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Talk:The Generation Game

Jim Davidson
I've hardly ever watched it, but wasn't Jim Davidson the most recent host? -- S
 * Yes. Fourohfour 22:54, 21 August 2006 (UTC)
 * Bit belated - that comment was written in 2003, when the article erroneously said Bruce Forsyth was the most recent host. Still, it's good that he finally has an answer after three years. BillyH 23:15, 21 August 2006 (UTC)

Trivia
I'm sure Trivia is discouraged anyway, but the last line of trivia regarding Matthew Kelly's comment to Jim Davidson is completely unsupported, and very odd. --Scottdavies 09:09, 1 November 2007 (UTC)

A Commons file used on this page or its Wikidata item has been nominated for speedy deletion
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for speedy deletion: You can see the reason for deletion at the file description page linked above. —Community Tech bot (talk) 18:08, 28 February 2023 (UTC)
 * Generation Jim.jpg

Nice to hear you. To hear you nice!
Such lovely comments - so much better than last week's! — Preceding unsigned comment added by 2.30.22.214 (talk) 20:08, 30 June 2023 (UTC)

title: Talk:The Generation Game


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India Unbound

India Unbound: From Independence to Global Information Age is a 2000 non-fiction book by Gurcharan Das. It is an account of India's economic journey after its Independence in 1947.

Overview
India Unbound is mainly about the transformation of India from birth of the writer in (1942) to 1999. The author majorly speaks about the Indian politics and the economy of India. He categorizes the complete timeline from 1942 to 1999 in three major sections: 'Spring of Hope (1942–65)', 'the Lost Generation (1966–91)' & 'Rebirth of Dream (1991–99)' and tell various stories(memoirs) and the historical facts of that time. The book is a great mixture of memoir, economic analysis, social investigation, political scrutiny and managerial outlook being thrown into the understanding of India. It begins shortly before independence and continues until the new millennium. As other authors cherish the revolution that began with independence in 1947, Gurcharan Das does not find full cause for jubilation until 1991, when India unleashed a series of economic reforms, the start of an "economic revolution" that he believes "may well be more important than the political revolution."

The book starts with a brief history of 18th and 19th century India. Gurucharan Das starts with the augmentation of the railways in India, which was presumed to usher India into the industrial revolution. The book also points to the factors which led to the decline in the Indian economy under the British Raj.

The ‘Spring of Hope’ (1942–65) describes the period in Indian history when it embarks upon as an independent country. The post independence economic policy followed the first Prime Minister of India Jawaharlal Nehru is socialist in nature which gives all the control over the industry to the public sector. Nehru was ably supported by some of the best economists from across India. He sets up a mixed economy which is a middle path between democratic rights followed by the Western countries and socialist ideology of the Soviet world.

The 'Lost Generation' (1966–91) refers to the phase of Indian Polity and economy which saw fading of the Indian dream. The economic policies followed by Indira Gandhi with the advice of her trusted economic advisers led India in to a very slow growth rate. The book cites an irony when it states that the 'Garibi Hatao' policies followed by Mrs. Indira Gandhi were actually working counter-productively and throwing millions of Indians further into poverty rather than alleviating their plight. The author marks the declaration of 'Emergency' as the lowest point of Indian Polity. Gurucharan Das gives a resounding account of his personal tryst with the Indian bureaucratic system prevailing at that time. He gives the account of incompetency spread in the Indian set-up which was choking the country’s economy to the death and License Raj which was the greatest hurdle to the growth of India.

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

title: India Unbound


text:
Talk:Rolan Bolan

Untitled
Umm. This page probably needs a bit more info! Besides being a son of those two folks, what else do we know about this person --Cimon Avaro on a pogo-stick 12:08 25 Jul 2003 (UTC)

I fancy him... and his father! :)

Redirect
Kingboyk, an AFD for this page resulted in the redirect to Marc Bolan. Per WP policy, that has been carried out, and it is not your place to override it. Ckessler 20:48, 8 December 2006 (UTC)
 * Yes it is. The AFD didn't result in a delete or a protection, but a soft redirect. I'm entitled to override that per WP:BOLD. I believe the guy is notable and the AFD only considered notability as a musician. --kingboyk 19:48, 9 December 2006 (UTC)

The AFD result was that he is not well known enough in his own right, per WP:Notability. That means there should not be a page about this person on Wikipedia. Ckessler 19:53, 9 December 2006 (UTC)

Rolan's own page
I really do believe he stands as an artist on his own right, so he's entitled to have his own wiki page. Rolan is a recording artist working on his first official release on 2007 September. So I think setting his page to Marc Bolan redirection again wouldn't be fair. (Goodlookingmanager 09:13, 28 April 2007 (UTC))

 * When he releases an album, of his own, on a major or indie record label, then the redirect can be removed. Until then, he has done nothing that points to notability. Ckessler 15:51, 28 April 2007 (UTC)

 * Oh, so you need a major label deal as a musician to be notable? Please, then go and start deleting all artists signed to independent labels as they are not important.. Please look around first and you'll see that thousands of artists are successful without a major label, let's name The Shins, franz ferdinand, Arctic monkeys, all with indies... Sorry, but it's not your call to deicide about whether an artist's work is notable or not at a certain point. I will keep adding back Rolan's profile. I'm sad to learn that there are such frustrated individuals out there who can't create something on their own and therefore they keep on hassling others. pathetic.(Goodlookingmanager 16:49, 28 April 2007 (UTC))

 * Those are artists who have had videos on MTV, done tours in large venues, had articles in mainstream press written about them, and are generally notable. Show me some proof that Mr. Bolan has done any of that, and meets the standards of WP:BIO, and I might stop reinstating the redirect. Personally attacking me is not going to help you make your case. Ckessler 17:26, 28 April 2007 (UTC)

 * It's not personal, but you seem to have little knowledge about the music industry. Being featured on MTV, having gigs in large venues are not the sole indicators of someone being relevant musician, artis. You can have an independent release, play small venues, as long as people take notice of your art your entitled to be called performing artist. Still I have no idea how you think you're able to judge an artist career. Can you please give me an explination what proplem you have with Rolan Bolan? Go and find other pages to delete if you have the mania to do so. (Goodlookingmanager 18:00, 28 April 2007 (UTC))

 * No, not by WP standards. Please see WP:BIO and WP:Notability. Wikipedia has hard and fast guidelines about what makes a person notable, and whether or not they should have an article here. Stop reverting the change, or I will bring an admin in. Ckessler 08:16, 29 April 2007 (UTC)

 * You misinterpret the rules. I'm happy to see an admin solving this, until then I do what I feel the right to be done according to the wikipedia rules, and I advise you to delete some other pages rather than the one of a notable person.(Goodlookingmanager 10:54, 29 April 2007 (UTC))

 * How am I misinterpreting the rules? The AFD decided redirect, and you have added nothing to the article that proves notability. Google showing nothing that proves notability. He has done nothing to be notable by WP standards, musically. Being born to two famous people is not notable, and I can point you to many AFD pages that prove it. Ckessler 17:30, 29 April 2007 (UTC)

 * The AFD decided redirect, and I agree with that. Make it a redirect. If you want, have a little section about him as an aspiring musician in the other article. No need for his own. He doesn't have a wide enough fan base yet. Maybe, once he career blossoms, he can warrant his own. -  hmwith  talk   17:41, 29 April 2007 (UTC)

My opinion: Wikipedia has a clear checklist at WP:MUSIC to determine if a musician is notable enough for an article. If Goodlookingmanager can supply reliable sources showing that Rolan Bolan meets at least one of those criteria, then an article should be written. Otherwise, the redirect should remain. Also, just in case it applies, I suggest reviweing WP:COI. Anomie 21:16, 29 April 2007 (UTC)

Redirect protected
I have protected the redirect to Marc Bolan. If you think the Articles for deletion/Rolan Bolan discussion needs a review, then Deletion review is the place.--Commander Keane 04:40, 1 May 2007 (UTC)

Thank you. Ckessler 04:43, 1 May 2007 (UTC)

title: Talk:Rolan Bolan


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User:Michael Larsen

Although I am primarily a number theorist, I have also worked in a number of other areas in mathematics, including algebraic geometry, representation theory, topology, and combinatorics. A number of my papers and preprints are available online. I am a Professor in the Mathematics Department at Indiana University.

New pages: adjoint representation, algebraically independent, binary Golay code, cokernel, Conway group, Coxeter group, Dynkin diagram, Fredholm operator, global field, Leech lattice, modular form, representation of Lie algebras, root system, transcendence degree, upper half plane, weight (representation theory), Weyl group

Modified pages: Bezout's theorem, braid theory, CW complex, fundamental theorem of algebra, Korteweg-de Vries equation, Lie algebra, local field, moduli space

title: User:Michael Larsen


text:
Triratna Buddhist Community

The Triratna Buddhist Community, formerly the Friends of the Western Buddhist Order (FWBO), is an international fellowship of Buddhists. It was founded in the UK in 1967 by Sangharakshita (born Dennis Philip Edward Lingwood) and describes itself as "an international network dedicated to communicating Buddhist truths in ways appropriate to the modern world". In keeping with Buddhist traditions, it also pays attention to contemporary ideas, particularly drawn from Western philosophy, psychotherapy, and art.

Worldwide, more than 100 groups are affiliated with the community, including in North America, Australasia and Europe. In the UK, it is one of the largest Buddhist movements, with some 30 urban centres and retreat centres. The UK based international headquarters is at Adhisthana retreat centre in Coddington, Herefordshire. Its largest following, however, is in India, where it is known as Triratna Bauddha Mahāsaṅgha (TBM), formerly the Trailokya Bauddha Mahasangha Sahayaka Gana (TBMSG).

The community has been described as "perhaps the most successful attempt to create an ecumenical international Buddhist organization". It has also been criticised for lacking "spiritual lineage" and has faced allegations of sexual exploitation and abuse during the 1970s and 1980s.

Practices and activities
Meditation is the common thread through activities. Order members teach two practices: The "mindfulness of breathing" (anapanasati), in which practitioners focus on the rise and fall of the breath; and metta bhavana, which approximately translates from the original Pali as "the cultivation of lovingkindness". These practices are felt to be complementary in promoting equanimity and friendliness towards others. Some friends of the Order may have little, if any, other involvement in its activities; but friendship, Sangha, and community are encouraged at all levels as essential contexts for meditation.

The founder, Sangharakshita, taught a system of practice emphasising five types of meditation. The first two according to his system ('integration' and 'positive emotion'), can be correlated to the traditional category of "calmingsamatha" practices, and the last two (spiritual death and spiritual rebirth) can be correlated to "insight" or vipassana practices. For those not ordained into the Triratna Buddhist Order, the practices associated with the first two are emphasised, though the spirit of the last two is also taught. The five types of meditation correspond to five 'stages' of the spiritual life.

