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Metro

Metro, short for metropolitan, may refer to:

Geography

 * Metro (city), a city in Indonesia
 * A metropolitan area, the populated region including and surrounding an urban center

Public transport

 * Rapid transit, a passenger railway in an urban area with high capacity and frequency
 * The public transport operator of city or metropolitan area
 * The transport authority of city or metropolitan area
 * The urban rail transit system of a city or metropolitan area

Africa

 * Algiers Metro in Algiers, Algeria
 * Cairo Metro in Cairo, Egypt
 * Lagos Rail Mass Transit in Lagos, Nigeria

Asia

 * Busan Metro, Republic of Korea (South Korea)
 * Daegu Metro, Republic of Korea (South Korea)
 * Dhaka Metro, Bangladesh
 * Doha Metro, Qatar
 * Dubai Metro, United Arab Emirates (U.A.E.)
 * Kaohsiung Rapid Transit, Taiwan
 * Lahore Metro, Pakistan
 * Manila Metro Rail Transit System, the Philippines
 * New Taipei Metro, Taiwan
 * Osaka Metro, Japan
 * Seoul Metropolitan Subway, Republic of Korea (South Korea)
 * Taichung Metro, Taiwan
 * Taipei Metro, Taiwan
 * Taoyuan Metro, Taiwan
 * Tokyo Metro, Japan

China (People's Republic of)

 * Shanghai Metro, in Shanghai and Kunshan, Jiangsu
 * Guangzhou Metro
 * Changsha Metro
 * Changzhou Metro
 * Dalian Metro
 * Foshan Metro
 * Fuzhou Metro
 * Guiyang Metro
 * Hangzhou Metro
 * Harbin Metro
 * Hefei Metro
 * Hohhot Metro
 * Jinan Metro
 * Kunming Metro
 * Lanzhou Metro
 * Nanchang Metro
 * Nanjing Metro
 * Nanning Metro
 * Qingdao Metro
 * Shenyang Metro
 * Shenzhen Metro
 * Shijiazhuang Metro
 * Tianjin Metro
 * Ürümqi Metro
 * Wenzhou Metro
 * Wuhan Metro
 * Wuxi Metro
 * Xi'an Metro, in Xi'an and Xianyang
 * Xiamen Metro
 * Xuzhou Metro
 * Zhengzhou Metro
 * Wuhu metro
 * Dongguan metro
 * Taizhou Metro
 * Luoyang Metro
 * Jinhua Metro

India

 * Chennai Metro
 * Delhi Metro
 * Kolkata Metro
 * Mumbai Metro
 * Hyderabad Metro
 * Kochi Metro
 * Nagpur Metro
 * Lucknow Metro
 * Noida Metro
 * Namma Metro
 * Rapid Metro Gurgaon
 * Jaipur Metro
 * Ahmedabad Metro
 * Kanpur Metro
 * Pune Metro

Iran

 * Tehran Metro
 * Mashhad Metro
 * Shiraz Metro
 * Tabriz Metro
 * Esfahan Metro

Europe

 * Amsterdam Metro, Netherlands
 * Athens Metro, Greece
 * Brussels Metro, Belgium
 * Bucharest Metro, Romania
 * Copenhagen Metro, Denmark
 * Helsinki Metro, Finland
 * Istanbul Metro, Turkey
 * Kyiv Metro, Ukraine
 * Madrid Metro, Spain
 * Milan Metro, Italy
 * Moscow Metro, Russia
 * Oslo Metro, Norway
 * Paris Métro, France
 * Prague Metro, Czech Republic
 * Rotterdam Metro, Netherlands
 * Saint Petersburg Metro, Russia
 * Sofia Metro, Bulgaria
 * Stockholm Metro, Sweden
 * Thessaloniki Metro, Greece
 * Tyne and Wear Metro, United Kingdom
 * Warsaw Metro, Poland

North America (outside the United States)

 * Mexico City Metro, in Mexico
 * Monterrey Metro, in Nuevo León, Mexico
 * Montreal Metro, in Montreal, Quebec, Canada
 * SkyTrain (Vancouver), automated light metro in Vancouver, British Columbia, Canada
 * Toronto Transit Commission, heavy rail rapid transit system in Toronto, Ontario, Canada

United States

 * Metro (Minnesota)
 * METRORail, a light rail system in Houston
 * Muni Metro, light rail system, San Francisco, California
 * Valley Metro Rail, mainly in Phoenix, Tempe and Mesa, Arizona
 * Washington Metro, heavy rail rapid transit system in Washington, D.C., and its Maryland and Virginia suburbs
 * Metropolitan Transportation Authority, heavy rail rapid transit in New York City

Oceania

 * Metro Trains Melbourne, a metropolitan commuter train operator in Australia
 * Metro Tunnel, a metropolitan rail infrastructure project currently under construction in Melbourne, Australia
 * Metro Transport Sydney, the former operator of Sydney Light Rail and Sydney Monorail
 * Sydney Metro, in Sydney, Australia
 * Tranz Metro, former commuter rail operator, Wellington
 * Adelaide Metro, an integrated network of public transport, Adelaide, Australia
 * AT Metro, an integrated network of public transport, Auckland, New Zealand
 * Brisbane Metro, a rapid transit network of high-capacity, fully electric, bi-articulated buses, Australia
 * Metro (Christchurch), co-ordinated public transport, Christchurch, New Zealand
 * Metro Tasmania, a state-owned bus company

South America

 * Caracas Metro, is a mass rapid transit system serving Caracas, Venezuela
 * Guarenas / Guatire Metro, Venezuela
 * Lima Metro, Lima, Peru
 * Los Teques Metro, Venezuela
 * Maracaibo Metro, Venezuela
 * Valencia Metro (Venezuela)
 * Medellin Metro, Medellin, Colombia
 * Metrovia, Ecuador
 * Panama Metro, Panama City, Panama
 * Belo Horizonte Metro, Belo Horizonte, MG, Brazil
 * Federal District Metro (Brazil), Federal District, Brazil
 * Fortaleza Metro, Fortaleza, CE, Brazil
 * Porto Alegre Metro, Porto Alegre, RS, Brazil
 * Recife Metro, Recife, PE, Brazil
 * Rio de Janeiro Metro, Rio de Janeiro, RJ, Brazil
 * Salvador Metro, Salvador, BA, Brazil
 * São Paulo Metro, São Paulo, SP, Brazil
 * Teresina Metro, Teresina, PI, Brazil
 * Metro de Santiago, the second largest in South America, serving Santiago, Chile

Bus and/or rail services, and other transit authorities

 * Metrobus (disambiguation), several bus operators

Asia

 * MetroMini, Jakarta, Indonesia
 * Lahore Metrobus, Pakistan

Europe

 * Metro (Belfast), bus services, Northern Ireland
 * Metrobus (Istanbul), Turkey]
 * West Yorkshire Metro or MetroTrain, England

North America

 * Metro Transit (disambiguation), several North American bus services
 * Metro (Cincinnati)
 * Greater Portland Metro Bus, Maine
 * King County Metro, Seattle, Washington
 * Madison Metro, former name of Metro Transit in Madison, Wisconsin
 * the Los Angeles County Metropolitan Transportation Authority, public transit system branded as "Metro"
 * Mountain Metropolitan Transit, Colorado Springs, Colorado
 * Metro Tasmania, a state-owned bus company

Airports

 * Detroit Metropolitan Airport, Michigan, United States
 * Metro International Airlines, a charter and scheduled passenger subsidiary of the defunct Flying Tiger Line

Education

 * Metro Academic and Classical High School, St Louis, Missouri, United States
 * Metropolitan State University of Denver, Denver, Colorado, United States
 * Metropolitan State University, Minneapolis–Saint Paul, Minnesota, United States

Broadcasting

 * Metro Broadcast Corporation, with a variety of Hong Kong‑based Metro‑branded radio stations
 * Metro FM, a national radio station, South Africa
 * Metro Radio, a radio station, Newcastle upon Tyne, North East England
 * Metro TV (disambiguation), several television stations
 * Metro 14, on air name of CJNT-DT, a television station, Montreal, Quebec
 * Radio Metro, a radio station, Argentina
 * Metro (Poland), a Polish television channel

Film

 * Metro (1997 film), starring Eddie Murphy
 * Metro (2007 film), an Indian Hindi-language by Anurag Basu
 * Metro (2024 film), its sequel by Basu
 * Metro (2013 film), Russian film
 * Metro (2016 film), an Indian Tamil-language film
 * The Metro (film), an Indian 2011 Malayalam-language film
 * Metro Pictures, one of Metro-Goldwyn-Mayer's predecessor companies

Artists

 * Metro (Hungarian band)
 * Metro (British band), active 1976–1980
 * Metro (Serbian band), active 1981–1985; 1994–1997
 * Metrô (band), Brazilian new wave band active 1978–1988; 2002–2004; 2014; 2015–present
 * Metro Boomin (born 1993), American songwriter, known simply as Metro
 * The Metros, an English indie band active 2006–2009

Titled works

 * Metro (album), 1977 self-titled album by the British band
 * Metro (musical), a 1991 Polish musical
 * "The Metro" (song), a 1983 single by Berlin
 * "Metro", a 2006 song by The Vincent Black Shadow
 * "Metro", a song by Tokyo Jihen from the 2007 album Variety

Venues

 * Metro Chicago, a music venue, Chicago, United States
 * The Metro Theatre, a music venue, Sydney, Australia
 * Metro Nightclub, the former name of the Palace theatre, Melbourne, Australia

Magazines

 * Metro (magazine), New Zealand lifestyle magazine
 * Metro Magazine, for bus and rail transit and motorcoach operators
 * Metro Weekly, a free weekly magazine-style publication for the LGBT community of Washington, D.C.

Newspapers

 * Harian Metro, Malaysia
 * Metro (Belgian newspaper)
 * Metro (British newspaper)
 * Metro (Dutch newspaper)
 * Metro (Italian newspaper)
 * Metro (Swedish newspaper)
 * Métro (Montreal newspaper), a French-language free newspaper
 * Metro (Philadelphia newspaper)
 * Metro New York (2004–2020), merged with AM New York, renamed AM New York Metro
 * AM New York Metro (2003, as AM New York), merged with Metro New York
 * Metro Daily, Hong Kong newspaper
 * Metro Santa Cruz, former name of a newspaper in Santa Cruz, California, renamed the Santa Cruz Weekly
 * Metro Silicon Valley, a newspaper in San Jose, California
 * StarMetro (newspaper), a chain of Canadian free newspapers (formerly called Metro until April 2018)
 * Zimbabwe Metro

Media Companies

 * Metro International, a Swedish media company
 * Metro Newspapers, a newspaper company in California

Others

 * Metro (franchise), a multimedia sci-fi franchise created by Russian author Dmitry Glukhovsky, including:
 * Books
 * Metro 2033 (novel) (2005)
 * Metro 2034 (2009)
 * Metro 2035 (2015)
 * Video games
 * Metro 2033 (video game) (2010)
 * Metro: Last Light (2013)
 * Metro Exodus (2019)

 * Metro (novel), 1933 novel by Vladimir Varankin
 * Metro (typeface)
 * Board games
 * Metro (board game), a 2000 board game

Business

 * Metro (department store), a chain based in Singapore
 * Metro (supermarket, Greece)
 * Metro (supermarket, Indonesia), a real estate and property management company
 * Metro AG, Germany's largest retailer
 * MetroCentre (shopping centre), indoor shopping mall, Gateshead, England, UK
 * Metro Cash and Carry, a hypermarket chain owned by Metro AG
 * METRO Foods Trading, Cyprus
 * Metro Inc., a Canadian supermarket chain
 * Metro (restaurant chain), an Icelandic fast food chain
 * Metropolitan New York Library Council (METRO), a non-profit organization
 * Metro by T-Mobile, a prepaid wireless service provider in the United States
 * Metro Retail Stores Group, a retail company based in Mandaue, Philippines

