{\alpha}(\vec{k})\,=\,0\,\,\,,
\end{equation}

\noindent
with

\begin{equation}
\label{219}
\Sigma^{\rho\alpha}(\vec{k})\,\equiv\,i\,\epsilon^{\rho\sigma\alpha}\,k_{\sigma}\,-\,
m\,g^{\rho\alpha}\,\,\,.
\end{equation}

\noindent
The vanishing of the determinant of the matrix $\Sigma^{\rho\alpha}$ is a
necessary and sufficient condition for Eq.(\ref{218}) to have solution
different from the trivial one. The computation of this determinant is
straightforward and yields

\begin{equation}
\label{220}
\det \| \Sigma \|\,=\,m\,(k^2\,-\,m^2)\,\,\,.
\end{equation}

\noindent
Therefore, the theory only propagates particles with
mass equal to $|m|$. To solve (\ref{218}) we adopt the
strategy presented in Ref.\cite{Gi2}, i.e., we start by relating
$\varepsilon^{\mu}(\vec{k})$ with $\varepsilon^{\mu}(0)$ through the corresponding Lorentz
transformation, namely,

\begin{mathletters}
\label{221}
\begin{eqnarray}
\varepsilon^{0}(\vec{k})\,& = &\, \frac{1}{|m|}\, \vec{k} \cdot \vec{\varepsilon}(0),
\label{mlett:a221} \\
\varepsilon^{j} (\vec{k})\,& = &\,\varepsilon^{j}(0)\,+\,
\frac{\vec{\varepsilon}(0) \cdot \vec{k}}{(\omega_{\vec{k}}\,+\,|m|)\,|m| }
\,k^{j}\,\,\,. \label{mlett:b221}
\end{eqnarray}
\end{mathletters}

\noindent
This form of the solution is particularly appealing because it shows,
explicitly, that $\varepsilon^{\mu}(\vec{k})$ goes continuously to the
corresponding value in the rest frame of reference. Now, from (\ref{218}) one
finds that

\begin{mathletters}
\label{222}
\begin{eqnarray}
\varepsilon^{0}(0)\,& = &\,0\,\,\,,\label{mlett:a222}\\
\varepsilon^{j}(0)\,& = &\,-i\,\frac{|m|}{m}\,\epsilon^{jl}\,\varepsilon_{l}(0)\,\,\,,
\label{mlett:b222}
\end{eqnarray}
\end{mathletters}

\noindent
which completes the determination of the polarization vector. One can check
that

\begin{mathletters}
\label{223}
\begin{eqnarray}
\varepsilon^{\mu}(\vec{k}) \varepsilon_{\mu}(\vec{k})
\,& = &\,-{\vec{\varepsilon}}(0) \cdot {\vec{\varepsilon}}(0)\,=\,0,
\label{mlett:a223} \\
\varepsilon^{\mu}(\vec{k}) \varepsilon^{\ast}_{\mu}(\vec{k})
\,& = &\,-{\vec{\varepsilon}}(0) \cdot {\vec{\varepsilon}}^{\ast}(0)\,=\,
-2\,|\varepsilon^{1}(0)|^{2}\label{mlett:b223},
\end{eqnarray}
\end{mathletters}

\noindent
where $|\varepsilon^{1}(0)|$ is to be fixed by normalization. This is just the
polarization vector of the MCS theory in the Landau gauge\cite{Gi2}.

By substituting Eq.(\ref{217}) into Eq.(\ref{mlett:b216}) one arrives to

\begin{equation}
\label{224}
\varepsilon^{\ast\mu}(\vec{k}) \varepsilon^{\nu}(\vec{k}\prime)[a^{(-)}(\vec{k})\,,\,a^{(+)}(\vec{k}\prime)]\,
=\,-\left( m^2\,g^{\mu\nu}\,
-\,k^{\mu}k^{\nu}\,+\,im\,\epsilon^{\mu\nu\rho}\,k_{\rho} \right)
\delta(\vec{k}\,-\,{\vec{k}}\prime)\,\,\,.
\end{equation}

\noindent
On the other hand, if we fix $|\varepsilon^{1}(0)|=m$ the polarization vector can be
seen to fulfill the relationship

\begin{equation}
\label{225}
\varepsilon^{\ast\mu}(\vec{k}) \varepsilon^{\nu}(\vec{k}) \,=\,-\left( m^2\,g^{\mu\nu}\,
-\,k^{\mu}k^{\nu}\,+\,im\,\epsilon^{\mu\nu\rho}\,k_{\rho} \right)\,\,\,.
\end{equation}

\noindent
By combining Eqs.(\ref{224}) and (\ref{225}) one obtains the commutator algebra
obeyed by the operators $a^{(\pm)}(\vec{k})$, i.e.,

\begin{equation}
\label{226}
[a^{(-)}(\vec{k})\,,\,a^{(+)}(\vec{k}\prime)]\,=\,\delta(\vec{k}\,-\,{\vec{k}}\prime)\,\,\,.
\end{equation}

\noindent
Thus, $a^{(-)}(\vec{k})$ and ($a^{(+)}(\vec{k})$) are, respectively, destruction and
creation operators and the space of states is a Fock space with positive
definite metric.

