OKLAHOMA CITY - Authorities have arrested one of two people accused of luring high school students into an Oklahoma City prostitution ring, according to KFOR.
On Thursday, officials announced that warrants had been issued for a man police say was "one of the most well-known pimps in Oklahoma City at one time," 45-year-old Germaine Coulter, and one of his main alleged prostitutes, 35-year-old Elizabeth Andrade.
Police said the two have been recruiting Edmond Santa Fe students to work for them from the inside.
According to court documents, police learned of their plan after arresting Andrade in a prostitution sting. When she was arrested, they discovered alarming photos of young girls on her cell phone. Some of those photos included two teenage Santa Fe students.
"He lures them in with jewelry, getting their nails done, buying them clothes, putting them into nice cars," Brian Bates, who exposes prostitution in Oklahoma City, said.
Andrade is currently on probation, and Coulter was recently released from jail.
"Edmond Public Schools was unaware of an OCPD investigation into allegations of human trafficking involving student(s) at Edmond Santa Fe High School. The School School Resource Officer (SRO) was contacted on March 1, 2018 to assist OCPD with an investigation involving the identification of a photo of a female student.
The SRO provided the identification sought by OCPD, and there was no further information provided to the SRO or Santa Fe High School until today, March 8, when news outlets began calling the school district to gather information germane to the probable cause affidavit filed in OK County district court earlier this week which referenced the female student."
When KFOR asked if there would be any further action taken at this time, the spokesperson said in a statement, "We have not determined what additional actions the school district will take because we were just made aware of this investigation late this afternoon."
On Friday, Oklahoma City police announced that Germaine Coulter was taken into custody.
Officials say Coulter was arrested by officers with the Wichita Police Department in Kansas on Thursday. However, they say Elizabeth Andrade is still on the run.
Police said there are three vehicles they may be driving, a 2013 Tan Cadillac XTS - OK Plate GBH236, a 2014 Gray Cadillac CTS - OK Plate CIT835, and a 2017 White Ford F250 - OK Plate EFD371.
If you have any information about Andrade's whereabouts, call CrimeStoppers at (405) 235-7300.


The New Acebil Road Jib KIT2 - Telescopic aluminium construction allows to extend arms of the crane in no time, thus giving you a possibility to spend more time on shooting instead of on setting-up the equipment. Its compact size makes it a perfect solution when shooting in most extreme places. The ACEBIL ROAD JIB KIT2 outperforms other camera cranes in terms of stability, rigidity and comfort of use. Remote Head Capacity 3kg.


The Dancehall King, Shatta Wale after some back and forth twitter exchanges with his former archrival, Samini drops this special dedication for him. He calls this one Oh Ghana and he managed to convince us that it is NOT a diss record to Samini. Well, check out the Young Kid Royal tune for yourself and let's know your take on it, enjoy.


JSA Authentication ticket is for one signature.
A JSA ticket must be purchased for each signature you would like to have authenticated.
Examples: four different guests sign one item – four JSA tickets required; one guest signs two different items – two JSA tickets required; one guest signs and inscribes one item – one JSA ticket required.


text: Q: Using While Loop . How to write a code that keeps reading numbers tell a specific number and find their avg So, By using c++ how can i write a code that keep reading numbers till a specific number and then find the avg of the previous numbers ?

redpajama_set_name: RedPajamaStackExchange


text:
Home / Watch Brands / Audemars Piguet
Used Audemars Piguet Watches – Raymond Lee Jewelers of Boca Raton, FL offers a wide variety of luxury timepieces from Audemars Piguet. Founded in 1983, the business continues to offer only the best service and values for its watches. Audemars Piguet is a Swiss watch manufacturer founded in 1875. The company is notable for competing with other high-end manufacturers such as Patek Philippe, Jaeger-LeCoultre and Vacheron Constantin. At its launch, the company did not experience much success. However, in 1889, the company opened its first branch in Geneva and began to create its own components under strict quality control. Between 1894 and 1899, the company was producing about 1,200 timepieces. In 1918 and 1919 when the company's founders died, the company steadily began to experience more success – with companies such as Tiffany & Co, Cartier, and Bulgari becoming big customers. These notable buyers rebranded and sold Audemars Piguet watches under their own house names. After several years, Audemars Piguet produced the world's smallest minute repeater and a jumping second hand. In 1925, the company also produced the thinnest pocket watch (1.32 mm) and 3 years after that, created the first skeleton watch.
Today, the company produces roughly 26,000 timepieces per year.
Audemars Piguet Royal Oak 18k White Gold All Diamond Watch
Audemars Piguet Royal Oak Quantieme Perpetuel Automatique
Audemars Piguet Quantieme Perpetual 18k Gold Chronograph Mens Watch
Audemars Piguet Royal Oak Offshore 18k Rose Gold Men's Watch
Audemars Piguet Royal Oak 18k White Gold Watch
Audemars Piguet Quantieme Perpetuel Automatique Watch
Audemars Piguet Royal Oak Quantieme Perpetual Calendar Automatique
Audemars Piguet Royal Oak Offshore with Diamond Bezel
Audemars Piguet Royal Oak 18k Gold Diamond Blue Dial Watch
Audemars Piguet Quantieme Perpetual Calendar 18k Rose Gold Watch
Audemars Piguet Royal Oak Mother of Pearl Diamond Dial/Bezel Watch
Audemars Piguet Royal Oak 30 Anniversary Alinghi City of Sails
Audemars Piguet Classic Day-Date Moonphase 18k Yellow Gold Watch
Audemars Piguet Quantieme Perpetuel Automatique 18k Yellow Gold Watch
Audemars Piguet Royal Oak Stainless Steel Diamond Bezel Watch
Audemars Piguet Royal Oak Dual Time Watch

redpajama_set_name: RedPajamaCommonCrawl


When you are current to Foreign Exchange trading Forex, you probably recognize how important it is to generate the correct trading Forex steps. Study to apply the cybercycle to raise your winning trades, and set more stock in your pocket. cybercycle is hence that he can indicate and manipulate any opportunities for maximum gain. The technical currency trader has to recognize the timeframe period that the cybercycle can be counted from.


text:
Biden Honors 'Giant of Our History' Bob Dole at US Capitol
The service is one of several over two days in Washington for the former Republican senator and presidential candidate
By Lisa Mascaro	• Published December 9, 2021 • Updated on December 9, 2021 at 10:08 am
Bob Dole lay in state Thursday at the U.S. Capitol, as the president and others gathered to pay tribute to a "giant of our history" who served the country in war and in politics with pragmatism, self-deprecating wit and a bygone era's sense of common civility and compromise.
President Joe Biden, a longtime friend, said Dole, the former Republican senator, presidential contender and World War II veteran, should have the final word as he read from his colleague of 25 years' own observation that he had served the country best when putting "principles over party."
"Truth of the matter is, as divided as we are, the only way forward for democracy is unity — consensus -- the only way," Biden told those gathered. "We may follow his wisdom and his timeless truth and reach consensus on the basic fundamental principles we all agree on."
The morning ceremony brought about 100 invited guests and congressional leaders as Dole's casket, draped with the American flag, sat under the dome. Dole, who served nearly 36 years in Congress, died Sunday at the age of 98.
Biden said Dole "belongs here, in this place, this temple to liberty and temple to possibility." The president said of Dole: "He, too, was a giant of our history."
More Bob Dole Coverage
Bob Dole Dec 5, 2021
Bob Dole, Former Senator and Republican Presidential Candidate, Dies at 98
Bob Dole to Lie in State at Capitol as Nation Honors Senator
The service is the first of several in Washington commemorating Dole's life and legacy. Thursday's event at the Capitol and Friday's funeral at the Washington National Cathedral are closed to the public. But Dole's funeral will be live streamed Friday at the World War II Memorial on the National Mall, and his motorcade is expected to stop by later at an event with actor Tom Hanks honoring his life and military service before the casket travels to his Kansas hometown and the state capital.
House Speaker Nancy Pelosi said Dole was being honored among the "pantheon of patriots" — his casket placed atop the Lincoln catafalque that has been used since 1865.
Pelosi kissed the hand of Dole's wife, Elizabeth Dole, and his daughter, Robin Dole, who were seated near Biden, before telling those gathered that Dole's "principled leadership" was long respected "on both sides of the aisle, both sides of the Capitol."
Black draperies hung on doorways under the Capitol dome in preparation for the service. A lectern was positioned in way that the statue of another Kansas statesman, Dwight Eisenhower, will likely be seen in the background behind the day's speakers.
Elizabeth Dole appeared to cry as she approached the casket and reached out her hand. She and first lady Jill Biden blew kisses to one another before the president spoke.
Sen. Mitch McConnell, now the longest-serving Republican Senate leader, said earlier that Dole idolized Eisenhower, calling the former president and general a hero who embodied "the finest qualities of the American people."
"We can say with certainty that Eisenhower isn't the only Kansan who meets those standards," McConnell said in a speech earlier this week.
McConnell had raised concerns that an event planner for the memorial service, Tim Unes, had been subpoenaed by the House committee investigating the deadly Jan. 6 attack on the Capitol about his role planning that day for Donald Trump, the former president, according to a person granted anonymity to discuss the private conversation. Trump had urged his followers to head to the Capitol to fight the certification of Biden's election victory.
Dole family spokesperson Steve Schwab said in a statement Thursday that Unes' role had been terminated.
Schwab said Unes, a former Dole campaign staffer, had "volunteered his time to serve on the advance team for this week's memorial events." Schwab said Elizabeth Dole, "was previously unaware" of Unes' participation and once he made her aware she "terminated his volunteer role."
Unes did not immediately respond to a request for comment. The story was first reported by The New York Times.
A look at some key moments in the life of Bob Dole.
Born a child of the Dust Bowl in Russell, Kansas, Dole suffered paralyzing and near-fatal wounds after being shot in World War II that sent him home with a severely damaged right arm that he could not use to shake hands. Instead, Dole held a pen in it and reached out with his left as a way to put greeters at ease.
After earning a law degree, he worked as county attorney and served as a Kansas state legislator before running for Congress in 1960, joining the House for eight years then going on to win the Senate seat. He was the GOP's presidential nominee in 1996, his third and final campaign for president — a race he never won.
Dole's quick wit was on display after losing the presidential contest to incumbent Democrat Bill Clinton, who awarded him the Presidential Medal of Freedom days before the 1997 inauguration.
McConnell said this week that when it was time for Dole's remarks, he stood at the podium and began: "I, Robert J. Dole … so solemnly swear … oh, sorry, wrong speech!"
But that humor was rarely seen on the campaign trail or in his public pronouncements, where it could have helped him win more votes.
Instead, Dole was seen as a GOP "hatchet man," a mentee of Richard Nixon and chairman of the Republican National Committee during the Watergate era. He went on to become the sharp-tongued vice presidential running mate to Gerald Ford, another lost race.
On Thursday, McConnell noted that Dole was "the last of the greatest generation to run for president."
But it was Dole's long career in the Senate where he grew to see the value of reaching across the aisle to Democrats and secured his more lasting achievements — most notably the Americans with Disabilities Act that to this day ensures a level of accessibility as a civil right.
At times, Dole bucked his own party, particularly on a landmark tax bill, and helped create the Martin Luther King Jr. federal holiday.
Senate Majority Leader Chuck Schumer praised Dole during Thursday's service as an American "who redefined and elevated what it means to serve country."
Current and former lawmakers from both parties joined the service. House Republican leader Kevin McCarthy joined but did not speak.
Dole announced in February 2021 a diagnosis of stage 4 lung cancer. Biden visited him at his home at the Watergate complex.
Bob DoleJoe BidenUS Capitol


Entering a space, shut away from external influences, with an ability to concentrate on what is evolving in this experimental surrounding – the connection of human senses. Synaēsthesia as the form of an event series intends to recreate this perceptual anomaly, to elevate the visitor into the perception of interlinked senses. Each event concentrates on a different main human sense connected with sound and visuals. For this event, the focus lies on scent and exploring the dimensions of Synaēsthesia in an experimental surrounding with explorative and dark sounds. Daimos Kore, Grinderteeth, Psychatrian, V3sta & EVA. Sculpture work by Alexandra Koumantaki in collaboration with Michal Andrysiak // Aroma concept by Berlin Scent Club. Poster design by Julia Parchimowicz.


Union Bank & Trust Company has been a sponsor of Lincoln Youth Football since 2001. The Union Bank Program was previously sponsored by the Moose Lodge.
The current day Union Bank & Trust Company was founded in 1917 as the Farmer's State Bank. In 1935, the name was changed to Union Bank, and in 1959 to Union Bank & Trust Company. In 1965, it was purchased by a Nebraska family, the Dunlaps. Today, over fifty years later, the Dunlaps still own and manage the bank, serving Nebraskans across the state as well as the Kansas City metro area. UBT has a rich history of community involvement, extending far beyond Lincoln Youth Football. They have established an annual share of earnings for contributions back into the community in the areas of education, the arts, and community betterment.
LYF is a nonprofit organization in Lincoln, Nebraska that provides an opportunity for 3rd-8th grade boys and girls to play football.


