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Dirac equation

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles such as electrons and quarks for which parity is a symmetry, it is consistent with both the principles of quantum mechanics and the theory of special relativity, was the first theory to account for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a rigorous way; the equation implied the existence of a new form of matter, antimatter unsuspected and unobserved and, experimentally confirmed several years later. It provided a theoretical justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin; the wave functions in the Dirac theory are vectors of four complex numbers, two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value.

Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation. Although Dirac did not at first appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represents one of the great triumphs of theoretical physics; this accomplishment has been described as on a par with the works of Newton and Einstein before him. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-1/2 particles; the Dirac equation in the form proposed by Dirac is: where ψ = ψ is the wave function for the electron of rest mass m with spacetime coordinates x, t. The p1, p2, p3 are the components of the momentum, understood to be the momentum operator in the Schrödinger equation. C is the speed of light, ħ is the reduced Planck constant; these fundamental physical constants reflect special quantum mechanics, respectively.

Dirac's purpose in casting this equation was to explain the behavior of the relativistically moving electron, so to allow the atom to be treated in a manner consistent with relativity. His rather modest hope was that the corrections introduced this way might have a bearing on the problem of atomic spectra. Up until that time, attempts to make the old quantum theory of the atom compatible with the theory of relativity, attempts based on discretizing the angular momentum stored in the electron's non-circular orbit of the atomic nucleus, had failed – and the new quantum mechanics of Heisenberg, Jordan, Schrödinger, Dirac himself had not developed sufficiently to treat this problem. Although Dirac's original intentions were satisfied, his equation had far deeper implications for the structure of matter and introduced new mathematical classes of objects that are now essential elements of fundamental physics; the new elements in this equation are the 4 × 4 matrices αk and β, the four-component wave function ψ.

There are four components in ψ because the evaluation of it at any given point in configuration space is a bispinor. It is interpreted as a superposition of a spin-up electron, a spin-down electron, a spin-up positron, a spin-down positron; the 4 × 4 matrices αk and β are all Hermitian and have squares equal to the identity matrix: α i 2 = β 2 = I 4 and they all mutually anticommute: α i α j + α j α i = 0 α i β + β α i = 0 The single symbolic equation thus unravels into four coupled linear first-order partial differential equations for the four quantities that make up the wave function. These matrices and the form of the wave function have a deep mathematical significance; the algebraic structure represented by the gamma matrices had been created some 50 years earlier by the English mathematician W. K. Clifford. In turn, Clifford's ideas had emerged from the mid-19th-century work of the German mathematician Hermann Grassmann in his Lineale Ausdehnungslehre; the latter had been regarded as well-nigh incomprehensible by most of his contemporaries.

The appearance of something so abstract, at such a late date, in such a direct physical manner, is one of the most remarkable chapters in the history of physics. The Dirac equation is superficially similar to the Schrödinger equation for a massive free particle: − ℏ 2 2 m ∇ 2 ϕ = i ℏ ∂ ∂ t ϕ; the left side represents the square of the momentum operator divided by twice the mass, the non-relativistic kinetic energy. Because relativity treats space and time as a whole, a relativistic generalization of this equation requires that space and time derivatives must enter symmetrically as they do in the Maxwell equations that govern the behavior of light — the equations must be differentially of the same order in space and time. In relat

Jim Walden (lawyer)

James Walden is an American lawyer. After serving in the U. S Department of Justice as an Assistant U. S. Attorney for the Eastern District of New York from 1993 to 2002, he entered private practice where he was involved in several prominent white-collar and antitrust cases in addition to a series of cases seeking governmental reform, he represents the former head of Russia's anti-doping laboratory. At one time Walden represented former UFC Lightweight champion Conor McGregor in McGregor's pending court case for felony criminal mischief and misdemeanor assault in Brooklyn, NY. Walden received his B. A. from Hamilton College. He graduated magna cum laude from Temple University School of Law in 1991 where he was first in his class. Following law school, Walden clerked for Anthony J. Scirica in the U. S Court of Appeals for the Third Circuit. Walden joined the U. S. Attorney's Office for the Eastern District of New York as an Assistant U. S. Attorney, he served in this capacity for nearly nine years before turning to private practice.