What remains to be done is to investigate whether the DBQP has, for the case of
the SD model, led to a relativistically invariant quantum theory and to
determine the spin of the corresponding particle excitations. The first of
these questions appears to be
trivial because the Lagrangian density (\ref{201}) is a Lorentz scalar.
However, one is to observe that the space-time and the space-space
components of the classical symmetric (Belinfante) energy-momentum tensor
derived in Ref.\cite{Jackiw1}
($\Theta^{\mu\nu}\,=\,f^{\mu}f^{\nu}\,-\,\frac{1}{2}\,g^{\mu\nu}\,f^{\alpha}
f_{\alpha}$),
become afflicted by ordering ambiguities when promoted to the quantum level
(Recall the equal-time commutation algebra (\ref{205})). To cope with this
problem we introduce the symmetric ordering prescription ($A\cdot B =
1/2(AB + BA)$) and take as the Poincar\'e densities of the quantum theory the
composite Hermitean operators

\begin{equation}
\label{227}
\Theta^{\mu\nu}\,=\,f^{\mu} \cdot f^{\nu}\,
-\,\frac{1}{2}\,g^{\mu\nu}\,f^{\alpha} \cdot f_{\alpha} \,\,\,.
\end{equation}

\noindent
After some algebra, one corroborates that the Dirac-Schwinger
algebra\cite{DiSch,Sch},

\begin{mathletters}
\label{228}
\begin{eqnarray}
& &[\Theta^{00}(x^{0}, \vec{x})\,,\,\Theta^{00}(x^{0}, \vec{y})]\,
=\,-i\,\left(\Theta^{0k}(x^{0}, \vec{x})\,
+\,\Theta^{0k}(x^{0},\vec{y})\right) \partial^{x}_{k}
\delta(\vec{x}-\vec{y}), \label{mlett:a228} \\
& &[\Theta^{00}(x^{0}, \vec{x})\,,\,\Theta^{0k}(x^{0},\vec{y})]\,=\,
-i\,\left(\Theta^{kj}(x^{0},\vec{x})\,
-\,g^{kj}\,\Theta^{00}(x^{0},\vec{y})\right)
\partial^{x}_{j}\delta(\vec{x}-\vec{y}), \label{mlett:b228} \\
& &[\Theta^{0k}(x^{0},\vec{x})\,,\,\Theta^{0j}(x^{0},\vec{y})]\,=\,-i\,
\left(\Theta^{0k}(x^{0},\vec{x})\partial_{j}^{x}\,+\,
\Theta^{0j}(x^{0},\vec{y})\partial^{x}_{k} \right) \delta(\vec{x}-\vec{y}),
\label{mlett:c228}
\end{eqnarray}
\end{mathletters}

\noindent
holds. Then, the generators of space-time translations
($P^{\mu}$), Lorentz boosts ($J^{0i}$), and spatial rotations ($J$),

\begin{mathletters}
\label{229}
\begin{eqnarray}
& &P^{0}\,\equiv\,\int d^2x\,\Theta^{00}(x^{0},\vec{x}) \,=\,H^{SD},
\label{mlett:a229} \\
& &P^{i}\,\equiv\,\int d^2x\,\Theta^{0i}(x^{0},\vec{x}),
\label{mlett:b229} \\
& &J^{0i}\,\equiv\,- x^{0}P^{i}\,+\,\int d^2x\,
[x^{j}\Theta^{00}(x^{0},\vec{x})], \label{mlett:c229} \\
& &J\,\equiv\,\epsilon_{lj} \int d^{2}x \,x^{l} \Theta^{0j}(x^{0},\vec{x}),
\label{mlett:d229}
\end{eqnarray}
\end{mathletters}

\noindent
fulfill the Poincar\'e algebra. To write the Poincar\'e
generators in terms of the operators $a^{(\pm)}(\vec{k})$ involves a
cumbersome calculation whose details will not be given here. We
only mention that the functional form of the polarization vector
$\varepsilon^{\mu}(\vec{k})$ (see Eqs.(\ref{221}) and (\ref{222})) plays a central role
for arriving at the following expressions

\begin{mathletters}
\label{230}
\begin{eqnarray}
& &P^{0}\,= \,H^{SD}\,=\,\int d^2k\,\omega_{\vec{k}}\,a^{(+)}(\vec{k})a^{(-)}(\vec{k})\,\,\,,
\label{mlett:a230}\\
& &P^{j}\,= \,\int d^2k\, k^j\,\,a^{(+)}(\vec{k})a^{(-)}(\vec{k})\,\,\,,
\label{mlett:b230}\\
& &J\,=\,\frac{m}{|m|} \int d^{2}k\,
a^{(+)}(\vec{k})a^{(-)}(\vec{k})\,+\,i \epsilon_{jl} \int d^{2}k\,
a^{(+)}(\vec{k})\,k^{j} \frac{\partial}{\partial k_{l}}\, a^{(-)}(\vec{k})\,\,\,.
\label{mlett:c230}
\end{eqnarray}
\end{mathletters}

\noindent
By acting with $P^{\mu}$ on the one particle state $a^{(+)}(\vec{k})|0>$ one learns
that $a^{(+)}(\vec{k})$ creates particles with three-momentum $k^{\mu}$
($k^0=\omega_{\vec{k}}$, $k^j$). The action of the operator $J$ on a single
particle state can also be readily derived. In particular, for the rest frame
of reference one finds that

\begin{equation}
\label{231}
J\,\{ a^{+}(\vec{k} = 0)|0> \} \,
=\,\frac{m}{|m|} \{ a^{+}(\vec{k} = 0)|0> \},
\end{equation}

\noindent
which sais that the spin of the SD quanta is $\pm 1$ depending upon the
sign of the mass factor.

