In Irish mythology, Boann or Boand was the goddess of the River Boyne. According to the Lebor Gabála Érenn she was the daughter of Delbáeth, son of Elada, of the Tuatha Dé Danann.

"Centuries" redirects here; for the cricketing term, see Century (cricket).
These pages contain the trends of millennia and centuries.

• The individual century pages contain lists of decades and years.


• See calendar and list of calendars for other groupings of years.


• See history, History by period, Periodization, etcetera for different organizations of historical events.


• For earlier time periods, see cosmological timeline, geologic timescale, evolutionary timeline, pleistocene, and logarithmic timeline.

Lists Overviews · Topics (basic) · Academic disciplines · Glossaries · Portals Countries · People · Timelines (centuries · decades) · 2007 · Anniversaries (today) · Current events · Deaths this year
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The individual century pages contain lists of decades and years.
See calendar and list of calendars for other groupings of years.
See history, History by period, Periodization, etcetera for different organizations of historical events.
For earlier time periods, see cosmological timeline, geologic timescale, evolutionary timeline, pleistocene, and logarithmic timeline.
Upper Paleolithic
10th millennium BC | 9th millennium BC | 8th millennium BC
7th millennium BC | 6th millennium BC | 5th millennium BC
5th millennium | 6th millennium | 7th millennium
8th millennium | 9th millennium | 10th millennium
11th millennium and beyond

ABC Radio National is an Australia-wide radio network broadcast by the Australian Broadcasting Corporation with many various programs, involving news and current affairs, arts, music, society, science, drama and comedy. Some programs are mirrored on Radio Australia, Australia's shortwave service.
The radio programs from many of the stations are available for real-time streaming over the Internet, or for downloading as MP3 podcast files. Some programs also have transcripts available on the Radio National website.
Radio National as it is known today has had a long history. RN's Sydney station 2FC first broadcast in 1923, the ABC taking over all existing "A" Class stations in 1932. 2FC stood for Farmer and Company, the original owner of the station before the ABC took it over. The Melbourne station, 3AR, began in 1924, followed soon after by 6WF Perth, 4QG Brisbane, 5CL Adelaide and 7ZL Hobart.
From 1947 until mid 1980s, "Radio 2" (as it became known) was broadcast to the major metropolitan centres with additional reception in adjacent areas. It contained most of the ABC's national programming. It began to take on a more serious tone in the 1970s.
In 1985 ABC renamed Radio 2 to Radio National. Since 1990, all RN stations have had the same callsign format - *RN, where * is the appropriate number for the state or territory.

Typical programs

AM, The World Today, PM: in-depth news and analysis (Mondays - Fridays at, respectively, 7.10, 12.00, and 17.00)
Asia Pacific: current regional affairs in the Asia Pacific region, mirrored on Radio Australia (Tuesdays - Saturdays at 1.00 and 5.00) News and analysis

Australia Talks: Mondays to Fridays 18.00 – 19.00, Tuesdays to Saturdays at 03.00) Talkback

Artworks: What's happening now in new music, art and culture in Australia and around the world (Sundays; 10.00, Tuesday 15.00)
Airplay: radio plays (Fridays 21.00, Sundays 15.00)
Poetica: poetry featured and produced for radio (Thursdays 21.00, Saturdays 15.00)
Music Deli: live music performances (Tuesdays 2.00, Fridays 20.00, Sundays 16.00)
The Music Show: latest developments in music, featured music and interviews with performers/composers (Saturdays 10.00 and 20.00)
Sound Quality: latest new music in the genre of electronica and others (Fridays 23.20 to Saturdays 1.00)
The Daily Planet / The Weekend Planet: music from all around the world (Mondays to Fridays 14.20 and 16.00, Mondays to Thursdays 2320, Saturdays and Sundays 22.00)
Radio Eye: Radio National's flagship documentary programme (Wednesdays 13.00, Saturdays 14.00)
The Night Air: a new kind of radio program – a kaleidoscopic composition of music, sounds, audio cut-ups, ideas and stories (Fridays 21.35, Sundays 0.00 and 20.30)
Exhibit A: a weekly discussion of arts issues, hosted by Julie Copeland, which is also downloadable from the ABC website. Arts and music
Radio National's religion unit provides reporting and analysis on religious and ethical issues for Australia. Following its ABC charter obligations, this unit forms a key part of the ABC's religion output, and is unique in providing the independent analysis of a public broadcaster.

