Mathematically speaking, the color charge of a particle is the value of a certain quadratic Casimir operator in the representation of the particle.

One of the best known examples of an observable physical effect that results from the vacuum expectation value of an operator is the Casimir effect.

The center of U ( L ) is called Z ( L ) and consists of the left-and right-invariant differential operators ; this in the case of G not commutative will often not be generated by first-order operators ( see for example Casimir operator of a semi-simple Lie algebra ).

In mathematics, a Casimir invariant or Casimir operator is a distinguished element of the centre of the universal enveloping algebra of a Lie algebra.

A prototypical example is the squared angular momentum operator, which is a Casimir invariant of the three-dimensional rotation group.

By carrying out this quantization and calculating the eigenvalues of the Casimir operator for the Kepler problem, Wolfgang Pauli was able to derive the energy levels of hydrogen-like atoms ( Figure 6 ) and, thus, their atomic emission spectrum.

A normalized first Casimir invariant operator can likewise be defined

Applying these ladder operators to the eigenstates of the total angular momentum, azimuthal angular momentum and energy operators, the eigenvalues of the first Casimir operator C < sub > 1 </ sub > are n < sup > 2 </ sup > − 1 ; importantly, they are independent of the l and m quantum numbers, making the energy levels degenerate.

It is an example of a Casimir operator.

Quantum effects such as the Casimir effect cannot violate the averaged null energy condition in any neighborhood of space with zero curvature, but calculations in semiclassical gravity suggest that quantum effects may be able to violate this condition in curved spacetime.

Casimir Perier, on being made President of the Conseil in March 1831, made Ferdinand Philippe's exclusion from the Conseil a condition of his taking the post.

In particular, the averaged null energy condition is satisfied in the Casimir effect.

Christopher M. Clark says that John Casimir of Poland was on the one hand " eager to separate Brandenburg from Sweden and to neutralize it as a military threat " when Poland-Lithuania was threatened by the Tsardom of Russia, and on the other hand was ready to accept the Hohenzollerns ' demands due to pressure by the House of Habsburg, who after the emperor's incidental death earlier that year needed to secure the elector's vote, and whose " urgings [...] carried a considerable weight, since the Poles were counting on Austrian assistance in the event of a renewed Swedish or Russian attack.

In modern theoretical physics, the Casimir effect plays an important role in the chiral bag model of the nucleon ; and in applied physics, it is significant in some aspects of emerging microtechnologies and nanotechnologies.

Nesterenko, G. Lambiase, G. Scarpetta, Calculation of the Casimir energy at zero and finite temperature: some recent results, arXiv: hep-th / 0503100 v2 13 May 2005

However, there is still some debate on whether vacuum energy is necessary to explain the Casimir effect.

Separately, a Casimir Pulaski Day is celebrated in Illinois and some other places on the first Monday of each March.

Similar festivals are also held in Hrodna, Belarus, the city where St. Casimir died, as well as in some cities in Poland following the fall of Communism in Poland.

The Polish king John II Casimir Vasa left Warsaw the same month to confront the Swedish army in the west, but after some skirmishes with the Swedish vanguard retreated southwards to Kraków.

In 2012, Wisconsin Public Schools are to observe in some appropriate fashion Casimir Pulaski Day on Monday, March 5, coinciding with the holiday in Illinois.

At the age of 70 ( or 85, according to some sources ), he was too old to resume his struggle for the Lithuanian throne and more importantly had no support from the Council of Lords led by Jonas Goštautas ( Jan Gastold ), that in June 1440 elected Casimir Jagiellon, brother of Polish King Władysław III as Grand Duke.

A member of the old Franconian noble family von Grumbach ( a branch of the Wolfskeel Uradel family ), he was born in Rimpar near Würzburg, and having passed some time at the court of the Hohenzollern margrave Casimir of Bayreuth, fought alongside the princes during the German Peasants ' War in 1524 and 1525.

The Polish-Lithuanian forces, commanded by King John II Casimir of Poland, comprised about 24, 000 regulars ( including only 950 Winged Hussars-8 banners ), 2, 000 Tatars and 10, 000 of the noble levy ( pospolite ruszenie ), altogether some 36, 000 men of which only about 4, 000 were infantry and the remainder cavalry & dragoons, with 18 artillery pieces.

At some time between 1038 – 1039 Casimir I the Restorer returned to Poland and it is supposed that Kraków became the royal residence and the capital of Poland at this time.

Casimir II, however, went a different route and, in order to gain the favor of the Duke of Racibórz given him the towns of Oświęcim and Bytom ( with the fortress of Oświęcim, Bytom, Mikołów, Siewierz and Pszczyna, although some historians estimate that these fortress had belonged to Mieszko only since 1179 ).

Casimir managed to have Anne and Cunigunde legitimated by Pope Urban V on 5 December 1369.

According to the Annales Cracovienses Compilati, this event took place in 1136 ; since it can be assumed that the Polish princess was younger than her betrothed, and also are known the birth dates of the youngest children of Bolesław III ( Agnes in 1137 and Casimir in 1138 ), Judith in consequence could have been born between 1130 and 1135.

The towns of Kvėdarna and Nemunaitis in Lithuania have Saint Casimir depicted on their Coat of Arms.

The development of the Barnes Industrial Park, Casimir Pulaski Industrial Park, Wharton Brook Industrial Park, and the South Turnpike Road area have greatly contributed to this transition.