Walden spent three years as a partner with O'Melveny & Myers LLP before joining the New York office of Gibson Dunn & Crutcher LLP in 2006. In 2015, Walden left Gibson Dunn to found Walden Macht & Haran LLP with fellow former prosecutors Timothy Macht and Sean Haran. While an Assistant U. S. Attorney in the Eastern District of New York, Walden led the prosecution of Li Yun-chung, a significant figure in an international heroin ring. U. S. Customs authorities conducted the then-largest seizure of heroin in U. S. history on June 20, 1991 in Hayward, California. Three-quarters of a ton of heroin was seized with a street value of $2.5-$3 billion. Li was indicted in U. S. District Court in May 1996. Following his work prosecuting heroin traffickers, Walden brought cases against members and associates of New York's most prominent Mafia families. In October 1999, Alphonse "Allie Boy" Persico, head of the Colombo crime family, was arrested on charges of loansharking. Walden helped build the cases against Persico.

These charges resurfaced in 2001, when Persico was indicted in Brooklyn the same day he completed a 15-month prison sentence on weapons charges in Florida. Walden successfully prosecuted Chris Paciello, aka Chris Ludwigsen, for his 1993 murder of Judith Shemtov during a robbery Paciello had planned in association with the Bonnanno crime family. Paciello turned himself in after murder charge were filed in November 1999, he pleaded guilty to those charges on October 13, 2000. Though Paciello only served a six-year sentence, Walden won guilty pleas from nearly 20 people related to the Shemtov murder. Benjamin Brafman, Paciello's attorney, "estimated that'more than 70 people' had been prosecuted directly and indirectly as a result of cooperation.'" This included testimony that Alphonse Persico plotted with Paciello in 1997 to kill a dissident mafioso. The identification of two made members of the Bonanno family led to the take-down of the entire crime organization. Walden appears in a 2018 "Vanity Fair Confidential" episode discussing the Shemtov murder and the Paciello prosecution and is cited extensively in the book Clubland: The Fabulous Rise and Murderous Fall of Club Culture.

In 2001, Walden led the successful prosecution of Anthony Spero, a soldier and one-time acting boss of the Bonanno crime family, on murder and racketeering charges. Spero was convicted on April 5, 2001 of ordering three murders during his 20 years serving the family. Walden's work prosecuting organized crime was profiled in the New York Times and featured in a documentary filmed by National Geographic. Following his work at the Department of Justice, in 2002 Walden entered private practice as a partner at O'Melveney & Myers LLP, he spent nearly four years there before joining Gibson Dunn & Crutcher in 2006. Walden co-chaired Gibson Dunn's White Collar-Criminal Defense & Investigations practice in addition to leading the office's pro bono efforts. In 2015, Walden started his own firm, Walden Macht & Haran LLP; the firm was founded with a focus on white-collar criminal defense, civil litigation and investigations. The firm gave Walden a platform to continue representing advocacy and community groups.

While a partner at Gibson Dunn & Crutcher, along with F. Joseph Warin represented Joseph Cassano, the CEO of AIG's Financial Products unit, for his alleged role in the 2008 financial crisis. U. S. Department of Justice investigators and prosecutors conducted an investigation into whether Cassano deliberately withheld information from investors and auditors. Walden utilized a proactive defense strategy by engaging prosecutors early in the process to present evidence, rather than engaging the public. Neither the Department of Justice nor the Securities and Exchange Commission brought charges against Cassano. In 2013, Walden represented Molly Bloom, arrested and charged as part of a $100 million illegal poker game in Los Angeles that attracted wealthy individuals and celebrities. In 2014, Walden secured a lenient sentence for Bloom, facing six months in federal prison for her involvement in the gambling ring. Bloom wrote a book about her experiences called Molly's Game, turned into a 2017 film by Aaron Sorkin.