The Ark : historians and authors about curious moments in religious history
Encounter : a radio documentary series exploring connections between religion and life
The Religion Report : religious current affairs (Wednesdays 0830 and 20.05)
The Rhythm Divine : "a musical journey through the world of belief"
The Spirit Of Things: an adventure into religion and spirituality, exploring contemporary values and beliefs (Mondays 21.00, Fridays 4.00, Sundays 18.00) Religion

All In The Mind investigates the mental universe, the mind, brain and behaviour (Wednesdays 21.00, Saturdays 13.00)
The Philosopher's Zone: your guide through the strange thickets of logic, metaphysics and ethics. (Saturdays 13:35, Mondays 13:35) Science

List of Australian radio stations
BBC Radio 4
Radio New Zealand National

Westfield Citrus Park, formerly Citrus Park Town Center, is a shopping mall in Citrus Park, Florida that opened in March 1999. Its four anchor stores are Dillard's, JCPenney, Macy's and Sears and also include a Regal Cinemas multiplex theater. Dick's Sporting Goods will open a store in 2007.
The Westfield Group acquired the shopping center in 2002, and renamed it "Westfield Shoppingtown Citrus Park", dropping the "Shoppingtown" name in June 2005.

He (IPA: [hi:]) is a third-person, singular personal pronoun (subject case) in Modern English.

People

Animals

Generic
A greeting phrase used by the Scooby Gang

Other

Main article: Gender in English Gender

Etymology
The reconstructed Indo-European language provides a demonstrative pronoun ko.

Indo-European
English is a development of the West Germanic language family.

Germanic
Speakers of Old English (OE) considered each noun to have a grammatical gender — masculine, feminine or neuter. selected to have the same grammatical gender as the noun they referred to. For example, dæg (IPA: [dæj], day) was masculine, so a masculine pronoun was used when referring to a day or days. The personal pronoun for a singular masculine subject was written he, just like Present-Day English (PrDE). However, OE he was probably pronounced like PrDE hay (IPA: [he:]). The vowel in hay is normally longer in duration than in the exlamation Hey! (IPA: [he]). Because the vowel sound of OE he was long in duration, scholars (and OE dictionaries) now write it as hē.

Middle English

Generic antecedents
Gender-specific pronoun
English personal pronouns

Hazrat Inayat Khan (July 5, 1882 – February 5, 1927) was the founder of Universal Sufism and the Sufi Order International. He initially came to the West as a representative of several traditions of classical Indian music, having received the title Tansen from the Nizam of Hyderabad. However, Khan's life mission was soon revealed to be the introduction and transmission of Sufi thought and practice to the West. His universal message of Divine Unity – Tawhid – focused on the themes of "Love, Harmony and Beauty" and evinced his distinctive and effective ability to transmit the highest spiritual truths of Sufism to Western audiences of his day.

Life
In 1922, during the summer school, Khan had a 'spiritual experience' in the South Dunes in Katwijk. He immediately told his students to meditate and proclaimed the place where he was on that moment holy. In 1969, a temple has been built on that specific place, the only Universal Soufi Temple in the world. Every year, the Soufi summer school takes place in this temple, and many Sufis from around the world visit the temple each summer.

Sayings

Edward Sapir (IPA: /səˈpɪər/), (January 26, 1884 – February 4, 1939) was an American anthropologist-linguist, a leader in American structural linguistics, and one of the creators of what is now called the Sapir-Whorf hypothesis. He is arguably the most influential figure in American linguistics, influencing several generations of linguists across several schools of linguistics.

Life and work

Selected publications
Sapir, Edward (1907). Herder's "Ursprung der Sprache". Chicago: University of Chicago Press. ASIN: B0006CWB2W. 
Sapir, Edward (1908), "On the etymology of Sanskrit asru, Avestan asru, Greek dakru", in Modi, Jivanji Jamshedji, Spiegel memorial volume. Papers on Iranian subjects written by various scholars in honour of the late Dr. Frederic Spiegel, Bombay: British India Press, pp. 156-159
Sapir, Edward (1909). Wishram texts, together with Wasco tales and myths. E.J. Brill. ASIN: B000855RIW. 
Sapir, Edward (1921). Language: An introduction to the study of speech. New York: Harcourt, Brace and company. ASIN: B000NGWX8I. 
Sapir, Edward; Swadesh, Morris (1939). Nootka Texts: Tales and ethnological narratives, with grammatical notes and lexical materials. Philadelphia: Linguistic Society of America. ASIN: B000EB54JC. 
Sapir, Edward (1949), Mandelbaum, David, ed., Selected writings in language, culture and personality, Berkeley: University of California Press, ASIN: B000PX25CS
Sapir, Edward; Irvine, Judith (2002). The psychology of culture: A course of lectures. Berlin: Walter de Gruyter. ISBN 978-3110172829. 

A Content Management System (CMS) is a software system used for content management. Content management systems are deployed primarily for interactive use by a potentially large number of contributors. For example, the software for the website Wikipedia is based on a wiki, which is a particular type of content management system. For the purposes of this page, Content Management means Web Content Management. Other related forms of content management are listed below.
The content managed includes computer files, image media, audio files, electronic documents and web content. The idea behind a CMS is to make these files available inter-office, as well as over the web. A Content Management System would most often be used as archival as well. Many companies use a CMS to store files in a non-proprietary form. Companies use a CMS to share files with ease, as most systems use server based software, even further broadening file availability. As shown below, many Content Management Systems include a feature for Web Content, and some have a feature for a "workflow process."
"Work flow" is the idea of moving an electronic document along for either approval, or for adding content. Some Content Management Systems will easily facilitate this process with email notification, and automated routing. This is ideally a collaborative creation of documents. A CMS facilitates the organization, control, and publication of a large body of documents and other content, such as images and multimedia resources.
A web content management system is a content management system with additional features to ease the tasks required to publish web content to web sites.