These measurements have been used to measure nanoscale contacts, atomic bonding, Van der Waals forces, and Casimir forces, dissolution forces in liquids and single molecule stretching and rupture forces.

in 1635 he might have served as a military adviser to John II Casimir Vasa, and possibly accompanied him to Vienna.

Besides the books which have been already mentioned he wrote: Organization in Daily Life, an Essay ( 1862 ), Casimir Maremma ( 1870 ), Brevia, Short Essays and Aphorisms ( 1871 ), Thoughts upon Government ( 1872 ), Life and Labors of Mr Thomas Brassey ( 1872 ), Iras de Biron ( 1874 ), Social Pressure ( 1875 ).

During the reign of Casimir III, many expelled German Jews from the Holy Roman Empire immigrated to the town, which grew to have one of the largest Jewish settlements in the kingdom.

Scientists have discovered a way of levitating ultra small objects by manipulating the so-called Casimir force, which normally causes objects to stick together due to forces predicted by quantum field theory.

King Casimir III ( 1310 – 1370 ) is known to have held court under Bartek.

Several Polish kings have borne this name, including Casimir III the Great and Saint Casimir, a patron saint of Poland and Lithuania.

To achieve this, the rumor was even spread that the charter claimed to have been granted to the Jews by Casimir III was a forgery, as a Catholic ruler would never have granted full civil rights to " unbelievers.

The original privileges are lost, but they have been mentioned and affirmed by King John II Casimir in 1658.

v .; A Favrot, Étude sur Casimir Delavigne ( 1894 ); and F Vuacheux, Casimir Delavigne ( 1893 ).

# to be an eigenfunction of certain Casimir operators on G ; and

Dutch physicists Hendrik B. G. Casimir and Dirk Polder at Philips Research Labs proposed the existence of a force between two polarizable atoms and between such an atom and a conducting plate in 1947, and, after a conversation with Niels Bohr who suggested it had something to do with zero-point energy, Casimir alone formulated the theory predicting a force between neutral conducting plates in 1948 ; the former is called the Casimir-Polder force while the latter is the Casimir effect in the narrow sense.

For assistance against the Order, the Confederation asked for help from King Casimir IV of Poland ; Casimir's subsequent claiming of Prussia led to the Thirteen Years ' War.

In the Baltic Sea region, Poland's struggle with the Teutonic Knights continued and included the Battle of Grunwald ( German: Battle of Tannenberg ; Lithuanian: Battle of Žalgiris ) ( 1410 ) and in 1466 the milestone Peace of Thorn under King Casimir IV Jagiellon ; the treaty created the future Duchy of Prussia.

* Earliest day on which Casimir Pulaski Day can fall, while March 7 is the latest ; celebrated on the first Monday in March.

The Jagiellonian University (, often shortened to UJ ; historical names:, University of Kraków, Kraków Academy, The Main Crown School, Main School of Kraków ) was established in 1364 by Casimir III the Great in Kazimierz ( district of Kraków ).

Soon after Jagiello accession to the Polish throne, Jagiello granted Vilnius a city charter like that of Kraków, modeled on the Magdeburg Law ; and Vytautas issued a privilege to a Jewish commune of Trakai on almost the same terms as privileges issued to the Jews of Poland in the reigns of Boleslaus the Pious and Casimir the Great.

; St John the Baptist church: founded by Casimir III, built in the years 1360 – 1370 in gothic style, and re-constructed many times

; Bernardine church and monastery: founded by Casimir IV of Poland, built in the years 1468 – 1507

Casimir Pulaski, or Kazimierz Pułaski in Polish (; full name Kazimierz Michał Wacław Wiktor Pułaski ; March 6, 1745 – October 11, 1779 ) was a Polish nobleman of Ślepowron coat-of-arms, soldier and military commander who has been called " the father of American cavalry ".

He knew he was cut off from his base ; so he ordered his cousin Ernst Casimir ( Ernst Casimir I of Nassau-Dietz ) with a force to delay the advancing Spanish while he was bringing the best part of the army to cross again the Yser and rejoin the rest of the army to face the Archduke.

; 1343: Jews persecuted in Western Europe are invited to Poland by Casimir the Great.

; V Corps Marshal Édouard Adolphe Casimir Joseph Mortier

* Charles Edward Louis Philip Casimir Stuart ( 1720 – 1788 ), aka " Bonnie Prince Charlie ", married Louise of Stolberg-Gedern and had no issue ; had affair with his cousin Marie Louise de La Tour d ' Auvergne and had issue ; had issue with his mistress Clementina Walkinshaw ;

John II Casimir (; ; ( 22 March 1609 – 16 December 1672 ) was King of Poland and Grand Duke of Lithuania during the era of the Polish – Lithuanian Commonwealth, Duke of Opole in Upper Silesia, and titular King of Sweden 1648 – 1660.

* English translation: John Casimir, by God's grace King of Poland, Grand Duke of Lithuania, Ruthenia, Prussia, Masovia, Samogitia, Livonia, Smolensk, Severia and, Chernihiv ; and also hereditary King of the Swedes, Goths and Vandals.

* Elisabeth of Austria ( 1436 – 1505 ), daughter of Albert II of Germany ; Queen consort of Casimir IV of Poland and Lithuania

# Casimir Pulaski ( 1745 – 1779 ), Polish military officer who fought on the side of the American colonists against the British in the American Revolutionary War ; member of the Polish – Lithuanian Commonwealth nobility, politician who has been called " The Father of the American Cavalry ", enacted on November 6, 2009, posthumously