These five stages are:

 * 1) Integration: The main practice at this stage is the mindfulness of breathing, which is intended to have the effect of "integrating the psyche" – improving mindfulness and concentration, and reducing psychological conflict.
 * 2) Positive emotion: The second aspect of samatha is developing positivity – an other-regarding, life-affirming attitude. The Brahmavihara meditations, especially the metta bhavana or cultivation of loving kindness meditations, are the key practices intended to foster the development of positive emotion.
 * 3) Spiritual death: The next stage is to develop insight into what is seen to be the emptiness of the self and reality. Meditations at this stage include considering the elements of which self and world are thought to be composed; contemplating impermanence (particularly of the body); contemplating suffering; and contemplating sunyata.
 * 4) Spiritual rebirth: Triratna teaches that, with the development of insight and the death of the limited ego-self, a person is spiritually reborn. Practices which involve the visualisation of Buddhas and bodhisattvas are among the main practices in this phase. At ordination, each dharmachari is given an advanced visualisation meditation on a particular figure.
 * 5) Receptivity and spontaneous compassionate activity: The practice associated with this phase is known as "just sitting" or "formless" meditation. Outside of meditation, this is the phase of "Dharmic responsiveness" - doing whatever needs to be done in any situation.

Centres also teach scripture, yoga and other methods of self-improvement, some of which are felt by some commentators to come from outside the Buddhist tradition. Recently, community activities have begun to include outdoor festivals, online meditation courses, arts festivals, poetry and writing workshops, tai chi, karate, and pilgrimages to Buddhist holy sites in India. For many years, the community charity Karuna Trust (UK) has raised money for aid projects in India.

As among Buddhists generally, Puja is a ritual practice at some events, intended to awaken the desire to liberate all beings from suffering. The most common ritual consists of a puja, derived and adapted from the Bodhicaryavatara of Shantideva.

Retreats provide a chance to focus on meditational practice more intensely, in a residential context outside of a retreatant's everyday life. Community retreats can be broadly categorised into meditation retreats, study retreats, and solitary retreats. Retreat lengths vary from short weekends to one or two weeks.

Businesses, said to operate on the principle of "right livelihood", generate funds for the movement, as well seeking to provide environments for spiritual growth through employment. Emphasis is placed on teamwork, and on contributing to the welfare of others: for example by funding social projects and by considering ethical matters such as fair trade. The largest community business was Windhorse: Evolution, a gift wholesaling business and a chain of gift shops. Windhorse: Evolution closed down in 2015.

Many cities with a Triratna centre also have a residential community. The first of these was formed after a retreat where some participants wanted to continue retreat-style living. Since it was felt that the most stable communities tended to be single sex, this has become the paradigm for communities. Support from fellow practitioners in a community is seen to be effective in helping members make spiritual progress.

The largest Triratna centre in the UK is the London Buddhist Centre in Bethnal Green, East London, which offers drop-in lunchtime meditation sessions each weekday, open to beginners, as well as courses and classes through the week. The centre's courses for depression, based on the mindfulness-based cognitive behavioural therapy methodology of Jon Kabat-Zinn at the University of Massachusetts Amherst, featured in the Financial Times in 2008. This initiative is supported by the local authority, the London borough of Tower Hamlets. The Times has also reported on the centre's work with those affected by alcohol dependency

Defining the movement
According to the community, six characteristics define it:
 * 1) An ecumenical movement. It is not identified with any particular strand or school of Buddhism, but draws inspiration from many. It calls itself "ecumenical" rather than "eclectic" because it is founded on the premise that there is an underlying unity to all schools.
 * 2) "Going for refuge" is central. "Going for Refuge to the Three Jewels" – meaning the Buddha, the Dharma, and the Sangha – is considered to be what makes someone a Buddhist.
 * 3) A unified Order. Unlike some sangha, the community does not propagate a monastic lineage. Sangharakshita devised a non-monastic ordination system, whilst also allowing the undertaking of the "anagarika" precept which enjoins celibacy. Identical ordination is open to both sexes. While the movement regards single-sex activities as important to spiritual growth, men and women are recognised as being equally able to practice and develop spiritually.
 * 4) An emphasis on spiritual friendship. There is a strong emphasis on the sangha, and spiritual friendship based on shared values. The community teaches that spending time with friends who share ideals, and engaging in ritual practice with them, supports ethical living and the arising of the bodhicitta.
 * 5) Teamwork. Working together in teams, in the spirit of generosity and with a focus on ethics, is considered a transformative spiritual practice.
 * 6) Importance of art. Engagement in, and an appreciation of, the arts are considered to be a valuable aspect of spiritual practice. The community teaches that a refinement of one's artistic tastes can help refine emotional sensitivity and provide a channel for the expression of right living, and spiritual growth. More broadly, the movement seeks ways to re-express Buddhism by making connections with sympathetic elements in the surrounding culture, regarding the arts as such an aspect of western culture.

"The FWBO's attitude to spreading the Dharma is one of heartfelt urgency," wrote Stephen Batchelor, a prominent British Buddhist author, in a book published in 1994. "For the FWBO, Western Society as such needs to be subject to the unflinching scrutiny of Buddhist values."

The Triratna Buddhist Order
The Triratna Buddhist Order is the focal-point of the community, and is a network of friendships between individuals who have made personal commitments to the Buddha, the dharma and the sangha, in communion with others. Members are known as dharmacharis (masculine) or dharmacharinis (feminine), and are ordained in accord with a ceremony formulated by the founder. At ordination they are given a religious name in Pali or Sanskrit. While there is an informal hierarchy within the order, there are no higher ordinations. A small number of members, however, take vows of celibacy and adopt a simpler lifestyle. Contrary to the traditional Buddhist structure of separating lay and monastic members, the order combines monastic and lay lifestyles under one ordination, a practice not dissimilar to that which evolved in some Japanese schools of Buddhism. When members are ordained, they are given a white kesa to signify their ordination; golden kesas may be taken later by those choosing to make a public commitment to greater simplicity of lifestyle, including celibacy.

As with followers of the Shingon school of Buddhism, order members observe ten precepts (ethical training rules). These precepts are different from monastic vows and do not appear in the Vinaya Pitaka, but were formulated on the basis of the so-called "dasa-kusala-dhammas" (ten wholesome actions). These are found in several places in the Pāli Canon, as well as in some Sanskrit sources. The karma sections of the fundamental meditation texts of all four schools of Tibetan Buddhism also list these acts as basic guidelines for lay or ordained practitioners intent on observing the law of cause and effect.

Beyond this, a commitment to personal dharma practice and to remain in communication with other members are the only expectations. Ordination confers no special status, nor any specific responsibilities, although many order members choose to take on responsibilities for such things as teaching meditation and dharma. In mid-2008, there were around 1,500 members of the order, in more than 20 countries.

The wider community
In the Triratna community, as in the Theravada, Mahayana and Vajrayana traditions, sangha is interpreted as the Buddhist community as a whole. Someone who regularly attends community activities is considered to be a "friend". Friends do not have to regard themselves as Buddhists, and can be of any faith, or none. Some choose, after some time, to participate in a formal ceremony of affiliation, and thus become a "mitra." "Mitra" is Sanskrit for "friend", which in this case denotes a person who considers themselves Buddhist, who makes an effort to live in accordance with the five ethical precepts, and who feels that this spiritual community is the appropriate one for them.

Those who wish to join the Order must request this in writing. It will then take several years to prepare for ordination. This is an informal process, the focus of which is to deepen one's commitment.

Some Friends, Mitras and Order members decide, at least for a while, to study teachings from outside the community, including non-Buddhist traditions such as Sufism.

History
As the Friends of the Western Buddhist Order, the community was founded in London in April 1967 by Sangharakshita. He had then recently returned to England after spending two decades as a Buddhist and monk in India, following demobilisation from the British army. He had been born in south London as Dennis Lingwood, in August 1925. He would lead the organisation until his formal retirement in 1995, and would continue to exert a decisive influence on its thinking and practices thereafter.

In the 1990s, the order grew in India, and, according to the Encyclopedia of Buddhism, Indian members now make up about half the movement's formal membership. In a book published in 2005, the FWBO's members and supporters were estimated to number 100,000, the majority of whom were in India.

In 1997, the responsibility for ordination and spiritual leadership passed to a "preceptor's college", based in Birmingham. In 2000, the first chair of a preceptor's council was chosen by Sangharakshita. In future, this position will be elected by the WBO to five-year terms.

In 2003, the public preceptors, responding to feedback, decided to move away from a formal relationship to the order and movement, and to concentrate on the ordination of new order members, teaching and dharma practice. At the same time, to increase flexibility, the number of preceptors was expanded.

Name change
In the spring of 2010, the Western Buddhist Order and Friends of the Western Buddhist Order changed their names to the Triratna Buddhist Order (which approximates in English to the name used in India - Triratna Bauddha Mahasangha) and the Triratna Buddhist Community. Since its western foundation the movement had spread to other parts of the world, including India where it was known by a different, non-western, name, and Indian members had long wished for one single worldwide name. An official history acknowledges this to have been controversial among some Order members. ("Triratna" is a Sanskrit term meaning Three Jewels)

Spiritual lineage
Although Sangharakshita studied under, and in some cases received initiations from, eminent Buddhist teachers during his two decades in India, including Jagdish Kashyap, Dhardo Rinpoche, HH Dudjom Rinpoche, HH Dilgo Khyentse, and Jamyang Khyentse Rinpoche, author James William Coleman has noted that "he never worked closely enough with any teacher to be recognized as a dharma-heir".

Rather than leadership by a guru, the community instead operates through what has been called a "friendly hierarchy," which some critics have said can cause problems. In 1997, Stephen Batchelor, a prominent Buddhist commentator, was quoted as saying that the FWBO operated as "a self-enclosed system" and that their writings "have the predictability of those who believe they have all the answers".

In 1980 Sangharakshita wrote of his "conviction that the less the FWBO is involved with 'Buddhist groups' and with individuals affiliated to existing Buddhist traditions, the better." Currently, however, the community is a member of the European Buddhist Union and the Network of Buddhist Organisations, individual members of the order serve on the board of the International Network of Engaged Buddhists, and the FWBO's former magazine, Dharma Life, frequently carried articles by Buddhists from other organisations.

1997 Guardian report
In October 1997, a report by Madeleine Bunting, then the religious affairs correspondent of the British Guardian newspaper, made wide-ranging allegations of sexual misconduct, dogmatism and misogyny within the movement during the 1970s and 1980s.

The most detailed complaints reported were claims by Mark Dunlop, who had lived with the movement's founder for a number of years in the early 1970s, and left the order in 1985. The report described intimate details of what Dunlop characterised as their relationship, and claimed that Sangharakshita, who declined to comment, had told him “that to develop spiritually he had to get over his anti-homosexual conditioning.”