The quantization of the free SD model is complete. We have
demonstrated that the theory only contains particles of mass $|m|$ and spin
$m/|m|$. This is just the particle content of the MCS theory
in the Coulomb (physical) gauge\cite{f5}. On the other hand, the
polarization vector in the SD theory is that of the MCS model
in the Landau gauge. To summarize, the SD model describes the gauge
invariant sector of the MCS model. Therefore, these theories are quantum
mechanically equivalent.

\section{The SD field minimally coupled to Fermions}
\label{sec:level3}

We now bring the fermions into de game\cite{f6}. The use of Dirac's
method\cite{Di,FVil,Su,Gi1} enables one to conclude that the dynamics of
the SD model minimally coupled to fermions is described by the canonical
Hamiltonian

\begin{eqnarray}
\label{301}
H_1\,=\,\int d^2x &&\left[\frac{i}{2}\,(\partial_k\bar{\psi})\gamma^k\psi\,-\,
\frac{i}{2}\,\bar{\psi} \gamma^k (\partial_k \psi)\,+\,M\,\bar{\psi}\psi\,-\,\frac{g}{m}\,\bar{\psi}\gamma_k\psi f^k
\right. \nonumber\\
 &-& \left. \frac{1}{2} f_{\mu}f^{\mu}\,
+\,f^0\,\left(\frac{1}{m} \epsilon_{ij}\partial^if^j\,-
\,\frac{g}{m}\,\bar{\psi}\gamma_0\psi\right) \right]\,\,\,,
\end{eqnarray}

\noindent
the primary constraints

\begin{mathletters}
\label{302}
\begin{eqnarray}
&&P_0\,=\,\pi_0\,\approx\,0\,\,\,, \label{mlett:a302} \\
&&P_i\,=\,\pi_i\,+\,\frac{1}{2m} \epsilon_{ij} f^j\,\approx\,0,\,\,i=1,2\,\,\,,
\label{mlett:b302}\\
&&\theta_a\,=\,\pi_{\bar{\psi}_a}\,-\,\frac{i}{2} \gamma^0_{ab}\psi_b\,\approx\,0,\,\,a=1,2
\,\,\,,\label{mlett:c302}\\
&&{\bar{\theta}}_a\,=\,\pi_{\psi_a}\,-\,\frac{i}{2} \bar{\psi}_b \gamma^0_{ba}\,\approx\,0,\,\,
a=1,2 \,\,\,,\label{mlett:d302}
\end{eqnarray}
\end{mathletters}

\noindent
and the secondary constraint

\begin{equation}
\label{303}
S^F\,=\,\frac{1}{m}\left(f^0\,-\,\frac{1}{m}\epsilon_{ij}\partial^if^j
\,+\,\frac{g}{m}\bar{\psi}\gamma_0\psi \right)
\,\approx\,0\,\,\,,
\end{equation}

\noindent
where $\pi_{\psi_a}$ and $\pi_{\bar{\psi}_a}$ are the momenta canonically conjugate to
$\psi_a$ and $\bar{\psi}_a$, respectively. Furthermore, all constraints are
second-class, as can be easily checked.

The next step consists in introducing Dirac brackets. The direct computations
of Dirac brackets with respect to all constraints is difficult. It is easier
to compute first partial Dirac brackets with respect to the
fermionic constraints, $\theta_a $ and ${\bar{\theta}}_a$, and then use this result
as input for computing the full Dirac brackets\cite{Su}. However, this
procedure is valid if and only if $\{\theta_a\,{\bar{\theta}}_a\} \cap \{P_0, P_j,
S^F \}=\{\phi\}$\cite{Su} which is not the case here, since $[\theta_a(\vec{x}),
S^F(\vec{y})]_P\neq 0$, $[{\bar{\theta}}_a(\vec{x}), S^F(\vec{y})]_P\neq 0$.
Nevertheless, we recall that any combination of constraints is also a
constraint\cite{Di} and we can, therefore, replace $S^F$ by ${\tilde{S}}^F$,
which is to be constructed such that
$[\theta_a(\vec{x}),{\tilde{S}}^F(\vec{y})]_P \approx 0$,
$[{\bar{\theta}}_a(\vec{x}),{\tilde{S}}^F(\vec{y})]_P \approx 0$, while keeping
$[P_0(\vec{x}), {\tilde{S}}^F(\vec{y})]_P \neq 0$. It has been shown
that\cite{Su,Fleck} that

\[
{\tilde{S}}^F(\vec{x})\,=\,S^F(\vec{x})\,+\,{\bar{\alpha}}_a(\vec{x})\, \theta_a(\vec{x})\,
+\,{\bar{\theta}}_a(\vec{x}) \, \alpha_a(\vec{x})\approx 0\,\,\,,
\]

\noindent
with $\alpha_a = -i\frac{g}{m^2}\psi_a$ and ${\bar{\alpha}}_a
= -i\frac{g}{m^2}\bar{\psi}_a$ verifies all the above requirements.