Web content management systems

Main article: Web content management system

(Julia Vipsania) Agrippina (PIR V 463) (14 BC – 18 October 33), most commonly known as Agrippina Major or Agrippina "the Elder", was one of the most prominent women in the Roman Empire in the early 1st century AD. She was the daughter of Marcus Vipsanius Agrippa by his third wife Julia the Elder, was a granddaughter of Augustus and wife of Germanicus.

Early life
The well regarded Germanicus was a candidate for the succession and had won fame campaigning in Germania and Gaul, where he was accompanied by Agrippina. This was most unusual for Roman wives, as convention required them to stay at home, and earned her a reputation as a model for heroic womanhood. She bore him two children in Gaul, a boy and Agrippina the Younger in the Rhine frontier.
Agrippina and Germanicus travelled to the Near East in 19, incurring the displeasure of the emperor Tiberius. Germanicus quarrelled with Gnaeus Calpurnius Piso, the governor of Syria, and died in Antioch in mysterious circumstances. It was widely suspected that Germanicus had been poisoned – perhaps on the orders of Tiberius himself – and Agrippina returned to Rome to avenge his death. She boldly accused Piso of the murder of Germanicus. According to Tacitus (Annals 3.14.1), the prosecution could not prove the poisoning charge, but other charges of treason seemed likely to stick, and Piso committed suicide.

Time in Rome
Agrippina and her sons Nero and Drusus were arrested in 29 on the orders of Tiberius. They were tried by the Senate and Agrippina was banished to the island of Pandataria (now called Ventotene) in the Tyrrhenian Sea off the coast of Campania where her mother had once been banished. There she was treated with great brutality, losing an eye from the blow of a centurion and later undergoing forcible feeding (Suetonius, Tib.53). She died on 18 October 33 in suspicious circumstances. Her death, according to Suetonius the result of voluntary starvation (ibid), was probably hastened by her realisation that the fall of Sejanus had "led to no abatement of horrors" (Tacitus, Annals 6.25). Tacitus also mentions malnutrition as a likely cause. After her death Tiberius accused her of "having had Asinius Gallus as a paramour and being driven by his death to loathe existence" (Annals 6.25). At Tiberius' prompting the Senate decreed that her birthday should be marked as a day of ill omen (Suet.ibid.).
Drusus died of starvation after being imprisoned in Rome and Nero Caesar either committed suicide or was murdered after his trial in 29. Only two of her children are of historical importance: Agrippina the Younger and Gaius Caesar, who succeeded Tiberius under the name of Caligula. Despite Tiberius' enmity towards Caligula's elder brothers, he nonetheless made Caligula and his cousin Tiberius Gemellus joint heirs to his property.
There is a portrait of her in the Capitoline Museums at Rome and a bronze medal in the British Museum showing her ashes being brought back to Rome by order of Caligula.

Personality
Tacitus, Annals i.-vi.
Suetonius, The Twelve Caesars
Julio-Claudian Family Tree

The agriculture of Thailand, may be traced through historical, scientific, and social aspects which produced modern Thailand's unique approach to agriculture. Following the Neolithic Revolution after society in the area evolved from hunting and gathering, it developed through phases of agro-cities, into state-religious empires, and with the immigration of the Tai produced a distinct approach to sustainable agriculture compared with most other agricultural practices in the world.
Rice is the country's most important crop; Thailand is a major exporter in the world rice market. Other agricultural commodities produced in significant amounts include fish and fishery products, tapioca, rubber, grain, and sugar. Exports of industrially processed foods such as canned tuna, pineapples, and frozen shrimp are on the rise.
From about 1000, the Tai wet glutinous rice culture determined administrative structures in a pragmatic society that regularly produced a salable surplus. Continuing today, these systems consolidate the importance of rice agriculture to national security and economic well being. Chinese and European influence later benefited agribusiness and initiated the demand that would expand agriculture through population increase until accessible land was expended.

Agriculture in transition
As agriculture declined in relative financial importance in terms of income with rising industrialization and Americanization of Thailand from the 1960s, but it continued to provide the benefits of employment and self-sufficiency, rural social support, and cultural custody. Technical and economic globalisation forces have continued to change agriculture to a food industry and thereby exposed smallholder farmers to such an extent the traditional environmental and human values have declined markedly in all but the poorer areas. Unregulated exploitation of natural resources is but one problem as values of sharing, producing just enough rather than for wealth creation, and integrity of purpose across wide income disparities have now disappeared in the face of unregulated profit making by urban-based investors.
Agribusiness, both privately and government-owned, expanded from the 1960s and subsistence farmers were partly viewed as a past relic from which agribusiness could modernise. However intensive integrated production systems of subsistence farming continued to offer efficiencies that were not financial, including social benefits which have now caused agriculture to be treated as both a social and financial sector in planning, with increased recognition of environmental and cultural values.