I am a big fan of Monkey World in Dorset. It provides homes for monkeys and apes that have been mistreated, illegally traded or have done their time in the laboratories. On most visits I find myself gazing at our distant cousins, the chimps, gorillas and orangutans, observing our similarities, speculating about what they're thinking. The movements of their arms and hands, especially when they are delicately picking away at the layers of a vegetable, are incredibly similar to humans. I never find I relate to any of the other species, in quite the same way.
When I visited Andasibe-Mantadia National Park in Madagascar, I was lucky enough to get very close to the Indri Indri indri, the largest of all lemurs. A group of four came down from the canopy to within just a few metres of us. They clung vertically to the trunks of the trees; like a human climbing a rope, and stared at us. These beautiful creatures are our very ancient cousins (twice removed) but looked just like fluffy people hanging from the trees. I was amazed how human such a distant relative could appear. And then after, what I will remember as a very strange but wonderful wildlife encounter, they hopped off from tree to tree; still remaining vertical; and still looking very human.
How the Indri got its name.
What is Participatory GIS (PGIS)?


The Buccaneer is situated in the heart of Filey town centre and on the first floor of a former cinema.
The large main room is divided into the bar area (with pool table and dart board) and restuarant area (Smugglers Cove). A local East Coast cask ale is always available. Home-cooked food is served throughout the day. Live music is a regular feature at weekends. Smokers are catered for under the entrance canopy. Children are welcome until 9pm.


text:
New pen forms to fit any hand, grip
Amy Resseguie
These days, just about every office supply can be adjusted or custom-ordered to fit its user like a glove.
Desk chairs can prevent lower back pain. Wrist pads stop sore hands. Ergonomics, the science of designing equipment to reduce user strain and fatigue, has become a commonplace term in offices.
One Greeley man has developed his own addition to the arsenal of ergonomic office supplies with a modified pen.
Bill Moxon watched the trend toward ergonomics a few years ago and began experimenting with a more comfortable pen design.
"After I started getting some ideas, I thought it would have some neat applications with the occupational therapy or rehab market," Moxon said.
His Pensation pens, which debuted at last weekend's Greeley Arts Picnic, are fitted to the user's hand so that the grip is looser and the pen feels lighter to hold.
The large pen grip is made of a thermo-formable plastic polymer, which, when heated in 140-degree water, can be shaped by the user to exactly fit their unique grip.
The shaft of the pen is bent to loop around the user's index finger, which allows the hand to support the weight of the pen.
"It takes the pen out of the domain of gripping to supporting," Moxon said. "You don't have to hang onto the pen to write, rather, your hand's going to fit right into it."
Moxon sells the premade pens, which users make the final adjustments on at home, for $9. He also sells kits for $15. The kits include all the parts of the pen, as well as different colors of the grip polymer, so that customers can custom-design the pen.
All of the orders are filled out of his home, and Moxon remains employed at Vehicle Systems in Fort Lupton.
"I still have a day job, so that tells you how far along in my dream I am," he said, laughing.
Moxon has created a company, Pengineering LLC, to sell the pens, and has obtained a patent. Though he felt the Arts Picnic was a success, he is still relying primarily on word-of-mouth advertising.
"I guess little independent inventors like myself just kind of throw it out there and see what comes back," Moxon said.
Moxon said his product is still new, and he understands that people are used to paying 99 cents for a pen and may be reluctant to invest in a much more expensive kind.
"My goal is not to be the next Bic pen," he said.
Ultimately, Moxon would like to team up with an existing pen company, one that already has distribution and marketing strategies in place, to help him meet his niche market of buyers.
"Americans really like different types of pens," he said.
Breakout:
To order, go to http://www.pengineering.com.


text:
October 20, 2014 October 20, 2014 lindacleary
Day 293- Valentin Serov- Carried Away by the Human Face
It's Day 293 and I fell in love with today's artist. I started it last night because I knew it would take a little longer. My piece didn't turn out exactly like the artist's work, but I think I captured his style and I did use his paintings as constant inspiration. 🙂 Join me in honoring Valentin Serov today! PS He has a planet named after him, a minor one…but still!
Valentin Serov
Portrait of Anna Benois- Valentin Serov
Valentin Alexandrovich Serov (Russian: Валенти́н Алекса́ндрович Серо́в; January 19, 1865 – December 5, 1911) was a Russian painter, and one of the premier portrait artists of his era.
Serov was born in St. Petersburg, son of the Russian composer Alexander Serov, and his wife Valentina Bergman, a composer of German-Jewish background. In his childhood he studied in
Portrait of Ida Rubinstein- Valentin Serov
Paris and Moscow under Ilya Repin and in the St. Petersburg Academy of Arts (1880–1885) under Pavel Chistyakov. Serov's early creativity was sparked by the realistic art of Repin and strict pedagogical system of Chistyakov.
Further influences on Serov were the old master paintings he viewed in the museums of Russia and Western Europe, friendships with Mikhail Vrubel and (later) Konstantin Korovin, and the creative atmosphere of theAbramtsevo Colony, to which he was closely connected.
The greatest works of Serov's early period were portraits: The Girl with Peaches (1887), and The Girl Covered by the Sun (1888), both in the Tretyakov Gallery. In these paintings Serov concentrated on spontaneity of perception of the model and nature. In the development of light and color, the complex harmony of reflections, the sense of atmospheric saturation, and the fresh picturesque perception of the world, there appeared the features of early Russian impressionism.
Portrait of Helena Ivanovna Roerich- Valentin Serov
From 1890 on, the portrait became the basic genre in Serov's art. It was in this field that his early style would become apparent, the paintings notable for the psychologically pointed characteristics of his subjects. Serov's favorite models were actors, artists, and writers (Konstantin Korovin, 1891, Isaac Levitan, 1893, Nikolai Leskov, 1894, Nikolai Rimsky-Korsakov, 1898, – all in the Tretyakov gallery).
Initially abstaining from the polychromatic, brightly colored painting style of the 1880s, Serov often preferred a dominant scale of black-grey or brown tones. Impressionistic features appeared sometimes in composite construction of a portrait, or to capture a sense of spontaneous movement.
As in the work of his contemporaries John Singer Sargent and Anders Zorn, the impressionism is
Portrait of Pobedonostsev- Valentin Serov
not doctrinaire, but derives as much from the study of Hals and Velázquez as from modern theory. Receiving wide popularity, in 1894 Serov joined with the Peredvizhniki (The Itinerants), and took on important commissions, among them portraits of grand duke Pavel Alexandrovich, (1897, Tretyakov Gallery), S.M. Botkin, 1899, and F.F. Yusupova, 1903 as well as Princess Olga Orlova (these in the Russian Museum in St. Petersburg).
In these truthful, compositionally skillful, and picturesque executions in the grand manner, Serov consistently used linear-rhythmic drawing coupled with decorative color combinations.
Child Portrait- Valentin Serov
At the same time, he developed a contrasting direction: he frequently produced intimate, heartfelt, chamber portraits, mainly of children and women. In portraits of children Serov aspired to capture pose and gesture, to reveal and emphasize a spontaneity of internal movement, sincere cleanliness and clearness of attitude of the child (Children, 1899, Russian Museum; Mika Morozov, 1901, Tretyakov gallery).
Serov frequently called upon various graphic techniques – watercolors, pastels, lithographs and so forth. Figures in Serov's portraits gradually became more and more graphically refined and economical, particularly during the late period (Vasily Kachalov, 1908, Tamara Karsavina, 1909; numerous figures from Ivan Krylov's fables, 1895–1911). From 1890 to 1900 Serov produced many landscape compositions on country themes, in which the artistic direction took a romantic turn.
Serov married Olga Trubnikova in 1887. His wife and his children were the subject of many of
Portrait of Maria Botkina- Valentin Serov
his works. Notably, his painting "Children" was of his sons Yura and Sasha.
During his late period, which began in 1900, Serov was a member of "The World of art", an influential Russian art association and magazine which grew, in part, out of dissatisfaction with the Itinerants movement. At the start of the 20th century, Serov was at a stylistic turning point: features of impressionism disappeared from his work, and his modernistic style developed, but the characteristic truthful and realistic comprehension of the nature of his subjects remained constant. In the early 20th century Serov created heroic portrait images; within the genre of the fashionable portrait, Serov focused on the dramatic depiction of creative artists, writers, actors, and musicians of import:Maxim Gorki's portraits (1904), A.M. Gorki's museum, Moscow; Maria Yermolova (1905), Feodor Chaliapin (charcoal, 1905) – both in the Tretyakov Gallery.
Self-Portrait- Valentin Serov
Serov's democratic beliefs were clearly shown during the Revolution from 1905 to 1907: he depicted a number of satirical figures exposing chastisers. A full member of theSt.Petersburg Academy of Arts since 1903, in 1905 he resigned as a gesture of protest against the execution of striking workers and their families on January 9, Bloody Sunday. His late creativity found expression in historical painting (Peter II departure and Empress Elizabeth Petrovna on hunting, 1900, Russian Museum), and depth of comprehension of the historical maintenance of an epoch (Peter I, distemper, 1907, Tretyakov Gallery).
The last years of Serov's life were marked by works on themes from classical mythology. While addressing images from the ancient tradition, Serov endowed classical subject matter with a personal interpretation.
Valentin died in Moscow on December 5, 1911. He is buried at the Novodevichy Cemetery.
The best works of Serov are among the greatest of Russian realistic art. He taught in the Moscow School of Painting, Sculpture and Architecture from 1897 to 1909), and among his students were Pavel Kuznetsov, N. N. Sapunov, Martiros Saryan, Kuzma Petrov-Vodkin, N.P. Ulyanov, and Konstantin Yuon.
A minor planet 3547 Serov, discovered by Soviet astronomer Lyudmila Zhuravlyova in 1978 is named after him.
Any human face is so complex and so unique that you can always find in it traits worthy of portrayal be they good or bad. For my part, each time I appraise a person's face I am inspired, you might even say carried away, not by his or her outer aspect which is trivial, but by the characterization it can be given on canvas.~ Valentin Serov
I hope you like my self portrait I did in tribute to Serov today! I love his style. I did my best with the time I had! I will see you tomorrow on Day 294.
Self-Portrait- Tribute to Valentin Serov
modernistic
portrait artist
Russian impressionists
Russian painter
Valentin Alexandrovich Serov
Previous Day 292- Philip Guston- We Work Until We Vanish
Next Day 294- Per Kirkeby- Landskaberne


Brrr! It was cold and rainy here yesterday and I had to pull out all my tricks to keep my kids entertained inside. Since it's just the beginning of bad weather season I thought I'd share some of my favorite easy indoor activities. These are perfect quick fixes to keep kids entertained during long days inside.
1. Have a picnic. Turn snack or meal time into an event! My toddlers find this so amusing. I just lay out a blanket in the living room and we pretend we are dining outside on a beautiful summer day. There's definitely a mess potential here but my boys are usually so taken with the novelty of eating on a blanket that they keep their food pretty well contained.
2. Make a fort. I'm all about intricate indoor forts but I promised this would be a list of easy activities. When I'm in a pinch, I just throw a sheet over the kitchen table and call it done. Kids have amazing imaginations and I love to let them use them! The sheet fort is always a hit.
3. Pillow mountain. This is one of our favorites. I just pile as many pillows as I can find - bed pillows, throw pillows, cushions, whatever - into a mountain in the living room and let them go. They have a ball climbing and diving on the "mountain." And its nice and soft so no one gets hurt.
4. Jumping on the bed. Similar to pillow mountain - we have an air mattress which I blow up and let them use as a trampoline. They love to bounce and tumble, and it's only a couple of inches off the ground so if they fall off it's OK. Just be sure to have plenty of space for this one so they don't roll off into a piece of furniture.
5. Sheet sleds. This one came about quite by accident when I was trying to fold laundry while my kids were playing (silly me!). Have them sit on the end of a sheet and hold on while you pull them around. My kids love this!
6. Turn off the lights. This works so well on these short days where it gets dark before dinner time. I turn off the lights, give them flashlights, and let them explore. Simple and fun!
7. Pipe cleaners, colanders, play dough. Look for fun things around the house that your kids can use to create. Mine love putting pipe cleaners into a colander and into play dough to create all sorts of wacky art.
8. Go swimming. In the bath tub! Letting them splash and play in the tub in the middle of the day is great fun. Add in some awesome toys, bath crayons, and bubbles and you're set.
9. Make a sensory bin. This might sound complicated but its just a tub filled with fun materials to explore. Check out my Farm Fun Bin for more details.
10. Give them a cardboard box. Seriously, this is all they need to be entertained sometimes. A huge box can become a car (push them around in it for major laughs!), a playhouse, an art canvas - let their imaginations go wild! Check out how we turned a cardboard box into an awesome reading fort here.
Hope these activities help you enjoy some of the long days ahead. I'd love to hear your favorite indoor boredom busters too!
These are great! I love the sheet sled idea! lol!