For the partial Dirac brackets with respect to the fermionic constraints
($\Delta$-brackets) one obtains

\begin{mathletters}
\label{304}
\begin{eqnarray}
\left[\psi_a(\vec{x}) , \psi_b(\vec{y})\right]_{\Delta}\,&=&\,0\,\,\,, \label{mlett:a304}\\
\left[\psi_a(\vec{x}) , \bar{\psi}_b(\vec{y})\right]_{\Delta}\,&=&\,-i\,\gamma^0_{ab}\, \delta(\vec{x}-\vec{y}) \,\,\,,
\label{mlett:b304}\\
\left[\bar{\psi}_a(\vec{x}) , \bar{\psi}_b(\vec{y})\right]_{\Delta}\,&=& \,0 \,\,\,, \label{mlett:c304}
\end{eqnarray}
\end{mathletters}

\noindent
while all $\Delta$-brackets involving bosonic variables equal the corresponding
Poisson brackets. For any pair of functionals, $\Lambda$ and $\Omega$, of the
phase space variables, the full Dirac bracket (D-bracket) is now to be computed
as follows\cite{Su}

\begin{equation}
\label{305}
[\Lambda , \Omega]_D\,=\,[\Lambda , \Omega]_{\Delta}\,-\,\sum_{q=1}^4
\sum_{p=1}^4 \int d^2u \int d^2v \,[\Lambda , \xi_p(\vec{u})]_{\Delta}\,R^{pq}(\vec{u},
\vec{v})\,[\xi_q(\vec{v}) , \Omega]_{\Delta}\,\,\,,
\end{equation}

\noindent
where $\xi_1\equiv P_0$, $\xi_j\equiv P_j, j=1,2$, $\xi_4\equiv S^F$,
$\parallel R\parallel \equiv \parallel Q^{-1}\parallel$ and $Q_{pq}(\vec{u},\vec{v})
\equiv [\xi_p(\vec{u}),\xi_q(\vec{v})]_{\Delta}$. Notice that the fermionic
constraints hold as strong identities within the $\Delta$-algebra
(\ref{304}) and, consequently, we can use $S^F$ instead of ${\tilde{S}}^F$
when computing the D-brackets through Eq.(\ref{305}).

As in the free field case, the quantization consists in promoting all
phase-space variables to operators obeying an equal-time commutation algebra
abstracted from the corresponding D-bracket algebra. After a lengthy
calculation one finds that the nonvanising ETC's and anticommutators read as
follows

\begin{mathletters}
\label{306}
\begin{eqnarray}
&&[f^0(\vec{x}),f^j(\vec{y})]\,=\,i\, \partial^j_x \delta(\vec{x}-\vec{y})\,\,\,,\label{mlett:a306}\\
&&[f^k(\vec{x}),f^j(\vec{y})]\,=\,-im\, \epsilon^{kj}\, \delta(\vec{x}-\vec{y})\,\,\,,\label{mlett:b306}\\
&&[f^0(\vec{x}),\pi_k(\vec{y})]\,=\,-\frac{i}{2m}\,\epsilon_{kj}\,
\partial^j_x \delta(\vec{x}-\vec{y})\,\,\,,\label{mlett:c306}\\
&&[f^j(\vec{x}),\pi_k(\vec{y})]\,=\,\frac{i}{2}\, g^j_k\, \delta(\vec{x}-\vec{y})\,\,\,,
\label{mlett:d306}\\
&&[\pi_j(\vec{x}),\pi_k(\vec{y})]\,=\,-\frac{i}{4m}\,\epsilon_{jk}\,
\delta(\vec{x}-\vec{y})\,\,\,,\label{mlett:e306}\\
&&[f^0(\vec{x}),\psi_a(\vec{y})]\,=\,\frac{g}{m}\,\psi_a(\vec{x})\,\delta(\vec{x}-\vec{y})\,\,\,,
\label{mlett:f306}\\
&&[f^0(\vec{x}),\bar{\psi}_a(\vec{y})]\,=\,-\frac{g}{m}\,\bar{\psi}_a(\vec{x})\,\delta(\vec{x}-\vec{y})\,\,\,,
\label{mlett:g306}\\
&&\{\psi_a(\vec{x}),\bar{\psi}_b(\vec{y})\}\,=\,\gamma^0_{ab}\,\delta(\vec{x}-\vec{y})\,\,\,. \label{mlett:h306}
\end{eqnarray}
\end{mathletters}