Radio frequency, or RF, is a frequency or rate of oscillation within the range of about 3 Hz and 30 GHz. This range corresponds to frequency of alternating current electrical signals used to produce and detect radio waves. Since most of this range is beyond the vibration rate that most mechanical systems can respond to, RF usually refers to oscillations in electrical circuits.

Special properties of RF electrical signals
Electrical currents or waves that oscillate at RF have properties that arise out of electromagnetic forces that do not affect direct current signals. One such property drives the RF current to the surface of conductors, known as the skin effect. Another property is the ability to appear to flow through paths that contain insulating material, like the dielectric insulator of a capacitor. The degree of effect of these properties depend on the frequency of the signals.

In theoretical astrophysics, the Nordtvedt effect refers to the relative motion between the Earth and the Moon which would be observed if the gravitational self-energy of a body contributed to its gravitational mass but not its inertial mass. If observed, the Nordtvedt effect would violate the strong equivalence principle, which shows that an object's movement in a gravitational field does not depend on its mass or composition.
The effect is named after Dr. Kenneth L. Nordtvedt, from Montana State University, who first demonstrated that some theories of gravity suggest that massive bodies should fall at different rates, depending upon their gravitational self-energy.
If gravity does in fact violate the strong equivalence principle, then the more-massive Earth should fall towards the Sun at a slightly different rate than the Moon. To test for the existence (or absence) of the Nordtvedt effect, scientists have used the Lunar Laser Ranging Experiment, which is capable of measuring the distance between the Earth and the Moon with near-millimetre accuracy. Thus far, the results have failed to find any evidence of the Nordtvedt effect, demonstrating that if it exists, the effect is exceedingly weak.

Frisia (West Frisian: Fryslân; North Frisian: Fraschlönj, Freesklöön, Freeskluin, Fresklun, and Friislön'; Saterfrisian (East Frisian): Fräislound; East Frisian Low Saxon: Freesland; Gronings: Fraislaand; German and Dutch: Friesland; Danish: Frisland) is a coastal region along the southeastern corner of the North Sea, i.e. the German Bight. Frisia is the traditional homeland of the Frisians, a Germanic people who speak Frisian, a language closely related to the English language. Frisia extends from the northwestern Netherlands across northwestern Germany and into a little part of southwestern Denmark (to the river Vidå).

Divisions
Frisia changed dramatically throughout time, both by floods and by a change in identity.

History
The Frisians had settled in Frisia from about 500 BC. According to Pliny the Younger, in Roman times, the Frisians (or, as it may be, their close neighbours, the Chauci) lived on terps, man-made islands. According to other sources, the Frisians lived along a broader expanse of the coast of the North Sea (or "Frisian Sea").
Frisia at this time comprised the present provinces of Friesland and North Holland. A large part of the population of the present Netherlands lived in present Friesland, because of the fertile grounds there.

Kingdom of Frisia
Frisians made polders in West Friesland, which moved further and further away from Friesland due to floods. The western part of Frisia became the county of Holland in 1101 after a few centuries of a different history than the other parts. Frisia began to identify itself as a country with free folk in the Middle Ages. The bishopric of Utrecht did not belong to this Frisia anymore. There were many floods in the 11th and 12th centuries, which led to the deaths of many, and the forming of the Zuiderzee. The largest flood was in 1322.

Loss of territory
The free Frisians (actually petty noblemen) and the city of Groningen founded the Opstalboom League to counter feudalism. It consisted of modern Friesland, Groningen, East Frisia and the German North Sea coast and parts of the Danish North Sea coast. But the Opstalboom league did not only consist of Frisians. The area Zevenwouden was Saxon and the city of Groningen as well. Some Frisians lived under the rule of the counts of Holland in West Friesland. The Opstalboom League was not a success. It collapsed after a few years because of continuous internal strife.

Opstalboom League
The 15th century saw the end of the free Frisians. The city of Groningen started to dominate Groningen. A petty nobleman in East Frisia managed to defeat the other petty noblemen and became count of East Frisia. The archbishop of Bremen-Hamburg and the king of Denmark conquered large areas of Frisia. Only Friesland remained for the Frisian Freedom. Friesland was conquered in the 1490s by duke Albert of Saxony-Meissen.

Frisian territories
Although the Frisian regions have their own separate flags, Frisia did not have a flag of its own until September 2006. The flag for united Frisia was made by the group of Auwerk, which supports a united Frisia as an official country.
As you can see, the flag has been inspired by the Scandinavic cross, like in the Norse and Icelandic flag. The four pompeblêden refer to the seven pompeblêden on the West-Frisian flag, but the amount of four means the four separated frisian regions.