Actor Idris Elba portrays a fictionalized character based loosely on Walden. In 2003, Walden represented Lloyd Silverstein, charged in Federal Court along with a number of other executives, with financial mismanagement at Computer Associates; as the case moved towards trial in 2004, Walden negotiated a plea arrangement with the prosecutor that helped Silverstein avoid jail time completely. In 2007, the former senior vice president of finance at Computer Associates, became the first executive to testify in what ultimatel

Foley & Lardner

Foley & Lardner LLP is an international law firm started in 1842. According to The American Lawyer, the firm ranked 39th on The American Lawyer's 2011 AmLaw 100 rankings of U. S. law firms, with $633,000,000 in gross revenue in 2010. Foley & Lardner has been in The American Lawyer's annual AmLaw 100 rankings of U. S. law firms by revenue since 1986. The oldest and largest law firm in Wisconsin, it was established in 1842 as Lynde, its founders were Asahel Finch, Jr. a Republican and former Michigan state representative, William Pitt Lynde, a Democrat who served in the United States House of Representatives, the Wisconsin state legislature, as mayor of Milwaukee. By 1970 the firm had changed its name 11 times, was beginning to grow substantially. In 2001, after absorbing firms in Chicago and Washington, D. C. it was the 11th largest firm in the United States. The firm's current name was adopted in 1969, refers to two name partners, both corporate lawyers: Leon Foley, who died at age 83 in 1978 after more than 50 years with the firm, Lynford Lardner, Jr. who died at age 58 in 1973 after drowning in the Milwaukee River.

When the firm merged with Gardere Wynne Sewell LLP in 2018, the plan -- -- was to use, in some cities, a firm name including the name partner name "Gardere". Foley & Lardner's primary practice areas include intellectual property, business law and regulatory. Notable clients of the firm include Johnson Controls, Harley Davidson, Major League Baseball, Nicholas Maduro Regime and Acciona. Barack Obama, Former President of the United States, was a summer associate in the Chicago office Russ Feingold, Former United States Senator from Wisconsin, was an associate in the Madison office Antonin Scalia, United States Supreme Court Justice, was a summer associate in the Milwaukee office Jim Doyle, Former Governor of Wisconsin, is of counsel in the Madison office Bob DuPuy, former president and chief operating officer of Major League Baseball, has been a partner in the Milwaukee and New York offices Thomas E. Fairchild, Seventh Circuit Court of Appeals Judge and Wisconsin Supreme Court justice, was an associate in the Milwaukee office from 1945 to 1948 Marcia Morales Howard, U.

S. District Court for the Middle District of Florida, was an associate in the Jacksonville office William Isaac, Chairman of the Federal Deposit Insurance Corporation from 1981 to 1985 and current Chairman of consulting firm LECG’s Global Financial Services Scott L. Klug, Former United States Congressman from Wisconsin's 2nd congressional district, is a public affairs director in the Madison office William M. Conley, U. S. District Court for the Western District of Wisconsin, was a partner in the Madison office Joan F. Kessler, Wisconsin Court of Appeals Judge and candidate for the Wisconsin Supreme Court, was a partner in the Milwaukee office Lisa S. Neubauer, Wisconsin Court of Appeals Judge, was a partner in the Milwaukee office Ulice Payne, Jr. Former CEO of the Milwaukee Brewers and first African-American CEO of a Major League Baseball franchise, was a partner in the Milwaukee office Manuel Rocha, Former U. S. Ambassador to Bolivia, is a Senior Advisor on International Business in the Miami office Fred Ridley, current Chairman of Augusta National.