\noindent
The Hamiltonian operator ($H$) is the quantum counterpart of $H_1$
(see Eq.(\ref{301}), i.e.,

\begin{eqnarray}
\label{307}
H\,=\,\int d^2x &&\left[\frac{i}{2}\,(\partial_k\bar{\psi})\cdot \gamma^k\psi\,-\,
\frac{i}{2}\,\bar{\psi} \cdot \gamma^k (\partial_k \psi)\,+\,M\,\bar{\psi} \cdot \psi\,
-\,\frac{g}{m}\,\bar{\psi} \cdot \gamma_k\psi f^k
\right. \nonumber\\
 &+& \left. \frac{1}{2}\,f^0f^0\,+\, \frac{1}{2} f^{i}f^{i}\,\,\right]\,\,\,.
\end{eqnarray}

\noindent
Clearly, Eq.(\ref{307}) follows from Eq.(\ref{301}) after appropriate
ordering of fermionic factors ($\psi_a\cdot\bar{\psi}_b \equiv 1/2
(\psi_a \bar{\psi}_b - \bar{\psi}_b \psi_a)$). The expression for $H$ was, furthermore,
simplified by using the constraint (\ref{303}). We are entitled to do so
because, within the algebra (\ref{306}), all constraints hold as strong
operator relations\cite{f7}.

The main observation, concerning the equal-time commutation algebra
(\ref{306}),
is that the bosonic field variable $f^0$ does not commute with the fermion
fields $\psi$ and $\bar{\psi}$. Hence, the implementation of the quantum dynamics in the
interaction picture will only be possible after the elimination of $f^0$. This
can be done by using the constraint relation (\ref{303}) which ultimately leads
to

\begin{equation}
\label{308}
H^D\,=\,H_0^D\,+\,H^D_I \,\,\,,
\end{equation}

\noindent
where

\begin{eqnarray}
\label{309}
H_0^D\,=\,&&\int d^2x \left[\frac{1}{2m^2}\epsilon^{ij}\,\epsilon^{kl}\,
(\partial_if^D_j)\,(\partial_kf^D_l)\,+\,\frac{1}{2}f^D_i f^D_i \right]
\nonumber\\
+\, &&\int d^2x \left[\frac{i}{2}\,(\partial_k\bar{\psi}^D)\cdot \gamma^k\psi^D\,-\,
\frac{i}{2}\,\bar{\psi}^D \cdot \gamma^k (\partial_k \psi^D)\,+\,M\,\bar{\psi}^D \cdot \psi^D \right]
\end{eqnarray}

\noindent
and

\begin{eqnarray}
\label{310}
H_I^D\,=\,\int d^2x && \left[ -\,\frac{g}{m}\,\bar{\psi}^D \cdot \gamma^k\psi^D f^D_k\,-\,
\frac{g}{m^2}\,\epsilon^{ij}\,\partial_if^D_j\,(\bar{\psi}^D\cdot \gamma^0\psi^D)\right.
\nonumber\\
&+& \left.
\frac{g^2}{2m^2}\,(\bar{\psi}^D\cdot \gamma^0\psi^D)\,(\bar{\psi}^D\cdot \gamma^0\psi^D)\right]\,\,\,.
\end{eqnarray}

\noindent
Here, the superscript $D$ denotes field operators belonging to the interaction
picture; the Heisenberg field operators, we were so far dealing with, bear no
picture superscript.

Since different pictures are connected by unitary
transformations, the equal-time commutation rules obeyed by the interaction
picture field operators can be read off directly from Eq.(\ref{306}). Then,
the equation of motion obeyed by the operator $\psi^D$ ($\partial_0 \psi^D
= i [H_0^D,\psi^D]$) is just the free Dirac equation and the corresponding
momentum space fermion propagator ($S(p)$) is well known to be

\begin{equation}
\label{311}
S(p)\,=\,i\,\frac{M\,+\,\gamma \cdot p}{p^2 - M^2 + i\epsilon}\,\,\,.
\end{equation}

\noindent
The equations of motion satisfied by the operators $f_i^D, i=1,2$ are exactly
those studied in detail in section 2 of this paper. Hence, from
Eqs.(\ref{215}), (\ref{217}), (\ref{221}), (\ref{222}) and (\ref{226}) one
finds that the momentum space Feynman propagator ($D_{lj}(k)$) is given by

\begin{equation}
\label{312}
D_{lj}(k)\,=\,\frac{i}{k^2 - m^2 +i\epsilon} \left( - m^2 g_{lj}\,+\,k_l k_j\,
-\,i m \epsilon_{lj} k_0\right)\,=\, D_{jl}(-k)\,\,\,.
\end{equation}

\noindent
To complete the derivation
of the Feynman rules, we turn into analyzing the vertices of the SD model
minimally coupled to fermions. We observe that $H_I^D$ in Eq.(\ref{310})
is made up by three different kind of monomial terms. The first monomial is
the spatial part of the standard field-current interaction, whereas the second
and third
monomials arised as by products in the process of elimination of $f_0$.
Unlike the cases of MCS minimally coupled to fermions\cite{Gi2} and Quantum
Electrodynamics, these extra terms are strictly local in space-time. Also,
they are non-renormalizable by power counting. The second term
in Eq.(\ref{310}) is the time component of a  magnetic coupling, while
the third one is the time component of a four-fermions (Thirring) interaction.