The report contained further allegations from an anonymous source, who said that he had been persuaded into a sexual relationship with the leader of FWBO's centre in Croydon, south of London. “The head of the community was a very powerful, intrusive personality and incredibly manipulative. He would intuitively become aware of people's vulnerabilities,” the source was reported to have said.

A third concern drew on complaints by the mother of a former FWBO member who had committed suicide in 1990, following a history of depression. A report by a clinical psychologist said, among other things: “He feels the community attempted to alienate him from his family and from women, and that direct attempts were made to encourage him to practise homosexuality. He stated that he did not indulge in homosexual practices, although attempts were made for him to do so both by using inducements and by using threats.”

Sexuality, women and the family
Following The Guardian report, a widespread debate ensued. Critics pointed to writings by Sangharakshita, and his senior advisor Dharmachari Subhuti, which placed such emphasis on single-sex activities, and what the Triratna Buddhist Community calls “spiritual friendship”, that the potential for misunderstandings or inappropriate behaviour appeared to some to be inevitable.

The FWBO had long been known for questioning assumptions about family life, and, according to the Encyclopedia of Buddhism:

 * "Among the unique characteristics of the FWBO has been the open acceptance of homosexuality among the members. Order members have concluded that precept rules against abusing sexuality do not relate to the formal structure of sexual relations so much as to the nature of the relationship itself."

In an official biography of Sangharakshita, published in 1994, and reissued in 2009, Subhuti says: “Sangharakshita believes that men must break down their fear of homosexuality by facing the fact that there may be some element of sexual attraction towards their friends.”

The Guardian report also raised controversy over statements by Sangharakshita which it interpreted as adverse to women and the family. Expressing views found in Buddhist texts from their earliest times,  he has argued, for example, that, at least in the early stages of their spiritual careers, men are more apt to commit themselves to the spiritual life than women. In 1986, he wrote that the couple and nuclear family can be sources of neurosis.
 * "A couple consists, in fact, of two half-people, each of whom unconsciously invests part of his or her total being in the other: each is dependent on the other for the kind of psychological security that can be found, ultimately, only within oneself."(Sangharakshita, 1986, Alternative Traditions).

Although scriptures and historians recognise that the Buddha himself had concerns over such issues, particularly after the birth of his son Rahula, when he left home convinced that "family life was incompatible with the highest forms of spirituality", critics cite Sangharakshita's conservative views as evidence that misogynystic attitudes persisted in the FWBO during the 1980s. Evidence that those ideas may have been more widely held is also found in the writings of Subhuti, who echoes the sutras when he says in his book, 'Women, men and angels,' that to be reborn as a woman is to be less spiritually able than to be reborn as a man.

As a movement, what was then the FWBO gave detailed responses and staunchly supported its founder, while in 2010 the renamed movement published an official history which acknowledged widespread concern among order members that, at least in the 1980s and before, the founder had misused his position as a Buddhist teacher to sexually exploit young men.

The controversies attracted little public interest, and between 2004 and 2008 both The Guardian, and its sister Sunday newspaper, The Observer, ran three supportive articles, recommending community activities. A Guardian Web directory listed the FWBO website as "a good starting point for children."

In a 2009 interview with a member of the group, Sangharakshita replied to a question about sex between him and students: “Perhaps in a very few cases they were not as willing as I had supposed at the time – that is possible." In reply to a follow-up question, he said: "I did not regard myself as a teacher with a capital T.” He added: “I have had many, many human encounters, the great majority non-sexual, and most of those encounters, including the sexual ones, have been satisfactory for both parties. If there were any encounters that were not satisfactory for the other person, whether at the time or in retrospect, then that is a pity and I am truly sorry that that should be the case."

2016 abuse report
In September 2016, BBC News reported that former members of the Triratna movement claimed that they had been subject to sexual abuse by Sangharakshita at the group's retreat centre in Norfolk. Following discussion on social media, another former member claimed that he had been groomed for sex by another senior member of the order at the group's centre in Croydon in 1980s when he was 16 years old.

On 4 January 2017, following his treatment in hospital for pneumonia, Sangharakshita issued a statement expressing “deep regret for all the occasions on which I have hurt, harmed or upset fellow Buddhists, and ask for their forgiveness." On 19 January 2017, Triratna's leadership issued a statement in response to Sangharakshita's apology: “Consideration of some aspects of Bhante [Sangharakshita]'s past has been difficult for some of us in the College, as it has been for many of our brothers and sisters in the Order and others associated with our community. Bhante is the founder of our Order and Movement, and we feel enormous appreciation and gratitude to him for his teachings and inspiration – and yet at the same time we must acknowledge the effects of some of his past actions."

2018 internal report
In October 2018, a group of members of the Triratna community released an internal report saying that more than one in 10 of them claim to have experienced or observed sexual misconduct while in the order, by Sangharakshita and other senior members. In July 2019, the group shared the report with The Observer newspaper.

Triratna Buddhist Community sites

 * Triratna Buddhist Community Official Site
 * Triratna Buddhist Community News News and information related to the Triratna Buddhist Community
 * Triratna Buddhist Community People Triratna Buddhist Community People
 * Sangharakshita.org Sangharakshita's home page
 * Dharma Life website of Triratna Buddhist Community-produced magazine
 * Buddhist free audio online online free Buddhist audio talks and discussions
 * Karuna Trust Buddhist Trust working against poverty and discrimination in South Asia
 * Windhorse Publications Publication and distribution of Buddhist literature
 * TBMSG renamed Triratna Baudha Mahasangha (TBM)

Outside views

 * ex-FWBO a site which provides information about some of the perceived harmful aspects of the UK FWBO.
 * FWBO Academic Bibliography lists books and academic articles by non-FWBO writers that discuss the FWBO
 * Many Bodies, One Mind: Movements in British Buddhism by Ken Jones in Buddhist Peace Fellowship (Page Not Found)
 * Journal of Global Buddhism Research summary by Sally A. McAra, (2000). Investigates Order members' narratives about their transformative relationship with the land, focusing on the retreat center Sudarshanaloka in New Zealand.
 * Land of Beautiful Vision: Making a Buddhist Sacred Place in New Zealand by Sally McAra (2007). This is an in-depth study of Sudarshanaloka, developed from McAra's MA Thesis. Through anthropological research methods, McAra explores beyond the level of public discourse on Buddhism to investigate group members' narratives about a stupa that they completed in 1997, and their changing sense of relationship with the land.
 * A Review of Extending the Hand of Fellowship by Sandra Bell, University of Durham. Journal of Buddhist Ethics (page not found)
 * Working in the Right Spirit by Martin Baumann, University of Hannover. Journal of Buddhist Ethics. The application of Buddhist Right Livelihood in the FWBO. (page not found)
 * Dangers in Devotion: Buddhist Cults and the Tasks of a Guru by John Crook, Western Chan Fellowship

title: Triratna Buddhist Community


text:
Talk:100 Mile House

I am the designer of the page I quoted and assembled the material myself. I thought it would be a good start though there is much more to be said about 100 Mile. However, until I or someone else can some up with more complete information, it is probably just as well to leave it blank and available for input.

The 100 Mile page referenced is in my (humble!) opinion the best presently available. I created it because there was no adequate page to which to refer people who wanted information on the town - in this case with special emphasis on older, more dependent people.

Having said that, I support and appreciate your scrupulous approach to copyright. Technically I should have had formal permission from the site's owner. While I know he'd be happy to give this, it's probably best to leave it for now.

Fort Alexandria
Fort Alexandria wasn't present day Quesnel, which was Quesnellemouth as it's article states. Fort Alexandria was between Quesnel and Soda Creek. So, that sentence "as a resting point for travellers moving between Kamloops and Fort Alexandria (present-day Quesnel)" should be changed to either Quesnel or Alexandria, as they are two seperate towns. An article is coming on Fort Alexandria with a redirect from Alexandria, British Columbia that should iron out some of these inconsistencies that keep coming up about Alexandria's location.CindyBo 02:00, 15 May 2007 (UTC)

Move/rename proposal
Move/rename IMO I've never heard "One Hundred Mile House" - if anything "Hundred Mile House", and nearly always it's "spelled" numerically, including in government documents. This article should be moved to 100 Mile House; if it's a city (is it a muni though?) there's no need to make it 100 Mile House, British Columbia also.Skookum1 (talk) 16:00, 22 February 2008 (UTC) It's been over a month since I posted that; I'm gonna unilaterally do it soon enouigh, I just havent' had time to sit down over all the redirects/links and do it. If anyone's gonna object do it sometime in the next week; but the official name, as in my inline comments, is numeric, not spelled out; it's NEVER spelled out, in newspapers, books, nada. Here is the BCGNIS proof.Skookum1 (talk) 14:13, 8 April 2008 (UTC)

Included photograph
The photograph that has been included and labelled '100 Mile House Museum' is a little confusing. That is the main building from the 108 Mile Heritage Site is not, to my knowledge, ever referred to as the '100 Mile Museum' —Preceding unsigned comment added by Scottfordshire (talk • contribs) 05:33, 3 April 2008 (UTC)

Demonym??
I removed the "BXer" addition, which I've never heard before and if it's a "new" thing from the town, isn't what people from 100 Mile are called outside of it; "100 Miler" is what I've heard and used....or "the folks from 100 Mile".Skookum1 (talk) 03:03, 11 July 2013 (UTC)

title: Talk:100 Mile House


text:
Senate of Finland

The Senate of Finland (Suomen senaatti, Senaten för Finland) combined the functions of cabinet and supreme court in the Grand Duchy of Finland from 1816 to 1917 and in the independent Finland from 1917 to 1918.

The body that would become the Senate was established on August 18, 1809, when Tsar Alexander I of Russia summoned the Diet of Porvoo and directed the Diet to draw up regulations for a Government Council. In 1816, Alexander renamed this body the Senate to demonstrate that it was equal to rather than inferior to its Russian equivalent. The Senate was headed by the Governor-General of Finland. The members of the Senate had to be Finnish citizens. The Senate was divided into the economic division and the judicial division. In 1822 both divisions were given a Finnish vice-chairman. From 1858 and onwards the members of the senate were formally known as senators. After the February Revolution in Russia the Vice Chairman of the Economic Division became the Chairman of the Senate. Due to the Civil War in 1918 the Senate was relocated to the town of Vaasa from January 29 to May 3.

In 1918 the economic division became the Cabinet and the judicial division became the Supreme Court and the Supreme Administrative Court of the independent Finland. The vice chairman of the economic division became the Prime Minister of Finland, and the other senators became ministers. Finland became a republic in 1919.