Harry Burnett (H.B.) Reese (born May 24, 1879, Frosty Hill, York County, Pennsylvania; died May 16, 1956, West Palm Beach, Florida) was the inventor of Reese's Peanut Butter Cups and founder of the H.B. Reese Candy Company.
Reese first tried his hand at candy making in Hummelstown and Palmyra, Pennsylvania, where he made Johnny Bars and Lizzy Bars. He first moved to Hershey, Pennsylvania in 1917, where by the mid-1920s he was manufacturing peanut butter cups (then called penny cups because they sold for one cent), among other small candies and assortments.
During World War II, economic constraints led him to discontinue his other candies and concentrate solely on his peanut butter cups. The chocolate for Reese's Peanut Butter Cups was supplied by Hershey in 10-pound blocks and Reese became Hershey's second largest chocolate customer after Mars.
Reese died in West Palm Beach, Florida eight days before his 77th birthday on May 16, 1956 of a heart attack.

Quantum field theory (QFT) provides a theoretical framework, widely used in particle physics and condensed matter physics, in which to formulate consistent quantum theories of many-particle systems, especially in situations where particles may be created and destroyed.
Non-relativistic quantum field theories are needed in condensed matter physics— for example in the BCS theory of superconductivity. Relativistic quantum field theories are indispensable in particle physics (see the standard model), although they are known to arise as effective field theories in condensed matter physics.

Origin of theory
Quantum mechanics in general deals with operators acting upon a (separable) Hilbert space. For a single nonrelativistic particle, the fundamental operators are its position and momentum,
and .
These operators are time dependent in the Heisenberg picture, but we may also choose to work in the Schrödinger picture or (in the context of perturbation theory) the interaction picture.
Quantum field theory is a special case of quantum mechanics in which the fundamental operators are an operator-valued field
.
A single scalar field describes a spinless particle. More fields are necessary for more types of particles, or for particles with spin. For example, particles with spin are usually described by higher order tensor or spinor-valued (or matrix-valued) tensor fields which in turn can be reinterpreted as a possibly large set of scalar fields with appropriate transformation rules as one changes the system of coordinates used.
In quantum field theory, the energy is given by the Hamiltonian operator, which can be constructed from the quantum fields; it is the generator of infinitesimal time translations. (Being able to construct the generator of infinitesimal time translations out of quantum fields means many unphysical theories are ruled out, which is a good thing.)In order for the theory to be sensible, the Hamiltonian must be bounded from below. The lowest energy eigenstate (which may or may not be degenerate) is called the vacuum in particle physics and the ground state in condensed matter physics (QFT appears in the continuum limit of condensed matter systems).

Explanation
Quantum field theory corrects several limitations of ordinary quantum mechanics. The time-dependent Schrödinger equation, in its most commonly encountered form, is
where denotes the quantum state (notation) of a particle with mass m, in the presence of a potential V.

QFT corrections to quantum mechanics
As described in the article on identical particles, quantum-mechanical particles of the same species are indistinguishable, in the sense that the state of the entire system must be symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged. These multi-particle states are extremely complicated to write. For example, the general quantum state of a system of N bosons is written as

where are the single-particle states, N_j is the number of particles occupying state j, and the sum is taken over all possible permutations p acting on N elements. In general, this is a sum of N! (N factorial) distinct terms, which quickly becomes unmanageable as N increases. Large numbers of particles are needed in condensed matter physics where typically the number of particles is on the order of Avogadro's number, approximately 10.

Large numbers of particles.
It is possible to modify the Schrödinger equation to include the rest energy of a particle, resulting in the Klein-Gordon equation or the Dirac equation. However, these equations have many unsatisfactory qualities; for instance, they possess energy eigenvalues which extend to –∞, so that there seems to be no easy definition of a ground state. Such inconsistencies occur because these equations neglect the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativity. Einstein's famous mass-energy relation predicts that sufficiently massive particles can decay into several lighter particles, and sufficiently energetic particles can combine to form massive particles. For example, an electron and a positron can annihilate each other to create photons. Such processes must be accounted for in a truly relativistic quantum theory. This problem brings to the fore the notion that a consistent relativistic quantum theory, even of a single particle, must be a many particle theory.

Schrödinger equation and special relativity

Quantizing a classical field theory
Quantum field theory solves these problems by consistently quantizing a field. By interpreting the physical observables of the field appropriately, one can create a (rather successful) theory of many particles. Here is how it is:
1. Each normal mode oscillation of the field is interpreted as a particle with frequency f.
2. The quantum number n of each normal mode (which can be thought of as a harmonic oscillator) is interpreted as the number of particles.
The energy associated with the mode of excitation is therefore which directly follows from the energy eigenvalues of a one dimensional harmonic oscillator in quantum mechanics. With some thought, one may similarly associate momenta and position of particles with observables of the field.
Having cleared up the correspondence between fields and particles (which is different from non-relativistic QM), we can proceed to define how a quantum field behaves.
Two caveats should be made before proceeding further:
The first method used to quantize field theory was the method now called canonical quantization (earlier known as second quantization). This method uses a Hamiltonian formulation of the classical problem. The later technique of Feynman path integrals uses a Lagrangian formulation. Many more methods are now in use; for an overview see the article on quantization.