Two Innovation Leaders Leave Louisville, Ky.
Replacements for Chief Innovation Officer Ted Smith and Theresa Reno-Weber, the city's chief of performance and technology, have already been named.
Two head innovators will depart and be replaced this week, Louisville, Ky., Mayor Greg Fischer announced Aug. 24.
Ted Smith, the city's chief innovation officer, will soon depart to assume a CEO role at Revon Systems, a health technology company. Smith will be replaced by Grace Simrall, founder of iGlass Analytics and executive director of innovation for Intel Care Innovations. Simrall begins working at the city this week, according to a press release.
Theresa Reno-Weber, the city's chief of performance and technology, will depart to become president and CEO of Metro United Way, a community-focused nonprofit. Reno-Weber will be replaced by Daro Mott, chief innovation officer for Cuyahoga County, Ohio.
"Daro and Grace will make a powerful team and will continue to create a culture of innovation and continuous improvement in Metro Government," Fischer said in the press release.
Smith is credited with leading several projects in the technology and innovation space, including AirLouisville, which distributed GPS-enabled asthma inhalers to identify air quality issues throughout the city.
"The opportunity to serve Mayor Fischer and the caring people of Louisville through the innovation leadership role was deeply rewarding and I look forward to serving further as a citizen-scientist and health-focused entrepreneur," said Smith, who was named as one of Government Technology's Top 25 Doers, Dreamers and Drivers in 2013.
And in 2015, Reno-Weber was named as one of Government Technology's Top 25 Doers, Dreamers and Drivers. She was hired in 2012, and spearheaded development of LouieStat, a performance platform that tracks progress on operational goals, and supplies city officials and residents with regular updates on progress while holding the metro government to a higher level of accountability.


How do you reward innovation?
Are you looking in the right places for innovation?
How are you motivating your team?
Sometimes it can be hard to see the real problem.
Make sure you have fully defined your requirements.
TRIZ can help you break psychological inertia.


text:
\section{Introduction}

In the last years there has been an explosion of interest about physics in the presence of a minimal length.
Reviews about the main approaches where a minimal length appears in the context of quantum  gravity
are given in \cite{ReviewQG}, with special reference to noncommutative geometry in \cite{ReviewNCG}.
Indeed it is expected that our universe, conventionally described in terms of a smooth differential manifold endowed
with a Riemannian structure, exhibits an uncertainty principle which prevents one from measuring positions to better accuracies
than the Planck length. The reason in support for this loss of resolution can be found in the fact that the momentum and the
energy required to perform such a measurement will itself dramatically modify the spacetime geometry at these scales
\cite{DeWitt}. In support of this general belief, there are all the current formulations of a quantum theory of gravity:
String Theory incorporates the concept of minimal length through the string tension, a fundamental parameter in the theory, while in Loop Quantum Gravity a minimal length naturally arises after quantizing the
gravitational field and formulating an area or a volume operator.
Along this line of thought, one can also implement a minimal length in spacetime by assuming nontrivial coordinate commutation
relations and thus postulating the existence of a Noncommutative Geometry as fundamental property of nature. While its original
formulation is dated back to early times \cite{Snyder:1946qz}, a model of Noncommutative Geometry has recently been obtained
within the theory of open strings, whose end points show a noncommutative behavior on D-branes \cite{Seiberg:1999vs}.
Alternatively one may assume that the conventional uncertainty principle among coordinates and momenta in Quantum Mechanics can
be generalized in order to get a modified dispersion relation at higher momenta and implement a minimal length in a quantum
theory \cite{GUP}.
More recently, the emergence of a minimal length has been considered in a variety of physical situations. The inclusion of
noncommutative effects in General Relativity has led to singularity free, thermodynamically stable black hole spacetimes
\cite{NCBHs} [for a review see \cite{review} and the references therein] and cosmological inflationary scenarios driven by
quantum geometry fluctuations rather than by an inflaton field \cite{NCcosmo}. A minimal length has been extensively
implemented to study black hole thermodynamics \cite{NCthermo}, to describe traversable wormholes sustained by noncommutative
geometry fluctuations \cite{NCwormhole} and to address the trans-planckian problem in particle production mechanisms in
accelerated reference frames \cite{NCApplications} and expanding universes \cite{NCparticle}. On the particle phenomenology
side, a lot of attention has been devoted to predict some signatures of an hypothetical production of microscopic regular black
holes \cite{NCBHs} in particle detectors due to extreme energy hadron collisions at the LHC \cite{NCpheno}. Finally, as a
warranty of theoretical consistency of effective models endowed with a minimal length, there is a recent result about the
spectral dimension of spacetime: the actual spacetime dimension $\mathbb{D}$ perceived by a random walker tends to the value
$\mathbb{D}=2$ for scales approaching the minimal length, suggesting the long sought possibility of improving the
renormalization properties of gravity at the Planck scale \cite{NCspectral}.

Against this background, there is the possibility for spacetime to be endowed not only with a minimal length but also with a
minimal momentum below which interactions, gravity included, cannot be resolved. This corresponds to some old ideas
about a the introduction of a natural infrared (IR) cutoff \cite{Infeld}, vivified by some  recent contributions. Indeed
attempts to provide an IR modification of General Relativity have
been formulated to describe the present acceleration of the universe. As a result a new dynamics of the gravitational field has
been derived, giving a small mass to the graviton and keeping unchanged the spacetime manifold \cite{Graviton}. In the
framework of semi-classical gravity, the description of an evaporating black hole is based on high frequency field mode
approximations to avoid unpleasant IR infinite contributions to the quantum stress energy tensor of matter fields \cite{HF}.
Instead of embarking in {\it ad hoc} approximations, it has been recently advocated the so called Ultra Strong Equivalence
Principle, according to which the vacuum expectation value of the (bare) energy momentum tensor has the same form as in flat
space, namely a quartically divergent term and no spacetime dependent (subleading) terms \cite{USEP}. If this principle is
true, we will need an IR completion of standard semi-classical gravity to get rid of all spacetime dependent subleading terms.
This fact is connected to the problem of the cosmological constant: even if only the fluctuations of the quantum vacuum energy
actually contribute to the value of the cosmological constant \cite{cosmoenergyfluc}, it has been recently underlined that
the value will still come too large without invoking an IR regulator \cite{Padmanabhan:2004qc}.

Recently, the introduction of an ultraviolet (UV) cutoff has been obtained in a clear and transparent way by averaging
noncommutative coordinate operators

\begin{eqnarray}
\left[\hat{q}^i,\hat{q}^j\right]&=&i\Theta^{ij}
\label{NCrel}
\end{eqnarray}
on coordinate coherent states \cite{NCQFT,NCGEOM,Banerjee:2009xx}. As a result every point-like structure is smeared out by the
presence of the new quantum of length in the manifold. For instance the Euclidean propagator in coordinate space can be found
by calculating the Fourier transform of the Schwinger representation

\begin{eqnarray}
\frac{e^{-\theta \vec p^2/2}}{\vec p^2+m^2}=e^{\theta m^2/2}\int_{\theta/2}^{\infty}ds\,e^{-s(\vec p^2+m^2)},
\label{Schwinger}
\end{eqnarray}
where $\sqrt\theta$ is the minimal length implemented by (\ref{NCrel}). Coming back to the Lorentzian signature
and considering the massless case, we find that the modified Wightman-Green function is

\begin{equation}
G_\theta(\Delta x)=-\frac{1}{ 4\pi^2 (\Delta  t^2-\Delta \vec{x\,}^2)}\left[1-e^{-(\Delta t^2-\Delta \vec{x\,}^2)/2\theta}\right],
\end{equation}
which is finite for any time-like interval $\Delta x$ in the coincidence limit \footnote{One must have care of the consistency
between the chosen signature and the sign of the exponent in the exponential damping factor.}.
On the other hand, the theory behaves nicely, as $G_\theta$ reduces to its conventional form in the limit $\theta\rightarrow
0$. We notice that the above smearing effect induced by the noncommutative character of the coordinates corresponds to a
modified integral measure in momentum space (\ref{Schwinger}), without invoking any deformed product among functions
\cite{ReviewNCG}. This new integral measure in momentum space has a akin in the context of the Generalized Uncertainty Principle
(GUP) approaches \cite{GUP}. Starting from an $n$ dimensional Euclidean manifold where the commutation relations are

\begin{eqnarray}
\left[\hat{q}^i, \hat{p}_j\right]&=&i\delta^i_{j}\left(1+ f(\hat{ \vec{p} }^2)\right),\nonumber \\
\left[\hat{p}_i, \hat{p}_j\right]&=&0,
\end{eqnarray}
one obtains indeed that coordinate operators fulfil the following noncommutative relation

\begin{eqnarray}
\left[\hat{q}^i, \hat{q}^j\right]&=&-2i\ f^\prime (\hat{ \vec{p} }^2)\left(1+f(\hat{ \vec{p} }^2)\right)\hat{L}^{ij},
\label{NCGUP}
\end{eqnarray}
where $\hat{L}_{ij}$ is the generator of rotations. The integration measure in momentum space

\begin{equation}
1=\int\frac{d^n p}{\left(1+f( \vec{p} ^2)\right)}\left|p\right>\left<p\right|
\end{equation}
becomes suppressed in the UV region, because $f(\vec{p}^2)$ depends on positive powers of the argument and usually is
$f(\vec{p}^2)=\beta {  \vec{p}}^2$. As a result the modification to the measure coming from (\ref{Schwinger}) turns out to be
even stronger, since it corresponds to the choice

\begin{equation}
f(\vec{p}^2)= e^{\theta \vec p^2/2}-1
\end{equation}
exhibiting an exponential suppression. This is in agreement with the fact that coherent states are the states of minimal
uncertainty on the noncommutative manifold, while the delocalization coming from (\ref{NCGUP}) can be arbitrarily large as the
momentum increases. For this reason, even if in the framework of GUP, both the UV and the IR cutoff has been already obtained
\cite{GUPM}, we will proceed in this paper towards an extension of the coherent state approach \cite{NCQFT} in order to endow
the spacetime of a minimal length as well as a minimal momentum/mass.

The paper is organized as follows: we start presenting the new algebra for the positions and momenta and we briefly review the
employment of coherent state for the case of a noncommutative plane (referring just to coordinates) \cite{NCQFT}.
In Section IV, we extend the noncommutative plane case to the presence of noncommutative momenta and we obtain plane waves
endowed with damping factors, depending on both the UV and the IR cutoffs. In Section V, we derive the new form of field
functions and we calculate modified propagators, considering the four dimensional spacetime too (Section VI).
In the last section we draw the conclusions.