Vice Chairman of the Economic Division (1822–1917)

 * Carl Erik Mannerheim, (1822–1826)
 * Samuel Fredrik von Born (acting), (1826–1828)
 * Anders Henrik Falck, (1828–1833)
 * Gustaf Hjärne, (1833–1841)
 * Lars Gabriel von Haartman, (1841–1858)
 * Johan Mauritz Nordenstam, (1858–1882)
 * Edvard Gustaf af Forselles, (1882–1885)
 * Samuel Werner von Troil, (1885–1891)
 * Sten Carl Tudeer, (1891–1900)
 * Constantin Linder, (1900–1905)
 * Emil Streng, (1905)
 * Leopold Henrik Stanislaus Mechelin, (1905–1908)
 * Edvard Immanuel Hjelt, (1908–1909)
 * August Johannes Hjelt, (1909)
 * Anders Wirenius, (1909)
 * Vladimir Ivanovich Markov, (1909–1913)
 * Mikhail Borovitinov, (1913–1917)
 * Anders Wirenius (acting), (1917)

Chairmen of the Senate (1917–1918)

 * Antti Oskari Tokoi, Social Democratic Party (1917)
 * Eemil Nestor Setälä, Young Finnish Party (1917)
 * Pehr Evind Svinhufvud, Young Finnish Party (1917–1918)
 * Juho Kusti Paasikivi, Finnish Party (1918)

title: Senate of Finland


text:
Fock state

In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play an important role in the second quantization formulation of quantum mechanics.

The particle representation was first treated in detail by Paul Dirac for bosons and by Pascual Jordan and Eugene Wigner for fermions. The Fock states of bosons and fermions obey useful relations with respect to the Fock space creation and annihilation operators.

Definition
One specifies a multiparticle state of N non-interacting identical particles by writing the state as a sum of tensor products of N one-particle states. Additionally, depending on the integrality of the particles' spin, the tensor products must be alternating (anti-symmetric) or symmetric products of the underlying one-particle Hilbert space. Specifically:

 * Fermions, having half-integer spin and obeying the Pauli exclusion principle, correspond to antisymmetric tensor products.
 * Bosons, possessing integer spin (and not governed by the exclusion principle) correspond to symmetric tensor products.

If the number of particles is variable, one constructs the Fock space as the direct sum of the tensor product Hilbert spaces for each particle number. In the Fock space, it is possible to specify the same state in a new notation, the occupancy number notation, by specifying the number of particles in each possible one-particle state.

Let $\left\{\mathbf{k}_{i}\right\}_{i \in I}$ be an orthonormal basis of states in the underlying one-particle Hilbert space. This induces a corresponding basis of the Fock space called the "occupancy number basis". A quantum state in the Fock space is called a Fock state if it is an element of the occupancy number basis.

A Fock state satisfies an important criterion: for each i, the state is an eigenstate of the particle number operator $$\widehat{N_{{\mathbf{k}}_i}}$$ corresponding to the i-th elementary state ki. The corresponding eigenvalue gives the number of particles in the state. This criterion nearly defines the Fock states (one must in addition select a phase factor).

A given Fock state is denoted by $$|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},..n_{{\mathbf{k}}_i}...\rangle$$. In this expression, $$n_{{\mathbf{k}}_i}$$ denotes the number of particles in the i-th state ki, and the particle number operator for the i-th state, $$\widehat{N_{{\mathbf{k}}_i}}$$, acts on the Fock state in the following way:

 * $$\widehat{N_{{\mathbf{k}}_i}}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},..n_{{\mathbf{k}}_i}...\rangle = n_{{\mathbf{k}}_i}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},..n_{{\mathbf{k}}_i}...\rangle$$

Hence the Fock state is an eigenstate of the number operator with eigenvalue $$n_{{\mathbf{k}}_i}$$.

Fock states often form the most convenient basis of a Fock space. Elements of a Fock space that are superpositions of states of differing particle number (and thus not eigenstates of the number operator) are not Fock states. For this reason, not all elements of a Fock space are referred to as "Fock states".

If we define the aggregate particle number operator $\widehat{N}$ as
 * $$\widehat{N} = \sum_i \widehat{N_{{\mathbf{k}}_i}},$$

the definition of Fock state ensures that the variance of measurement $$\operatorname{Var}\left(\widehat{N}\right) = 0$$, i.e., measuring the number of particles in a Fock state always returns a definite value with no fluctuation.

Example using two particles
For any final state $$|f\rangle$$, any Fock state of two identical particles given by $$|1_, 1_\rangle$$, and any operator $$ \widehat{\mathbb{O}} $$, we have the following condition for indistinguishability:

\left|\left\langle f\left|\widehat{\mathbb{O}}\right|1_{\mathbf{k}_1}, 1_{\mathbf{k}_2}\right\rangle\right|^2 = \left|\left\langle f\left|\widehat{\mathbb{O}}\right|1_{\mathbf{k}_2}, 1_{\mathbf{k}_1}\right\rangle\right|^2 $$.

So, we must have $$\left\langle f\left|\widehat{\mathbb{O}}\right|1_, 1_\right\rangle = e^{i\delta}\left\langle f\left|\widehat{\mathbb{O}}\right|1_, 1_\right\rangle$$

where $$e^{i\delta} = +1$$ for bosons and $$-1$$ for fermions. Since $$\langle f| $$ and $$\widehat{\mathbb{O}}$$ are arbitrary, we can say,
 * $$\left|1_{\mathbf{k}_1}, 1_{\mathbf{k}_2}\right\rangle = +\left|1_{\mathbf{k}_2}, 1_{\mathbf{k}_1}\right\rangle$$ for bosons and
 * $$\left|1_{\mathbf{k}_1}, 1_{\mathbf{k}_2}\right\rangle = -\left|1_{\mathbf{k}_2}, 1_{\mathbf{k}_1}\right\rangle$$ for fermions.

Note that the number operator does not distinguish bosons from fermions; indeed, it just counts particles without regard to their symmetry type. To perceive any difference between them, we need other operators, namely the creation and annihilation operators.

Bosonic Fock state
Bosons, which are particles with integer spin, follow a simple rule: their composite eigenstate is symmetric under operation by an exchange operator. For example, in a two particle system in the tensor product representation we have $$\hat{P}\left|x_1, x_2\right\rangle = \left|x_2, x_1\right\rangle$$.

Boson creation and annihilation operators
We should be able to express the same symmetric property in this new Fock space representation. For this we introduce non-Hermitian bosonic creation and annihilation operators, denoted by $$b^{\dagger}$$ and $$b$$ respectively. The action of these operators on a Fock state are given by the following two equations:

 * Creation operator $b^{\dagger}_{{\mathbf{k}}_l} $ :
 * $$b^{\dagger}_{{\mathbf{k}}_l}|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}},...\rangle=\sqrt{n_{{\mathbf{k}}_{l}} +1 } |n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}+1 ,...\rangle $$
 * Annihilation operator $b_{{\mathbf{k}}_l} $ :
 * $$b_{{\mathbf{k}}_l}|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}},n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}},...\rangle=\sqrt{n_{{\mathbf{k}}_{l}}} |n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}}, n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}-1 ,...\rangle $$

Non-Hermiticity of creation and annihilation operators
The bosonic Fock state creation and annihilation operators are not Hermitian operators. $$

Operator identities
The commutation relations of creation and annihilation operators in a bosonic system are
 * $$\left[b^{\,}_i, b^\dagger_j\right] \equiv b^{\,}_i b^\dagger_j - b^\dagger_jb^{\,}_i = \delta_{i j},$$
 * $$\left[b^\dagger_i, b^\dagger_j\right] = \left[b^{\,}_i, b^{\,}_j\right] = 0,$$

where $$[\ \, \ \ ]$$ is the commutator and $$\delta_{i j}$$ is the Kronecker delta.

Action of number operators
The number operators $\widehat{N_{{\mathbf{k}}_l}}$ for a bosonic system are given by $$\widehat{N_{{\mathbf{k}}_l}}=b^{\dagger}_{{\mathbf{k}}_l}b_{{\mathbf{k}}_l}$$, where $$\widehat{N_{{\mathbf{k}}_l}}|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}...\rangle=n_{{\mathbf{k}}_{l}}  |n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}...\rangle $$

Number operators are Hermitian operators.

Symmetric behaviour of bosonic Fock states
The commutation relations of the creation and annihilation operators ensure that the bosonic Fock states have the appropriate symmetric behaviour under particle exchange. Here, exchange of particles between two states (say, l and m) is done by annihilating a particle in state l and creating one in state m. If we start with a Fock state $$|\psi\rangle = \left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, .... n_{\mathbf{k}_m} ... n_{\mathbf{k}_l} ... \right\rangle$$, and want to shift a particle from state $$k_l$$ to state $$k_m$$, then we operate the Fock state by $$b_{\mathbf{k}_m}^\dagger b_{\mathbf{k}_l}$$ in the following way:

Using the commutation relation we have, $$b_{\mathbf{k}_m}^\dagger.b_{\mathbf{k}_l} = b_{\mathbf{k}_l}.b_{\mathbf{k}_m}^\dagger$$
 * $$\begin{align}

b_{\mathbf{k}_m}^\dagger.b_{\mathbf{k}_l} \left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, .... n_{\mathbf{k}_m} ... n_{\mathbf{k}_l} ... \right\rangle &= b_{\mathbf{k}_l}.b_{\mathbf{k}_m}^\dagger \left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, .... n_{\mathbf{k}_m} ... n_{\mathbf{k}_l} ... \right\rangle \\   &= \sqrt{n_{\mathbf{k}_m} + 1}\sqrt{n_{\mathbf{k}_l}} \left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, .... n_{\mathbf{k}_{m}} + 1 ... n_{\mathbf{k}_l} - 1 ...\right\rangle \end{align}$$

So, the Bosonic Fock state behaves to be symmetric under operation by Exchange operator.

Fermion creation and annihilation operators
To be able to retain the antisymmetric behaviour of fermions, for Fermionic Fock states we introduce non-Hermitian fermion creation and annihilation operators, defined for a Fermionic Fock state $$ |\psi\rangle = |n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}},...\rangle$$ as:

 * The creation operator $$c^{\dagger}_{{\mathbf{k}}_l} $$ acts as:
 * $$c^{\dagger}_{{\mathbf{k}}_l}|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}},...\rangle=\sqrt{n_{{\mathbf{k}}_{l}} +1 } |n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}+1 ,...\rangle $$
 * The annihilation operator $c_{{\mathbf{k}}_l} $ acts as:
 * $$c_{{\mathbf{k}}_l}|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}},...\rangle=\sqrt{n_{{\mathbf{k}}_{l}}} |n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}-1 ,...\rangle $$

These two actions are done antisymmetrically, which we shall discuss later.

Operator identities
The anticommutation relations of creation and annihilation operators in a fermionic system are,
 * $$\begin{align}

\left\{c^{\,}_i, c^\dagger_j\right\} \equiv c^{\,}_i c^\dagger_j + c^\dagger_jc^{\,}_i &= \delta_{ij}, \\ \left\{c^\dagger_i, c^\dagger_j\right\} = \left\{c^{\,}_i, c^{\,}_j\right\} &= 0, \end{align}$$

where $${\{ \, \ \} }$$ is the anticommutator and $$\delta_{i j}$$ is the Kronecker delta. These anticommutation relations can be used to show antisymmetric behaviour of Fermionic Fock states.