Each of these "particles" obeys the usual uncertainty principle of quantum mechanics. The "field" is an operator defined at each point of spacetime.
Quantum field theory is not a wildly new theory. Classical field theory is the same as classical mechanics of an infinite number of dynamical quantities (say, tiny elements of rubber on a rubber sheet). Quantum field theory is the quantum mechanics of this infinite system. Canonical quantization
Suppose we have a system of N bosons which can occupy mutually orthogonal single-particle states and so on. The usual method of writing a multi-particle state is to assign a state to each particle and then impose exchange symmetry. As we have seen, the resulting wavefunction is an unwieldy sum of N! terms. In contrast, in the second quantized approach we will simply list the number of particles in each of the single-particle states, with the understanding that the multi-particle wavefunction is symmetric. To be specific, suppose that N = 3, with one particle in state and two in state. The normal way of writing the wavefunction is
In second quantized form, we write this as

which means "one particle in state 1, two particles in state 2, and zero particles in all the other states."
Though the difference is entirely notational, the latter form makes it easy for us to define creation and annihilation operators, which add and subtract particles from multi-particle states. These creation and annihilation operators are very similar to those defined for the quantum harmonic oscillator, which added and subtracted energy quanta. However, these operators literally create and annihilate particles with a given quantum state. The bosonic annihilation operator a₂ and creation operator have the following effects:


We may well ask whether these are operators in the usual quantum mechanical sense, i.e. linear operators acting on an abstract Hilbert space. In fact, the answer is yes: they are operators acting on a kind of expanded Hilbert space, known as a Fock space, composed of the space of a system with no particles (the so-called vacuum state), plus the space of a 1-particle system, plus the space of a 2-particle system, and so forth. Furthermore, the creation and annihilation operators are indeed Hermitian conjugates, which justifies the way many have written them.
The bosonic creation and annihilation operators obey the commutation relation
where δ stands for the Kronecker delta. These are precisely the relations obeyed by the "ladder operators" for an infinite set of independent quantum harmonic oscillators, one for each single-particle state. Adding or removing bosons from each state is therefore analogous to exciting or de-exciting a quantum of energy in a harmonic oscillator.
The final step toward obtaining a quantum field theory is to re-write our original N-particle Hamiltonian in terms of creation and annihilation operators acting on a Fock space. For instance, the Hamiltonian of a field of free (non-interacting) bosons is

where E_k is the energy of the k-th single-particle energy eigenstate. Note that
.

Canonical quantization for bosons
It turns out that the creation and annihilation operators for fermions must be defined differently, in order to satisfy the Pauli exclusion principle. For fermions, the occupation numbers N_i can only take on the value 0 or 1, since particles cannot share quantum states. We then define the fermionic annihilation operators c and creation operators by




The fermionic creation and annihilation operators obey an anticommutation relation,

One may notice from this that applying a fermionic creation operator twice gives zero, so it is impossible for the particles to share single-particle states, in accordance with the exclusion principle.

Canonical quantization for fermions
When we re-write a Hamiltonian using a Fock space and creation and annihilation operators, as in the previous example, the symbol N, which stands for the total number of particles, drops out. This means that the Hamiltonian is applicable to systems with any number of particles. Of course, in many common situations N is a physically important and perfectly well-defined quantity. For instance, if we are describing a gas of atoms sealed in a box, the number of atoms had better remain a constant at all times. This is certainly true for the above Hamiltonian. Viewing the Hamiltonian as the generator of time evolution, we see that whenever an annihilation operator a_k destroys a particle during an infinitesimal time step, the creation operator to the left of it instantly puts it back. Therefore, if we start with a state of N non-interacting particles then we will always have N particles at a later time.
On the other hand, it is often useful to consider quantum states where the particle number is ill-defined, i.e. linear superpositions of vectors from the Fock space that possess different values of N. For instance, it may happen that our bosonic particles can be created or destroyed by interactions with a field of fermions. Denoting the fermionic creation and annihilation operators by and c_k, we could add a "potential energy" term to our Hamiltonian such as:

This describes processes in which a fermion in state k either absorbs or emits a boson, thereby being kicked into a different eigenstate k + q. In fact, this is the expression for the interaction between phonons and conduction electrons in a solid. The interaction between photons and electrons is treated in a similar way; it is a little more complicated, because the role of spin must be taken into account. One thing to notice here is that even if we start out with a fixed number of bosons, we will generally end up with a superposition of states with different numbers of bosons at later times. On the other hand, the number of fermions is conserved in this case.
In condensed matter physics, states with ill-defined particle numbers are also very important for describing the various superfluids. Many of the defining characteristics of a superfluid arise from the notion that its quantum state is a superposition of states with different particle numbers.