Sports

 * Metro (wrestler), a Mexican masked luchador
 * Metro Conference, a defunct NCAA Division I athletic conference
 * Metro FC (New Zealand), association football club based in Auckland
 * Little League World Series (Metro Region), a Little League World Series United States region

Technology

 * Metro (design language), developed by Microsoft for interfaces
 * Eclipse Metro, an open source web service stack that is part of Eclipse Enterprise for Java (EE4J)
 * Metro Ethernet, an Ethernet-based computer network that covers a metropolitan area
 * Metro, former code name of the XML Paper Specification, Microsoft's document framework

Aircraft

 * Fairchild Metro, a commuter airliner

Automobiles

 * Austin Metro, a British supermini hatchback, also called MG Metro
 * Geo Metro or Chevrolet Metro, an American subcompact car, later called Chevrolet Metro
 * International Metro Van, an American step van

Other uses

 * El Metro 4, a Mexican high-ranking drug trafficker
 * Gregorio Sauceda-Gamboa, Mexican drug lord nicknamed Metro 2
 * Samuel Flores Borrego (1972–2011), Mexican drug lord nicknamed Metro 3
 * Metro (Oregon regional government)
 * Metro Vancouver Regional District
 * Las Vegas Metropolitan Police Department

title: Metro


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Evangelical United Brethren Church

The Evangelical United Brethren Church (EUB) was a North American Protestant denomination from 1946 to 1968 with Arminian theology, roots in the Mennonite and German Reformed communities, and close ties to Methodism. It was formed by the merger of a majority of the congregations of the Evangelical Church founded by Jacob Albright (excluding those that became the Evangelical Church of North America, along with the Evangelical Congregational Church) and the Church of the United Brethren in Christ (New Constitution) (as opposed to the Church of the United Brethren in Christ (Old Constitution), still extant without the parenthetical). The United Brethren and the Evangelical Association had considered merging off and on since the early 19th century because of their common emphasis on holiness and evangelism and their common German heritage.

In 1968, the United States section of the EUB merged with the Methodist Church to form the United Methodist Church, while the Canadian section joined the United Church of Canada.

History
United Brethren in Christ was an American religious denomination which originated in the last part of the 18th century. Though not formally organized until 1800, the roots of the church reach back to 1767. In May of that year, a "Great Meeting" (part of an interdenominational revival movement) was held at a barn belonging to Isaac Long in Lancaster, Pennsylvania, near the village of Oregon. Martin Boehm (1725–1812), a Mennonite preacher, spoke of his becoming a Christian through crying out to God while plowing in the field. Philip William Otterbein (1726–1813), a German Reformed pastor at York, Pennsylvania, (and later of Baltimore), left his seat, embraced Boehm and said to him, "Wir sind Brüder!" (we are brethren!). The followers of Boehm and Otterbein formed a loose movement for many years. It spread to include German-speaking churches supplemented later by English-speaking followers in Pennsylvania, Virginia, Maryland, and later spread west into Ohio.

Pastor Otterbein, returned to Baltimore to the newly organized German Reformed congregation and built a brick Georgian-styled church with a bell tower southwest of Baltimore Town at South Sharp and West Conway Streets, later named Old Otterbein United Methodist Church, which had continued for two centuries in the Brethren denomination before their 1968 merger with the Methodists and is now the oldest active church and building in the city, whose name was also given to the surrounding restored historic neighborhood.

By 1800, they began a yearly conference. Thirteen ministers attended the first conference at the home of Peter Kemp in Frederick, Maryland. At that conference in 1800, they adopted a name, the United Brethren in Christ, and elected Boehm and Otterbein as bishops of the conference. The United Brethren Church claims this organization in 1800 as the first denomination to actually begin in the United States. The first delegated general conference met at Mount Pleasant, Pennsylvania, in 1815, and adopted a confession of faith (similar to one written by Otterbein in 1789), rules of order and a book of discipline, which were revised in 1885–1889, when women were first admitted to ordination.

The ecclesiastical polity of the church was Wesleyan and its theology Arminian. It variously practised believer's baptism or infant baptism. Bishops were elected for four years.

The United Brethren took a strong stand against slavery, beginning around 1820. After 1837, slave owners were no longer allowed to remain as members of the United Brethren Church. The Evangelical United Brethren churches sustained a strong fellowship with Nazarene (believing) Jews. In 1853, the Home, Frontier, and Foreign Missionary Society was organized. Expansion occurred into the western United States, but the church's stance against slavery limited expansion to the south.

By 1889, the United Brethren had grown to over 200,000 members with six bishops. In that same year they experienced a division. Denominational leaders desired to make three changes: to give local conferences proportional representation at the General Conference; to allow laymen to serve as delegates to General Conference; and to allow United Brethren members to hold membership in secret societies such as the Freemasons. The denominational leadership made these changes, but the minority felt the changes violated the constitution because they were not made by the majority vote of all United Brethren members. One of the bishops, Milton Wright (the father of aviation pioneers Wilbur Wright and Orville Wright), disagreed with the actions of the majority. Bishop Wright and other conference delegates left the meeting and resumed the session elsewhere. They believed that the other delegates had violated the constitution (and, in effect, withdrawn from the denomination), and deemed themselves to be the true United Brethren Church. Therefore, the body initially known as the United Brethren in Christ of the Old Constitution, now called the Church of the United Brethren in Christ.

The majority branch had 3,732 organizations in 1906 with a total membership of 274,649. The denomination merged with the Evangelical Church in 1946 to form a new denomination known as the Evangelical United Brethren Church (EUB). This in turn merged in 1968 with The Methodist Church to form the United Methodist Church (UMC).

Missionary and educational establishments
This body carried on missions in West Africa (since 1855), Japan, China, the Philippines and Puerto Rico. It had a publishing house (1834) and two seminaries Bonebrake Theological Seminary (1871) at Dayton, Ohio, and Evangelical Theological Seminary (1873) in Naperville, IL. The EUB supported several colleges and universities including Otterbein University (1847) at Westerville, Ohio; Plainfield College (now North Central College) (1861) at Naperville, IL; Westfield College (1865) at Westfield, Illinois; Leander Clark College (1857) at Toledo, Iowa; York College (1890) at York, Nebraska; Western Union College (1900), renamed Westmar College (1948) at Le Mars, Iowa; Philomath College (1867) at Philomath, Oregon; Lebanon Valley College (1867) at Annville, Pennsylvania; Campbell College (1864) at Holton, Kansas, and Indiana Central College (later Indiana Central University and now the University of Indianapolis) (1907) at Indianapolis, Indiana. In 1946, with cooperation of three other denominations, it formed the United Andean Indian Mission, an agency that sent missionaries to Ecuador.

Outside the United States
The EUB congregations in Canada joined the United Church of Canada in 1968, the largest Protestant denomination in Canada formed in 1925 by Presbyterians (70% came in), Methodists, and Congregationalists. In the Philippines, the EUB congregations joined the Philippine Methodist Church, Christian Church (Disciples), Presbyterian Church, Congregational Church, Iglesia Evangelica Unida de Cristo, Iglesia Evangelica Nacional and some segments of the Iglesia Evangelica Metodista En Las Islas Filipinas (IEMELIF) to form the United Church of Christ in the Philippines Lewis, Charles (May 14, 2011).

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Evangelical United Brethren Church


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Tiger Electronics

Tiger Electronics Ltd. (also known as Tiger and Tiger Toys) is an American toy manufacturer best known for its handheld electronic games, the Furby, the Talkboy, Giga Pets, the 2-XL robot, and audio games such as Brain Warp and the Brain Shift. When it was an independent company, Tiger Electronics Inc., its headquarters were in Vernon Hills, Illinois. It has been a subsidiary of Hasbro since 1998.

History
Gerald Rissman, Randy Rissman and Arnold Rissman founded the company in June 1978. It started with low-tech items like phonographs, then began developing handheld electronic games and educational toys. Prominent among these was the 2-XL Robot in 1978, and K28, Tiger's Talking Learning Computer (1984) which was sold worldwide by Kmart and other chain stores. Tiger also achieved success with many simple handheld electronics games like Electronic Bowling and titles based on licenses, such as RoboCop, Terminator, and Spider-Man. An early 1990s hit was the variable-speed portable cassette player and recorder, the Talkboy (first seen in the 1992 movie Home Alone 2: Lost in New York), followed by the Brain family of games which include games like Brain Bash, Brain Warp, and Brain Shift. It also licensed the Lazer Tag brand from its inventors, Shoot the Moon Products, which was born from the remnants of the Worlds of Wonder company.

The company's cash cow through much of the 1990s was their line of licensed handheld LCD games. In a 1993 feature on these games, GamePro attributed their success to the following three factors:
 * Tiger's effective licensing. Director of marketing Tamara Lebovitz stated, "We read all the magazines and talk to all the studios to keep on the cutting edge of what's hot with kids." As a fairly small company at the time, Tiger was able to pursue desirable licenses quickly and aggressively. This allowed them to release licensed games while the properties they were licensed from were still at the peak of their popularity.
 * The low price per game. Tiger handheld games sold for roughly $20 each. By comparison, most handheld games of the time cost over $30, and required a separately sold system (an additional $50 or more) to play it on.
 * The simplistic, addictive gameplay of the games. While older gamers tended to find Tiger handheld games one-dimensional and boring, for kids aged five to twelve years old, their simple and easy-to-learn mechanics were more appealing than other video games of the time, which were often frustratingly difficult and dauntingly complex for younger children.

In the fall of 1994, Tiger introduced a specialized line of their handheld LCD games, called Tiger Barcodzz. These were barcode games which read any barcode and used it to generate stats for the player character. The line was a major success in Japan, where there were even reality shows based around gamers competing to find the best barcodes to defeat other players. Tiger produced a version of Lights Out around 1995. In 1997 it produced a quaint fishing game called Fishing Championship, in the shape of a reduced fishing rod. Another 1990s creation was Skip-It.

In 1995, Tiger acquired the Texas Instruments toy division. Tiger agreed to manufacture and market electronic toys for Hasbro and Sega.

Merging with Hasbro
Tiger Electronics has been part of the Hasbro toy company since 1998. Hasbro paid approximately $335 million for the acquisition. In 2000, Tiger was licensed to provide a variety of electronics with the Yahoo! brand name, including digital cameras, webcams, and a "Hits Downloader" that made music from the Internet (mp3s, etc.) accessible through Tiger's assorted "HitClips" players. Tiger also produces the long-lasting iDog Interactive Music Companion, the ZoomBox—a portable 3-in-1 home entertainment projector that will play DVDs, CDs and connects to most gaming systems—, the VideoNow personal video player, the VCamNow digital camcorder, the ChatNow line of kid-oriented two-way radios and the TVNow, a personal handheld DVR player. They released an electronic tabletop version of Who Wants to Be a Millionaire? with voice recordings by host Chris Tarrant. Tiger also released an electronic version of The Weakest Link with voice recordings by Anne Robinson.