Action of number operators
Number operators $\widehat{N_{{\mathbf{k}}_l}}$ for Fermions are given by $$\widehat{N_{{\mathbf{k}}_l}}=c^{\dagger}_{{\mathbf{k}}_l}.c_{{\mathbf{k}}_l}$$.
 * $$\widehat{N_{{\mathbf{k}}_l}}|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}...\rangle=n_{{\mathbf{k}}_{l}} |n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}...\rangle $$

Maximum occupation number
The action of the number operator as well as the creation and annihilation operators might seem same as the bosonic ones, but the real twist comes from the maximum occupation number of each state in the fermionic Fock state. Extending the 2-particle fermionic example above, we first must convince ourselves that a fermionic Fock state $$|\psi\rangle = \left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} ... n_{\mathbf{k}_l} ... \right\rangle$$ is obtained by applying a certain sum of permutation operators to the tensor product of eigenkets as follows:

\left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} ... n_{\mathbf{k}_l} ...\right\rangle = S_-\left|i_1, i_2, i_3 ... i_l ...\right\rangle = \frac{1}{\sqrt{N!}}\begin{vmatrix} \left|i_1\right\rangle_1 & \cdots & \left|i_1\right\rangle_N \\ \vdots & \ddots &                  \vdots \\ \left|i_N\right\rangle_1 & \cdots & \left|i_N\right\rangle_N \end{vmatrix}$$

This determinant is called the Slater determinant. If any of the single particle states are the same, two rows of the Slater determinant would be the same and hence the determinant would be zero. Hence, two identical fermions must not occupy the same state (a statement of the Pauli exclusion principle). Therefore, the occupation number of any single state is either 0 or 1. The eigenvalue associated to the fermionic Fock state $$\widehat{N_{{\mathbf{k}}_l}}$$ must be either 0 or 1.

Antisymmetric behaviour of Fermionic Fock state
Antisymmetric behaviour of Fermionic states under Exchange operator is taken care of the anticommutation relations. Here, exchange of particles between two states is done by annihilating one particle in one state and creating one in other. If we start with a Fock state $$|\psi\rangle = \left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, ... n_{\mathbf{k}_m}... n_{\mathbf{k}_l} ...\right\rangle$$ and want to shift a particle from state $$k_l$$ to state $$k_m$$, then we operate the Fock state by $$c_{\mathbf{k}_m}^{\dagger}.c_{\mathbf{k}_l}$$ in the following way:

Using the anticommutation relation we have
 * $$c_{\mathbf{k}_m}^\dagger.c_{\mathbf{k}_l} = -c_{\mathbf{k}_l}.c_{\mathbf{k}_m}^\dagger$$

c_{\mathbf{k}_m}^{\dagger}.c_{\mathbf{k}_l} \left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, .... n_{\mathbf{k}_m} ... n_{\mathbf{k}_l} ... \right\rangle = \sqrt{n_{\mathbf{k}_m} + 1}\sqrt{n_{\mathbf{k}_l}} \left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, .... n_{\mathbf{k}_m} + 1 ... n_{\mathbf{k}_l} - 1 ...\right\rangle$$

but, $$\begin{align} &c_{{\mathbf{k}}_{l}}.c_{{\mathbf{k}}_{m}}^{\dagger}|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}}, ....n_{{\mathbf{k}}_{m}}... n_{{\mathbf{k}}_{l}}...\rangle \\ ={} -&c_{{\mathbf{k}}_{m}}^{\dagger}.c_{{\mathbf{k}}_{l}}|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}}, .... n_{{\mathbf{k}}_{m}}... n_{{\mathbf{k}}_{l}}...\rangle \\ ={} -&\sqrt{n_{{\mathbf{k}}_{m}} +1 }\sqrt{n_{{\mathbf{k}}_{l}}}|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}}, .... n_{{\mathbf{k}}_{m}} + 1 ... n_{{\mathbf{k}}_{l}} - 1...\rangle \end{align}$$

Thus, fermionic Fock states are antisymmetric under operation by particle exchange operators.

Fock states are not energy eigenstates in general
In second quantization theory, the Hamiltonian density function is given by
 * $$\mathfrak{H} = \frac{1}{2m} \nabla_{i}\psi^{*}(x)\, \nabla_{i}\psi(x)$$

The total Hamiltonian is given by
 * $$\begin{align}

\mathcal{H} &= \int d^3 x\,\mathfrak{H} = \int d^{3}x \psi^{*}(x)\left(-\frac{\nabla^2}{2m}\right)\psi(x) \\ \therefore \mathfrak{H} &= -\frac{\nabla^2}{2m} \end{align}$$

In free Schrödinger theory,
 * $$\mathfrak{H}\psi_{n}^{(+)}(x) = -\frac{\nabla^2}{2m}\psi_{n}^{(+)}(x) = E_{n}^{0}\psi_{n}^{(+)}(x)$$

and
 * $$\int d^3 x\, \psi_{n}^{(+)^{*}}(x)\, \psi_{n'}^{(+)}(x) = \delta_{nn'}$$

and
 * $$\psi(x) = \sum_n a_n \psi_{n}^{(+)}(x)$$,

where $$a_n$$ is the annihilation operator.

 * $$\therefore \mathcal{H} = \sum_{n,n'}\int d^{3}x\, a^{\dagger}_{n'}\psi_{n'}^{(+)^{*}}(x)\, \mathfrak{H}a_n \psi_{n}^{(+)}(x)$$

Only for non-interacting particles do $$\mathfrak{H}$$ and $$a_n$$ commute; in general they do not commute. For non-interacting particles,
 * $$\mathcal{H} =

\sum_{n,n'}\int d^3 x\, a^{\dagger}_{n'}\psi_{n'}^{(+)^{*}}(x)\, E^{0}_{n}\psi_{n}^{(+)}(x)a_n = \sum_{n,n'}E^{0}_{n} a^{\dagger}_{n'}a_n\delta_{nn'} = \sum_{n}E^{0}_{n}a^{\dagger}_n a_n = \sum_{n}E^{0}_{n}\widehat{N} $$

If they do not commute, the Hamiltonian will not have the above expression. Therefore, in general, Fock states are not energy eigenstates of a system.

Vacuum fluctuations
The vacuum state or $$|0\rangle$$ is the state of lowest energy and the expectation values of $$a$$ and $$a^\dagger$$ vanish in this state:

 * $$a|0\rangle = 0 = \langle 0 |a^\dagger$$

The electrical and magnetic fields and the vector potential have the mode expansion of the same general form:

 * $$F\left(\vec{r}, t\right) = \varepsilon a e^{i\vec{k}x - \omega t} + h.c. $$

Thus it is easy to see that the expectation values of these field operators vanishes in the vacuum state:

 * $$\langle 0|F|0 \rangle = 0$$

However, it can be shown that the expectation values of the square of these field operators is non-zero. Thus there are fluctuations in the field about the zero ensemble average. These vacuum fluctuations are responsible for many interesting phenomenon including the Lamb shift in quantum optics.

Multi-mode Fock states
In a multi-mode field each creation and annihilation operator operates on its own mode. So $$a_{\mathbf{k}_l}$$ and $$a^{\dagger}_{\mathbf{k}_l}$$ will operate only on $$\left|n_{\mathbf{k}_l}\right\rangle$$. Since operators corresponding to different modes operate in different sub-spaces of the Hilbert space, the entire field is a direct product of $$|n_{\mathbf{k}_l}\rangle$$ over all the modes:

\left|n_{\mathbf{k}_1}\right\rangle \left|n_{\mathbf{k}_2}\right\rangle \left|n_{\mathbf{k}_3}\right\rangle \ldots \equiv \left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3}... n_{\mathbf{k}_l}... \right\rangle \equiv \left|\{n_\mathbf{k}\}\right\rangle $$

The creation and annihilation operators operate on the multi-mode state by only raising or lowering the number state of their own mode:

 * $$\begin{align}

a_{{\mathbf{k}}_l} |n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}}, n_{{\mathbf{k}}_{3}}... n_{{\mathbf{k}}_{l}}, ...\rangle &= \sqrt{n_{{\mathbf{k}}_{l}}} |n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}}, n_{{\mathbf{k}}_{3}}... n_{{\mathbf{k}}_{l}}-1, ...\rangle \\ a^{\dagger}_{{\mathbf{k}}_l} |n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}}, n_{{\mathbf{k}}_{3}}... n_{{\mathbf{k}}_{l}},...\rangle &= \sqrt{n_{{\mathbf{k}}_{l}} +1 } |n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}}, n_{{\mathbf{k}}_{3}}... n_{{\mathbf{k}}_{l}} + 1, ...\rangle \end{align}$$

We also define the total number operator for the field which is a sum of number operators of each mode:
 * $$ \hat{n}_{\mathbf{k}} = \sum \hat{n}_{\mathbf{k}_l} $$

The multi-mode Fock state is an eigenvector of the total number operator whose eigenvalue is the total occupation number of all the modes
 * $$ \hat{n}_{\mathbf{k}} |\{n_{\mathbf{k}}\}\rangle = \left( \sum n_{\mathbf{k}_l} \right) |\{n_{\mathbf{k}}\}\rangle $$

In case of non-interacting particles, number operator and Hamiltonian commute with each other and hence multi-mode Fock states become eigenstates of the multi-mode Hamiltonian

 * $$ \hat{H} \left|\{n_{\mathbf{k}}\}\right\rangle = \left( \sum \hbar \omega \left(n_{\mathbf{k}_l} + \frac{1}{2} \right)\right) \left|\{n_{\mathbf{k}}\}\right\rangle $$

Source of single photon state
Single photons are routinely generated using single emitters (atoms, ions, molecules, Nitrogen-vacancy center, Quantum dot ). However, these sources are not always very efficient, often presenting a low probability of actually getting a single photon on demand; and often complex and unsuitable out of a laboratory environment.

Other sources are commonly used that overcome these issues at the expense of a nondeterministic behavior. Heralded single photon sources are probabilistic two-photon sources from whom the pair is split and the detection of one photon heralds the presence of the remaining one. These sources usually rely on the optical non-linearity of some materials like periodically poled Lithium niobate (Spontaneous parametric down-conversion), or silicon (spontaneous Four-wave mixing) for example.