The Feynman rules derived in this section are non-manifestly covariant. Then,
we must elucidate whether or not this set of rules
leads to a relativistically invariant $S$-matrix. As in section 2,
one may first build the symmetric energy-momentum tensor for the interacting
theory and then use the equal-time commutation rules (\ref{306}) to
check the fulfillment of the Dirac-Schwinger algebra. However, the validity
of such procedure would now be dubious, since the components of the
energy-momentum
tensor necessarily involve (ill defined) products of Heisenberg field
operators evaluated at the same space-time point. The next section is
dedicated to test the relativistic invariance of the coupled theory in
connection with the specific process of elastic fermion-fermion scattering.
Since we are dealing with a non-renormalizable theory, our computations will be
restricted to the tree approximation.

\section{Lowest order elastic fermion-fermion scattering amplitude}
\label{sec:level4}

From the inspection of Eq.(\ref{310}) follows that the contributions of order
$g^2$ to the lowest order elastic fermion-fermion scattering amplitude
($R^{(2)}$) can be grouped into four different kind of terms,

\begin{equation}
\label{401}
R^{(2)}\,=\,\sum_{\alpha=1}^{4}R^{(2)}_{\alpha},
\end{equation}

\noindent
where

\begin{mathletters}
\label{402}
\begin{eqnarray}
R^{(2)}_1 & = & -\frac {g^2}{2}(\gamma^k)_{ab}(\gamma^l)_{cd}
\int d^3x \int d^3y
<\Phi _f|T \left\{ \frac{1}{m} :\bar{\psi}^D_a(x)\psi^D_b(x) f^D_k(x): \right.
\nonumber \\ & \times & \left.
\frac{1}{m} :\bar{\psi}^D_c(y) \psi^D_d (y)f^D_l(y): \right\}|\Phi _i>
\,\,\,, \label{mlett:a402} \\
R^{(2)}_2 & = & \frac {-ig^2}{2}(\gamma^0)_{ab}(\gamma^0)_{cd}
\int d^3x\int d^3y \delta (x - y) \nonumber \\
& \times & <\Phi _f|\frac{1}{m^2}:\bar{\psi}^D_a (x) \psi^D_b (x)
\bar{\psi}^D_c (y) \psi^D_d (y):|\Phi _i>
\,\,\,, \label{mlett:b402}\\
R^{(2)}_3 & = & - g^2 (\gamma^k)_{ab}(\gamma^0)_{cd}
\int d^3x \int d^3y
<\Phi _f| T\left\{ \frac{1}{m}:\bar{\psi}^D_a(x) \psi^D_b(x) f^D_k(x): \right.
\nonumber  \\
& \times & \left.  :\frac{1}{m^2} \epsilon^{li} \left(\partial_l^y f^D_i(y)\right)
\bar{\psi}^D_c(y) \psi^D_d(y):\right\} | \Phi _i> \,\,\,,
\label{mlett:c402} \\
R^{(2)}_4 & = & -\frac {g^2}{2} (\gamma ^0)_{ab} (\gamma ^0)_{cd} \nonumber \\
& \times & \int d^3x \int d^3y
< \Phi _f| T\left\{ :\frac{1}{m^2} \epsilon^{kj} \left(\partial_k^xf^D_j(x)\right)
\bar{\psi}^D_a(x) \psi^D_b(x): \right.
\nonumber \\ & \times & \left. :\frac{1}{m^2} \epsilon^{li}
\left(\partial_l^yf^D_i(y)\right)
\bar{\psi}^D_c(y) \psi^D_d(y):\right\} |\Phi _i>\,\,\,. \label{mlett:d402}
\end{eqnarray}
\end{mathletters}

\noindent
Here, $T$ is the chronological ordering operator, whereas $|\Phi_i>$ and
$|\Phi_f>$ denote the initial and final state of the reaction, respectively.
For the case under analysis, both $|\Phi_i>$ and $|\Phi_f>$ are two-electron
states. Fermion states obeying the free Dirac equation in
2+1-dimensions were explicitly constructed in Ref.\cite{Gi3}, where the
notation $v^{(-)}(\vec{p})\, (\bar{v}^{(+)}(\vec{p}))$ was employed to
designate the two-component spinor describing a free electron of two-momentum
$\vec{p}$, energy $p^{0}=+({\vec{p}}^{\,2}+m^{2})^{1/2}$ and spin $s= M/|M|$ in
the initial (final) state. The plane wave expansion of the free fermionic
operators $\psi$ and $\bar{\psi}$ in terms of these spinors and of the
corresponding creation and annihilation operators goes as usual.