Standalone handhelds
Tiger is most well known for their low-end handheld electronic game systems with LCD screens. Each unit contains a fixed image printed onto the handheld that can be seen through the screen. Static images then light up individually in front of the background that represent characters and objects, similar to numbers on a calculator or digital clock. In addition to putting out some of its own games, Tiger was able to secure licenses from many of the time's top selling companies to sell their own versions of games such as Capcom's Street Fighter II, Sega's Sonic 3D Blast, and Konami's Castlevania II: Simon's Quest. Later, Tiger introduced what they called "wrist games". These combined a digital watch with a scaled-down version of a Tiger handheld game.

In 1995, Tiger introduced Super Data Blasters, a line of sports-themed handhelds. Each featured the contemporary statistics for players in a specific sport, the ability to record new sports statistics, a built-in electronic game for the sport, and typical electronic organizer features such as an address book and calculator.

In 1998, Tiger released 99X Games, a series of handhelds fitted with a dot-matrix screen, allowing a wide variety of backgrounds and different gameplay for a single game. Although running a software program stored in ROM, those systems were dedicated consoles, similarly to the plug-and-play TV games of the 2000s decade. Two systems running the same game could be linked with the included cable to allow two players to challenge each other.

Cartridge-based handhelds
Tiger made three notable cartridge-based systems. The first was the Quiz Wiz, a highly popular interactive quiz game system. Players inserted a cartridge and played using the corresponding quiz book. The second was the R-Zone. It employed red LCD cartridges, much like Nintendo's Virtual Boy, which were projected via backlight onto a reflective screen that covered one of the player's eyes. The third was the Game.com handheld system, which was meant to compete with Nintendo's Game Boy and Game Boy Color, as well as Sega's Game Gear and Genesis Nomad, and boasted such novel features as a touchscreen and limited Internet connectivity. However, the R-Zone and Game.com were commercial failures and garnered a negative reception.

Furby
Hasbro, previously shy of high-tech toys, was interested in the development of the Furby. With Hasbro's support, Tiger was able to rush through the development process and get the Furby on the shelves for the 1998 holiday season, during which it was a runaway hit — the "it" toy of the 1998 and 1999 seasons. The continuing development of Furby-type technology led to the release of the FurReal line of toys in 2003, the more modern iteration of the Furby toyline in 2012, and also the high-tech Furby Connect in 2016. Now there is a new 2023 furby with voice commands.

Brain Family
From 1994–1999, Tiger invented the Brain Family, which are a line of electronic handheld audio games. In 1994, Tiger released the Brain Bash. It has four inner purple buttons and four yellow buttons outside the unit. It features five game modes. Game One is called Touch Command, where the electronic voice issues a command like "one touch one" and the corresponding player has to press purple one and yellow one.

In 1996, Tiger released the Brain Warp. This game is a spherical unit that has six colored knobs sticking out. There were three different revisions of the circuit board of Brain Warp resulting in audio changes and pitch differences. Two revisions were made in a blue base. Revision 2.0 has a different hidden sound sampling mode to the first revision. When Hasbro re-released Brain Warp in 2002, they took the programming from Revision 2.0 and placed it on a new circuit board with an enhanced speaker which reduced the loudness of the device. This game is very similar to Bop It. A voice that was recorded for the game says a color or a number, or a sequence of colors or numbers, or both depending on the game selected, and the correct knob must be shown facing upwards. In 1997, a Star Wars version called Death Star Escape was released. The game order is different and comes with six Star Wars characters.

In 1998, Tiger released Brain Shift. This game has six colored LED lights. It is known for its distinctive low pitched "Orange!" voice which is heard on the last color of a pattern in Stick Shift and in Memory Shift and Who Shifts It? The player has to use the stick shift to follow the voice commands. There is a memory game, and both Brain Shift and Brain Warp have a code buster game where the player has to find a certain number of colors in sixty seconds. Some Brain Shift game units had a bad chip in the game which causes the game to mess up the audio on low batteries, and in rare cases, the voice in the game will start counting, going through a list of numbers and skipping a few.

Making toys and games for other brands
The company became one of the most prominent producers of electronic toys based on a wide variety of licenses, including Star Trek, Star Wars, Barney & Friends, Arthur, Teletubbies, Winnie the Pooh, Franklin, Neopets, Jeopardy!, Wheel of Fortune, Weakest Link, Who Wants to Be a Millionaire?, Batman Returns, The Lost World: Jurassic Park, and Sonic the Hedgehog.

In 1996, Tiger produced replicas of the Turbo Man doll, which was featured in the 1996 holiday comedy Jingle All the Way. It retained most of the features of the film version, including the disk shooter, boomerang accessory, light and sound jetpack, and a voice box. Despite being advertised as having five phrases in the movie, the actual toy only possessed four.

Boogey Ball
In 1999, Tiger Electronics released an electronic LED light game called Boogey Ball. There were 2 versions of the game released. The first version was buggy and it had issues playing several games (games 2, 3, and 5). In games 2 and 3, the player failed automatically sometime after 20 seconds due to the speed of the red light being impossible to stay away from. In game 5, the light patterns went in different directions and it was harder to play. Also, the game had a loud voice but quiet background music. In version 2.0, all the issues were fixed in the audio and game modes. The gameplay is similar to Pac-Man, in that the player maneuvers a green LED light through a maze of 30 LED lights and has to either avoid (ditch) the red light or catch (snag) the yellow light. The game was known for its Austin Powers and Melle Mel style voices; the electronic voice would often say "baby". When the game was first turned on, it said "Oh you turned me on baby, let's boogey!" When the player failed, the game said "Oh drat!". This game was also published by Hasbro. The game also suffered from a glitch: it would become stuck, playing every sound from the game, and pressing the power button would not turn the game off. The loud crackling over the top of the rapidfire used to scare people. The cause for this glitch is unknown but it might be trying to go through an automatic test mode.

Harry Potter Challenge Wand
In 2001, Tiger Electronics released a memory game called the Harry Potter Magic Spell Challenge, simply known as the 'Challenge Wand', on which the game sees the player up against an Evil Wizard as he casts a spell on the wand unit which the player must memorize in order. The first game in the unit is called 'Compete Against an Evil Wizard'. In this game, the Evil Wizard says "Try and stop this!" (or "me") and starts to cast a spell on the wand. The game has 4 auditory command sounds and 2 vocal commands "Wingardium" which requires the player to tilt the wand down 90 degrees, and "Leviosa!" which requires the player to tilt it up 90 degrees. The game has 8 levels; at each level, the patterns get longer and longer. If the player makes a mistake trying to memorize the pattern, the Evil Wizard will say something like "Now the pain begins!" or "No match for me!" If the player makes 3 mistakes in a round, the game is over and the Evil Wizard will say, "Your powers are now mine!". The announcer will announce how many rounds the player has completed. The game also includes a Simon-style game called "2 Wizards are Better Than One" which involves sticking with one pattern and adding an extra command on it each time. Game 3 has different music to the other 2 games on which requires 2 wands. One player makes a pattern and sends it to the other wand and then the other player has to repeat it back.

Test modes/demo modes
Tiger Electronics and Hasbro are known to include a hidden test mode (also known as a demo mode) in all their electronic games. These test modes signal either a sine wave or a square wave tone usually at 1000hz as a way of testing the speaker and then play through all of the sounds that are pre-programmed in the device either manually (by pushing a button), or automatic (playing every sound by itself). Games like Brain Warp, Brain Shift, Boogey Ball, and Brain Bash have these test modes, as do tabletop games (such as Who Wants to Be a Millionaire?).

title: Tiger Electronics


text:
Bleem!

Bleem! (styled as bleem!) is a commercial PlayStation emulator released by the '''Bleem! Company''' in 1999 for IBM-compatible PCs using Microsoft Windows and the Dreamcast. It is notable for being one of the few commercial software emulators to be aggressively marketed during the emulated console's lifetime, and was the center of multiple controversial lawsuits.

History
Bleem! was a PlayStation emulator designed to allow people to play original PlayStation games on Windows 95 or 98 or on the Dreamcast (the Dreamcast version was called Bleemcast!). It was released in March 1999. The company that developed and commercialized Bleem! initially consisted of just two people, David Herpolsheimer (president) and Randy Linden, but in the commercial phase included Will Kempe, Scott Karol, Sean Kauppinen, Bryan Stokes, James Sinclair, and Paul Chen, later of Rovio Entertainment.

Context
To allow for full-speed emulation on lower-end computers of what was at the time a current generation console, the authors coded Bleem! in assembly. This allowed them to create precise optimizations. Unlike Connectix's commercial Virtual Game Station, it made use of a PC's 3D graphics hardware for rendering, allowing for enhanced resolutions and filtered textures not possible in real-time software rendering of the time.

Bleem! used low-level memory emulation and other real-mode technology. It did not function on operating systems using the Windows NT kernel, including Windows 2000. In fact, Bleem!'s statement at the time was that Bleem! would never support running on Windows NT-based systems, as Windows 98 was the dominant operating system at the time.

Sony, despite having lost its case with Connectix, continued to pursue legal action against Bleem!. Bleem!, financially unable to defend itself, was forced to go out of business.

As of 2005, two members of the team were working for Sony: Randy Linden was working for SCEA on porting titles and looking at the possibility of emulation of previous generation titles for the next PlayStation, and Sean Kauppinen was promoting EverQuest II and Star Wars Galaxies for Sony Online Entertainment.

Copy protection
To combat redistribution of the small downloadable emulator, the user had to buy the Bleem!-CD, containing about 35 MB of data: a DirectX distributable and the actual version of Bleem! available at the time of the CD's printing. The rest of the CD was only for copy protection and was impossible to copy by conventional means; nevertheless, the copy protection was cracked within two weeks of the release.

Further updates to the emulator were free until the company ceased operation several years later.

Bleemcast!
Bleemcast! is an independently developed commercial emulator by Bleem! that allows one to load and play PlayStation discs on the Sega Dreamcast. It is compatible with most Dreamcast controllers and steering wheels, and leverages the Dreamcast's superior processing power for enhanced graphics. It was created by using the MIL-CD security hole found in the Dreamcast BIOS.

History
Originally, Bleem! was planning to have the disc able to run any PlayStation game on the Dreamcast, but due to technical difficulties, they developed the concept of the "Bleempak", in which the software would boot only 100 specific games each. New Bleempaks would have to be purchased if one game was not available to boot in a Bleempak. Due to the Dreamcast controller's fewer buttons compared to the PlayStation, there were plans to release a Bleem! controller somewhat similarly designed to the PlayStation controller, and a PlayStation-to-Dreamcast controller adapter, which would allow one to use a PlayStation controller on the Dreamcast. As technical difficulties grew further, all these ideas were scrapped, with no "Bleempak" and no hardware releases.

However, they managed to release individual Bleemcast! bootdiscs for three popular games: Gran Turismo 2, Tekken 3, and Metal Gear Solid. WWF SmackDown! was also being planned for a release, but was not completed, while a couple of screenshots of Final Fantasy IX were surfacing during this time, but was never announced as a planned release. As promised from the beginning, the games ran in a 640×480 resolution, as opposed to the PS1's 320×240 resolution, and featured anti-aliasing and bilinear filtering. This drastically improved the games' graphics, but also brought out some graphical imperfections that were originally hidden in the lower resolution.

Sony lawsuit
Two days after Bleem! started taking preorders for their emulator, Sony filed suit over violations of copyright. Sony had accused Bleem! of engaging in unfair competition by allowing PlayStation BIOSs to be used on a personal computer as this would ultimately damage Sony's sales of the PlayStation. The Judge had rejected the notion, and issued a protective order to "protect David from Goliath". Sony's second copyright allegation regarded the use of screenshots on their advertisements comparing the native PlayStation and emulated Bleem! versions. The district court had held in favor of Sony regarding the allegation and issued a preliminary injunction against Bleem!; however, Bleem! later appealed the decision providing their use of copyrighted material was protected under fair use. The appeal was successful, with the court stating that the use of screenshots of Sony's video games rather constituted comparative advertising.