Non-classical behaviour
The Glauber–Sudarshan P-representation of Fock states shows that these states are purely quantum mechanical and have no classical counterpart. The $$\scriptstyle\varphi(\alpha) \,$$ of these states in the representation is a $$2n$$'th derivative of the Dirac delta function and therefore not a classical probability distribution.

title: Fock state


text:
Chidambaram Subramaniam

Chidambaram Subramaniam (commonly known as CS) (30 January 1910 – 7 November 2000), was an Indian politician and independence activist. He served as Minister of Finance and Minister of Defence in the union cabinet. He later served as the Governor of Maharashtra. As the Minister for Food and Agriculture, he ushered the Indian Green Revolution, an era of self-sufficiency in food production along with M. S. Swaminathan, B. Sivaraman and Norman E. Borlaug. He was awarded Bharat Ratna, Indian's highest civilian award, in 1998, for his role in ushering Green Revolution.

Early life and education
Subramaniam was born in Senguttaipalayam a village near Pollachi in Coimbatore district, Presidency of Fort St. George (now the Tamil Nadu state). Subramaniam completed his early education in Pollachi before moving to Chennai where he did his B.Sc in Physics at the Presidency College, Chennai (affiliated to the University of Madras). Later he graduated with degree in law from Madras Law college, Chennai (then affiliated to the same university). During his college days, he started Vanamalar Sangam and published a magazine called Pithan from Gobichettipalayam along with Periyasaamy Thooran, K. M. Ramasami Gounder, O. V. Alagesan and Justice Palanisami. His inspiration was his uncle Swami Chidbhavananda.

Early years
Subramaniam was an active member of the Civil disobedience movement against the British during his college days. He was imprisoned during the Quit India Movement in 1942. He was later elected to the Constituent Assembly and had a hand in the framing of the Constitution of India. He was a minister of Education, Law and Finance for Madras State from 1952 to 1962 under chief ministers Rajaji and K. Kamaraj. He was the First Leader of the House in the Madras Legislative Assembly for the entire duration. He was elected to the Lok Sabha in 1962 and was the Minister for Steel and Mines. Subsequently, he served as the Minister for Food and Agriculture. He also worked as the Deputy Chairman of the Planning Commission from 2 May 1971 to 22 July 1972.

Green Revolution
Along with M. S. Swaminathan and B. Sivaraman, Subramaniam was the architect of India's modern agricultural development policy, after the success of his programme which led to a record production of wheat in 1972 termed as the Indian Green Revolution. As Minister for Food and Agriculture, he introduced high-yielding varieties of seeds and more intensive application of fertilizers which paved the way for increased output of cereals and attainment of self-sufficiency in food-grains in the country. About his contribution, Dr. Norman E. Borlaug, writes:

"The vision and influence of Mr. Subramaniam in bringing about agricultural change and in the very necessary political decisions needed to make the new approach effective, should never be under-emphasized. The groundwork for this advance (in the production of wheat) was solidly laid during that period (1964–67) when Mr. Subramaniam was the guiding political force instituting change."

He appointed M. S. Swaminathan, who played a major role in green revolution and Verghese Kurien as the chairman of National Dairy Development Board when he ushered the Indian White Revolution. Kurien says, that the key role played by Subramaniam in the whole thing (Operation Flood) is hardly mentioned. He founded the National Agro Foundation, Chennai and Bharathidasan Institute of Management, Tiruchirappalli.

Finance ministry and emergency
When the Indian National Congress split in 1969, he became the interim president of the Congress (R) faction led by Indira Gandhi. Later, he was appointed Minister of Finance in the union cabinet by Indira Gandhi. He advised her to devalue Indian rupee and was the finance minister during the Emergency in 1976. After the Emergency, he parted ways with Indira and joined the breakaway Congress faction led by Devraj Urs and Kasu Brahmananda Reddy.

Later years
He was appointed the union Minister of Defence by Charan Singh in 1979. He became the Governor of Maharashtra in 1990. He resigned after his criticism of the style of functioning of the then Indian Prime Minister P. V. Narasimha Rao.

Subramaniam died on 7 November 2000 at the age of 90 in Chennai. At his death, he was the last surviving cabinet minister who had served under Jawaharlal Nehru, as well as the last surviving cabinet minister from the Shastri and Nanda cabinets.

Awards

 * Bharat Ratna, India's highest civilian honor, 1998
 * Y. B. Chavan National Integration Award
 * U Thant peace award, 1996
 * Norman Borlaug award, 1996
 * Anuvrat award, 1988

Publications

 * The New Strategy in Indian Agriculture
 * Some Countries which I visited Round The World
 * The India of My Dreams

Legacy
A commemorative coin in his honour, was released by the Government of India in August 2010. A commemorative postage stamp was also released in his honour in 2010.

Shri Chidambaram Subramaniam Award - For Excellence in Character has been instituted by Bharatiya Vidya Bhavans which is awarded to its students annually.

title: Chidambaram Subramaniam


text:
Root system

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory (such as singularity theory). Finally, root systems are important for their own sake, as in spectral graph theory.

Definitions and examples
As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right; call them roots. These vectors span the whole space. If you consider the line perpendicular to any root, say β, then the reflection of R2 in that line sends any other root, say α, to another root. Moreover, the root to which it is sent equals α + nβ, where n is an integer (in this case, n equals 1). These six vectors satisfy the following definition, and therefore they form a root system; this one is known as A2.

Definition
Let E be a finite-dimensional Euclidean vector space, with the standard Euclidean inner product denoted by $$(\cdot,\cdot)$$. A root system $$\Phi$$ in E is a finite set of non-zero vectors (called roots) that satisfy the following conditions:

 * 1) The roots span E.
 * 2) The only scalar multiples of a root $$\alpha\in\Phi$$ that belong to $$\Phi$$ are $$\alpha$$ itself and $$-\alpha$$.
 * 3) For every root $$\alpha\in\Phi$$, the set $$\Phi$$ is closed under reflection through the hyperplane perpendicular to $$\alpha$$.
 * 4) (Integrality) If $$\alpha$$ and $$\beta$$ are roots in $$\Phi$$, then the projection of $$\beta$$ onto the line through $$\alpha$$ is an integer or half-integer multiple of $$\alpha$$.

An equivalent way of writing conditions 3 and 4 is as follows:

 * 1) For any two roots $$\alpha,\beta \in \Phi $$, the set $$\Phi$$ contains the element $$\sigma_\alpha(\beta):=\beta-2\frac{(\alpha,\beta)}{(\alpha,\alpha)}\alpha.$$
 * 2) For any two roots $$\alpha,\beta\in\Phi$$, the number $$ \langle \beta, \alpha \rangle := 2 \frac{(\alpha,\beta)}{(\alpha,\alpha)}$$ is an integer.

Some authors only include conditions 1–3 in the definition of a root system. In this context, a root system that also satisfies the integrality condition is known as a crystallographic root system. Other authors omit condition 2; then they call root systems satisfying condition 2 reduced. In this article, all root systems are assumed to be reduced and crystallographic.

In view of property 3, the integrality condition is equivalent to stating that β and its reflection σα(β) differ by an integer multiple of α. Note that the operator $$ \langle \cdot, \cdot \rangle \colon \Phi \times \Phi \to \mathbb{Z}$$ defined by property 4 is not an inner product. It is not necessarily symmetric and is linear only in the first argument. The rank of a root system Φ is the dimension of E. Two root systems may be combined by regarding the Euclidean spaces they span as mutually orthogonal subspaces of a common Euclidean space. A root system which does not arise from such a combination, such as the systems A2, B2, and G2 pictured to the right, is said to be irreducible.

Two root systems (E1, Φ1) and (E2, Φ2) are called isomorphic if there is an invertible linear transformation E1 → E2 which sends Φ1 to Φ2 such that for each pair of roots, the number $$ \langle x, y \rangle$$ is preserved.

The  of a root system Φ is the Z-submodule of E generated by Φ. It is a lattice in E.

Weyl group
The group of isometries of E generated by reflections through hyperplanes associated to the roots of Φ is called the Weyl group of Φ. As it acts faithfully on the finite set Φ, the Weyl group is always finite. The reflection planes are the hyperplanes perpendicular to the roots, indicated for $$A_2$$ by dashed lines in the figure below. The Weyl group is the symmetry group of an equilateral triangle, which has six elements. In this case, the Weyl group is not the full symmetry group of the root system (e.g., a 60-degree rotation is a symmetry of the root system but not an element of the Weyl group).

Rank one example
There is only one root system of rank 1, consisting of two nonzero vectors $$\{\alpha, -\alpha\}$$. This root system is called $$A_1$$.

Rank two examples
In rank 2 there are four possibilities, corresponding to $$\sigma_\alpha(\beta) = \beta + n\alpha$$, where $$n = 0, 1, 2, 3$$. The figure at right shows these possibilities, but with some redundancies: $$A_1\times A_1$$ is isomorphic to $$D_2$$ and $$B_2$$ is isomorphic to $$C_2$$.

Note that a root system is not determined by the lattice that it generates: $$A_1 \times A_1$$ and $$B_2$$ both generate a square lattice while $$A_2$$ and $$G_2$$ both generate a hexagonal lattice.

Whenever Φ is a root system in E, and S is a subspace of E spanned by Ψ = Φ ∩ S, then Ψ is a root system in S. Thus, the exhaustive list of four root systems of rank 2 shows the geometric possibilities for any two roots chosen from a root system of arbitrary rank. In particular, two such roots must meet at an angle of 0, 30, 45, 60, 90, 120, 135, 150, or 180 degrees.

Root systems arising from semisimple Lie algebras
If $$\mathfrak{g}$$ is a complex semisimple Lie algebra and $$\mathfrak{h}$$ is a Cartan subalgebra, we can construct a root system as follows. We say that $$\alpha\in\mathfrak{h}^*$$ is a root of $$\mathfrak{g}$$ relative to $$\mathfrak{h}$$ if $$\alpha\neq 0$$ and there exists some $$X\neq 0\in\mathfrak{g}$$ such that $$[H,X]=\alpha(H)X$$ for all $$H\in\mathfrak{h}$$. One can show that there is an inner product for which the set of roots forms a root system. The root system of $$\mathfrak{g}$$ is a fundamental tool for analyzing the structure of $$\mathfrak{g}$$ and classifying its representations. (See the section below on Root systems and Lie theory.)

History
The concept of a root system was originally introduced by Wilhelm Killing around 1889 (in German, Wurzelsystem ). He used them in his attempt to classify all simple Lie algebras over the field of complex numbers. Killing originally made a mistake in the classification, listing two exceptional rank 4 root systems, when in fact there is only one, now known as F4. Cartan later corrected this mistake, by showing Killing's two root systems were isomorphic.

Killing investigated the structure of a Lie algebra $$L$$, by considering what is now called a Cartan subalgebra $$\mathfrak{h}$$. Then he studied the roots of the characteristic polynomial $$\det (\operatorname{ad}_L x - t)$$, where $$x \in \mathfrak{h}$$. Here a root is considered as a function of $$\mathfrak{h}$$, or indeed as an element of the dual vector space $$\mathfrak{h}^*$$. This set of roots form a root system inside $$\mathfrak{h}^*$$, as defined above, where the inner product is the Killing form.