Brian Hagedorn, Wisconsin Court of Appeals Judge, Candidate for the Wisconsin Supreme Court, worked at the Milwaukee office


Saint-Quentin-en-Yvelines is a new town and a communauté d'agglomération in the French département of Yvelines. It is one of the original five villes nouvelles of Paris and was named after the Saint Quentin Pond, chosen to become the town's centre; the town was built from a greenfield site starting in the 1960s. In 2014, Saint-Quentin-en-Yvelines had a population of 231,057, it is part of the much larger Paris metropolitan area, is around 25 km west of the centre of Paris. The communauté d'agglomération is made up of 12 communes: Les Clayes-sous-Bois Coignières Élancourt Guyancourt Magny-les-Hameaux Maurepas Montigny-le-Bretonneux Plaisir Trappes La Verrière Villepreux Voisins-le-BretonneuxOf these communes, Montigny-le-Bretonneux is the most centrally located and has the largest population; the 2 most importants communes are Montigny. The population of the whole communauté d'agglomération has grown tenfold in the last 40 years, from a population of 15,118 in 1962, its president is Robert Cadalbert.

Until 2019, Saint-Quentin-en-Yvelines served as the start and finish point of the famous Paris–Brest–Paris bicycling endurance event. Europcar has its head office in the business park of Val Saint-Quentin at Voisins-le-Bretonneux, it had its head office in the Immeuble Les Quadrants in Saint-Quentin-en-Yvelines. At one time Bouygues had its head office in the Kevin Roche-designed Challenger complex in Saint-Quentin-en-Yvelines, it is now occupied by one of the group's subsidiaries. Le Golf National is a private golf course in Saint-Quentin, it is hosting the 2018 Ryder Cup. The city has the Vélodrome de Saint-Quentin-en-Yvelines, it was built between 2011 and 2014 and hosted the 2015 UCI Track Cycling World Championships and the 2016 UEC European Track Championships. Next to the velodrome is a BMX track. Between 1991 and 2015, Saint-Quentin-en-Yvelines was the start and finish location for the Paris-Brest-Paris randonnée. Universities: Versailles Saint-Quentin-en-Yvelines UniversitySenior high schools/sixth form colleges: Lycée les Sept Mares Lycée Sonia Delaunay Lycée Dumont d'Urville Lycée Descartes de Montigny le Bretonneux Lycée des métiers Louis Blériot Lycée d'hôtellerie et de tourisme de Guyancourt Lycée des métiers Henri Matisse Lycée de la Plaine de Neauphle Lycée de Villaroy Lycée Jean-Vilar International schools: Institut culturel franco-japonais Media related to Saint-Quentin-en-Yvelines at Wikimedia Commons The communauté d'agglomération's website The communauté d'agglomération's website

Ivonne Ortega Pacheco

Ivonne Aracely Ortega Pacheco is a Mexican politician from Yucatán affiliated with the Institutional Revolutionary Party, former Governor of Yucatán and current federal deputy to the LXIII Legislature of the Mexican Congress. She is Yucatan's first elected female governor, although the second to serve, after Dulce María Sauri who served as interim governor from 1991 to 1994, she is the sixth woman to serve as governor of a Mexican state. Ortega has occupied different positions inside the PRI of Yucatán. In 1998 she was elected municipal president of Yucatán. In 2001 she served as local deputy in the Congress of Yucatán and in 2003 she was elected as a federal deputy representing Yucatán's Second District. In 2006 she was elected senator, to serve from 2006 to 2012, but she left that position after two months to run for Governor of Yucatán in the 2007 Yucatán gubernatorial election. Ortega won the election to serve as governor from 2007 to 2012, she was inaugurated as governor on August 1, 2007.

