text:
Harrison & Harrison

Harrison & Harrison Ltd is a British company based in Durham that makes and restores pipe organs. It was established in Rochdale in 1861. It is well known for its work on instruments such as King's College, Cambridge, Westminster Abbey, and the Royal Festival Hall.

History of the firm
Thomas Harrison established an organ building company in 1861 in Rochdale, then moved to Durham in 1872. The company was moderately successful but did not achieve real success until 1896 when Thomas's sons Arthur and Harry took over. Harry designed the organs and Arthur proved to be a particularly gifted voicer, resulting in commissions for rebuilds of several great organs including Durham Cathedral, the Grand Organ at the Royal Albert Hall and new commissions including Westminster Abbey, and Rossall School (1925). Between 1890 and 1996, Harrisons was located on Cross Street (now Hawthorn Terrace), Durham in a former paper mill. The building is now called Harrison House.

Arthur Harrison died in 1936 and Harry retired in 1946. The company passed to Harry's son Cuthbert. The firm was led by Mark Venning from 1975. David Hirst joined as Deputy Managing Director in 2008 and led the firm as Managing Director from 2009. In 2010, Mark Venning returned to maintain operations until the arrival of Dr Christopher Batchelor in 2011. The firm was run by Dr Batchelor from 2011 to June 2017. In October 2017, Andrew Reid left his job as Director of the Royal School of Church Music and joined Harrisons as the new managing director. In 2022, Andrew Scott took over as managing director, the first to work his way up through the firm from an apprenticeship. Mark Venning remains active as chairman.

In the postwar period Harrisons contributed significantly to the renaissance of classical organ building; their instruments for the Royal Festival Hall (designed in close collaboration with Ralph Downes) and St Albans Abbey (for Peter Hurford) are of particular importance. To those two landmark instruments may be added the organs of Coventry Cathedral, the chapel of Somerville College, Oxford and St George's Chapel, Windsor Castle, all of these instruments being built during the tenure of Cuthbert Harrison and under his direction.

Harrisons continued to innovate and push the boundaries of organ building, even as they maintained their traditional craftsmanship. In the late 20th century, the firm expanded its operations to include electronic and digital organs. They also worked to adapt their instruments to modern performance styles and spaces, designing organs for concert halls and recording studios.

Under the leadership of Mark Venning, the company continued to receive numerous commissions for high-profile instruments. These included the organs of several British cathedrals, and of King's College Chapel in Cambridge. Harrisons also worked on international projects, including the restoration of the organ in St. Stephen's Cathedral, Vienna, and the construction of a new organ for the National Concert Hall in Dublin. These builds were detailed in "The Harrison Story", recounting the history of the firm, was published in 1974 by Laurence Elvin.

title: Harrison & Harrison


text:
Portal:Ancient Egypt/Selected Egyptologist

Professor Sir William Matthew Flinders Petrie FRS (3 June 1853 – 28 July 1942), known as Flinders Petrie, was an English Egyptologist and a pioneer of systematic methodology in archaeology. He excavated at many of the most important archaeological sites in Egypt such as Abydos and Amarna.

Petrie was encouraged to pursue his interest in archaeology from an early age, and the accurate surveying techniques he learned from his father were put to use in his 1880 survey of the Great Pyramid at Giza, making him the first to investigate properly how they were constructed. On that visit he was appalled by the rate of destruction of monuments and he described Egypt as "a house on fire, so rapid was the destruction" and felt his duty to be that of a "salvage man, to get all I could, as quickly as possible and then, when I was 60, I would sit and write it all down."

Petrie took complete control over his excavations and removed the pressure on the workmen to find treasures quickly but sloppily. Though he was regarded as an amateur and dilettante by more established Egyptologists, his careful attention to detail and investigation of the seriation of pottery allowed him to accurately establish the chronologies of ancient sites. In 1923, Petrie was knighted for his services. Petrie's innovative and careful excavations cataloged in his 97 books revolutionized the study of ancient Egypt and cemented his legacy as the founder of modern Egyptology.

More... Archives

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


text: User talk:68.6.93.222

title: User talk:68.6.93.222


text: Category:Theatre in Finland

title: Category:Theatre in Finland


text:
Talk:Nie Haisheng

Biography assessment rating comment
WikiProject Biography Summer 2007 Assessment Drive

The article may be improved by following the WikiProject Biography 11 easy steps to producing at least a B article. -- Yamara 23:53, 26 June 2007 (UTC)

Discussion
Born October 16, but celebrate birthday on October 13???

Married someone with the same surname?
That would be highly unusual (not to mention borderline gross). Especially considering Nie is a rather rare family name. -- 我♥中國 08:10, 10 December 2006 (UTC)

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External links modified
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title: Talk:Nie Haisheng


text:
ICD-10 Procedure Coding System

The ICD-10 Procedure Coding System (ICD-10-PCS) is a US system of medical classification used for procedural coding. The Centers for Medicare and Medicaid Services, the agency responsible for maintaining the inpatient procedure code set in the U.S., contracted with 3M Health Information Systems in 1995 to design and then develop a procedure classification system to replace Volume 3 of ICD-9-CM. ICD-9-CM contains a procedure classification; ICD-10-CM does not. ICD-10-PCS is the result. ICD-10-PCS was initially released in 1998. It has been updated annually since that time. Despite being named after the WHO's International Classification of Diseases, it is a US-developed standard which is not used outside the United States.