Elementary consequences of the root system axioms

The cosine of the angle between two roots is constrained to be one-half of the square root of a positive integer. This is because $$ \langle \beta, \alpha \rangle$$ and $$\langle \alpha, \beta \rangle$$ are both integers, by assumption, and $$ \begin{align} \langle \beta, \alpha \rangle \langle \alpha, \beta \rangle &= 2 \frac{(\alpha,\beta)}{(\alpha,\alpha)} \cdot 2 \frac{(\alpha,\beta)}{(\beta,\beta)} \\ &= 4 \frac{(\alpha,\beta)^2}{\vert \alpha \vert^2 \vert \beta \vert^2} \\ &= 4 \cos^2(\theta) = (2\cos(\theta))^2 \in \mathbb{Z}. \end{align}$$

Since $$2\cos(\theta) \in [-2,2]$$, the only possible values for $$\cos(\theta)$$ are $$0, \pm \tfrac{1}{2}, \pm\tfrac{\sqrt{2}}{2}, \pm\tfrac{\sqrt{3}}{2}$$ and $$\pm\tfrac{\sqrt{4}}{2} = \pm 1$$, corresponding to angles of 90°, 60° or 120°, 45° or 135°, 30° or 150°, and 0° or 180°. Condition 2 says that no scalar multiples of α other than 1 and −1 can be roots, so 0 or 180°, which would correspond to 2α or −2α, are out. The diagram at right shows that an angle of 60° or 120° corresponds to roots of equal length, while an angle of 45° or 135° corresponds to a length ratio of $$\sqrt{2}$$ and an angle of 30° or 150° corresponds to a length ratio of $$\sqrt{3}$$.

In summary, here are the only possibilities for each pair of roots.
 * Angle of 90 degrees; in that case, the length ratio is unrestricted.
 * Angle of 60 or 120 degrees, with a length ratio of 1.
 * Angle of 45 or 135 degrees, with a length ratio of $$\sqrt 2$$.
 * Angle of 30 or 150 degrees, with a length ratio of $$\sqrt 3$$.

Positive roots and simple roots
Given a root system $$\Phi$$ we can always choose (in many ways) a set of positive roots. This is a subset $$\Phi^+$$ of $$\Phi$$ such that
 * For each root $$\alpha\in\Phi$$ exactly one of the roots $$\alpha$$, $$-\alpha$$ is contained in $$\Phi^+$$.
 * For any two distinct $$\alpha, \beta\in \Phi^+$$ such that $$\alpha+\beta$$ is a root, $$\alpha+\beta\in\Phi^+$$.

If a set of positive roots $$\Phi^+$$ is chosen, elements of $$-\Phi^+$$ are called negative roots. A set of positive roots may be constructed by choosing a hyperplane $$V$$ not containing any root and setting $$\Phi^+$$ to be all the roots lying on a fixed side of $$V$$. Furthermore, every set of positive roots arises in this way.

An element of $$\Phi^+$$ is called a simple root (also fundamental root) if it cannot be written as the sum of two elements of $$\Phi^+$$. (The set of simple roots is also referred to as a base for $$\Phi$$.) The set $$\Delta$$ of simple roots is a basis of $$E$$ with the following additional special properties:
 * Every root $$\alpha\in\Phi$$ is a linear combination of elements of $$\Delta$$ with integer coefficients.
 * For each $$\alpha\in\Phi$$, the coefficients in the previous point are either all non-negative or all non-positive.

For each root system $$\Phi$$ there are many different choices of the set of positive roots—or, equivalently, of the simple roots—but any two sets of positive roots differ by the action of the Weyl group.

The dual root system
If Φ is a root system in E, the coroot α∨ of a root α is defined by $$\alpha^\vee= {2\over (\alpha,\alpha)}\, \alpha.$$

The set of coroots also forms a root system Φ∨ in E, called the dual root system (or sometimes inverse root system). By definition, α∨ ∨ = α, so that Φ is the dual root system of Φ∨. The lattice in E spanned by Φ∨ is called the coroot lattice. Both Φ and Φ∨ have the same Weyl group W and, for s in W, $$ (s\alpha)^\vee= s(\alpha^\vee).$$

If Δ is a set of simple roots for Φ, then Δ∨ is a set of simple roots for Φ∨.

In the classification described below, the root systems of type $$A_n$$ and $$D_n$$ along with the exceptional root systems $$E_6,E_7,E_8,F_4,G_2$$ are all self-dual, meaning that the dual root system is isomorphic to the original root system. By contrast, the $$B_n$$ and $$C_n$$ root systems are dual to one another, but not isomorphic (except when $$n=2$$).

Integral elements
A vector $$\lambda$$ in E is called integral if its inner product with each coroot is an integer: $$2\frac{(\lambda,\alpha)}{(\alpha,\alpha)}\in\mathbb Z,\quad\alpha\in\Phi.$$ Since the set of $$\alpha^\vee $$ with $$\alpha\in\Delta$$ forms a base for the dual root system, to verify that $$\lambda$$ is integral, it suffices to check the above condition for $$\alpha\in\Delta$$.

The set of integral elements is called the weight lattice associated to the given root system. This term comes from the representation theory of semisimple Lie algebras, where the integral elements form the possible weights of finite-dimensional representations.

The definition of a root system guarantees that the roots themselves are integral elements. Thus, every integer linear combination of roots is also integral. In most cases, however, there will be integral elements that are not integer combinations of roots. That is to say, in general the weight lattice does not coincide with the root lattice.

Classification of root systems by Dynkin diagrams

A root system is irreducible if it cannot be partitioned into the union of two proper subsets $$\Phi=\Phi_1\cup\Phi_2$$, such that $$(\alpha,\beta)=0$$ for all $$\alpha\in\Phi_1$$ and $$\beta\in\Phi_2$$.

Irreducible root systems correspond to certain graphs, the Dynkin diagrams named after Eugene Dynkin. The classification of these graphs is a simple matter of combinatorics, and induces a classification of irreducible root systems.

Constructing the Dynkin diagram
Given a root system, select a set Δ of simple roots as in the preceding section. The vertices of the associated Dynkin diagram correspond to the roots in Δ. Edges are drawn between vertices as follows, according to the angles. (Note that the angle between simple roots is always at least 90 degrees.) The term "directed edge" means that double and triple edges are marked with an arrow pointing toward the shorter vector. (Thinking of the arrow as a "greater than" sign makes it clear which way the arrow is supposed to point.)
 * No edge if the vectors are orthogonal,
 * An undirected single edge if they make an angle of 120 degrees,
 * A directed double edge if they make an angle of 135 degrees, and
 * A directed triple edge if they make an angle of 150 degrees.

Note that by the elementary properties of roots noted above, the rules for creating the Dynkin diagram can also be described as follows. No edge if the roots are orthogonal; for nonorthogonal roots, a single, double, or triple edge according to whether the length ratio of the longer to shorter is 1, $$\sqrt 2$$, $$\sqrt 3$$. In the case of the $$G_2$$ root system for example, there are two simple roots at an angle of 150 degrees (with a length ratio of $$\sqrt 3$$). Thus, the Dynkin diagram has two vertices joined by a triple edge, with an arrow pointing from the vertex associated to the longer root to the other vertex. (In this case, the arrow is a bit redundant, since the diagram is equivalent whichever way the arrow goes.)

Classifying root systems
Although a given root system has more than one possible set of simple roots, the Weyl group acts transitively on such choices. Consequently, the Dynkin diagram is independent of the choice of simple roots; it is determined by the root system itself. Conversely, given two root systems with the same Dynkin diagram, one can match up roots, starting with the roots in the base, and show that the systems are in fact the same.

Thus the problem of classifying root systems reduces to the problem of classifying possible Dynkin diagrams. A root systems is irreducible if and only if its Dynkin diagrams is connected. The possible connected diagrams are as indicated in the figure. The subscripts indicate the number of vertices in the diagram (and hence the rank of the corresponding irreducible root system).

If $$\Phi$$ is a root system, the Dynkin diagram for the dual root system $$\Phi^\vee$$ is obtained from the Dynkin diagram of $$\Phi$$ by keeping all the same vertices and edges, but reversing the directions of all arrows. Thus, we can see from their Dynkin diagrams that $$B_n$$ and $$C_n$$ are dual to each other.

Weyl chambers and the Weyl group
If $$\Phi\subset E$$ is a root system, we may consider the hyperplane perpendicular to each root $$\alpha$$. Recall that $$\sigma_\alpha$$ denotes the reflection about the hyperplane and that the Weyl group is the group of transformations of $$E$$ generated by all the $$\sigma_\alpha$$'s. The complement of the set of hyperplanes is disconnected, and each connected component is called a Weyl chamber. If we have fixed a particular set Δ of simple roots, we may define the fundamental Weyl chamber associated to Δ as the set of points $$v\in E$$ such that $$(\alpha,v)>0$$ for all $$\alpha\in\Delta$$.

Since the reflections $$\sigma_\alpha,\,\alpha\in\Phi$$ preserve $$\Phi$$, they also preserve the set of hyperplanes perpendicular to the roots. Thus, each Weyl group element permutes the Weyl chambers.

The figure illustrates the case of the $$A_2$$ root system. The "hyperplanes" (in this case, one dimensional) orthogonal to the roots are indicated by dashed lines. The six 60-degree sectors are the Weyl chambers and the shaded region is the fundamental Weyl chamber associated to the indicated base.

A basic general theorem about Weyl chambers is this:
 * Theorem: The Weyl group acts freely and transitively on the Weyl chambers. Thus, the order of the Weyl group is equal to the number of Weyl chambers.

In the $$A_2$$ case, for example, the Weyl group has six elements and there are six Weyl chambers.

A related result is this one:
 * Theorem: Fix a Weyl chamber $$C$$. Then for all $$v\in E$$, the Weyl-orbit of $$v$$ contains exactly one point in the closure $$\bar C$$ of $$C$$.

Root systems and Lie theory
Irreducible root systems classify a number of related objects in Lie theory, notably the following: In each case, the roots are non-zero weights of the adjoint representation.
 * simple complex Lie algebras (see the discussion above on root systems arising from semisimple Lie algebras),
 * simply connected complex Lie groups which are simple modulo centers, and
 * simply connected compact Lie groups which are simple modulo centers.

We now give a brief indication of how irreducible root systems classify simple Lie algebras over $$\mathbb C$$, following the arguments in Humphreys. A preliminary result says that a semisimple Lie algebra is simple if and only if the associated root system is irreducible. We thus restrict attention to irreducible root systems and simple Lie algebras.