In spite of the loss, the release of the Bleemcast! caused Sony to file another lawsuit accusing them of unfair competition and patent infringement regarding the use of PlayStation BIOSs on the Sega Dreamcast. This approach had become problematic for Bleem!, despite no actual court ruling against them. The main issue regarded the financial problems Bleem! had faced as they had to deal with defense costs of $1 million per patent. This had caused Bleem!'s work to decline, so that they had only managed to release three games: Metal Gear Solid, Gran Turismo 2, and Tekken 3, for the Bleemcast!. At this point, Sony had obstructed Bleem! from developing further video games for the Bleemcast! and had even threatened retailers selling these products.

The legal fees forced the company out of business and eBay auctions of some of the company's possessions were held soon after – including a huge library of worldwide game releases used for compatibility testing.

Closure of Bleem!
Although Sony ultimately did not win any of its lawsuits against them, Bleem! had to shut down when the huge court costs became too much for the small company to handle. Bleem! shut down in November 2001, the same year Sega announced that they would discontinue the Dreamcast in North America. Bleem! closed their website, with only an image on their front page showing Sonic the Hedgehog tearfully holding a flower next to a Bleem! gravestone. The image was later altered and Sonic was removed, ironically to avoid a lawsuit from Sega. Sega themselves had also been indirectly involved in the obstruction of Bleemcast! by the way of the removal of support for the unpopular (for Sega's intended use) MIL-CD format on later Dreamcast models.

Beta leak
A beta build of Bleemcast! was eventually leaked. Even though it was buggy and incomplete, it would run some PlayStation games, though not all the games it ran would be playable. Using this beta, hackers were able to create "Bleemed games" – discs of a PlayStation title with the Bleemcast! emulator built in. ISO images of many of these discs were known to circulate on file-sharing networks.

After the leak, Rod Maher, one of the developers of Bleemcast!, made a public statement regarding the beta, providing some insight into the development process. He revealed that the leaked beta predated the beta that had been shown at E3 and that the leaked beta was 30% complete. The commercial Bleemcast! release was notable as the only release on the Dreamcast that had not been pirated, as it had a complex copy-protection scheme. All three Bleempaks were finally cracked and made available online in December 2009, eight years after their introduction.

Bleem Powered
On January 4, 2021, Piko Interactive announced that they had acquired the Bleem! brand name, with plans to start a retrogaming-focused online storefront titled Bleem Powered, which as of late 2022 still seems to be under development.

title: Bleem!


text:
Representation of a Lie group

In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras.

Representations
A complex representation of a group is an action by a group on a finite-dimensional vector space over the field $$\mathbb C$$. A representation of the Lie group G, acting on an n-dimensional vector space V over $$\mathbb C$$ is then a smooth group homomorphism
 * $$\Pi:G\rightarrow\operatorname{GL}(V)$$,

where $$\operatorname{GL}(V)$$ is the general linear group of all invertible linear transformations of $$V$$ under their composition. Since all n-dimensional spaces are isomorphic, the group $$\operatorname{GL}(V)$$ can be identified with the group of the invertible, complex $n\times n$ matrices, generally called $\operatorname{GL}(n;\mathbb C).$ Smoothness of the map $$\Pi$$ can be regarded as a technicality, in that any continuous homomorphism will automatically be smooth.

We can alternatively describe a representation of a Lie group $$G$$ as a linear action of $$G$$ on a vector space $$V$$. Notationally, we would then write $$g\cdot v$$ in place of $$\Pi(g)v$$ for the way a group element $$g\in G$$ acts on the vector $$v\in V$$.

A typical example in which representations arise in physics would be the study of a linear partial differential equation having symmetry group $$G$$. Although the individual solutions of the equation may not be invariant under the action of $$G$$, the space $$V$$ of all solutions is invariant under the action of $$G$$. Thus, $$V$$ constitutes a representation of $$G$$. See the example of SO(3), discussed below.

Basic definitions
If the homomorphism $$\Pi$$ is injective (i.e., a monomorphism), the representation is said to be faithful.

If a basis for the complex vector space V is chosen, the representation can be expressed as a homomorphism into general linear group $$\operatorname{GL}(n;\mathbb C)$$. This is known as a matrix representation. Two representations of G on vector spaces V, W are equivalent if they have the same matrix representations with respect to some choices of bases for V and W.

Given a representation $$\Pi:G\rightarrow\operatorname{GL}(V)$$, we say that a subspace W of V is an invariant subspace if $$\Pi(g)w\in W$$ for all $$g\in G$$ and $$w\in W$$. The representation is said to be irreducible if the only invariant subspaces of V are the zero space and V itself. For certain types of Lie groups, namely compact and semisimple groups, every finite-dimensional representation decomposes as a direct sum of irreducible representations, a property known as complete reducibility. For such groups, a typical goal of representation theory is to classify all finite-dimensional irreducible representations of the given group, up to isomorphism. (See the Classification section below.)

A unitary representation on a finite-dimensional inner product space is defined in the same way, except that $$\Pi$$ is required to map into the group of unitary operators. If G is a compact Lie group, every finite-dimensional representation is equivalent to a unitary one.

Lie algebra representations
Each representation of a Lie group G gives rise to a representation of its Lie algebra; this correspondence is discussed in detail in subsequent sections. See representation of Lie algebras for the Lie algebra theory.

An example: The rotation group SO(3)
In quantum mechanics, the time-independent Schrödinger equation, $$\hat{H}\psi=E\psi$$ plays an important role. In the three-dimensional case, if $$\hat{H}$$ has rotational symmetry, then the space $$V_E$$ of solutions to $$\hat{H}\psi=E\psi$$ will be invariant under the action of SO(3). Thus, $$V_E$$ will—for each fixed value of $$E$$—constitute a representation of SO(3), which is typically finite dimensional. In trying to solve $$\hat{H}\psi=E\psi$$, it helps to know what all possible finite-dimensional representations of SO(3) look like. The representation theory of SO(3) plays a key role, for example, in the mathematical analysis of the hydrogen atom.

Every standard textbook on quantum mechanics contains an analysis which essentially classifies finite-dimensional irreducible representations of SO(3), by means of its Lie algebra. (The commutation relations among the angular momentum operators are just the relations for the Lie algebra $$\mathfrak{so}(3)$$ of SO(3).) One subtlety of this analysis is that the representations of the group and the Lie algebra are not in one-to-one correspondence, a point that is critical in understanding the distinction between integer spin and half-integer spin.

Ordinary representations
The rotation group SO(3) is a compact Lie group and thus every finite-dimensional representation of SO(3) decomposes as a direct sum of irreducible representations. The group SO(3) has one irreducible representation in each odd dimension. For each non-negative integer $$k$$, the irreducible representation of dimension $$2k+1$$ can be realized as the space $$V_k$$ of homogeneous harmonic polynomials on $$\mathbb{R}^3$$ of degree $$k$$. Here, SO(3) acts on $$V_k$$ in the usual way that rotations act on functions on $$\mathbb{R}^3$$:
 * $$(\Pi(R)f)(x)=f(R^{-1}x)\quad R\in \operatorname{SO}(3).$$

The restriction to the unit sphere $$S^2$$ of the elements of $$V_k$$ are the spherical harmonics of degree $$k$$.

If, say, $$k=1$$, then all polynomials that are homogeneous of degree one are harmonic, and we obtain a three-dimensional space $$V_1$$ spanned by the linear polynomials $$x$$, $$y$$, and $$z$$. If $$k=2$$, the space $$V_2$$ is spanned by the polynomials $$xy$$, $$xz$$, $$yz$$, $$x^2-y^2$$, and $$x^2-z^2$$.

As noted above, the finite-dimensional representations of SO(3) arise naturally when studying the time-independent Schrödinger equation for a radial potential, such as the hydrogen atom, as a reflection of the rotational symmetry of the problem. (See the role played by the spherical harmonics in the mathematical analysis of hydrogen.)

Projective representations
If we look at the Lie algebra $$\mathfrak{so}(3)$$ of SO(3), this Lie algebra is isomorphic to the Lie algebra $$\mathfrak{su}(2)$$ of SU(2). By the representation theory of $\mathfrak{su}(2)$, there is then one irreducible representation of $$\mathfrak{so}(3)$$ in every dimension. The even-dimensional representations, however, do not correspond to representations of the group SO(3). These so-called "fractional spin" representations do, however, correspond to projective representations of SO(3). These representations arise in the quantum mechanics of particles with fractional spin, such as an electron.

Operations on representations
In this section, we describe three basic operations on representations. See also the corresponding constructions for representations of a Lie algebra.

Direct sums
If we have two representations of a group $$G$$, $$\Pi_1:G\rightarrow GL(V_1)$$ and $$\Pi_2:G\rightarrow GL(V_2)$$, then the direct sum would have $$V_1\oplus V_2$$ as the underlying vector space, with the action of the group given by
 * $$\Pi(g)(v_1,v_2)=(\Pi_1(g)v_1,\Pi_2(g)v_2),$$

for all $$v_1\in V_1,$$ $$v_2\in V_2$$, and $$g\in G$$.

Certain types of Lie groups—notably, compact Lie groups—have the property that every finite-dimensional representation is isomorphic to a direct sum of irreducible representations. In such cases, the classification of representations reduces to the classification of irreducible representations. See Weyl's theorem on complete reducibility.

Tensor products of representations
If we have two representations of a group $$G$$, $$\Pi_1:G\rightarrow GL(V_1)$$ and $$\Pi_2:G\rightarrow GL(V_2)$$, then the tensor product of the representations would have the tensor product vector space $$V_1\otimes V_2$$ as the underlying vector space, with the action of $$G$$ uniquely determined by the assumption that

 * $$\Pi(g)(v_1\otimes v_2)=(\Pi_1(g)v_1)\otimes(\Pi_2(g)v_2)$$

for all $$v_1\in V_1$$ and $$v_2\in V_2$$. That is to say, $$\Pi(g)=\Pi_1(g)\otimes\Pi_2(g)$$.

The Lie algebra representation $$\pi$$ associated to the tensor product representation $$\Pi$$ is given by the formula:

 * $$\pi(X)=\pi_1(X)\otimes I+I\otimes\pi_2(X).$$

The tensor product of two irreducible representations is usually not irreducible; a basic problem in representation theory is then to decompose tensor products of irreducible representations as a direct sum of irreducible subspaces. This problem goes under the name of "addition of angular momentum" or "Clebsch–Gordan theory" in the physics literature.

Dual representations
Let $$G$$ be a Lie group and $$\Pi:G\rightarrow GL(V)$$ be a representation of G. Let $$V^*$$ be the dual space, that is, the space of linear functionals on $$V$$. Then we can define a representation $$\Pi^*:G\rightarrow GL(V^*)$$ by the formula
 * $$\Pi^*(g)=(\Pi(g^{-1}))^\operatorname{tr},$$

where for any operator $$A:V\rightarrow V$$, the transpose operator $$A^\operatorname{tr}:V^*\rightarrow V^*$$ is defined as the "composition with $$A$$" operator:
 * $$(A^\operatorname{tr}\phi)(v)=\phi(Av).$$

(If we work in a basis, then $$A^{\operatorname{tr}}$$ is just the usual matrix transpose of $$A$$.) The inverse in the definition of $$\Pi^*$$ is needed to ensure that $$\Pi^*$$ is actually a representation of $$G$$, in light of the identity $$(AB)^\operatorname{tr}=B^\operatorname{tr}A^\operatorname{tr}$$.