\section{General Assumption of noncommuting positions and momenta}

The starting point is the extension of the Heisenberg algebra for the operators describing the spacetime positions
$\hat Q^\mu$ and the corresponding momenta $\hat P_\mu$

\begin{equation}
\left[\hat Q^\mu,\hat Q^\nu\right]=i\Theta^{\mu\nu} \quad,\quad
\left[\hat Q^\mu,\hat P_\nu\right]=i\delta^{\mu}_{\nu} \quad,\quad
\left[\hat P_\mu,\hat P_\nu\right]=i\Sigma_{\mu\nu},
\label{Algebra}
\end{equation}
where $\Theta^{\mu\nu}$ and $\Sigma_{\mu\nu}$ are second rank antisymmetric tensors which do not depend on coordinates and
momenta. This means that the commutation relations ($\ref{Algebra}$) represent a canonical algebra.
Besides the usual uncertainty relation between position and momenta, $\Delta Q^\mu \Delta P_\nu \geq
\frac{1}{2}\delta^\mu_\nu$, additional relations descend from ($\ref{Algebra}$), i.e.

\begin{equation}
\Delta Q^\mu \Delta Q^\nu \geq \frac{1}{2}\left|\Theta^{\mu\nu}\right|
\quad,\quad \Delta P_\mu \Delta P_\nu \geq \frac{1}{2}\left|\Sigma_{\mu\nu}\right|.
\label{relation_xx_pp}
\end{equation}
The relations ($\ref{relation_xx_pp}$) give rise to the appearance of a smallest scale for positions and momenta. Position and
momentum operators obeying the canonical algebra ($\ref{Algebra}$) can be expressed through  a couple of four-vector operators,
$\hat \alpha^\mu$ and $\hat \beta^\mu$, which fulfil the following commutation relations

\begin{equation}
\left[\hat \alpha^\mu,\hat \alpha^\nu\right]=i\Theta^{\mu\nu}\quad,\quad
\left[\hat \alpha^\mu,\hat \beta_\nu\right]=0\quad,\quad
\left[\hat \beta_\mu,\hat \beta_\nu\right]=i\Sigma_{\mu\nu},
\label{algebra_alpha_beta}
\end{equation}
if there are further operators $\hat q^\mu$ and $\hat p_\mu$ which behave like usual position and momentum operators

\begin{equation}
\left[\hat q^\mu,\hat q^\nu\right]=0\quad,\quad
\left[\hat q^\mu,\hat p_\nu\right]=i\delta^{\mu}_{\nu} \quad,\quad
\left[\hat p_\mu,\hat p_\nu\right]=0.
\label{commutation_qk}
\end{equation}
According to ($\ref{algebra_alpha_beta}$) and ($\ref{commutation_qk}$) the position and momentum operators $\hat Q^\mu$ and
$\hat P_\mu$ can be expressed as follows

\begin{equation}
\hat Q^\mu=\hat q^\mu+\hat \alpha^\mu \quad,\quad \hat P_\mu=\hat p_\mu+\hat \beta_\mu.
\label{representation_operators}
\end{equation}
This requires that the following relations have to be satisfied

\begin{eqnarray}
\left[\hat \alpha^\mu,\hat q^\nu\right]+\left[\hat q^\mu,\hat \alpha^\nu\right]=0\quad,\quad
\left[\hat \beta_\mu,\hat p_\nu\right]+\left[\hat p_\mu,\hat \beta_\nu\right]=0\quad,\quad
\left[\hat \alpha^\mu,\hat p_\nu\right]+\left[\hat q^\mu,\hat \beta_\nu\right]=0.
\label{alphabeta}
\end{eqnarray}

\section{The noncommutative plane and modifications of plane waves}

We start recalling the introduction of a UV cutoff due to the noncommutative character of coordinate operators along the lines
of \cite{NCQFT}. The core of the treatment lies in the possibility of defining a set of quasi classical coordinates, averaging
the corresponding quantum operators on suitable coherent states. The commutation relations concern the case of a
($2+1$)-dimensional manifold, with spatial noncommutative coordinates and commutative time, namely

\begin{equation}
\left[\hat Q^i,\hat Q^j\right]=i\Theta^{ij} \quad,\quad
\left[\hat Q^i,\hat P_j\right]=i\delta^{i}_{j} \quad,\quad
\left[\hat P_i,\hat P_j\right]=0,
\label{algebra_commuting_momenta}
\end{equation}
where

\begin{equation}
\Theta=\left(\begin{matrix}0&0&0\\0&0&\theta\\0&-\theta&0\end{matrix}\right)
\end{equation}
is a constant antisymmetric matrix.
The above relations imply that the operator $\hat Q^0$ commutes with the other coordinate operators.
One starts by defining the following ladder operators

\begin{eqnarray}
\hat A=\frac{1}{\sqrt{2\theta}}(\hat Q^1+i\hat Q^2) \quad,\quad \hat A^{\dagger}=\frac{1}{\sqrt{2\theta}}(\hat Q^1-i\hat Q^2)
\end{eqnarray}
satisfying the commutation relation

\begin{eqnarray}
\left[\hat A,\hat A^{\dagger}\right]=1,
\end{eqnarray}
which corresponds to the algebra of usual bosonic creation and annihilation operators. As a result the noncommutative
configuration space is isomorphic to the the Fock space spanned by the states
$|n\rangle_\theta\equiv \frac{1}{\sqrt{n!}}\left(A^\dagger\right)^n|0\rangle_\theta$

\begin{equation}
{\cal H}_\theta=\mathrm{Span}\{|n\rangle_\theta\}_{n=0}^{\infty},
\end{equation}
where the ground state $|0\rangle_\theta$ is the state annihilated by $A$.
To distinguish the conventional quantum phase space from the above Fock space, we will introduce the notation

\begin{equation}
{\cal H}_\hbar=\mathrm{Span}\{|n\rangle_\hbar\}_{n=0}^{\infty},
\end{equation}
where the states $|n\rangle_\hbar$ are built starting from the commutation relation $\left[\hat q^\mu,\hat
p_\nu\right]=i\delta^{\mu}_{\nu}$ between coordinates and momenta. As stressed in \cite{Banerjee:2009xx}
Quantum Mechanics works on ${\cal H}_\hbar$, while Noncommutative Geometry works on ${\cal H}_\theta$. In both cases
observables are defined as self-adjoint operators acting respectively on ${\cal H}_\hbar$ and ${\cal H}_\theta$ and they form
algebras denoted as ${\cal A}({\cal H}_\hbar)$ and ${\cal A}({\cal H}_\theta)$. In analogy to Quantum Mechanics one can
introduce an ``Hamiltonian'' $\hat H_\theta\in {\cal A}({\cal H}_\theta)$ as

\begin{equation}
\hat H_\theta\equiv \frac{(\hat Q^1)^2}{2}+\frac{ (\hat  Q^2)^2}{2}=\theta\left(\hat N_\theta +\frac{1}{2}\right),
\end{equation}
where the corresponding ``number´´ operator $\hat N_\theta=A^\dagger A$ identifies the raising and lowering operators

\begin{equation}
\left[\hat N_\theta,\hat A\right]=-\hat A \quad,\quad \left[\hat N_\theta,\hat A^\dagger \right]=+\hat A^\dagger .
\end{equation}
It is straightforward to define coherent states $|a\rangle_\theta$ as eigenstates of $A$  according to

\begin{equation}
\hat A|a\rangle_\theta=a|a\rangle_\theta,
\end{equation}
where

\begin{equation}
|a\rangle_\theta =e^{-{\frac{|a|^2}{2}}}\sum_{n=0}^{\infty}{\frac{a^n}{\sqrt{n!}}}|n\rangle_\theta
=e^{-{\frac{|a|^2}{2}}}e^{a\hat A^\dagger}|0\rangle_\theta .
\end{equation}
To calculate the modification to the plane waves due to the presence of noncommutative coordinate operators we need to recall
that $e^{ip_1q^1+ip_2q^2}$ is a representation of the eigenstates of the momentum operator $\hat p_i$, namely

\begin{equation}
\hat p_i\left|p_i\right>=p_i\left|p_i\right>,
\end{equation}
with
\begin{equation}
\psi_\hbar\equiv\left<q^i|p_i\right>=e^{ip_iq^i}.
\label{usual_plane_waves}
\end{equation}
This is true only if we work in the framework of conventional Quantum Mechanics. Indeed $p_i$ and $q^i$ appear as usual numbers
within the expressions for the plane waves and represent the corresponding eigenvalues of the states.
In the Noncommutative Geometry case  the quantities appearing in the expression of the plane waves which correspond to
$\hat Q^i$, are still operators, since we have an additional Fock space ${\cal H}_\theta$. They behave like usual numbers with
respect to the usual Hilbert space of Quantum Mechanics ${\cal H}_\hbar$, but they are operators with respect to the additional
Hilbert space ${\cal H}_\theta$. Therefore it is necessary to define a set of classical coordinates, which are usual numbers
with respect to the complete Hilbert space $\mathcal{H}=\mathcal{H}_\hbar \otimes \mathcal{H}_\theta$.
This can be done by averaging coordinate operators on the above coherent states, which turn out to be the states of minimal
uncertainty on the noncommutative plane

\begin{equation}
Q^1\equiv {}_\theta\langle a|\hat Q^1 |a\rangle_\theta = \sqrt{2\theta}\ \mathrm{Re}\ a\quad,\quad
Q^2\equiv {}_\theta\langle a|\hat Q^2 |a\rangle_\theta = \sqrt{2\theta}\ \mathrm{Im}\ a.
\label{coordinates}
\end{equation}
According to ($\ref{representation_operators}$) we can represent the coordinate operators as

\begin{equation}
\hat Q^i=\hat q^i+\hat \alpha^i.
\end{equation}
It is important to emphasize that conventional coordinate operators $\hat q^i$
have no influence on the Fock space ${\cal H}_\theta $ because they fulfil the relation
$[\hat \alpha^i,\hat q^j]+[\hat q^i,\hat \alpha^j]=0$ of equation ($\ref{alphabeta}$).
Since in the usual expressions for plane waves ($\ref{usual_plane_waves}$) there are eigenvalues $q^i$ and $p_i$ of
the usual operators $\hat q^i$ and $\hat p_i$ and not the operators themselves, in the generalized case of this paper it is
necessary to define quantities $\mathcal{Q}^i$ which are eigenvalues with respect to the Hilbert space of usual quantum
mechanics $\mathcal{H}_{\hbar}$ but operators with respect to the additional Hilbert space $\mathcal{H}_{\theta}$

\begin{eqnarray}
\mathcal{Q}^i &=& q^i \hat 1+\hat \alpha^i,
\end{eqnarray}
where $q^i$ is no longer an operator. Therefore the Fock space ${\cal H}_\theta $ refers to the noncommutative character
of $\hat \alpha^\mu$ only. Now we have that the coordinates (\ref{coordinates})

\begin{equation}
Q^1=q^1+{}_\theta\langle a|\hat \alpha^1 |a\rangle_\theta = \sqrt{2\theta}\ \mathrm{Re}\ a
\quad,\quad Q^2=q^2+{}_\theta\langle a|\hat \alpha^2 |a\rangle_\theta = \sqrt{2\theta}\ \mathrm{Im}\ a
\end{equation}
turn out to be the sum of the conventional classical coordinates $q^i$ and the mean values of the noncommutative fluctuations
$\langle a|\hat \alpha^i|a \rangle$. Since we are not yet considering the complete algebra ($\ref{Algebra}$) but the simpler
case of ($\ref{algebra_commuting_momenta}$), the  momenta $\hat P_i$ are still assumed to commute with each other and of course
they act as usual numbers on ${\cal H}_\theta $. This implies that the corresponding quantities appearing in the expression for
the extended plane waves, using an analogue representation to $\mathcal{Q}^i$ have the form $\mathcal{P}_i=P_i\hat 1$. Therefore for the operator

\begin{eqnarray}
\hat A =\frac{1}{\sqrt{2\theta}}[( q^1+i q^2)\hat1+\hat \alpha^1+i\hat \alpha^2],
\end{eqnarray}
we have the relations $\left[\mathcal{P}_i,\hat A \right]=0$ meaning that

\begin{equation}
P_1={}_\theta\langle a|\mathcal{P}_1 |a\rangle_\theta, \quad\quad P_2={}_\theta\langle a|\mathcal{P}_2 |a\rangle_\theta.
\end{equation}
Thus if we consider as new expression for plane waves

\begin{eqnarray}
{}_\theta\langle a|e^{i\mathcal{P}_1 \mathcal{Q}^1+i\mathcal{P}_2\mathcal{Q}^2} |a\rangle_\theta,
\end{eqnarray}
where now the symbol ``$\ \hat{}\ $'' indicates an operator acting only on ${\cal H}_\theta$, we will have

\begin{eqnarray}
{}_\theta\langle a|e^{i\mathcal{P}_1\mathcal{Q}^1+i\mathcal{P}_2\mathcal{Q}^2}|a\rangle_\theta
={}_\theta\langle a|e^{i P_1 \mathcal{Q}^1+i P_2 \mathcal{Q}^2} |a\rangle_\theta
={}_\theta\langle a|e^{i\sqrt{\theta}\hat A^\dagger P_+ +i\sqrt{\theta}\hat AP_-} |a\rangle_\theta
\label{NCplanewave}
\end{eqnarray}
with $P_\pm=\frac{1}{\sqrt{2}}(P_1\pm iP_2)$. By means of the Baker Campbell Hausdorff formula one obtains

\begin{equation}
{}_\theta\langle a|e^{i\sqrt{\theta}\hat A^\dagger P_+ +i\sqrt{\theta}\hat AP_-} |a\rangle_\theta
={}_\theta\langle a|e^{i\sqrt{\theta}\hat A^\dagger P_+}
e^{i\sqrt{\theta}\hat AP_-}e^{-(1/2)\theta P_+P_-[\hat A, \hat A^\dagger]} |a\rangle_\theta
=e^{iP_1 Q^1+iP_2 Q^2}e^{-\frac{\theta}{4}(P^2_1+P^2_2)}.
\end{equation}
As a result, we have that according to \cite{NCQFT} the plane wave is damped by an exponential factor coming from the
noncommutativity of coordinate operators,  which means that the plane wave expression is modified according to

\begin{eqnarray}
\psi_{\hbar}=e^{i \vec p\cdot \vec q}\longrightarrow \psi_{\hbar\theta}=e^{i \vec p\cdot \vec q}e^{-\frac{\theta}{4} \vec p^2}.
\end{eqnarray}
This corresponds to having a response from the noncommutative manifold which provides a natural UV cutoff for higher momenta
modes as a consequence of the loss of space resolution at short distances.