Significance of creation and annihilation operators
We can now define field operators that create or destroy a particle at a particular point in space. In particle physics, these are often more convenient to work with than the creation and annihilation operators, because they make it easier to formulate theories that satisfy the demands of relativity.
Single-particle states are usually enumerated in terms of their momenta (as in the particle in a box problem.) We can construct field operators by applying the Fourier transform to the creation and annihilation operators for these states. For example, the bosonic field annihilation operator is

The bosonic field operators obey the commutation relation
where δ(x) stands for the Dirac delta function. As before, the fermionic relations are the same, with the commutators replaced by anticommutators.
It should be emphasized that the field operator is not the same thing as a single-particle wavefunction. The former is an operator acting on the Fock space, and the latter is just a scalar field. However, they are closely related, and are indeed commonly denoted with the same symbol. If we have a Hamiltonian with a space representation, say

where the indices i and j run over all particles, then the field theory Hamiltonian is

This looks remarkably like an expression for the expectation value of the energy, with φ playing the role of the wavefunction. This relationship between the field operators and wavefunctions makes it very easy to formulate field theories starting from space-projected Hamiltonians.

Field operators
So far, we have shown how one goes from an ordinary quantum theory to a quantum field theory. There are certain systems for which no ordinary quantum theory exists. These are the "classical" fields, such as the electromagnetic field. There is no such thing as a wavefunction for a single photon in classical electromagnetism, so a quantum field theory must be formulated right from the start.
The essential difference between an ordinary system of particles and the electromagnetic field is the number of dynamical degrees of freedom. For a system of N particles, there are 3N coordinate variables corresponding to the position of each particle, and 3N conjugate momentum variables. One formulates a classical Hamiltonian using these variables, and obtains a quantum theory by turning the coordinate and position variables into quantum operators, and postulating commutation relations between them such as
For an electromagnetic field, the analogue of the coordinate variables are the values of the electrical potential and the vector potential at every point . This is an uncountable set of variables, because is continuous. This prevents us from postulating the same commutation relation as before. The way out is to replace the Kronecker delta with a Dirac delta function. This ends up giving us a commutation relation exactly like the one for field operators! We therefore end up treating "fields" and "particles" in the same way, using the apparatus of quantum field theory. Only by accident electrons were not regarded as de Broglie waves and photons governed by geometrical optics were not the dominant theory when QFT was developed.

Quantization of classical fields

Path integral methods
There have been many attempts to put quantum field theory on a firm mathematical footing by formulating a set of axioms for it. These attempts fall into two broad classes.
The first class of axioms (most notably the Wightman, Osterwalder-Schrader, and Haag-Kastler systems) tried to formalize the physicists' notion of an "operator-valued field" within the context of functional analysis. These axioms enjoyed limited success. It was possible to prove that any QFT satisfying these axioms satisfied certain general theorems, such as the spin-statistics theorem and the PCT theorems. Unfortunately, it proved extraordinarily difficult to show that any realistic field theory (e.g. quantum chromodynamics) satisfied these axioms. Most of the theories which could be treated with these analytic axioms were physically trivial: restricted to low-dimensions and lacking in interesting dynamics. Constructive quantum field theory is the construction of theories which satisfy one of these sets of axioms. Important work was done in this area in the 1970s by Segal, Glimm, Jaffe and others.
In the 1980s, a second wave of axioms were proposed. These axioms (associated most closely with Atiyah and Segal, and notably expanded upon by Witten, Borcherds, and Kontsevich) are more geometric in nature, and more closely resemble the path integrals of physics. They have not been exceptionally useful to physicists, as it is still extraordinarily difficult to show that any realistic QFTs satisfy these axioms, but have found many applications in mathematics, particularly in representation theory, algebraic topology, and geometry.
Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics. In fact, one of the Clay Millennium Prizes offers $1,000,000 to anyone who proves the existence of a mass gap in Yang-Mills theory. It seems likely that we have not yet understood the underlying structures which permit the Feynman path integrals to exist.

The axiomatic approach
Some of the problems and phenomena eventually addressed by renormalization actually appeared earlier in the classical electrodynamics of point particles in the 19th and early 20th century. The basic problem is that the observable properties of an interacting particle cannot be entirely separated from the field that mediates the interaction. The standard classical example is the energy of a charged particle. To cram a finite amount of charge into a single point requires an infinite amount of energy; this manifests itself as the infinite energy of the particle's electric field. The energy density grows to infinity as one gets close to the charge.
A single particle state in quantum field theory incorporates within it multiparticle states. This is most simply demonstrated by examining the evolution of a single particle state in the interaction picture—

Taking the overlap with the initial state, one retains the even powers of H_I. These terms are responsible for changing the number of particles during propagation, and are therefore quintessentially a product of quantum field theory. Corrections such as these are incorporated into wave function renormalization and mass renormalization. Similar corrections to the interaction Hamiltonian, H_I, include vertex renormalization, or, in modern language, effective field theory.