The dual of an irreducible representation is always irreducible, but may or may not be isomorphic to the original representation. In the case of the group SU(3), for example, the irreducible representations are labeled by a pair $$(m_1,m_2)$$ of non-negative integers. The dual of the representation associated to $$(m_1,m_2)$$ is the representation associated to $$(m_2,m_1)$$.

Overview
In many cases, it is convenient to study representations of a Lie group by studying representations of the associated Lie algebra. In general, however, not every representation of the Lie algebra comes from a representation of the group. This fact is, for example, lying behind the distinction between integer spin and half-integer spin in quantum mechanics. On the other hand, if G is a simply connected group, then a theorem says that we do, in fact, get a one-to-one correspondence between the group and Lie algebra representations.

Let $G$ be a Lie group with Lie algebra $$\mathfrak g$$, and assume that a representation $$\pi$$ of $$\mathfrak g$$ is at hand. The Lie correspondence may be employed for obtaining group representations of the connected component of the $G$. Roughly speaking, this is effected by taking the matrix exponential of the matrices of the Lie algebra representation. A subtlety arises if $G$ is not simply connected. This may result in projective representations or, in physics parlance, multi-valued representations of $G$. These are actually representations of the universal covering group of $G$.

These results will be explained more fully below.

The Lie correspondence gives results only for the connected component of the groups, and thus the other components of the full group are treated separately by giving representatives for matrices representing these components, one for each component. These form (representatives of) the zeroth homotopy group of $G$. For example, in the case of the four-component Lorentz group, representatives of space inversion and time reversal must be put in by hand. Further illustrations will be drawn from the representation theory of the Lorentz group below.

The exponential mapping

If $$G$$ is a Lie group with Lie algebra $$\mathfrak g$$, then we have the exponential map from $$\mathfrak g$$ to $$G$$, written as
 * $$X\mapsto e^X,\quad X\in\mathfrak g.$$

If $$G$$ is a matrix Lie group, the expression $$e^X$$ can be computed by the usual power series for the exponential. In any Lie group, there exist neighborhoods $$U$$ of the identity in $$G$$ and $$V$$ of the origin in $$\mathfrak g$$ with the property that every $$g$$ in $$U$$ can be written uniquely as $$g=e^X$$ with $$X\in V$$. That is, the exponential map has a local inverse. In most groups, this is only local; that is, the exponential map is typically neither one-to-one nor onto.

Lie algebra representations from group representations
It is always possible to pass from a representation of a Lie group $G$ to a representation of its Lie algebra $$\mathfrak{g}.$$ If $Π : G → GL(V)$ is a group representation for some vector space $V$, then its pushforward (differential) at the identity, or Lie map, $$\pi : \mathfrak{g} \to \text{End} V$$ is a Lie algebra representation. It is explicitly computed using

A basic property relating $$\Pi$$ and $$\pi$$ involves the exponential map:
 * $$\Pi(e^X)=e^{\pi(X)}.$$

The question we wish to investigate is whether every representation of $$\mathfrak g$$ arises in this way from representations of the group $$G$$. As we shall see, this is the case when $$G$$ is simply connected.

Group representations from Lie algebra representations
The main result of this section is the following:
 * Theorem: If $$G$$ is simply connected, then every representation $$\pi$$ of the Lie algebra $$\mathfrak g$$ of $$G$$ comes from a representation $$\Pi$$ of $$G$$ itself.

From this we easily deduce the following:
 * Corollary: If $$G$$ is connected but not simply connected, every representation $$\pi$$ of $$\mathfrak g$$ comes from a representation $$\Pi$$ of $$\tilde G$$, the universal cover of $$G$$. If $$\pi$$ is irreducible, then $$\Pi$$ descends to a projective representation of $$G$$.

A projective representation is one in which each $$\Pi(g),\,g\in G,$$ is defined only up to multiplication by a constant. In quantum physics, it is natural to allow projective representations in addition to ordinary ones, because states are really defined only up to a constant. (That is to say, if $$\psi$$ is a vector in the quantum Hilbert space, then $$c\psi$$ represents the same physical state for any constant $$c$$.) Every finite-dimensional projective representation of a connected Lie group $$G$$ comes from an ordinary representation of the universal cover $$\tilde G$$ of $$G$$. Conversely, as we will discuss below, every irreducible ordinary representation of $$\tilde G$$ descends to a projective representation of $$G$$. In the physics literature, projective representations are often described as multi-valued representations (i.e., each $$\Pi(g)$$ does not have a single value but a whole family of values). This phenomenon is important to the study of fractional spin in quantum mechanics.

We now outline the proof of the main results above. Suppose $$\pi : \mathfrak{g} \to \mathfrak{gl}(V)$$ is a representation of $$\mathfrak g$$ on a vector space $$. If there is going to be an associated Lie group representation $$\Pi$$, it must satisfy the exponential relation of the previous subsection. Now, in light of the local invertibility of the exponential, we can define a map $$\Pi$$ from a neighborhood $$U$$ of the identity in $$G$$ by this relation:
 * $$\Pi(e^X)=e^{\pi(X)},\quad g=e^X\in U.$$

A key question is then this: Is this locally defined map a "local homomorphism"? (This question would apply even in the special case where the exponential mapping is globally one-to-one and onto; in that case, $$\Pi$$ would be a globally defined map, but it is not obvious why $$\Pi$$ would be a homomorphism.) The answer to this question is yes: $$\Pi$$ is a local homomorphism, and this can be established using the Baker–Campbell–Hausdorff formula.

If $$G$$ is connected, then every element of $$G$$ is at least a product of exponentials of elements of $$\mathfrak g$$. Thus, we can tentatively define $$\Pi$$ globally as follows.

Note, however, that the representation of a given group element as a product of exponentials is very far from unique, so it is very far from clear that $$\Pi$$ is actually well defined.

To address the question of whether $$\Pi$$ is well defined, we connect each group element $$g\in G$$ to the identity using a continuous path. It is then possible to define $$\Pi$$ along the path, and to show that the value of $$\Pi(g)$$ is unchanged under continuous deformation of the path with endpoints fixed. If $$G$$ is simply connected, any path starting at the identity and ending at $$g$$ can be continuously deformed into any other such path, showing that $$\Pi(g)$$ is fully independent of the choice of path. Given that the initial definition of $$\Pi$$ near the identity was a local homomorphism, it is not difficult to show that the globally defined map is also a homomorphism satisfying $V$.

If $$G$$ is not simply connected, we may apply the above procedure to the universal cover $$\tilde G$$ of $$G$$. Let $$p:\tilde G\rightarrow G$$ be the covering map. If it should happen that the kernel of $$\Pi:\tilde G\rightarrow \operatorname{GL}(V)$$ contains the kernel of $$p$$, then $$\Pi$$ descends to a representation of the original group $$G$$. Even if this is not the case, note that the kernel of $$p$$ is a discrete normal subgroup of $$\tilde G$$, which is therefore in the center of $$\tilde G$$. Thus, if $$\pi$$ is irreducible, Schur's lemma implies that the kernel of $$p$$ will act by scalar multiples of the identity. Thus, $$\Pi$$ descends to a projective representation of $$G$$, that is, one that is defined only modulo scalar multiples of the identity.

A pictorial view of how the universal covering group contains all such homotopy classes, and a technical definition of it (as a set and as a group) is given in geometric view.

For example, when this is specialized to the doubly connected $GL(V)$, the universal covering group is $$\text{SL}(2,\Complex)$$, and whether its corresponding representation is faithful decides whether $Π$ is projective.

Classification in the compact case
If G is a connected compact Lie group, its finite-dimensional representations can be decomposed as direct sums of irreducible representations. The irreducibles are classified by a "theorem of the highest weight." We give a brief description of this theory here; for more details, see the articles on representation theory of a connected compact Lie group and the parallel theory classifying representations of semisimple Lie algebras.

Let T be a maximal torus in G. By Schur's lemma, the irreducible representations of T are one dimensional. These representations can be classified easily and are labeled by certain "analytically integral elements" or "weights." If $$\Sigma$$ is an irreducible representation of G, the restriction of $$\Sigma$$ to T will usually not be irreducible, but it will decompose as a direct sum of irreducible representations of T, labeled by the associated weights. (The same weight can occur more than once.) For a fixed $$\Sigma$$, one can identify one of the weights as "highest" and the representations are then classified by this highest weight.

An important aspect of the representation theory is the associated theory of characters. Here, for a representation $$\Sigma$$ of G, the character is the function
 * $$\chi_G:G\rightarrow\mathbb{C}$$

given by
 * $$\chi_G(g)=\operatorname{trace}(\Sigma(g)).$$

Two representations with the same character turn out to be isomorphic. Furthermore, the Weyl character formula gives a remarkable formula for the character of a representation in terms of its highest weight. Not only does this formula gives a lot of useful information about the representation, but it plays a crucial role in the proof of the theorem of the highest weight.

Unitary representations on Hilbert spaces
Let V be a complex Hilbert space, which may be infinite dimensional, and let $$U(V)$$ denote the group of unitary operators on V. A unitary representation of a Lie group G on V is a group homomorphism $$\Pi:G\rightarrow U(V)$$ with the property that for each fixed $$v\in V$$, the map
 * $$g\mapsto \Pi(g)v$$

is a continuous map of G into V.

Finite-dimensional unitary representations
If the Hilbert space V is finite-dimensional, there is an associated representation $$\pi$$ of the Lie algebra $$\mathfrak g$$ of $$G$$. If $$G$$ is connected, then the representation $$\Pi$$ of $$G$$ is unitary if and only if $$\pi(X)$$ is skew-self-adjoint for each $$X\in\mathfrak g$$.

If $$G$$ is compact, then every representation $$\Pi$$ of $$G$$ on a finite-dimensional vector space V is "unitarizable," meaning that it is possible to choose an inner product on V so that each $$\Pi(g),\, g\in G$$ is unitary.

Infinite-dimensional unitary representations
If the Hilbert space V is allowed to be infinite dimensional, the study of unitary representations involves a number of interesting features that are not present in the finite dimensional case. For example, the construction of an appropriate representation of the Lie algebra $$\mathfrak g$$ becomes technically challenging. One setting in which the Lie algebra representation is well understood is that of semisimple (or reductive) Lie groups, where the associated Lie algebra representation forms a (g,K)-module.

Examples of unitary representations arise in quantum mechanics and quantum field theory, but also in Fourier analysis as shown in the following example. Let $$G=\mathbb R$$, and let the complex Hilbert space V be $$L^2(\mathbb R)$$. We define the representation $$\psi:\mathbb R\rightarrow U(L^2(\mathbb R))$$ by
 * $$[\psi(a)(f)](x)=f(x-a).$$

Here are some important examples in which unitary representations of a Lie group have been analyzed.
 * The Stone–von Neumann theorem can be understood as giving a classification of the irreducible unitary representations of the Heisenberg group.
 * Wigner's classification for representations of the Poincaré group plays a major conceptual role in quantum field theory by showing how the mass and spin of particles can be understood in group-theoretic terms.
 * The representation theory of SL(2,R) was worked out by V. Bargmann and serves as the prototype for the study of unitary representations of noncompact semisimple Lie groups.