In terms of the initial ($p_{1}, p_{2}$) and final momenta
($p_{1}^{\prime}, p_{2}^{\prime}$), the partial amplitudes in (\ref{402}) are
found to read

\begin{mathletters}
\label{403}
\begin{eqnarray}
R^{(2)}_{1} & = & \frac {1}{2\pi}\delta^{(3)}(p_1'+p_2'-p_1-p_2)
\nonumber \\
& \times & \left\{ [\bar v^{(+)} (\vec{p}_1^{\,\prime})
(ig\gamma ^j)v^{(-)}(\vec p_1)][\bar v^{(+)}(\vec{p}_2^{\,\prime})
(ig\gamma^l)v^{(-)}(\vec p_2)] \frac{1}{m^2} D_{jl}(k)\,- \, p_1^{\,\prime}
\longleftrightarrow p_2^{\,\prime} \right\}\,\,\,, \label{mlett:a403} \\
R^{(2)}_{2} & = & \frac{1}{2\pi}\delta ^{(3)}(p_1'+p_2'-p_1-p_2)
\nonumber \\
& \times & \left\{ \frac{i}{m^2}
[\bar v^{(+)}(\vec p_1^{\,\prime}) (ig\gamma ^0) v^{(-)}(\vec p_1)]
[ \bar v^{(+)}(\vec p_2^{\,\prime})(ig\gamma ^0)v^{(-)}(\vec p_2)]
\,-\, p_1^{\,\prime}
\longleftrightarrow p_2^{\,\prime} \right\}\,\,\,, \label{mlett:b403} \\
R^{(2)}_{3} & = & \frac{1}{2\pi}\delta^{(3)}(p_1'+p_2'-p_1-p_2)
\nonumber \\
& \times & \left\{ [\bar v^{(+)}(\vec p_1^{\,\prime}) (ig\gamma ^j)
v^{(-)}(\vec p_1)]
[ \bar v^{(+)}(\vec p_2^{\,\prime})(ig\gamma ^0)v^{(-)}(\vec p_2)]
\frac{1}{m^2} \Gamma_j(k) \right.
\nonumber \\
& + & \left. [\bar v^{(+)}(\vec p_1^{\,\prime}) (ig\gamma ^0)
v^{(-)}(\vec p_1)]
[ \bar v^{(+)}(\vec p_2^{\,\prime})(ig\gamma ^j)v^{(-)}(\vec p_2)]
\frac{1}{m^2} \Gamma_j(-k)\,
- \,p_1^{\,\prime} \longleftrightarrow p_2^ {\,\prime} \right\}\,\,\,,
\label{mlett:c403} \\
R^{(2)}_{4} & = & \frac{1}{2\pi}\delta ^{(3)}(p_1'+p_2'-p_1-p_2)
\nonumber \\
& \times & \left\{ [\bar v^{(+)} (\vec p_1^{\,\prime})(ig\gamma ^0)
v^{(-)}(\vec p_1)]
[ \bar v^{(+)}(\vec p_2^{\,\prime})(ig\gamma ^0)v^{(-)}(\vec p_2)]
\frac{1}{m^2} \Lambda(k)\,
- \,\,p_1^{\,\prime} \longleftrightarrow p_2^ {\,\prime} \right\}\,\,\,,
\label{mlett:d403}
\end{eqnarray}
\end{mathletters}

\noindent
where

\begin{mathletters}
\label{404}
\begin{eqnarray}
&& \frac{1}{m^2} D_{jl}(k)\,
=\,\frac {i}{k^2- m^2 + i\epsilon}\left(- g_{jl}\, + \, \frac{k_j k_l}
{m^2}\,-\,\frac{i}{m} \epsilon_{jl} k_0 \right)\,\,\,, \label{mlett:a404} \\
&& \frac{1}{m^2}\Gamma_j(k)\,=\,\frac {i}{k^2- m^2 + i\epsilon }
\left(\frac{ i \epsilon_{jl} k^l}{m} \,+\,\frac{k_j k_0}{m^2} \right)\,\,\,,
\label{mlett:b404} \\
&&\frac{1}{m^2} \Lambda (k)\,=\,\frac{i}{k^2-\theta ^2 + i\epsilon }
\left( \frac{-k^l k_l}{m^2} \right)\,\,\,,\label{mlett:c404}
\end{eqnarray}
\end{mathletters}

\noindent
and

\begin{equation}
\label{405}
k \equiv  p_1'-p_1\,=\,-(p_2'-p_2)
\end{equation}

\noindent
is the momentum transfer (our convention for the Fourier
integral representation is
$f(x)= 1/(2\pi)^{3}\int d^{3}k f(k) \exp(- ik\cdot x)$). By substituting
(\ref{403}) into (\ref{401}) and after taking into account (\ref{404}) one
arrives to

\begin{eqnarray}
\label{406}
R^{(2)}\,& = &\,\left( -\frac{g^2}{2\pi}\right) \delta^{(3)}(p_1'+p_2'-p_1-p_2)
\nonumber \\
& \times & \left\{ [\bar v^{(+)}(\vec p_1^{\,\prime})
\gamma^{\mu} v^{(-)}(\vec p_1)]\,[\bar v^{(+)}(\vec p_2^{\,\prime})
\gamma^{\nu} v^{(-)}(\vec p_2)]\,\frac{1}{m^2}\,D_{\mu\nu}(k)\,
- p_1^{\,\prime} \longleftrightarrow p_2^ {\,\prime} \right\}\,\,\,,
\end{eqnarray}

\noindent
where

\begin{equation}
\label{407}
\frac{1}{m^2}\,D_{\mu\nu}(k)\,
=\,-\,\frac{i}{k^2 - m^2 + i\epsilon}\,\left(g_{\mu\nu}\,
+\,i\,\epsilon_{\mu\nu\sigma} \frac{k^{\sigma}}{m} \right)\,\,\,.
\end{equation}