Renormalization
A gauge theory is a theory which admits a symmetry with a local parameter. For example, in every quantum theory the global phase of the wave function is arbitrary and does not represent something physical, so the theory is invariant under a global change of phases (adding a constant to the phase of all wave functions, everywhere); this is a global symmetry. In quantum electrodynamics, the theory is also invariant under a local change of phase, that is - one may shift the phase of all wave functions so that in every point in space-time the shift is different. This is a local symmetry. However, in order for a well-defined derivative operator to exist, one must introduce a new field, the gauge field, which also transforms in order for the local change of variables (the phase in our example) not to affect the derivative. In quantum electrodynamics this gauge field is the electromagnetic field. The change of local change of variables is termed gauge transformation.
In quantum field theory the excitations of fields represent particles. The particle associated with excitations of the gauge field is the gauge boson, which is the photon in the case of quantum electrodynamics.
The degrees of freedom in quantum field theory are local fluctuations of the fields. The existence of a gauge symmetry reduces the number of degrees of freedom, simply because some fluctuations of the fields can be transformed to zero by gauge transformations, so they are equivalent to having no fluctuations at all, and they therefore have no physical meaning. Such fluctuations are usually called "non-physical degrees of freedom" or gauge artifacts; usually some of them have a negative norm, making them inadequate for a consistent theory. Therefore, if a classical field theory has a gauge symmetry, then its quantized version (i.e. the corresponding quantum field theory) will have this symmetry as well. In other words, a gauge symmetry cannot have a quantum anomaly. If a gauge symmetry is anomalous (i.e. not kept in the quantum theory) then the theory is non-consistent: for example, in quantum electrodynamics, had there been a gauge anomaly, this would require the appearance of photons with longitudinal polarization and polarization in the time direction, the latter having a negative norm, rendering the theory inconsistent; another possibility would be for these photons to appear only in intermediate processes but not in the final products of any interaction, making the theory non unitary and again inconsistent (see optical theorem).
In general, the gauge transformations of a theory consist several different transformations, which may not be commutative. These transformations are together described by a mathematical object known as a gauge group. Infinitesimal gauge transformations are the gauge group generators. Therefore the number of gauge bosons is the group dimension (i.e. number of generators forming a basis).
All the fundamental interactions in nature are described by gauge theories. These are:

Quantum electrodynamics, whose gauge transformation is a local change of phase, so that the gauge group is U(1). The gauge boson is the photon.
Quantum chromodynamics, whose gauge group is SU(3). The gauge bosons are eight gluons.
The electroweak Theory, whose gauge group is (a direct product of U(1) and SU(2)).
Gravity, whose classical theory is general relativity, admits the equivalence principle which is a form of gauge symmetry. Gauge theories
Supersymmetry assumes that every fundamental fermion has a superpartner which is a boson and vice versa. It was introduced in order to solve the so-called Hierarchy Problem, that is, to explain why particles not protected by any symmetry (like the Higgs boson) do not receive radiative corrections to its mass driving it to the larger scales (GUT, Planck...). It was soon realized that supersymmetry has other interesting properties: its gauged version is an extension of general relativity (Supergravity), and it is a key ingredient for the consistency of string theory.
The way supersymmetry protects the hierarchies is the following: since for every particle there is a superpartner with the same mass, any loop in a radiative correction is cancelled by the loop corresponding to its superpartner, rendering the theory UV finite.
Since no superpartners have yet been observed, if supersymmetry exists it must be broken (through a so-called soft term, which breaks supersymmetry without ruining its helpful features). The simplest models of this breaking require that the energy of the superpartners not be too high; in these cases, supersymmetry is expected to be observed by experiments at the Large Hadron Collider.

Supersymmetry

List of quantum field theories
Feynman path integral
Quantum chromodynamics
Quantum electrodynamics
Schwinger-Dyson equation
Relationship between string theory and quantum field theory
Abraham-Lorentz force
Photon polarization
Theoretical and experimental justification for the Schrödinger equation
Invariance mechanics Notes

Wilczek, Frank ; Quantum Field Theory, Review of Modern Physics 71 (1999) S85-S95. Review article written by a master of Q.C.D., Nobel laureate 2003. Full text available at : hep-th/9803075
Ryder, Lewis H. ; Quantum Field Theory (Cambridge University Press, 1985), [ISBN 0-521-33859-X]. Introduction to relativistic Q.F.T. for particle physics.
Zee, Anthony ; Quantum Field Theory in a Nutshell, Princeton University Press (2003) [ISBN 0-691-01019-6].
Peskin, M and Schroeder, D. ;An Introduction to Quantum Field Theory (Westview Press, 1995) [ISBN 0-201-50397-2]
Weinberg, Steven ; The Quantum Theory of Fields (3 volumes) Cambridge University Press (1995). A monumental treatise on Q.F.T. written by a leading expert, Nobel laureate 1979.
Loudon, Rodney ; The Quantum Theory of Light (Oxford University Press, 1983), [ISBN 0-19-851155-8]
Greiner, Walter and Müller, Berndt (2000). Gauge Theory of Weak Interactions. Springer. ISBN 3-540-67672-4. 
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Gordon L. Kane (1987). Modern Elementary Particle Physics. Perseus Books. ISBN 0-201-11749-5.