Projective representations
In quantum physics, one is often interested in projective unitary representations of a Lie group $$G$$. The reason for this interest is that states of a quantum system are represented by vectors in a Hilbert space $$\mathbf H$$—but with the understanding that two states differing by a constant are actually the same physical state. The symmetries of the Hilbert space are then described by unitary operators, but a unitary operator that is a multiple of the identity does not change the physical state of the system. Thus, we are interested not in ordinary unitary representations—that is, homomorphisms of $$G$$ into the unitary group $$U(\mathbf H)$$—but rather in projective unitary representations—that is, homomorphisms of $$G$$ into the projective unitary group
 * $$PU(\mathbf H):=U(\mathbf H)/\{e^{i\theta}I\}.$$

To put it differently, for a projective representation, we construct a family of unitary operators $$\rho(g),\,\, g\in G$$, where it is understood that changing $$\rho(g)$$ by a constant of absolute value 1 is counted as "the same" operator. The operators $$\rho(g)$$ are then required to satisfy the homomorphism property up to a constant:
 * $$\rho(g)\rho(h)=e^{i\theta_{g,h}}\rho(gh).$$

We have already discussed the irreducible projective unitary representations of the rotation group SO(3) above; considering projective representations allows for fractional spin in addition to integer spin.

Bargmann's theorem states that for certain types of Lie groups $$G$$, irreducible projective unitary representations of $$G$$ are in one-to-one correspondence with ordinary unitary representations of the universal cover of $$G$$. Important examples where Bargmann's theorem applies are SO(3) (as just mentioned) and the Poincaré group. The latter case is important to Wigner's classification of the projective representations of the Poincaré group, with applications to quantum field theory.

One example where Bargmann's theorem does not apply is the group $$\mathbb R^{2n}$$. The set of translations in position and momentum on $$L^2(\mathbb R^n)$$ form a projective unitary representation of $$\mathbb R^{2n}$$ but they do not come from an ordinary representation of the universal cover of $$\mathbb R^{2n}$$—which is just $$\mathbb R^{2n}$$ itself. In this case, to get an ordinary representation, one has to pass to the Heisenberg group, which is a one-dimensional central extension of $$\mathbb R^{2n}$$. (See the discussion here.)

The commutative case
If $$G$$ is a commutative Lie group, then every irreducible unitary representation of $$G$$ on complex vector spaces is one dimensional. (This claim follows from Schur's lemma and holds even if the representations are not assumed ahead of time to be finite dimensional.) Thus, the irreducible unitary representations of $$G$$ are simply continuous homomorphisms of $$G$$ into the unit circle group, U(1). For example, if $$G=\mathbb R$$, the irreducible unitary representations have the form
 * $$\Pi(x)=[e^{iax}]$$,

for some real number $$a$$.

See also Pontryagin duality for this case.

title: Representation of a Lie group


text:
Josip Runjanin

Josip Runjanin (Јосиф Руњанин; 8 December 1821 – 2 February 1878) was a Croatian Serb soldier and composer from the Austrian Empire best known for composing the melody of Lijepa naša domovino, which later became the Croatian national anthem.

Life
Runjanin was born Josif Runjanin to a Serb  family on 8 December 1821 and baptized in the Serbian Orthodox Church of Pentecost in Vinkovci. He received his education in Vinkovci, and then Sremski Karlovci. He served in the Imperial Army as a cadet in the town of Glina along the Military Frontier in the Croatian Military Frontier district.

While serving in Glina, he attained the rank of captain, and became proficient in playing the piano, being taught by the military bandmaster of Glina. There, he was introduced to the Illyrist circles, where he met noted poet Antun Mihanović. It is generally agreed that Runjanin, an amateur musician, composed the music for Mihanović's patriotic Croatian poem "Horvatska domovina" in 1846 using inspiration from Gaetano Donizetti's aria "O sole piu ratto a sorger t'appresta" from the third act of his opera Lucia di Lammermoor, according to Croatian musicologist Josip Andreis.

His song "Ljubimo te naša diko" was composed using motives from Donizetti's L'elisir d'amore. "Ljubimo te naša diko (Hrvati svome banu)" was composed in honour of the Ban of Croatia, Josip Jelačić.

The Croatian anthem by Mihanović and Runjanin would be first played in the streets of Zagreb in 1891 during the Croatian-Slavonian exhibit, so both men only achieved postmortem fame. An obedient soldier, Runjanin was later made colonel.

In 1864 at the age of 43, Runjanin married the daughter of the pensioned captain Toma Perković. As a representative of the First Banate regiment, he entered the Croatian Assembly in 1865.

After retirement, Runjanin moved to Novi Sad where he died at the age of 57 on 2 February 1878 and was buried at the Serbian Orthodox cemetery.

Legacy
There are several schools in Croatia named after Runjanin, most notably the Elementary Music School of Josip Runjanin (Osnovna glazbena škola Josipa Runjanina) in Vinkovci.

title: Josip Runjanin


text:
Unitary representation

In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in the case that G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.

The theory has been widely applied in quantum mechanics since the 1920s, particularly influenced by Hermann Weyl's 1928 book Gruppentheorie und Quantenmechanik. One of the pioneers in constructing a general theory of unitary representations, for any group G rather than just for particular groups useful in applications, was George Mackey.

Context in harmonic analysis
The theory of unitary representations of topological groups is closely connected with harmonic analysis. In the case of an abelian group G, a fairly complete picture of the representation theory of G is given by Pontryagin duality. In general, the unitary equivalence classes (see below) of irreducible unitary representations of G make up its unitary dual. This set can be identified with the spectrum of the C*-algebra associated with G by the group C*-algebra construction. This is a topological space.

The general form of the Plancherel theorem tries to describe the regular representation of G on L2(G) using a measure on the unitary dual. For G abelian this is given by the Pontryagin duality theory. For G compact, this is done by the Peter–Weyl theorem; in that case, the unitary dual is a discrete space, and the measure attaches an atom to each point of mass equal to its degree.

Formal definitions
Let G be a topological group. A strongly continuous unitary representation of G on a Hilbert space H is a group homomorphism from G into the unitary group of H,

 * $$ \pi: G \rightarrow \operatorname{U}(H) $$

such that g → π(g) ξ is a norm continuous function for every ξ ∈ H.

Note that if G is a Lie group, the Hilbert space also admits underlying smooth and analytic structures. A vector ξ in H is said to be smooth or analytic if the map g → π(g) ξ is smooth or analytic (in the norm or weak topologies on H). Smooth vectors are dense in H by a classical argument of Lars Gårding, since convolution by smooth functions of compact support yields smooth vectors. Analytic vectors are dense by a classical argument of Edward Nelson, amplified by Roe Goodman, since vectors in the image of a heat operator e–tD, corresponding to an elliptic differential operator D in the universal enveloping algebra of G, are analytic. Not only do smooth or analytic vectors form dense subspaces; but they also form common cores for the unbounded skew-adjoint operators corresponding to the elements of the Lie algebra, in the sense of spectral theory.

Two unitary representations π1: G → U(H1), π2: G → U(H2) are said to be unitarily equivalent if there is a unitary transformation A:H1 → H2 such that π1(g) = A* ∘ π2(g) ∘ A for all g in G. When this holds, A is said to be an intertwining operator for the representations $$(\pi_1,H_1),(\pi_2,H_2)$$.

If $$\pi$$ is a representation of a connected Lie group $$G$$ on a finite-dimensional Hilbert space $$H$$, then $$\pi$$ is unitary if and only if the associated Lie algebra representation $$d\pi:\mathfrak{g}\rightarrow\mathrm{End}(H)$$ maps into the space of skew-self-adjoint operators on $$H$$.

Complete reducibility
A unitary representation is completely reducible, in the sense that for any closed invariant subspace, the orthogonal complement is again a closed invariant subspace. This is at the level of an observation, but is a fundamental property. For example, it implies that finite-dimensional unitary representations are always a direct sum of irreducible representations, in the algebraic sense.

Since unitary representations are much easier to handle than the general case, it is natural to consider unitarizable representations, those that become unitary on the introduction of a suitable complex Hilbert space structure. This works very well for finite groups, and more generally for compact groups, by an averaging argument applied to an arbitrary hermitian structure. For example, a natural proof of Maschke's theorem is by this route.

Unitarizability and the unitary dual question
In general, for non-compact groups, it is a more serious question which representations are unitarizable. One of the important unsolved problems in mathematics is the description of the unitary dual, the effective classification of irreducible unitary representations of all real reductive Lie groups. All irreducible unitary representations are admissible (or rather their Harish-Chandra modules are), and the admissible representations are given by the Langlands classification, and it is easy to tell which of them have a non-trivial invariant sesquilinear form. The problem is that it is in general hard to tell when the quadratic form is positive definite. For many reductive Lie groups this has been solved; see representation theory of SL2(R) and representation theory of the Lorentz group for examples.

title: Unitary representation


text:
List of Madrid Metro stations

This is a list of the stations of the Madrid Metro.

Line 1
{{legend|#|Pinar de Chamartín - Valdecarros}}

Line 2
{{legend|#|Las Rosas - Cuatro Caminos}}

Line 3
{{legend|#|Villaverde Alto - Moncloa}}

Line 4
{{legend|#|Argüelles - Pinar de Chamartín}}

Line 5
{{legend|#| Alameda de Osuna - Casa de Campo}}

Line 6
{{legend|#|Circular}}

Line 7
{{legend|#|Pitis - Hospital del Henares}}

Line 8
{{legend|#|Nuevos Ministerios - Aeropuerto T4}}

Line 9
{{legend|#|Paco de Lucía - Arganda del Rey}}

Line 10
{{legend|#|Hospital Infanta Sofía - Puerta del Sur}}

Line 11
{{legend|#|Plaza Elíptica - La Fortuna}}

Line 12
{{legend|#|MetroSur}}

Ramal
{{legend|#|Ópera - Príncipe Pío}}

title: List of Madrid Metro stations


text:
Sidecar (cocktail)

The sidecar is any cocktail traditionally made with cognac, orange liqueur (Cointreau, Grand Marnier, dry curaçao, or a triple sec), plus lemon juice. In its ingredients, the drink is perhaps most closely related to the older brandy crusta, which differs both in presentation and in proportions of its components.

Description
Like the daiquiri, the sidecar evolved from the original sour formula, but sidecars are often drier than sours, combining liqueurs like curaçao with citrus. Sidecars are considered more of a challenge for bartenders because the proportion of ingredients is more difficult to balance for liqueurs of variable sweetness.

Origin
The exact origin of the sidecar is unclear, but it is thought to have been invented around the end of World War I in either London or Paris. The drink was directly named for the motorcycle attachment, which was very commonly used at the time.

The Ritz Hotel in Paris claims origin of the drink. The first recipes for the sidecar appear in 1922, in Harry MacElhone's Harry's ABC of Mixing Cocktails and Robert Vermeire's Cocktails and How to Mix Them. It is one of six basic drinks listed in David A. Embury's The Fine Art of Mixing Drinks (1948).

In early editions of MacElhone's book, he cites the inventor as Pat MacGarry, "the popular bartender at Buck's Club, London", but in later editions he cites himself. While Vermiere states that the drink was "very popular in France. It was first introduced in London by MacGarry, the celebrated bartender of Buck's Club." Embury credits the invention of the drink to an American army captain in Paris during World War I and named after the motorcycle sidecar that the captain used.

Journalist O.O. McIntyre reports in his 1937 summary of a visit to New York City that bartenders there attributed the drink to American expatriates Erskine Gywnne and Basil Woon.

Both MacElhone and Vermiere state the recipe as equal parts cognac, Cointreau, and lemon juice, now known as "the French school". Later, an "English school" of sidecars emerged, as found in the Savoy Cocktail Book (1930), which call for two parts cognac and one part each of Cointreau and lemon juice.