In 2006 she expressed her intention to be candidate of her party for Governor of Yucatán in the 2007 elections and asked permission to leave her duties as a senator. Her nomination was a surprise, because she had occupied the post of senator for only two months, faced candidates with more political career than her. On December 4, 2006, the PRI's National Executive Committee was presented the results of the Consulta Mitofsky poll conducted between Ortega and 5 other candidates in the PRI party of Yucatán: Carlos Sobrino Sierra, Erick Rubio Barthell, Dulce María Sauri, Orlando Paredes Lara and Rubén Calderón Cecilio. Ortega was the most favored by the Yucatecans, was declared a candidate of unity with the agreement of the rest of the candidates. Again her victory was considered a surprise because the former interim governor Dulce María Sauri had been seen as the favorite. On January 13, 2007 she was sworn as the PRI candidate for the governorship of Yucatan. On February 26 her candidacy was endorsed by the Ecologist Green Party of Mexico and the Alliance for Yucatan Party, which formed a three-party coalition with PRI for the 25 local deputations and 106 municipalities.

The PRI placed Ortega Pacheco on its list for proportional representation federal deputies from the third electoral region in 2015, sending her to the Chamber of Deputies for the LXIII Legislature. She presides over the Communicaciones Commission and sits on the Constitutional Points and Rules and Parliamentary Practices Commissions. In February 2015 the ex governor of Yucatan Ivonne Ortega Pacheco was accused by the ex governor of Yucatan Patricio Patrón Laviada of deviating funds for the construction of a hospital in Tekax in the amount of at least 112 million Mexican pesos. Patrón Laviada started the construction of the hospital in 2006, with had an estimated cost of 52.07 million Mexican pesos. Leaving the hospital with a progress of 67%, Patrón Laviada left office in 2007, year in which Ortega Pacheco became governor of Yucatan. By 2010 governor Ortega Pacheco had spent more than 100 million Mexican pesos in the further construction of the hospital, which she never finished. In 2016 the governor of Yucatán Rolando Zapata Bello made an investment of 80 million Mexican pesos in order to finish the hospital, which should start operations at the end of 2016.

The Palace of Mayan Civilization, conceived as the first piece of a “Mayan Disneyland” in Yucatan has been abandoned since 2012. About 90 million Mexican pesos were spent in the development of the project. Located 11.5 kilometres of Pisté, the project had 520,000 square meters for its development. In the first stage a parking lot and the foundations for 5 building would be built; this stage counted with 70 million Mexican pesos of federal funds. The second and third stages would have had an estimated cost of 140 million Mexican pesos, but in 2012 the project was postponed until 2014, which had a 50% progress. However, in November 2014, the 13 thousand square meters of construction, which started on December 21 of 2009, were abandoned and have started to be swallowed by the jungle. With a progress of 90%, the Gran Museo del Mundo Maya was inaugurated in September 2012 by Ivonne Ortega Pacheco; the museum, one of Ortega Pacheco's main projects, proved to be a financial disaster. Ortega Pacheco informed.

However, the state government will have to pay for the construction and operation of the museum 4,643 million Mexican pesos during 21 years at a yearly rate of 221.1 million Mexican pesos, according to information based on official documents obtained by Mayaleaks

Río Lagartos

Río Lagartos is a town in the state of Yucatán, Mexico. The town lies 42 kilometres north of Tizimín. Mérida is 230 kilometres further. Río Lagartos is located at a lagoon, the Ria Lagartos, part of a natural reserve; this makes it an ideal place for birdwatching. This lagoon is part of the Petenes mangroves ecoregion, the Ria Lagartos has been designated as an internationally recognized Important Bird Area. In 2004, UNESCO designated the area as Ría Lagartos Biosphere Reserve; the creek where Francisco Hernandez's 1517 expedition tried to obtain water, was named El Estero de los Lagartos, because of the "many large alligators". World Wildlife Fund. Eds. Mark McGinley, C. Michael Hogan & C. Cleveland. 2010. Petenes mangroves. Encyclopedia of Earth. National Council for Science and the Environment. Washington DC Official Website Río Lagartos Photo Essay