\section{Incorporation of noncommutative momenta}

We proceed now with the extension of the above procedure for a ($2+1$)-dimensional manifold to the case in which momentum
operators do not commute anymore as well as the position operators, i.e.

\begin{equation}
\left[\hat Q^i,\hat Q^j\right]=i\Theta^{ij} \quad,\quad
\left[\hat Q^i,\hat P_j\right]=i\delta^{i}_{j} \quad,\quad
\left[\hat P_i,\hat P_j\right]=i\Sigma_{ij},
\end{equation}
where $\Theta^{ij}$ and $\Sigma_{ij}$ are antisymmetric matrices which do not depend on the coordinates or the momenta.
For later convenience we assume the following form of nonvanishing commutators

\begin{equation}
[\hat Q^1,\hat Q^2]=i \theta \quad,\quad [\hat P_2, \hat P_1]=i \sigma,
\label{algebra_two_dimensions}
\end{equation}
where we have set

\begin{equation}
\Theta=\left(\begin{matrix}0&0&0\\0&0&\theta\\0&-\theta&0\end{matrix}\right)\quad,\quad
\Sigma=\left(\begin{matrix}0&0&0\\0&0&-\sigma\\0&\sigma&0\end{matrix}\right).
\label{algebra_two_dimensions_matrices}
\end{equation}
Along the line of reasoning followed for the noncommutative plane, we introduce

\begin{equation}
\hat A= \frac{1}{\sqrt{2\theta}}(\hat Q^1+i\hat Q^2) \quad,\quad \hat A^{\dagger}
=\frac{1}{\sqrt{2\theta}}(\hat Q^1-i\hat Q^2), \quad,\quad
\hat B=\frac{1}{\sqrt{2\sigma}}(\hat P_2+i\hat P_1) \quad,\quad \hat B^{\dagger}
=\frac{1}{\sqrt{2\sigma}}(\hat P_2-i\hat P_1),
\end{equation}
satisfying the commutation relations

\begin{eqnarray}
\left[\hat A,\hat A^{\dagger}\right]=1 \quad,\quad \left[\hat B,\hat B^{\dagger}\right]=1.\nonumber\\
\label{commutation_AB}
\end{eqnarray}
Again both, $\hat A$ and $\hat B$, define ``Hamiltonian'' operators as

\begin{equation}
\hat H_\theta\equiv \frac{ (\hat Q^1)^2}{2}+\frac{ (\hat Q^2)^2}{2}=\theta\left(\hat N_q +\frac{1}{2}\right)\quad,\quad
\hat H_\sigma\equiv \frac{\hat P_1^2}{2}+\frac{\hat P_2^2}{2}=\sigma\left(\hat N_p +\frac{1}{2}\right),
\end{equation}
where the corresponding ``number'' operators $\hat N_\theta=A^\dagger A$ and $\hat N_\sigma=B^\dagger B$ identify raising and
lowering operators

\begin{equation}
\left[\hat N_\theta,\hat A \right]=-\hat A \quad,\quad \left[\hat N_\theta,\hat A^\dagger \right]
=\hat A^\dagger\quad,\quad\left[\hat N_\sigma,\hat B \right]=
-\hat B \quad\quad \left[\hat N_\sigma,\hat B^\dagger\right]=\hat B^\dagger.
\end{equation}
Since now we are endowed not only with a quantum of action $\hbar$ and a quantum of length $\sqrt{\theta}$ but also with a
quantum of momentum $\sqrt{\sigma}$, we have three corresponding Fock spaces ${\cal H}_\hbar $, ${\cal H}_\theta $ and

\begin{equation}
{\cal H}_\sigma=\mathrm{Span}\{|n\rangle_\sigma\}_{n=0}^{\infty},
\end{equation}
where $|n\rangle_\sigma\equiv \frac{1}{\sqrt{n!}}B^\dagger|0\rangle_\sigma$.
 We can follow the same procedure as above and consider the following representation of the operators
$\mathcal{Q}^i\in{\cal H}_\theta$

\begin{equation}
\mathcal{Q}^i=q^i\hat 1+\hat\alpha^i
\label{coordinates_plane_waves}
\end{equation}
and $\mathcal{P}_i\in{\cal H}_\sigma $

\begin{equation}
\mathcal{P}_i=p_i\hat 1+\hat\beta_i.
\label{momenta_plane_waves}
\end{equation}
provided that $[\hat \alpha^i,\hat q^j]+[\hat q^i,\hat \alpha^j]=[\hat \beta_i,\hat p_j]+[\hat p_i,\hat \beta_j]=0$ as in ($\ref{alphabeta}$). This means that the $\mathcal{Q}^i$ are operators with respect to the additional Hilbert
space ${\cal H}_\theta$ and the $\mathcal{P}_i$ are operators with respect to the additional Hilbert space ${\cal H}_\sigma$.
However both, the $\mathcal{Q}^i$ and the $\mathcal{P}_i$, as they appear in the plane wave expression, act as usual numbers on
${\cal H}_\hbar$. Therefore the coordinates (\ref{coordinates}) read

\begin{equation}
Q^1=q^1+{}_\theta\langle a|\hat \alpha^1 |a\rangle_\theta = \sqrt{2\theta}\ \mathrm{Re}\ a\quad,\quad
Q^2=q^2+{}_\theta\langle a|\hat \alpha^2 |a\rangle_\theta = \sqrt{2\theta}\ \mathrm{Im}\ a\\
\end{equation}
and the momenta read

\begin{equation}
P_1=p_1+{}_\sigma\langle b|\hat \beta_1 |b\rangle_\sigma = \sqrt{2\sigma}\ \mathrm{Im} \ b,\quad,\quad
P_2=p_2+{}_\sigma\langle b|\hat \beta_2 |b\rangle_\sigma = \sqrt{2\sigma}\ \mathrm{Re}\ b,\\
\end{equation}
where the coherent states $|a\rangle_\theta$ and $|b\rangle_\sigma$ are eigenstates of the operators

\begin{equation}
\hat A=\frac{1}{\sqrt{2\theta}}[(q^1+iq^2)\hat 1+\hat \alpha^1+i\hat \alpha^2]\quad,\quad
\hat B=\frac{1}{\sqrt{2\sigma}}[(p^1+ip^2)\hat 1+\hat \beta^1+i\hat \beta^2].
\end{equation}
Since $\left[\hat \alpha^\mu,\hat \beta_\nu\right]=0$, we can conclude that ${\cal H}_\theta\cong {\cal H}_\sigma$ are
isomorphic, being $\left[\hat A,\hat B\right]=0$ and admit common eigenstates $\left. |a\ b\right>_{\theta\sigma}$.
We are now ready to compute the modifications to plane waves due to the presence of both, noncommutative positions and momenta,
corresponding to the parameters $\sqrt{\theta}$ and $\sqrt{\sigma}$.

\begin{eqnarray}
{}_{\theta\sigma}\langle a\ b |\exp\left[i\mathcal{P}_1 \mathcal{Q}^1+i\mathcal{P}_2 \mathcal{Q}^2\right]
|a \ b\rangle_{\theta\sigma}
&=&{}_{\theta\sigma}\langle a\ b |\exp\left[-t \hat A^{\dagger}\hat B^\dagger + t\hat A \hat B\right]
|a \ b\rangle_{\theta\sigma},
\label{plane_waves_AB}
\end{eqnarray}
where the dimensionless parameter $t=\sqrt{\theta\sigma}$ is the product of the quantum of length and the quantum of momentum.
Therefore we can safely assume that $t\ll 1$. Again we stress that now the symbol ``$\ \hat{}\ $'' indicates an operator only
with respect to ${\cal H}_\sigma$ and ${\cal H}_\theta$, spanned by $\{|a \ b\rangle_{\theta\sigma}\}$. To evaluate the
expression ($\ref{plane_waves_AB}$), we can use the Baker Campbell Hausdorff formula

\begin{equation}
e^{tX} e^{tY}=e^{tX+tY+\frac{t^2}{2}[X,Y]+\frac{t^3}{12}[X,[X,Y]]-\frac{t^3}{12}[Y,[X,Y]]+...}
\label{BCH}
\end{equation}
with $X=-\hat A^{\dagger}\hat B^{\dagger}$ and $Y=\hat A \hat B$. The above formula can be written in terms of the following
combinatoric expansion

\begin{equation}
e^{t(X+Y)}= e^{tX}~ e^{tY} ~e^{-\frac{t^2}{2} [X,Y]} ~ e^{\frac{t^3}{6}(2[Y,[X,Y]]+ [X,[X,Y]] )}
~ e^{\frac{-t^4}{24}([[[X,Y],X],X] + 3[[[X,Y],X],Y] + 3[[[X,Y],Y],Y]) } \cdots
\end{equation}
provided $t\ll 1$. We notice that

\begin{eqnarray}
&&[X,Y]=\hat A^{\dagger}\hat A +\hat B^{\dagger}\hat B, \quad\quad [X,[X,Y]]=-2X,  \quad\quad [Y,[X,Y]]=2Y
\end{eqnarray}
and thus we can see that any odd term will be a linear combination of $t^{2n+1} X$ and $t^{2n+1} Y$ and every even term will be
$\sim t^{2n}[X,Y]$. Applying recursively the Baker Campbell Hausdorff formula to each term, we get

\begin{equation}
e^{t(X+Y)}= e^{(t+{\cal O}(t^3))X}~ e^{(t+{\cal O}(t^3))Y} ~e^{-(\frac{t^2}{2}+{\cal O}(t^4)) [X,Y]}.
\end{equation}
As a result, we can safely keep just the leading order in the expansion

\begin{equation}
e^{t(X+Y)}\simeq e^{tX}e^{tY}e^{-\frac{t^2}{2}[X,Y]}
\label{approximation_BCH}
\end{equation}
and write the plane wave as

\begin{eqnarray}
&&{}_{\theta\sigma}\langle a\ b |\exp\left[i\mathcal{P}_1 \mathcal{Q}^1+i\mathcal{P}_2 \mathcal{Q}^2\right]
|a \ b\rangle_{\theta\sigma}
={}_{\theta\sigma}\langle a\ b |\exp\left[-t \hat A^{\dagger}\hat B^\dagger
+t\hat A \hat B\right]|a \ b\rangle_{\theta\sigma}=\nonumber\\
&&={}_{\theta\sigma}\langle a\ b |\exp\left[-t \hat A^{\dagger}\hat B^\dagger\right]\exp\left[ t\hat A \hat B\right]
\exp\left[-(t^2/2)(\hat A^{\dagger}\hat A +\hat B^{\dagger}\hat B)\right]|a \ b\rangle_{\theta\sigma}=\nonumber
\\ &&=\exp\left[-t a^{\ast} b^\ast\right]\exp\left[ t a b\right]
\exp\left[-t^2( |a|^2 + |b|^2)/2\right]=\nonumber
\\ &&=\exp\left[i\mathcal{P}_1\mathcal{Q}^1+i\mathcal{P}_2\mathcal{Q}^2\right]
\exp\left\{-t^2\left[ \sigma (Q^1)^2+\sigma (Q^2)^2+\theta P_1^2+\theta P_2^2\right]/4\right\}
\end{eqnarray}
In summary the modifications to the conventional plane waves are of the form

\begin{equation}
\psi_\hbar = e^{i \vec p\cdot \vec q}\longrightarrow \psi_{\hbar\theta\sigma}
=e^{i \vec p\cdot \vec q-(\sigma/4) \vec q^2 -(\theta/4) \vec p^2}
\label{ampl}
\end{equation}
exhibiting both  UV and IR damping terms. This result is in agreement with what has been found within the framework of a the
Generalized Uncertainty Principle, once a relation like

\begin{eqnarray}
\left[\hat{q}^i, \hat{p}_j\right]&=&i\delta^i_{j}\left(1+ f(\hat{ \vec{p} }^2)+ g(\hat{ \vec{q} }^2)\right)
\end{eqnarray}
is assumed as in \cite{GUPM}. As a comment we have to remember that the above result has been obtained up to subleading terms
$\sim e^{i\sigma\theta\vec q\cdot\vec p}$, which are responsible of a modified dispersion relation and are extremely suppressed
in the regime $\theta\sigma\ll 1$.