According to Embury, the original sidecar had several ingredients, which were "refined away". Embury also states the drink is simply a daiquiri with brandy as its base rather than rum, and with Cointreau as the sweetening agent rather than sugar syrup. He recommends the same proportions (8:2:1) for both, making a much-less-sweet sidecar. However, Simon Difford, in his book Encyclopedia of Cocktails, notes Harry Craddock's ratio of 2:1:1 in The Savoy Cocktail Book, and then suggests a middle ground between Craddock's recipe and the "French school" equal parts recipe of 3:2:2, calling Embury's daiquiri formula "overly dry" for a sidecar.

The earliest mention of sugaring the rim on a sidecar glass is 1934, in three books: Burke's Complete Cocktail & Drinking Recipes, Gordon's Cocktail & Food Recipes, and Drinks As They Are Mixed (a revised reprint of Paul E. Lowe's 1904 book).

title: Sidecar (cocktail)


text:
Mina al Fahal

Mina' al Fahal (ميناء الفحل, also known as Miana al Fahl, also in the past Saih al Malieh, and Saih al-Malih (سيح المالح)) is a coastal area in the northeast of Oman, near to the country's capital, Muscat. It was renamed from Saih al Maleh as the petroleum processing plant was developed. It is a key area for the country's petroleum operations. Petroleum Development Oman (PDO) is based at Mina al Fahal, and the Oman Refinery Company (ORC) has a 104,000 b/d oil refinery located here. Crude oil and refinery products are loaded onto tankers off Mina Al Fahal by subsea pipeline and SBM (Single Buoy Mooring) Systems. Two of the SBMs are owned by PDO for crude export and the third is owned by SOM (Shell Oman Marketing).

title: Mina al Fahal


text:
Dirac spinor

In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain combination of two Weyl spinors, specifically, a bispinor that transforms "spinorially" under the action of the Lorentz group.

Dirac spinors are important and interesting in numerous ways. Foremost, they are important as they do describe all of the known fundamental particle fermions in nature; this includes the electron and the quarks. Algebraically they behave, in a certain sense, as the "square root" of a vector. This is not readily apparent from direct examination, but it has slowly become clear over the last 60 years that spinorial representations are fundamental to geometry. For example, effectively all Riemannian manifolds can have spinors and spin connections built upon them, via the Clifford algebra. The Dirac spinor is specific to that of Minkowski spacetime and Lorentz transformations; the general case is quite similar.

This article is devoted to the Dirac spinor in the Dirac representation. This corresponds to a specific representation of the gamma matrices, and is best suited for demonstrating the positive and negative energy solutions of the Dirac equation. There are other representations, most notably the chiral representation, which is better suited for demonstrating the chiral symmetry of the solutions to the Dirac equation. The chiral spinors may be written as linear combinations of the Dirac spinors presented below; thus, nothing is lost or gained, other than a change in perspective with regards to the discrete symmetries of the solutions.

The remainder of this article is laid out in a pedagogical fashion, using notations and conventions specific to the standard presentation of the Dirac spinor in textbooks on quantum field theory. It focuses primarily on the algebra of the plane-wave solutions. The manner in which the Dirac spinor transforms under the action of the Lorentz group is discussed in the article on bispinors.

Definition
The Dirac spinor is the bispinor $$u\left(\vec{p}\right)$$ in the plane-wave ansatz $$\psi(x) = u\left(\vec{p}\right)\; e^{-i p \cdot x} $$ of the free Dirac equation for a spinor with mass $$m$$, $$\left(i\hbar\gamma^\mu \partial_\mu - mc\right)\psi(x) = 0$$ which, in natural units becomes $$\left(i\gamma^\mu \partial_\mu - m\right)\psi(x) = 0$$ and with Feynman slash notation may be written $$\left(i\partial\!\!\!/ - m\right)\psi(x) = 0$$

An explanation of terms appearing in the ansatz is given below.
 * The Dirac field is $$\psi(x)$$, a relativistic spin-1/2 field, or concretely a function on Minkowski space $$\mathbb{R}^{1,3}$$ valued in $$\mathbb{C}^4$$, a four-component complex vector function.
 * The Dirac spinor related to a plane-wave with wave-vector $$\vec{p}$$ is $$u\left(\vec{p}\right)$$, a $$\mathbb{C}^4$$ vector which is constant with respect to position in spacetime but dependent on momentum $$\vec{p}$$.
 * The inner product on Minkowski space for vectors $$p$$ and $$x$$ is $$p \cdot x \equiv p_\mu x^\mu \equiv E_\vec{p} t - \vec{p} \cdot \vec{x}$$.
 * The four-momentum of a plane wave is $p^\mu = \left(\pm\sqrt{m^2 + \vec{p}^2},\, \vec{p}\right) := \left(\pm E_\vec{p}, \vec{p}\right)$ where $$\vec{p}$$ is arbitrary,
 * In a given inertial frame of reference, the coordinates are $$x^\mu$$. These coordinates parametrize Minkowski space. In this article, when $$x^\mu$$ appears in an argument, the index is sometimes omitted.

The Dirac spinor for the positive-frequency solution can be written as $$ u\left(\vec{p}\right) = \begin{bmatrix} \phi \\ \frac{\vec{\sigma} \cdot \vec{p}}{E_\vec{p} + m} \phi \end{bmatrix} \,, $$ where
 * $$\phi$$ is an arbitrary two-spinor, concretely a $$\mathbb{C}^2$$ vector.
 * $$\vec{\sigma}$$ is the Pauli vector,
 * $$E_\vec{p}$$ is the positive square root $E_\vec{p} = + \sqrt{m^2 + \vec{p}^2}$ . For this article, the $$\vec{p}$$ subscript is sometimes omitted and the energy simply written $$E$$.

In natural units, when $m^{2}$ is added to $p^{2}$ or when $m$ is added to $${p\!\!\!/}$$, $m$ means $mc$ in ordinary units; when $m$ is added to $E$, $m$ means $mc^{2}$ in ordinary units. When m is added to $$\partial_\mu$$ or to $$\nabla$$ it means $\frac{mc}{\hbar}$ (which is called the inverse reduced Compton wavelength) in ordinary units.

Derivation from Dirac equation
The Dirac equation has the form $$\left(-i \vec{\alpha} \cdot \vec{\nabla} + \beta m \right) \psi = i \frac{\partial \psi}{\partial t} $$

In order to derive an expression for the four-spinor $ω$, the matrices $α$ and $β$ must be given in concrete form. The precise form that they take is representation-dependent. For the entirety of this article, the Dirac representation is used. In this representation, the matrices are $$ \vec\alpha = \begin{bmatrix} \mathbf{0} & \vec{\sigma} \\ \vec{\sigma} & \mathbf{0} \end{bmatrix} \quad \quad \beta = \begin{bmatrix} \mathbf{I} & \mathbf{0} \\ \mathbf{0} & -\mathbf{I} \end{bmatrix} $$

These two 4×4 matrices are related to the Dirac gamma matrices. Note that $0$ and $I$ are 2×2 matrices here.

The next step is to look for solutions of the form $$\psi = \omega e^{-i p \cdot x} = \omega e^{ -i \left(E t - \vec{p} \cdot \vec{x}\right) },$$ while at the same time splitting $ω$ into two two-spinors: $$\omega = \begin{bmatrix} \phi \\ \chi \end{bmatrix} \,.$$

Results
Using all of the above information to plug into the Dirac equation results in $$ E \begin{bmatrix} \phi \\ \chi \end{bmatrix} = \begin{bmatrix} m \mathbf{I} & \vec{\sigma}\cdot\vec{p} \\ \vec{\sigma}\cdot\vec{p} & -m \mathbf{I} \end{bmatrix}\begin{bmatrix} \phi \\ \chi \end{bmatrix}. $$ This matrix equation is really two coupled equations: $$\begin{align} \left(E - m \right) \phi &= \left(\vec{\sigma} \cdot \vec{p} \right) \chi \\ \left(E + m \right) \chi &= \left(\vec{\sigma} \cdot \vec{p} \right) \phi \end{align}$$

Solve the 2nd equation for $χ$ and one obtains $$\omega = \begin{bmatrix} \phi \\ \frac{\vec{\sigma} \cdot \vec{p}}{E + m} \phi \end{bmatrix} .$$

Note that this solution needs to have $E = +\sqrt{\vec p^2 + m^2}$ in order for the solution to be valid in a frame where the particle has $$\vec p = \vec 0$$.

Derivation of the sign of the energy in this case. We consider the potentially problematic term $\frac{\vec\sigma\cdot \vec{p}}{E + m} \phi$.

 * If $E = +\sqrt{p^2 + m^2}$, clearly $\frac{\vec\sigma\cdot\vec p}{E + m} \rightarrow 0$ as $$\vec p \rightarrow \vec 0$$.
 * On the other hand, let $E = -\sqrt{p^2 + m^2}$, $$\vec p = p\hat{n}$$ with $$\hat n$$ a unit vector, and let $$p \rightarrow 0$$.

$$\begin{align} E = -m\sqrt{1 + \frac{p^2}{m^2}} &\rightarrow -m\left(1 + \frac{1}{2}\frac{p^2}{m^2}\right) \\ \frac{\vec\sigma\cdot\vec p}{E + m} &\rightarrow p\frac{\vec\sigma\cdot\hat n}{-m - \frac{p^2}{2m} + m} \propto \frac{1}{p} \rightarrow \infty \end{align}$$

Hence the negative solution clearly has to be omitted, and $E = +\sqrt{p^2 + m^2}$. End derivation.

Assembling these pieces, the full positive energy solution is conventionally written as $$\psi^{(+)} = u^{(\phi)}(\vec p)e^{-i p \cdot x} = \textstyle \sqrt{\frac{E + m}{2m}} \begin{bmatrix} \phi \\ \frac{\vec{\sigma} \cdot \vec{p}}{E + m} \phi \end{bmatrix} e^{-i p \cdot x}$$ The above introduces a normalization factor $ \sqrt{\frac{E+m}{2m}},$ derived in the next section.

Solving instead the 1st equation for $$\phi $$ a different set of solutions are found: $$\omega = \begin{bmatrix} -\frac{\vec{\sigma} \cdot \vec{p}}{-E + m} \chi \\ \chi \end{bmatrix} \,.$$

In this case, one needs to enforce that $E = -\sqrt{\vec p^2 + m^2}$ for this solution to be valid in a frame where the particle has $$\vec p = \vec 0$$. The proof follows analogously to the previous case. This is the so-called negative energy solution. It can sometimes become confusing to carry around an explicitly negative energy, and so it is conventional to flip the sign on both the energy and the momentum, and to write this as $$\psi^{(-)} = v^{(\chi)}(\vec p) e^{i p \cdot x} = \textstyle\sqrt{\frac{E + m}{2m}} \begin{bmatrix} \frac{\vec{\sigma} \cdot \vec{p}}{E + m} \chi \\ \chi \end{bmatrix} e^{i p \cdot x}$$

In further development, the $$\psi^{(+)}$$-type solutions are referred to as the particle solutions, describing a positive-mass spin-1/2 particle carrying positive energy, and the $$\psi^{(-)}$$-type solutions are referred to as the antiparticle solutions, again describing a positive-mass spin-1/2 particle, again carrying positive energy. In the laboratory frame, both are considered to have positive mass and positive energy, although they are still very much dual to each other, with the flipped sign on the antiparticle plane-wave suggesting that it is "travelling backwards in time". The interpretation of "backwards-time" is a bit subjective and imprecise, amounting to hand-waving when one's only evidence are these solutions. It does gain stronger evidence when considering the quantized Dirac field. A more precise meaning for these two sets of solutions being "opposite to each other" is given in the section on charge conjugation, below.