 * First, we must establish that for each simple algebra $$\mathfrak g$$ there is only one root system. This assertion follows from the result that the Cartan subalgebra of $$\mathfrak g$$ is unique up to automorphism, from which it follows that any two Cartan subalgebras give isomorphic root systems.
 * Next, we need to show that for each irreducible root system, there can be at most one Lie algebra, that is, that the root system determines the Lie algebra up to isomorphism.
 * Finally, we must show that for each irreducible root system, there is an associated simple Lie algebra. This claim is obvious for the root systems of type A, B, C, and D, for which the associated Lie algebras are the classical Lie algebras. It is then possible to analyze the exceptional algebras in a case-by-case fashion. Alternatively, one can develop a systematic procedure for building a Lie algebra from a root system, using Serre's relations.

For connections between the exceptional root systems and their Lie groups and Lie algebras see E8, E7, E6, F4, and G2.

Properties of the irreducible root systems
Irreducible root systems are named according to their corresponding connected Dynkin diagrams. There are four infinite families (An, Bn, Cn, and Dn, called the classical root systems) and five exceptional cases (the exceptional root systems). The subscript indicates the rank of the root system.

In an irreducible root system there can be at most two values for the length $Φ$, corresponding to short and long roots. If all roots have the same length they are taken to be long by definition and the root system is said to be simply laced; this occurs in the cases A, D and E. Any two roots of the same length lie in the same orbit of the Weyl group. In the non-simply laced cases B, C, G and F, the root lattice is spanned by the short roots and the long roots span a sublattice, invariant under the Weyl group, equal to r2/2 times the coroot lattice, where r is the length of a long root.

In the adjacent table, $|Φ|$ denotes the number of short roots, $I$ denotes the index in the root lattice of the sublattice generated by long roots, D denotes the determinant of the Cartan matrix, and |W| denotes the order of the Weyl group.

An
Let E be the subspace of Rn+1 for which the coordinates sum to 0, and let Φ be the set of vectors in E of length $D$ and which are integer vectors, i.e. have integer coordinates in Rn+1. Such a vector must have all but two coordinates equal to 0, one coordinate equal to 1, and one equal to −1, so there are n2 + n roots in all. One choice of simple roots expressed in the standard basis is $|Φ^{<}|$ for $|W|$.

The reflection σi through the hyperplane perpendicular to αi is the same as permutation of the adjacent ith and (i + 1)th coordinates. Such transpositions generate the full permutation group. For adjacent simple roots, σi(αi+1) = αi+1 + αi = σi+1(αi) = αi + αi+1, that is, reflection is equivalent to adding a multiple of 1; but reflection of a simple root perpendicular to a nonadjacent simple root leaves it unchanged, differing by a multiple of 0.

The An root lattice – that is, the lattice generated by the An roots – is most easily described as the set of integer vectors in Rn+1 whose components sum to zero.

The A2 root lattice is the vertex arrangement of the triangular tiling.

The A3 root lattice is known to crystallographers as the face-centered cubic (or cubic close packed) lattice. It is the vertex arrangement of the tetrahedral-octahedral honeycomb.

The A3 root system (as well as the other rank-three root systems) may be modeled in the Zometool construction set.

In general, the An root lattice is the vertex arrangement of the n-dimensional simplicial honeycomb.

Bn
Let E = Rn, and let Φ consist of all integer vectors in E of length 1 or $I$. The total number of roots is 2n2. One choice of simple roots is $A_{n} (n ≥ 1)$ for $n(n + 1)$ (the above choice of simple roots for An−1), and the shorter root $n + 1$.

The reflection σn through the hyperplane perpendicular to the short root αn is of course simply negation of the nth coordinate. For the long simple root αn−1, σn−1(αn) = αn + αn−1, but for reflection perpendicular to the short root, σn(αn−1) = αn−1 + 2αn, a difference by a multiple of 2 instead of 1.

The Bn root lattice—that is, the lattice generated by the Bn roots—consists of all integer vectors.

B1 is isomorphic to A1 via scaling by $\sqrt{2}$, and is therefore not a distinct root system.

Cn

Let E = Rn, and let Φ consist of all integer vectors in E of length $\sqrt{2}$ together with all vectors of the form 2λ, where λ is an integer vector of length 1. The total number of roots is 2n2. One choice of simple roots is: αi = ei − ei+1, for 1 ≤ i ≤ n − 1 (the above choice of simple roots for An−1), and the longer root αn = 2en. The reflection σn(αn−1) = αn−1 + αn, but σn−1(αn) = αn + 2αn−1.

The Cn root lattice—that is, the lattice generated by the Cn roots—consists of all integer vectors whose components sum to an even integer.

C2 is isomorphic to B2 via scaling by $\sqrt{2}$ and a 45 degree rotation, and is therefore not a distinct root system.

Dn
Let $(n + 1)!$, and let Φ consist of all integer vectors in E of length $\sqrt{2}$. The total number of roots is $B_{n} (n ≥ 2)$. One choice of simple roots is $C_{n} (n ≥ 3)$ for $2n(n − 1)$ (the above choice of simple roots for $D_{n} (n ≥ 4)$) together with $2n(n − 1)$.

Reflection through the hyperplane perpendicular to αn is the same as transposing and negating the adjacent n-th and (n − 1)-th coordinates. Any simple root and its reflection perpendicular to another simple root differ by a multiple of 0 or 1 of the second root, not by any greater multiple.

The Dn root lattice – that is, the lattice generated by the Dn roots – consists of all integer vectors whose components sum to an even integer. This is the same as the Cn root lattice.

The Dn roots are expressed as the vertices of a rectified n-orthoplex, Coxeter–Dynkin diagram: .... The $(α, α)^{1/2}$ vertices exist in the middle of the edges of the n-orthoplex.

D3 coincides with A3, and is therefore not a distinct root system. The twelve D3 root vectors are expressed as the vertices of, a lower symmetry construction of the cuboctahedron.

D4 has additional symmetry called triality. The twenty-four D4 root vectors are expressed as the vertices of, a lower symmetry construction of the 24-cell.

E6, E7, E8
D_8 \cup \left\{ \frac 1 2 \left( \sum_{i=1}^8 \varepsilon_i \mathbf e_i \right) : \varepsilon_i = \pm1, \, \varepsilon_1 \cdots \varepsilon_8 = +1 \right\}. $$
 * The E8 root system is any set of vectors in R8 that is congruent to the following set:$$

The root system has 240 roots. The set just listed is the set of vectors of length $\sqrt{2}$ in the E8 root lattice, also known simply as the E8 lattice or Γ8. This is the set of points in R8 such that: Thus, $$ E_8 = \left\{ \alpha\in\mathbb Z^8 \cup \left(\mathbb Z + \tfrac 1 2\right)^8 : |\alpha|^2 = \sum\alpha_i^2 = 2,\, \sum\alpha_i \in 2\mathbb Z. \right\} $$
 * 1) all the coordinates are integers or all the coordinates are half-integers (a mixture of integers and half-integers is not allowed), and
 * 2) the sum of the eight coordinates is an even integer.

 * The root system E7 is the set of vectors in E8 that are perpendicular to a fixed root in E8. The root system E7 has 126 roots.
 * The root system E6 is not the set of vectors in E7 that are perpendicular to a fixed root in E7, indeed, one obtains D6 that way. However, E6 is the subsystem of E8 perpendicular to two suitably chosen roots of E8. The root system E6 has 72 roots.

An alternative description of the E8 lattice which is sometimes convenient is as the set Γ'8 of all points in R8 such that The lattices Γ8 and Γ'8 are isomorphic; one may pass from one to the other by changing the signs of any odd number of coordinates. The lattice Γ8 is sometimes called the even coordinate system for E8 while the lattice Γ'8 is called the odd coordinate system.
 * all the coordinates are integers and the sum of the coordinates is even, or
 * all the coordinates are half-integers and the sum of the coordinates is odd.

One choice of simple roots for E8 in the even coordinate system with rows ordered by node order in the alternate (non-canonical) Dynkin diagrams (above) is:
 * αi = ei − ei+1, for 1 ≤ i ≤ 6, and
 * α7 = e7 + e6

(the above choice of simple roots for D7) along with $$ \boldsymbol\alpha_8 = \boldsymbol\beta_0 = -\frac{1}{2} \sum_{i=1}^8\mathbf{e}_i = (-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2).$$ One choice of simple roots for E8 in the odd coordinate system with rows ordered by node order in alternate (non-canonical) Dynkin diagrams (above) is
 * αi = ei − ei+1, for 1 ≤ i ≤ 7

(the above choice of simple roots for A7) along with
 * α8 = β5, where
 * $\boldsymbol\beta_j = \frac{1}{2} \left(- \sum_{i=1}^j e_i + \sum_{i=j+1}^8 e_i\right).$

(Using β3 would give an isomorphic result. Using β1,7 or β2,6 would simply give A8 or D8. As for β4, its coordinates sum to 0, and the same is true for α1...7, so they span only the 7-dimensional subspace for which the coordinates sum to 0; in fact −2β4 has coordinates (1,2,3,4,3,2,1) in the basis (αi).)

Since perpendicularity to α1 means that the first two coordinates are equal, E7 is then the subset of E8 where the first two coordinates are equal, and similarly E6 is the subset of E8 where the first three coordinates are equal. This facilitates explicit definitions of E7 and E6 as

Note that deleting α1 and then α2 gives sets of simple roots for E7 and E6. However, these sets of simple roots are in different E7 and E6 subspaces of E8 than the ones written above, since they are not orthogonal to α1 or α2.

F4
For F4, let E = R4, and let Φ denote the set of vectors α of length 1 or $\sqrt{2}$ such that the coordinates of 2α are all integers and are either all even or all odd. There are 48 roots in this system. One choice of simple roots is: the choice of simple roots given above for B3, plus $\boldsymbol\alpha_4 = -\frac{1}{2} \sum_{i=1}^4 e_i$.

The F4 root lattice—that is, the lattice generated by the F4 root system—is the set of points in R4 such that either all the coordinates are integers or all the coordinates are half-integers (a mixture of integers and half-integers is not allowed). This lattice is isomorphic to the lattice of Hurwitz quaternions.

G2
The root system G2 has 12 roots, which form the vertices of a hexagram. See the picture above.

One choice of simple roots is (α1, β = α2 − α1) where αi = ei − ei+1 for i = 1, 2 is the above choice of simple roots for A2.

The G2 root lattice—that is, the lattice generated by the G2 roots—is the same as the A2 root lattice.

The root poset
The set of positive roots is naturally ordered by saying that $$\alpha \leq \beta$$ if and only if $$\beta-\alpha$$ is a nonnegative linear combination of simple roots. This poset is graded by $\deg\left(\sum_{\alpha \in \Delta} \lambda_\alpha \alpha\right) = \sum_{\alpha \in \Delta}\lambda_\alpha$, and has many remarkable combinatorial properties, one of them being that one can determine the degrees of the fundamental invariants of the corresponding Weyl group from this poset. The Hasse graph is a visualization of the ordering of the root poset.

title: Root system