\section{Derivation of the propagator of a quantum field in a (2+1)-dimensional spacetime}

We are now ready to define a quantum field.
 As shown in \cite{NCQFT},  the modifications to plane waves due to the employment of coherent state mean coordinates do not affect the conventional
algebra of creation and annihilation operators of quantum fields, since the modification of commutation relations
($\ref{Algebra}$) refers to the Hilbert space of single particles which is extended according to the above
transition $\mathcal{H}_\hbar \rightarrow \mathcal{H}_\hbar \otimes \mathcal{H}_\theta \otimes \mathcal{H}_\sigma$. Therefore the construction of a Fock space of many particles follows the conventional procedure.
Therefore by introducing the notation $x^\mu=(t,x^1,x^2)=(t,{\bf x})$ and $k_\mu=(\omega_k,k_1,k_2)=(\omega_k,{\bf k})$, we can
express a scalar field as follows

\begin{equation}
\phi({\bf x},t)=\int \frac{d^2 k}{\left(2\pi\right)^2 2\omega_k}
\left[a({\bf k}){}_{\theta\sigma}\langle a\ b|e^{i{\bf \hat k}{\bf \hat x}-i\omega_k t}|a\ b\rangle_{\theta\sigma}
+a^{\dagger}({\bf k}){}_{\theta\sigma}\langle a\ b|e^{-i{\bf \hat k}{\bf \hat x}
+i\omega_k t}|a\ b\rangle_{\theta\sigma}\right]
\label{scalar_field}
\end{equation}
whose quantization is achieved in the usual way by postulating

\begin{equation}
\left[a({\bf k}),a^{\dagger}({\bf k})\right]=\delta^2\left({\bf k}-{\bf k^{\prime}}\right).
\end{equation}
Again the symbol ``$\ \hat{}\ $'' in (\ref{scalar_field}) indicates operators acting on the Fock space ${\cal H}_\theta\cong {\cal H}_\sigma$.
If the expectation values are calculated, we get

\begin{equation}
\phi({\bf x},t)=\int \frac{d^2 k}{\left(2\pi\right)^2 2\omega_k}
\left[a({\bf k})e^{i{\bf kx}-i\omega_k t}+a^{\dagger}({\bf k})e^{-i{\bf kx}+i\omega_k t}\right]
\exp\left[-\frac{1}{4}\theta{\bf k}^2\right]\exp\left[-\frac{1}{4}\sigma{\bf x}^2\right].
\end{equation}
The propagator is obtained in the usual way by considering the expectation value of the time
ordered product of the quantum field operator at two different positions with respect to
the vacuum $|0 \rangle$ state of the quantum field

\begin{equation}
G(x-x^\prime)=i\Delta_F(x-x^\prime)=\langle 0|T[\phi(x)\phi(x^\prime)]|0 \rangle,
\label{expression_propagator}
\end{equation}
where $T$ denotes the time ordering operator.
Without loss of generality the coordinate system can always be chosen in such a way that one of the coordinates between $x$ and
$x^{\prime}$ lies at the origin. As a result, the expression of the propagator only depends on the difference of the variables
$z=x-x^{\prime}$ if the coordinates are accordingly chosen as $x=z$ and $x^{\prime}=0$.
Thus the propagator can be formulated in such a way that it only depends on the difference of the coordinates

\begin{eqnarray}
\langle 0|T[\phi(z)\phi(0)]|0 \rangle
=\int \frac{d^2 k}{(2\pi)^2 2 \omega_k}
\left[e^{i{\bf kz}-i\omega_k t_z}\theta(t_z)+e^{-i{\bf kz}+i\omega_k t_z}\theta(-t_z)\right]
\exp\left[-\frac{1}{2}\theta{\bf k}^2\right]\exp\left[-\frac{1}{4}\sigma{\bf z}^2\right].
\label{expression_operator_product2}
\end{eqnarray}
From now on, we can follow the usual procedure of the derivation of a propagator. By using

\begin{equation}
\theta(t)e^{-i\omega_k t}=\frac{1}{2\pi i}\int^{\infty}_{-\infty}dE \frac{e^{-iEt}}{\omega_k-E-i\epsilon},
\label{step_function}
\end{equation}
equation ($\ref{expression_operator_product2}$) can be transformed to

\begin{equation}
\langle 0|T[\phi(z)\phi(0)]|0 \rangle=G(z)=i\int \frac{d^3 k}{(2\pi)^3}
\frac{e^{-ikz}}{k^2-m^2+i\epsilon}\exp\left[-\frac{1}{2}\theta{\bf k}^2\right]\exp\left[-\frac{1}{4}\sigma{\bf z}^2\right]
\label{expression_operator_product3}
\end{equation}
which is the propagator for a scalar field in the case of a (2+1) dimensional spacetime with the special
manifestation ($\ref{algebra_two_dimensions}$) and ($\ref{algebra_two_dimensions_matrices}$)
of ($\ref{Algebra}$). The propagator in momentum space is obtained by a Fourier transformation as usual,
since the above expression is already expressed in terms of expectation values.
As a result, the propagator in momentum space reads

\begin{eqnarray}
G(k)&=&\int \frac{d^2 k^{\prime}}{(2\pi)^2}\frac{\rho_\sigma({\bf k,k^{\prime}})\ e^{-\frac{1}{2}\theta{\bf k}^{\prime 2}}}
{\omega_k^2-{\bf k^{\prime 2}}-m^2+i\epsilon},
\label{propagator_momentum_space_integrated}
\end{eqnarray}
where

\begin{eqnarray}
\rho_\sigma({\bf k,k^{\prime}})&=&\frac{1}{\pi\sigma}\ e^{-\frac{(\bf k-k^{\prime})^2}{\sigma}}.
\end{eqnarray}
We notice that the IR behavior of the propagator is now controlled by the function $\rho_\sigma({\bf k,k^{\prime}})$,
which in the limit $\sigma\to 0$ becomes a delta function, leading to the result

\begin{equation}
G(k)=\frac{ e^{-\frac{1}{2}\theta{\bf k}^{2}}}
{\omega_k^2-{\bf k^{ 2}}-m^2+i\epsilon}\\
\end{equation}
for the UV  finite propagator \cite{NCQFT}. Conversely, for $\sigma\neq 0$, ($\ref{propagator_momentum_space_integrated}$)
provides a Gaussian modulation of the propagator, preventing the resolution of momenta smaller that $\sqrt\sigma$.
In other words the distribution of $\bf k^\prime$ has a bell shape of width $\sqrt\sigma$ and it is no longer peaked on the
value $\bf k$ due to the intrinsic uncertainty we have introduced in the formulation.

\section{The case of a (3+1)-dimensional spacetime}

The ($2+1$)-dimensional case can be extended to the more realistic case of a ($3+1$)-dimensional Minkowski spacetime.
Let us start from the algebra ($\ref{Algebra}$). As a first remark, we proceed along the line of \cite{NCQFT}
recalling that our matrices describing the noncommutativity are Lorentz tensors.
For later convenience, we are initially working with an Euclidean signature without loss of generality.
Then one can exploit the property for which any antisymmetric tensor can be written in a block diagonal form as

\begin{equation}
\Theta=\left(\begin{matrix}0&\theta_1&0&0\\-\theta_1&0&0&0\\0&0&0&\theta_2\\0&0&-\theta_2&0\end{matrix}\right)\quad,\quad
\Sigma=\left(\begin{matrix}0&-\sigma_1&0&0\\\sigma_1&0&0&0\\0&0&0&-\sigma_2\\0&0&\sigma_2&0\end{matrix}\right).
\end{equation}
In a more compact form this can be written as

\begin{equation}
\Theta=\mathrm{diag}\left(\ \hat{\theta}^1,\  \hat{\theta}^2\ \right)\quad,\quad
\Sigma=\mathrm{diag}\left(\ \hat{\sigma}_1,\  \hat{\sigma}_2\ \right),
\end{equation}
where

\begin{equation}
\hat\theta^{i}=\theta^i\left( \begin{array}{cc}
0 & 1  \\
-1 & 0 \\
\end{array} \right)\quad,\quad
\hat\sigma_{j}=-\sigma_j\left( \begin{array}{cc}
0 & 1  \\
-1 & 0 \\
\end{array} \right)
\end{equation}
with $i,j=1,2$. As a result the noncommutative relations actually foliate the spacetime in two planes. Lorentz covariance
implies that different observers get different noncommutativity  due to different projections of noncommutative plane
in their reference frames. Conversely what is invariant, namely a fact recognized by any observer, is the presence of
these two planes which foliate the spacetime. Thus the problem is reduced to an effective two-dimensional problem
\begin{equation}
\hat A_i=\frac{1}{\sqrt{2\theta_i}}\left(\hat Q_i^1+i\hat Q_i^2\right) \quad,\quad \hat A^{\dagger}_i
=\frac{1}{\sqrt{2\theta_i}}\left(\hat Q^1_i-i\hat Q_i^2\right)\quad,\qua
\hat B_j=\frac{1}{\sqrt{2\theta_j}}\left(\hat P_{j2}+i\hat P_{j1}\right) \quad,\quad \hat B^{\dagger}_j
=\frac{1}{\sqrt{2\theta_j}}\left(\hat P_{j2}-i\hat P_{j1}\right),
\end{equation}
where $(\hat Q_i^1,\hat Q_i^2)$ are coordinate operators in the $i$-th plane, while $(\hat P_{j1},\hat P_{j2})$ are
the corresponding ones in momentum space. These operators fulfil the following commutation relations

\begin{eqnarray}
\left[\hat A_i,\hat A_j^{\dagger}\right]=\delta_{ij}
\quad,\quad \left[\hat B_i,\hat B_j^{\dagger}\right]=\delta_{ij}.
\end{eqnarray}
Coherent states can be defined as

\begin{equation}
|a\rangle_{\theta} =\prod_i e^{-{\frac{|a_i|^2}{2}}}\sum_{n=0}^{\infty}{\frac{a_i^n}{\sqrt{n!}}}|n\rangle_{\theta_i}
=\prod_i e^{-{\frac{|a_i|^2}{2} }}e^{a_i\hat A_i^\dagger}|0\rangle_{\theta_i},
\end{equation}
with

\begin{equation}
\hat A_i|a\rangle_{\theta}=a_i|a\rangle_{\theta}
\end{equation}
and

\begin{equation}
|b\rangle_{\sigma}=\prod_j e^{-{\frac{|b_j|^2}{2} }}\sum_{n=0}^{\infty}{\frac{b_j^n}{\sqrt{n!}}}|n\rangle_{\sigma_j}
=\prod_j e^{-{\frac{|b_j|^2}{2}}}e^{b_j\hat B_j^\dagger}|0\rangle_{\sigma_j},
\end{equation}
with
\begin{equation}
\hat B_i|b\rangle_\sigma=b_i|b\rangle_\sigma.
\end{equation}
With the help of these states, we can define expectation values for the positions and momenta according to

\begin{eqnarray}
{}_\theta\langle a|\hat Q_i^1 |a\rangle_\theta=Q_i^1\quad,\quad
{}_\theta\langle a|\hat Q_i^2 |a\rangle_\theta=Q_i^2\quad,\quad
{}_\sigma\langle b|\hat P_{j1} |b\rangle_\sigma=P_{j1}\quad,\quad
{}_\sigma\langle b|\hat P_{j2} |b\rangle_\sigma=P_{j2}.
\label{expectation_values_3+1}
\end{eqnarray}
As a result, the expectation values of the plane wave have the following shape

\begin{eqnarray}
{}_{\theta\sigma}\langle a \ b |\exp\left[i\sum_i\mathcal{Q}_i\cdot \mathcal{P}_i \right]|a\ b\rangle_{\theta\sigma}
=\exp\left[i\sum_i \vec P_i\cdot \vec Q_i\right]\exp\left[-\frac{1}{4}\sum_i\left(\theta_i \vec P_i^2+\sigma_i \vec Q_i^2 \right)\right].
\label{modified_plane_waves_3+1}
\end{eqnarray}
The above expression clearly depends on the orientations of the noncommutative plane due to the presence of different
parameters $\theta_i$ and $\sigma_i$. To restore a Lorentz covariant formalism, we follow the lines in \cite{NCQFT} and we set
$\theta_i=\theta$ and $\sigma_i=\sigma$. Given this point and in analogy to the (2+1)-dimensional scenario we are ready to
calculate the scalar field propagator. We will use the following notation for the position
\ $x^\mu=(t,x^1,x^2,x^3)=(t,{\bf x})=(Q^1_1,Q^2_1,Q^1_2,Q^2_2)$ and the momentum
\ $k_\mu=(k_0,k_1,k_2,k_3)=(k_0,{\bf k})=(P_{11},P_{12},P_{21},P_{22})$.
Starting from the scalar field written as

\begin{equation}
\phi({\bf x},t)=\int \frac{d^3 k}{\left(2\pi\right)^3 2 k_0}
\left[a({\bf k}){}_{\theta\sigma}\langle a\ b |e^{i\hat k_\mu \hat x^\mu}|a\ b\rangle_{\theta\sigma}+a^{\dagger}({\bf k})
{}_{\theta\sigma}\langle a\ b| e^{-i\hat k_\mu \hat x^\mu}|a\ b \rangle_{\theta\sigma}\right],
\end{equation}
with