Chiral basis
In the chiral representation for $$\gamma^\mu$$, the solution space is parametrised by a $$\mathbb{C}^2$$ vector $$\xi$$, with Dirac spinor solution $$u(\mathbf{p}) = \begin{pmatrix}\sqrt{\sigma \cdot p}\,\xi\\ \sqrt{\bar\sigma \cdot p}\,\xi\end{pmatrix}$$ where $$\sigma^\mu = (I_2, \sigma^i),~ \bar\sigma^\mu = (I_2, -\sigma^i)$$ are Pauli 4-vectors and $$\sqrt{\cdot}$$ is the Hermitian matrix square-root.

Two-spinors
In the Dirac representation, the most convenient definitions for the two-spinors are: $$ \phi^1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \quad \quad \phi^2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} $$ and $$ \chi^1 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \quad \quad \chi^2 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} $$ since these form an orthonormal basis with respect to a (complex) inner product.

Pauli matrices
The Pauli matrices are $$ \sigma_1 = \begin{bmatrix} 0 & 1\\   1 & 0  \end{bmatrix} \quad \quad \sigma_2 = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} \quad \quad \sigma_3 = \begin{bmatrix} 1 & 0 \\    0 & -1  \end{bmatrix} $$

Using these, one obtains what is sometimes called the Pauli vector: $$\vec{\sigma}\cdot\vec{p} = \sigma_1 p_1 + \sigma_2 p_2 + \sigma_3 p_3 = \begin{bmatrix} p_3        & p_1 - i p_2 \\ p_1 + i p_2 & - p_3 \end{bmatrix}$$

Orthogonality
The Dirac spinors provide a complete and orthogonal set of solutions to the Dirac equation. This is most easily demonstrated by writing the spinors in the rest frame, where this becomes obvious, and then boosting to an arbitrary Lorentz coordinate frame. In the rest frame, where the three-momentum vanishes: $$\vec p = \vec 0,$$ one may define four spinors $$u^{(1)}\left(\vec{0}\right) = \begin{bmatrix} 1 \\   0 \\    0 \\    0  \end{bmatrix} \qquad u^{(2)}\left(\vec{0}\right) = \begin{bmatrix} 0 \\   1 \\    0 \\    0   \end{bmatrix} \qquad v^{(1)}\left(\vec{0}\right) = \begin{bmatrix} 0 \\   0 \\    1 \\    0   \end{bmatrix} \qquad v^{(2)}\left(\vec{0}\right) = \begin{bmatrix} 0 \\   0 \\    0 \\    1   \end{bmatrix} $$

Introducing the Feynman slash notation $${p\!\!\!/} = \gamma^\mu p_\mu$$

the boosted spinors can be written as $$u^{(s)}\left(\vec{p}\right) = \frac{{p\!\!\!/} + m}{\sqrt{2m(E+m)}} u^{(s)}\left(\vec{0}\right) = \textstyle \sqrt{\frac{E+m}{2m}} \begin{bmatrix} \phi^{(s)}\\ \frac {\vec\sigma \cdot \vec p} {E+m} \phi^{(s)} \end{bmatrix} $$ and $$ v^{(s)}\left(\vec{p}\right) = \frac{-{p\!\!\!/} + m}{\sqrt{2m(E+m)}} v^{(s)}\left(\vec{0}\right) = \textstyle \sqrt{\frac{E+m}{2m}} \begin{bmatrix} \frac {\vec\sigma \cdot \vec p} {E+m} \chi^{(s)} \\ \chi^{(s)} \end{bmatrix} $$

The conjugate spinors are defined as $$\overline \psi = \psi^\dagger \gamma^0$$ which may be shown to solve the conjugate Dirac equation $$\overline \psi (i{\partial\!\!\!/} + m) = 0$$

with the derivative understood to be acting towards the left. The conjugate spinors are then $$ \overline u^{(s)}\left(\vec{p}\right) = \overline u^{(s)}\left(\vec{0}\right) \frac{{p\!\!\!/} + m}{\sqrt{2m(E+m)}} $$ and $$ \overline v^{(s)}\left(\vec{p}\right) = \overline v^{(s)}\left(\vec{0}\right) \frac{-{p\!\!\!/} + m}{\sqrt{2m(E+m)}} $$

The normalization chosen here is such that the scalar invariant $$\overline\psi \psi$$ really is invariant in all Lorentz frames. Specifically, this means $$ \begin{align} \overline u^{(a)} (p) u^{(b)} (p) &= \delta_{ab} & \overline u^{(a)} (p) v^{(b)} (p) &= 0 \\ \overline v^{(a)} (p) v^{(b)} (p) &= -\delta_{ab} & \overline v^{(a)} (p) u^{(b)} (p) &= 0 \end{align} $$

Completeness
The four rest-frame spinors $$u^{(s)}\left(\vec{0}\right),$$ $$\;v^{(s)}\left(\vec{0}\right)$$ indicate that there are four distinct, real, linearly independent solutions to the Dirac equation. That they are indeed solutions can be made clear by observing that, when written in momentum space, the Dirac equation has the form $$({p\!\!\!/} - m)u^{(s)}\left(\vec{p}\right) = 0$$ and $$({p\!\!\!/} + m)v^{(s)}\left(\vec{p}\right) = 0$$

This follows because $$ {p\!\!\!/}{p\!\!\!/} = p^\mu p_\mu = m^2 $$ which in turn follows from the anti-commutation relations for the gamma matrices: $$\left\{\gamma^\mu, \gamma^\nu\right\} = 2\eta^{\mu\nu}$$ with $$\eta^{\mu\nu}$$ the metric tensor in flat space (in curved space, the gamma matrices can be viewed as being a kind of vielbein, although this is beyond the scope of the current article). It is perhaps useful to note that the Dirac equation, written in the rest frame, takes the form $$\left(\gamma^0 - 1\right)u^{(s)}\left(\vec{0}\right) = 0$$ and $$\left(\gamma^0 + 1\right)v^{(s)}\left(\vec{0}\right) = 0$$ so that the rest-frame spinors can correctly be interpreted as solutions to the Dirac equation. There are four equations here, not eight. Although 4-spinors are written as four complex numbers, thus suggesting 8 real variables, only four of them have dynamical independence; the other four have no significance and can always be parameterized away. That is, one could take each of the four vectors $$u^{(s)}\left(\vec{0}\right),$$ $$\;v^{(s)}\left(\vec{0}\right)$$ and multiply each by a distinct global phase $$e^{i\eta}.$$ This phase changes nothing; it can be interpreted as a kind of global gauge freedom. This is not to say that "phases don't matter", as of course they do; the Dirac equation must be written in complex form, and the phases couple to electromagnetism. Phases even have a physical significance, as the Aharonov–Bohm effect implies: the Dirac field, coupled to electromagnetism, is a U(1) fiber bundle (the circle bundle), and the Aharonov–Bohm effect demonstrates the holonomy of that bundle. All this has no direct impact on the counting of the number of distinct components of the Dirac field. In any setting, there are only four real, distinct components.

With an appropriate choice of the gamma matrices, it is possible to write the Dirac equation in a purely real form, having only real solutions: this is the Majorana equation. However, it has only two linearly independent solutions. These solutions do not couple to electromagnetism; they describe a massive, electrically neutral spin-1/2 particle. Apparently, coupling to electromagnetism doubles the number of solutions. But of course, this makes sense: coupling to electromagnetism requires taking a real field, and making it complex. With some effort, the Dirac equation can be interpreted as the "complexified" Majorana equation. This is most easily demonstrated in a generic geometrical setting, outside the scope of this article.

Energy eigenstate projection matrices
It is conventional to define a pair of projection matrices $$\Lambda_{+}$$ and $$\Lambda_{-}$$, that project out the positive and negative energy eigenstates. Given a fixed Lorentz coordinate frame (i.e. a fixed momentum), these are $$\begin{align} \Lambda_{+}(p) = \sum_{s=1,2}{u^{(s)}_p \otimes \bar{u}^{(s)}_p} &= \frac{{p\!\!\!/} + m}{2m} \\ \Lambda_{-}(p) = -\sum_{s=1,2}{v^{(s)}_p \otimes \bar{v}^{(s)}_p} &= \frac{-{p\!\!\!/} + m}{2m} \end{align}$$

These are a pair of 4×4 matrices. They sum to the identity matrix: $$\Lambda_{+}(p) + \Lambda_{-}(p) = I$$ are orthogonal $$\Lambda_{+}(p) \Lambda_{-}(p) = \Lambda_{-}(p) \Lambda_{+}(p)= 0$$ and are idempotent $$\Lambda_{\pm}(p) \Lambda_{\pm}(p) = \Lambda_{\pm}(p) $$

It is convenient to notice their trace: $$\operatorname{tr} \Lambda_{\pm}(p) = 2 $$

Note that the trace, and the orthonormality properties hold independent of the Lorentz frame; these are Lorentz covariants.

Charge conjugation
Charge conjugation transforms the positive-energy spinor into the negative-energy spinor. Charge conjugation is a mapping (an involution) $$\psi\mapsto\psi_c$$ having the explicit form $$\psi_c = \eta C \left(\overline\psi\right)^\textsf{T}$$ where $$(\cdot)^\textsf{T}$$ denotes the transpose, $$C$$ is a 4×4 matrix, and $$\eta$$ is an arbitrary phase factor, $$\eta^*\eta = 1.$$ The article on charge conjugation derives the above form, and demonstrates why the word "charge" is the appropriate word to use: it can be interpreted as the electrical charge. In the Dirac representation for the gamma matrices, the matrix $$C$$ can be written as $$C = i\gamma^2\gamma^0 = \begin{pmatrix} 0 & -i\sigma_2 \\ -i\sigma_2 & 0 \end{pmatrix} $$ Thus, a positive-energy solution (dropping the spin superscript to avoid notational overload) $$\psi^{(+)} = u\left(\vec{p}\right) e^{-ip\cdot x} = \textstyle \sqrt{\frac{E + m}{2m}} \begin{bmatrix} \phi\\ \frac{\vec{\sigma} \cdot \vec{p}}{E + m} \phi \end{bmatrix} e^{-ip\cdot x} $$ is carried to its charge conjugate $$\psi^{(+)}_c = \textstyle \sqrt{\frac{E + m}{2m}} \begin{bmatrix} i\sigma_2 \frac{\vec{\sigma}^* \cdot \vec{p}}{E + m} \phi^*\\ -i\sigma_2 \phi^* \end{bmatrix} e^{ip\cdot x} $$ Note the stray complex conjugates. These can be consolidated with the identity $$\sigma_2 \left(\vec\sigma^* \cdot \vec k\right) \sigma_2 = - \vec\sigma\cdot\vec k$$ to obtain $$\psi^{(+)}_c = \textstyle \sqrt{\frac{E + m}{2m}} \begin{bmatrix} \frac{\vec{\sigma} \cdot \vec{p}}{E + m} \chi \\ \chi \end{bmatrix} e^{ip\cdot x} $$ with the 2-spinor being $$\chi = -i\sigma_2 \phi^*$$ As this has precisely the form of the negative energy solution, it becomes clear that charge conjugation exchanges the particle and anti-particle solutions. Note that not only is the energy reversed, but the momentum is reversed as well. Spin-up is transmuted to spin-down. It can be shown that the parity is also flipped. Charge conjugation is very much a pairing of Dirac spinor to its "exact opposite".

title: Dirac spinor