The amplitude in Eq.(\ref{406}) is a Lorentz scalar. The theory has then
passed the test on relativistic invariance. Also, one is to observe
that $D_{\mu\nu}(k)/m^2 $ in Eq.(\ref{406}) is contracted into conserved
currents and, hence, terms proportional to $k^{\mu}$ can be added at will.
Thus, we can replace $D_{\mu\nu}(k)/m^2$, given at Eq.(\ref{407}), by

\begin{equation}
\label{408}
\frac{1}{m^2}\,D_{\mu\nu}^F (k)\,
=\,-\,\frac{i}{k^2 - m^2 + i\epsilon}\,\left(g_{\mu\nu}\,
-\,\frac{k_{\mu}k_{\nu}}{m^2}\,
+\,i\,\epsilon_{\mu\nu\sigma} \frac{k^{\sigma}}{m} \right)\,\,\,,
\end{equation}

\noindent
which, up to contact terms, is the free field propagator arising from
(\ref{212}). In the tree approximation one is, therefore,
allowed to replace all the non-covariant terms in $H^D_I$ (see Eq.(\ref{310})
by the minimal covariant interaction
$-\frac{g}{m} \bar{\psi}^D \cdot \gamma^{\mu}\psi^D f^D_{\mu}$.

We close this section by noticing that the high energy behavior of
the propagator in (\ref{408}) is radically different from that of the MCS
theory in the Landau gauge\cite{Gi2,Jackiw2},

\begin{equation}
\label{409}
D_{\mu\nu}^L (k)\,
=\,-\,\frac{i}{k^2 - m^2 + i\epsilon}\,\left(g_{\mu\nu}\,-\,\frac{k_{\mu} k_{\nu}}
{k^2} \,+\,i\,m\,\epsilon_{\mu\nu\sigma} \frac{k^{\sigma}}{k^2} \right)\,\,\,.
\end{equation}

\noindent
In fact, the SD model coupled to fermions is a non-renormalizable theory.
This corroborates the arguments in the previous section, based on power
counting.

\section{Final remarks}
\label{sec:level5}

The equivalence between the free SD and MCS models was first observed,
at the level of
the field equations, in Ref.\cite{Jackiw1}. As known\cite{Jackiw2},
the Lagrangian density describing the dynamics of the free MCS theory,

\begin{equation}
\label{501}
{\cal L}^{MCS} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{m}{4}
\epsilon^{\mu\nu\alpha}F_{\mu\nu}A_{\alpha},
\end{equation}

\noindent
where $F_{\mu\nu}\equiv \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, leads to the
equation of motion

\begin{equation}
\label{502}
\partial_{\alpha}F^{\alpha\beta}\,+\,\frac{m}{2}\,\epsilon^{\alpha\mu\alpha}
F_{\mu\alpha}\,=\,0\,\,\,.
\end{equation}

\noindent
It is now easy to see that the mapping

\begin{equation}
\label{503}
F^{\beta}\,\equiv\,\frac{1}{2}\,\epsilon^{\beta\mu\alpha}
F_{\mu\alpha}\,\longrightarrow \,f^{\beta}\,\,\,,
\end{equation}

\noindent
carries Eq.(\ref{502}) into Eq.(\ref{209}). In section 2 we proved that this
equivalence holds rigorously at the quantum level. Since the SD
field $f^{\beta}$ is identified with the vector $F^{\beta}$, dual of the tensor
$F^{\mu\nu}$, only the gauge invariant excitations of the MCS theory must
show up in the SD model. As demonstrated in this work, this is the case.

The proof of equivalence of the quantized versions of the free SD and MCS
models, given in this paper, was entirely carried out within the operator
approach.
One the main advantages of our presentation is that relativistic invariance
is kept operational in all stages. This is to be compared with
the outcomes in Refs.\cite{Rothe1,Kim,Rothe2,Rothe3}. Frequently, the
embedding of a second-class system in a larger phase-space, with the aim of
making it first-class\cite{BFT}, implies in loosing relativistic
invariance\cite{f8}.

The formulation of the interaction picture dynamics for the SD model minimally
coupled to fermions was possible only after the elimination
of the degree of freedom $f^0$, whose ETC's with the fermionic fields do not
vanish. As consequence, the Hamiltonian formulation did not retain any relic
of relativistic covariance. Nonetheless, we proved
that the non-covariant pieces in $H^D_I$ are equivalent to the
minimal covariant field-current interaction. The high energy behavior of the
$f$-propagator signalizes that the coupled theory is non-renormalizable.

Thus, the SD model minimally coupled to fermions bears no resemblance
with the renormalizable model defined by the MCS field minimally coupled
to fermions. To give further support to this conclusion we recall that it
has recently been shown\cite{Malacarne} that the SD
model minimally coupled to fermions is equivalent to the MCS model with
{\it non-minimal} magnetic coupling to fermions. Also, for the fermionic
sectors of the two theories to agree one is to add a Thirring like interaction
in one of the models. These are, of course, non-renormalizable
field theories.

\newpage

redpajama_set_name: RedPajamaArXiv


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