\begin{equation}
\left[a({\bf k}),a^{\dagger}({\bf k})\right]=\delta^3\left({\bf k}-{\bf k^{\prime}}\right).
\end{equation}
we get

\begin{equation}
\phi({\bf x},t)=\int \frac{d^3 k}{\left(2\pi\right)^3 2 k_0}
\left[a({\bf k})e^{ik_\mu x^\mu}+a^{\dagger}({\bf k})
e^{-i k_\mu x^\mu}\right]\exp\left[-\frac{1}{4}\theta k_\mu k^\mu\right]\exp\left[-\frac{1}{4}\sigma x_\mu x^\mu\right].
\label{scalar_field_3+1}
\end{equation}
The related expression of the propagator reads

\begin{eqnarray}
G(z)&=&\langle 0|T[\phi(z)\phi(0)]|0 \rangle\\&=&\int \frac{d^3 k}{(2\pi)^3 2 k_0}
\left[e^{i k_\mu z^\mu}\theta(z_0)+e^{-i k_\mu z^\mu}\theta(-z_0)\right]
\exp\left[-\frac{1}{2}\theta k_\mu k^\mu\right]\exp\left[-\frac{1}{4}\sigma z_\mu z^\mu\right]\nonumber
\label{expression_operator_product_3+1}
\end{eqnarray}
where, without loss of generality, we have again chosen the coordinate system in such a way that one of the coordinates
lies at the origin. This means that we assume that $x=z$ and $x^\prime=0$ and therefore the propagator turns out to depend
only on the variable $z=x-x^\prime$. By using (\ref{step_function}) and following the usual procedure, one can write

\begin{eqnarray}
G(z)=\langle 0|T[\phi(z)\phi(0)]|0 \rangle=i\int \frac{d^4 k}{(2\pi)^4}\frac{e^{-ikz}}{k^2+m^2+i\epsilon}
\exp\left[-\frac{1}{2}\theta k^2-\frac{1}{4}\sigma z^2\right].
\label{propagator_position_space_3+1}
\end{eqnarray}
As a result the corresponding propagator in momentum space reads

\begin{eqnarray}
G(k)=\int \frac{d^4 k^{\prime}}{(2\pi)^4}
\frac{\rho_\sigma (k,k^\prime)}{k^{\prime 2}+m^2+i\epsilon}
e^{-\frac{1}{2}\theta k^{\prime 2}},
\label{finitepropag}
\end{eqnarray}
where the modulation function

\begin{equation}
\rho_\sigma(k,k^\prime)=\frac{1}{\pi^2\sigma^2}e^{-\frac{(k-k^{\prime})^2}{\sigma}}
\end{equation}
accounts for the uncertainty of the momentum.
In the limit $\sigma\to 0$, there is no longer any uncertainty of momentum, which is exactly peaked on the value $k$.
The modulation function $\rho_\sigma(k,k^\prime)$ becomes a Dirac delta $\rho_\sigma(k,k^\prime)\to\delta(k,k^\prime)$
and the propagator reads

\begin{equation}
G(k)=\frac{e^{-\frac{1}{2}\theta k^{2}}}{k^{2}+m^2+i\epsilon},
\end{equation}
reproducing the UV finite propagator as in \cite{NCQFT}. Conversely for $\sigma\neq 0$,
the modulation function $\rho(k,k^\prime)$ prevents better resolutions in momentum space than $\sqrt\sigma$.
The integral (\ref{finitepropag}) is actually endowed with a IR cutoff. As a result, if we consider the limit
$m\to 0$, the propagator converges even for vanishing values of $k$. As shown in the appendix we have

\begin{equation}
G(0)\approx\frac{1}{\sigma}
\end{equation}
for $m\to 0$. As a conclusion, the propagator is finite both in the UV and IR regime.
In support of this claim, we also provide  the following integral representation for the Green's function

\begin{equation}
G(x-x')=\frac{e^{\theta m^2/2}}{16\pi^{2}} \int_{\theta/(2+\sigma\theta)}^{1/\sigma}\frac{ds}{s^2}
\ e^{-m^2s/(1-\sigma s)}\ e^{-(x-x')^2/4s}
\label{integral}
\end{equation}
that after manipulations can be cast in the more familiar form

\begin{equation}
G(x-x')=\int_0^\infty ds \ e^{-m^2s}\ \frac{e^{-(\sigma+\frac{1}{s+\theta/2})(x-x')^2/4}}{[4\pi(s+\theta/2)]^{2}}.
\end{equation}
In the massless case, we obtain after integration

\begin{equation}
G(x-x')=\frac{e^{-\frac{1}{4}\sigma (x-x')^2}}{4\pi^{2}(x-x')^{2}} \ \left(1-e^{-(x-x')^2/2\theta}\right).
\end{equation}
In addition, if the coordinates are Wick rotated back to Minkowski signature, we can express the momentum space
propagator as \footnote{One must have care of the correct definition of the damping terms within the chosen signature.}

\begin{eqnarray}
G(k)=\frac{i}{\pi^2\sigma^2}\int \frac{d^4 k^{\prime}}{(2\pi)^4}
\frac{e^{-\frac{(k-k^{\prime})^2}{\sigma}}}{k^{\prime 2}-m^2+i\epsilon}
e^{-\frac{1}{2}\theta k^{\prime 2}}.
\end{eqnarray}
For completeness we study also the case of a fermionic field. Since the deformation we are introducing concerns the shape
of the Fourier modes only, a fermionic field can be treated in analogy to a scalar field ($\ref{scalar_field_3+1}$) and reads

\begin{eqnarray}
\psi({\bf x},x_0)=\sum_{\pm s}\int \frac{d^3 k}{\left(2\pi\right)^3}\sqrt{\frac{m}{k_0}}
\left[b_s({\bf k})u_s({\bf k})e^{i k_\mu x^\mu}
+d^{\dagger}_s({\bf k})v_s({\bf k})e^{-i k_\mu x^\mu}\right]
\exp\left[-\frac{1}{4}\theta k_\mu k^\mu\right]\exp\left[-\frac{1}{4}\sigma x_\mu x^\mu\right],\nonumber\\
\psi^{\dagger}({\bf x},x_0)
=\sum_{\pm s}\int \frac{d^3 k}{\left(2\pi\right)^3}\sqrt{\frac{m}{k_0}}
\left[b^{\dagger}_s({\bf k}) u^{\dagger}_s({\bf k}) e^{-i k_\mu x^\mu}
+d_s({\bf k}) v^{\dagger}_s({\bf k}) e^{i k_\mu x^\mu}\right]
\exp\left[-\frac{1}{4}\theta k_\mu k^\mu\right]\exp\left[-\frac{1}{4}\sigma x_\mu x^\mu\right],\nonumber\\
\label{fermion_field_3+1}
\end{eqnarray}
where the coefficients of the quantized fields fulfil as usual the anti-commutation relations

\begin{equation}
\{b_s \left({\bf k}\right), b_{s^{\prime}}^{\dagger}\left(\bf k^{\prime}\right)\}
=\delta^3\left({\bf k-\bf k^{\prime}}\right)\delta_{s,s^{\prime}}\quad,\quad
\{d_s \left({\bf k}\right), d_{s^{\prime}}^{\dagger}\left(\bf k^{\prime}\right)\}
=\delta^3\left({\bf k-\bf k^{\prime}}\right)\delta_{s,s^{\prime}}.
\end{equation}
As usual there is a factor $\left[k\!\!\!/+m\right]$ with respect the calculation for the scalar field

\begin{eqnarray}
G(z)=\langle 0|T[\psi(z)\bar \psi(0)]|0 \rangle
=i\int \frac{d^4 k}{(2\pi)^3}\frac{e^{-ikz}\left[k\!\!\!/+m\right]}{k^2+m^2+i\epsilon}
\exp\left[-\frac{1}{2}\theta k^2-\frac{1}{4}\sigma z^2\right],
\label{propagator_position_space}
\end{eqnarray}
where $\bar \psi(x)=\psi^{\dagger}(x)\gamma^0 $ and $k\!\!\!/=\gamma^\mu k_\mu$.
The corresponding propagator in momentum space reads

\begin{eqnarray}
G(k)=\frac{i}{\pi^2\sigma^2}\int \frac{d^4 k^{\prime}}{(2\pi)^4}
\frac{e^{-\frac{(k-k^{\prime})^2}{\sigma}}}{k\!\!\!/^{\prime}+m+i\epsilon}
e^{-\frac{1}{2}\theta k^{\prime 2}}.
\end{eqnarray}
If the coordinates are now Wick rotated back, the propagator in Minkowski space reads

\begin{equation}
G(k)=\frac{i}{\pi^2\sigma^2}\int \frac{d^4 k^{\prime}}{(2\pi)^4}
\frac{e^{-\frac{(k-k^{\prime})^2}{\sigma}}}{k\!\!\!/^{\prime}-m+i\epsilon}
e^{-\frac{1}{2}\theta k^{\prime 2}}.
\label{Minkowski_propagator_momentum_space}
\end{equation}
Here the IR finiteness can be shown in an analogous way of the scalar field case.
Since momentum vectors of particles are time-like or light-like,
assuming  the signature (1,-1,-1,-1) for the Minkowski metric, we maintain the damping
behaviour of our new factors within the integral of the
expression for the propagator in momentum space ($\ref{Minkowski_propagator_momentum_space}$).
However even if it is nice to display formulas in Minkowski space, our general prescription is to perform virtual momentum integrations in Euclidean spacetime, where the exponential terms have unambiguous sign. Only the
final results are, then, continued into Minkowski spacetime and the physical implications
are discussed.

\section{Summary and Discussion}

In this paper, we considered an extension of the usual Heisenberg algebra, assuming nontrivial commutation relations for
position as well as for momentum operators. As a result we have introduced a minimal length and a minimum momentum as the
smallest scales of resolution in coordinate and momentum space. To define suitable coordinates and momenta, we have extended
the approach in \cite{NCQFT}, calculating expectation values of operators on coherent states referring to two Fock spaces,
one built starting from coordinate operators and the other from momentum operators. As a result we obtained that conventional
plane waves are modulated by two factors, $e^{-\frac{1}{4}\theta p^2}$ and $e^{-\frac{1}{4}\sigma x^2}$, corresponding to
damping terms in position and momentum space. This result is in agreement with the one which was found in \cite{GUPM}. As a
remark we stress that these modulations are physically consistent, since the modified plane waves become square integrable
functions and reliably describe physical states. After this investigation, we have considered the consequences for quantum
field theory, which now is endowed with two cutoffs, an UV cutoff and an IR one. We derived the propagator, which shows
regularity at higher momenta and at lower momenta in the massless case, too. Finally we showed that this property holds both,
for the scalar and the fermionic field. In the case of  a (3+1)-dimensional space-time, Lorentz invariance of the plane waves can be maintained only, if the noncommutativity parameters
referring to the two planes are assumed to be equal: $\theta_1=\theta_2$ and $\sigma_1=\sigma_2$.
With respect to this, we think that the appearance of UV as well as IR cutoffs will deserve further investigations to make
compelling predictions for current research activities in phenomenological and experimental frameworks.
For instance, IR Effects on Quantum Mechanics are currently under investigation \cite{Mirza:2009mh}. It would also be very interesting to study within our approach the phenomenological
consequences which are so far just explored in other formulations of noncommutative geometry like that based on the use of the star product \cite{PhenomenologicalConsequencesNCG}.\\
In contrast to the star product approach to noncommutative geometry
the coherent state approach implies no modification of the interaction
structure between fields. Therefore it changes the kinematics of quantum
mechanics and thus of quantum field theory, but not the dynamics. This
is related to the fact that the extended algebra has to be interpreted
as a modification of the properties of the operators $\hat q^i$ and $\hat p^i$
referring to the state of a particle. In such a way we can still define in a quasi-classical way the concept of coordinate and momentum. On the contrary, in the star product approach the
noncommutative coordinates modify the structure of the product among functions, a fact that has
 an influence on interaction terms (see  \cite{ChangingInteractionStructureNCG} for
example). Since this is not the case in the coherent state approach, we have that
 within the Feynman rules just the propagator changes but the vertices
remain completely unchanged. Because of this, we conclude that no UV/IR mixing occurs in the sense
of \cite{Minwalla:1999px} according to our scenario. This conclusion is supported by analogous arguments in \cite{NCQFT} for the specific case of noncommutative coordinates and commuting momenta.

\textit{Acknowledgement}: M. K. would like to thank the Messer Stiftung for financial support. P.N. is supported by the
Helmholtz International Center for FAIR within the framework of the LOEWE program (Landesoffensive zur Entwicklung
Wissenschaftlich-\"{O}konomischer Exzellenz) launched by the State of Hesse.

redpajama_set_name: RedPajamaArXiv


text:
Cape Wolfe är en udde i Kanada.   Den ligger i provinsen Prince Edward Island, i den östra delen av landet,  km öster om huvudstaden Ottawa.

Terrängen inåt land är platt. Havet är nära Cape Wolfe åt nordväst. Den högsta punkten i närheten är  meter över havet,  km öster om Cape Wolfe. Runt Cape Wolfe är det mycket glesbefolkat, med  invånare per kvadratkilometer.. Närmaste större samhälle är O'Leary,  km öster om Cape Wolfe.

Omgivningarna runt Cape Wolfe är en mosaik av jordbruksmark och naturlig växtlighet.  Inlandsklimat råder i trakten. Årsmedeltemperaturen i trakten är  °C. Den varmaste månaden är augusti, då medeltemperaturen är  °C, och den kallaste är februari, med  °C. Genomsnittlig årsnederbörd är  millimeter. Den regnigaste månaden är december, med i genomsnitt  mm nederbörd, och den torraste är maj, med  mm nederbörd.

Kommentarer

Källor

Halvöar i Prince Edward Island
Kontrollbehov inkommande wikilänkar

redpajama_set_name: RedPajamaWikipedia


text:
Database Snapshots
This topic describes the database snapshot feature, which was new in Microsoft SQL Server 2005. Database snapshots are available only in SQL Server 2005 Enterprise Edition and later versions. All recovery models support database snapshots.
A database snapshot is a read-only, static view of a database (the source database). Multiple snapshots can exist on a source database and always reside on the same server instance as the database. Each database snapshot is transactionally consistent with the source database as of the moment of the snapshot's creation. A snapshot persists until it is explicitly dropped by the database owner.
Unlike default behavior for user databases, a database snapshot is created with the ALLOW_SNAPSHOT_ISOLATION database option set ON regardless of the setting of this option on the primary database or the model system database.
Snapshots can be used for reporting purposes. Also, in the event of a user error on a source database, you can revert the source database to the state it was in when the snapshot was created. Data loss is confined to updates to the database since the snapshot's creation.
Reverting does not work in an offline or corrupted database. Therefore, taking regular backups and testing your restore plan are necessary to protect a database.
Database snapshots are unrelated to snapshot backups, snapshot isolation of transactions, or snapshot replication.
How Database Snapshots Work
Provides a conceptual overview of the technology of database snapshots.
Typical Uses of Database Snapshots
Presents a variety of situations in which database snapshots are useful.
Limitations and Requirements of Database Snapshots
Discusses the impact of database snapshots on the source database and the system environment, as well as limitations on the snapshots themselves.
Creating a Database Snapshot
Considers some best practices for creating database snapshots and provides a link to a description of how to create them.
Understanding Sparse File Sizes in Database Snapshots
Discusses the sparse files created for database snapshots and how to view their actual and maximum sizes.
Reverting to a Database Snapshot
Discusses how reverting to a snapshot works and the restrictions on revert operations, and provides a link to a description of how to revert.
Dropping a Database Snapshot
Discusses deleting snapshots and provides a link to a description of how to delete them.


Darrell Heinz has helped 2 people.
Darrell Heinz has written 1 review.
Waxspace provided me best VPS service at affordable price since the first day when I switched my website to their servers. Once or twice I've had contact to their technical support team both times they have responded me and resolved my issues with high priority. I am looking forward to add more my online business on their servers. Thanks a lot.


Clicking the close application button does not always close the settings application.
The bug occurs when I open the settings application and navigate to the advanced tab before trying to close the settings application, following these steps recreates the issue every time. The close button works properly on other tabs.


Pamela Approved by the ABOTA National Board as a New Member!
Pamela Was Confirmed By The National Board in Lake Tahoe as a new member of the Orange County Chapter of ABOTA! We are so proud of her accomplishments!


text:
March 2020Reviews
Reviews – March 2020
Tramps like us
Directed by Scott Graham
Certificate tbc, 77 minutes
Released 13 March
Finnie (Mark Stanley) hates his job in a fish factory in Fraserburgh, Aberdeenshire. He and his wife Katie (Amy Manson) have Springsteen's legend 'Born to Run' tattooed on his chest and her ankle, but as he says to her: 'We never did run very far, did we?' This is a story about regret and longing, being born somewhere unremarkable and not getting out. And it's also about family and relationships.
Finnie's older son Kid (Anders Hayward) has just got his girlfriend Kelly (Marli Siu) pregnant, isn't talking to her and has been dumped by her. Kid throws a wobbly at work, in the same plant as Finnie, and loses his job. When Finnie comes in from work and Katie gives him the come on, he just wants to escape into the shower because he stinks of fish. His broken-down car won't start. Out of nowhere, he borrows Kid's car keys and takes his son's car out for a spot of illegal nighttime street racing, something he used to do when younger. At the local leisure centre where she works, Kelly gets into the car thinking Kid is driving…
The Scottish writer director Scott Graham brings his trademark intensity to this low budget drama, extracting consistently fantastic and very believable performances from his cast. Once Finnie gets into his son's car, things suddenly rev into incredible driving and racing sequences, a headlit wonderland well above the speed limit, culminating in a magnificent, full throttle race along a harbour's edge where huge waves spray over a warehouse wall to crash down onto the windscreen, momentarily obliterating our vision.
Is this just a joyride though? Or will anyone run? Is there a road to freedom, or do they all lead to the entrapment of their small town existence? After taking place mostly in the dark, the final scenes in daylight suggest that something is worth salvaging. The pedal-to-the-floor lifestyle is exhilarating while it lasts but there's something bigger out there beyond that.
Jeremy Clarke is a film critic
Green Bible wisdom
Ruth Valerio
This book was commissioned by the Archbishop of Canterbury as his Lent book for 2020. Ruth Valerio is a theologian and environmentalist working for the charity Tearfund as Global Advocacy and Influencing Director. She has been involved in global development for many years and uses a wide range of information in her book, as she investigates the current environmental situation.
Cleverly, Valerio uses the days of creation (Genesis) to consider seven themes: light; water; land; the seasons; other creatures; humankind; Sabbath rest and resurrection hope. She relates these themes to environmental, ethical and social concerns, including key factors of present environmental debates. The book provides one chapter each week, to be read when preparing for Easter. It could be used equally by Lent groups or by individuals.
Following each chapter, questions are provided which relate closely to the section's theme. For most chapters, there are video links – to an interview with an environmental expert, for example – which makes the book particularly valuable for study groups.
Valerio's personal anecdotes demonstrate her broad experience. They are non-sentimental and always relevant. The book also offers examples from others around the world who are concerned about creation, highlighting the practical action they've taken to address those concerns. This kept my interest, and informed me further about the range of environmental protection activities being undertaken. The examples also challenged any apathy I had due to lack of awareness of, or insight into, the needs of others.
Prayers, at the end of each chapter, serve as a conclusion to Valerio's explanations. Her use of the scriptures is superb. Many texts are included in full, which makes the book truly user friendly.
The book's extensive bibliography includes a long list of websites, and there are 18 pages of notes, all adding great value. Resources based on Valerio's book are available from www.churchofengland.org/livelent. Saying Yes to Life makes a significant contribution to popular Christian environmental studies. Intelligently crafted, it will continue to be relevant long after Lent 2020.
Tony Shepherd studies theology at St John's College, Durham
On Christians in war
One Word of Truth: The Cold War Memoir of Michael Bourdeaux and Keston College
Michael Bourdeaux
Darton, Longman and Todd
£19.99 (hardback)
This is book needed to be written but I would not recommend it to many. It covers the the different responses of Christians to Cold War challenges. Events have moved on considerably, yet it is important that lessons are learned and remembered. On the other hand, the author goes into a level of detail that at times is tedious.
The book is a memoir of the life of an Anglican priest and of the college he founded. Surprising events, in which he sees the hand of God, led Michael Bourdeaux to learn Russian, study in the country for a year, and then take up the cause of persecuted Christians in Russia and in other countries in the Soviet bloc. Bourdeaux sees himself as a researcher, a recorder of facts, and declares himself uninvolved in politics. However, because facts were disputed in that situation, and because Russian Christians were divided in their response, there is, in my view, a clear political element to the work of the college and its founder. This is to be seen in his heavy criticism of Russian Orthodox Church leadership and also of the ecumenical position adopted by both the World Council of Churches and the British Council of Churches at that time.
This is a book for historians but it has contemporary relevance. There are still places where Christians have to decide how to respond to governments that restrict them. We also have to decide how best to exercise our calling to support those fellow Christians. These are not straightforward decisions.
John Waller is a retired church minister living in Hythe, Kent
The cricketing bishop's story
David Sheppard: Batting for the poor
Andrew Bradstock
The cover's photograph says it all: a bright-eyed David Sheppard, dog-collared and determined, bat in hand, preparing to face fast bowler Ray Lindwall. And then the clever title intrudes. This was someone who could have been one of the greatest English batsmen but instead became a Church leader who arguably changed the priorities of Anglicanism by his application of a deep, personal faith in a creative, justice-requiring, hope-sustaining,
open-hearted, inclusive God.
Andrew Bradstock, a former United Reformed Church Secretary for Church and Society, has written a clear and thorough account. It is the product of meticulous archival research and personal reminiscences. Especially effective is the final chapter, in which, 'warts and all', he shows how Sheppard's instincts and praxis evolved.
As an ordinand, and then a curate in Islington, Sheppard used his fame to speak frequently about the faith which had transformed him, while scoring many more runs in test matches than his time for practice warranted! Then he was called to the Mayflower Centre in Canning Town, which he described as 'like a second conversion, conversion to Christ in the city'. He later wrote: 'There is a divine bias to the disadvantaged and the Church needs to be much more faithful in reflecting it.' This is still as true now as then.
The book's latter half spells out in detail Sheppard's time in Liverpool, where he went after six years as Suffragan Bishop of Woolwich. These were the years of partnership with Liverpool's Catholic Archbishop Derek Worlock, a personal and political pairing: 'Like fish and chips, better together and always in the newspapers!' Bradstock sums it up: 'Over those 22 years, he would help bring lasting change in the region while redefining the Church's role in the public square.'
The book also focuses on Grace, Sheppard's wife, whose struggles with agoraphobia and cancer are dealt with most sensitively, as is her contribution to what her husband achieved. This is a splendid biography of a most remarkable man of God.
Peter Brain is a minister who served in Liverpool, and as United Reformed Church Secretary for Church and Society
The radicals' story
The Journey to the Mayflower: God's outlaws and the invention of freedom
Stephen Tomkins
Hodder and Stoughton
£20 (hardback) £14 from www.urcshop.co.uk
Look elsewhere for definitive histories of the Mayflower voyage and the establishment of Plymouth Colony and Massachusetts. Stephen Tomkins has chosen instead to tell the unlikely, disquieting, disturbing story of the soft underbelly of Tudor Protestant dissent. He journeys from the shadowy world of the Marian underground church, to the Elizabethan separatists Robert Browne, Henry Barrow, Thomas Wolsey and their successors under James I, who were holed up in grinding exile.
This is a story of abject failure: of Elizabethan Puritans to budge the Queen from her defiant Erastianism, of the separatists to convince the wider (especially Dutch) Reformed world of the justice of their cause, and of their Jacobean successors to sway James' regime towards any kind of toleration. All that was left was yet more exodus – the promised land across the Atlantic. This is indeed the journey 'to' the ironically named Speedwell (which leaked and limped back to England) and the Mayflower. There was nowhere left to go.
Cussed, argumentative, querulous,
these radicals engaged in excommunication and belligerent theological pamphlet warfare. Their congregational life included more than a few examples of what we would call bullying and sexual harassment. Yet their commitment was extraordinary. Position, power and wealth were forsaken for the ideal of the 'true', 'pure' church. They endured the foulest of London prisons and were subject to interrogation, maltreatment and, in three cases, judicial execution.
Tomkins threads together the story through the paradox that the '… most illiberal ideal of purity led eventually to the conception of religious freedom' because a 'pure' church is necessarily voluntary, which includes the freedom to leave. He traces variations on that theme in the writings of Browne, Barrow, Smyth, Helwys and Robinson. From such unpromising soil grew Congregationalism, the Baptists, and (with many more twists and turns) US Protestantism. All United Reformed Church members should read this book, if only not to recognise ourselves in the mirror it holds up.
Tomkins tells the story compellingly, wearing a good deal of learning with a beguiling lightness of touch. The book would therefore benefit from a proper bibliography and more extensive footnotes.
David Cornick is a retired church minister living in Cambridge
This article was published in the March 2020 edition of Reform
On the pilgrim way: 'January blues have hit'
Interview: It was good