Section structure
Each code consists of seven alphanumeric characters. The first character is the 'section'. The second through seventh characters mean different things in each section. Each character can be any of 34 possible values the ten digits 0-9 and the 24 letters A-H, J-N and P-Z may be used in each character. The letters O and I are excluded to avoid confusion with the numbers 0 and 1. There are no decimals in ICD-10-PCS

Of the 72,081 codes in ICD-10-PCS, 62,022 are in the first section, "Medical and surgical".

Root operations
For medical/surgical, these are the root operation codes:

00 alteration; 01 bypass; 02 change; 03 control; 04 creation; 05 destruction; 06 detachment; 07 dilation; 08 division; 09 drainage; 0B excision; 0C extirpation; 0D extraction; 0F fragmentation; 0G fusion; 0H insertion; 0J inspection; 0K map; 0L occlusion; 0M reattachment; 0N release; 0P removal; 0Q repair; 0R replacement; 0S reposition; 0T resection; 0U supplement ; 0V restriction; 0W revision; 0X transfer; 0Y transplantation

They can be grouped into several categories:

 * take out or eliminate all or a portion of a body part: excision (sigmoid polypectomy), resection (total nephrectomy), extraction (toenail extraction), destruction (rectal polyp fulguration), detachment (below knee amputation). For biopsies, "extraction" is used when force is required (as with endometrial biopsy), and "excision" is used when minimal force is involved (as with liver biopsy). See also ectomy.
 * involve putting in or on, putting back, or moving living body part: transplantation (heart transplant), reattachment (finger reattachment), reposition (reposition undescended testicle), transfer (tendon transfer)
 * take out or eliminate solid matter, fluids, or gases from a body part: drainage (incision and drainage), extirpation (thrombectomy), fragmentation (lithotripsy of gallstones)
 * only involve examination of body parts and regions: inspection (diagnostic arthroscopy), map (cardiac mapping)
 * involve putting in or on, putting back, or moving living body part: bypass (gastrojejunal bypass), dilation (coronary artery dilation), occlusion (fallopian tube ligation), restriction (cervical cerclage)
 * always involve devices: insertion (pacemaker insertion), replacement (total hip replacement), supplement (herniorrhaphy using mesh), removal (cardiac pacemaker removal), change (drainage tube change), revision (hip prosthesis adjustment)
 * involve cutting and separation only: division (osteotomy), release (peritoneal adhesiolysis)
 * involving other repair: control (control of postprostatectomy bleeding), repair (suture of laceration)
 * with other objectives: alteration (face lift), creation (artificial vagina creation), fusion (spinal fusion)

title: ICD-10 Procedure Coding System


text:
Portal:Ancient Egypt/Selected Dynasty

The Fourth dynasty of Egypt was the second of the four dynasties considered forming the Old Kingdom. The Pharaohs of this dynasty include some of the best-known kings of ancient Egypt, known for constructing pyramids, perhaps the hallmark of Egypt. All of the kings of this dynasty commissioned at least one pyramid to serve as a tomb or cenotaph. Like the Third dynasty, these kings maintained their capital at Memphis.

title: Portal:Ancient Egypt/Selected Dynasty


text:
Computational Chemistry List

The Computational Chemistry List (CCL) was established on January 11, 1991, as an independent electronic forum for chemistry researchers and educators from around the world. According to the forum's web site, it is estimated that more than 3000 members in more than 50 countries are reading CCL messages regularly, and the discussions cover all aspects of computational chemistry. The list is widely supported and used by the computational chemistry community.

History
The CCL is a mailing list, portal, and community which brings together people interested in computational chemistry. It was formed in 1991 by initiative of Jan Labanowski, at the time a computational chemistry specialist in the Ohio Supercomputing Center, as a mailing list for the hundred persons who had participated in a workshop he had organized together with one of the founding fathers of the field, Charles Bender. The purpose of the list as first created was to continue the lively discussions and encounters that had taken place in the workshop and help grow the field which was accelerating due to the recent availability of maturing quantum, classical and semi-empirical methods, of supercomputers and their power, of personal computers and their flexibility and interoperability, of promising software packages bound to occupy a market niche in the chemical and pharmaceutical industry.

The list has undergone many transformations and survived through them. It went from the original hundred to several thousands members; from a strict, ASCII mailing list to a combination of mailing list, online forum, and portal containing document and software repositories, event announcements, and other communicational resources for the practitioners.

The CCL is thus a typical scholar mailing list of its time as many other flourished in the eighties and nineties in various scientific fields (though a majority of them eventually withered). The genealogy of mailing lists as a communication tool between scientists can be traced back to the times of the fledgling Arpanet. The aim of the computer scientists involved in this project was to develop protocols for the communication between computers. In so doing, they have also built the first tools of human computer-mediated communication. Broadly speaking, the scholarly mailing lists can even be seen as the modern version of the salons of the Enlightenment ages, designed by scholars for scholars.

Pisanty and Labanowski had led a survey of membership from which some of the above conclusions were extracted, and Labanowski published about the difficult role of moderator on the CCL. Hocquet and Wieber have more recently discussed the history and activity of the CCL. They have for example explored aspects such as the performative functions of language in the list on the one hand and discourse structure on the other.

Community
At its core the CCL still is a mailing list, a quaint survivor from early Internet time, in which discussions take place about general principles, practical interpretations of theory, and computational methods. The membership of the list has evolved but continues to hold a mix of theoreticians, computationalists, and experimentalists; of highly experienced practitioners, well known in the field and sometimes founders of it, blended together with young researchers and researchers from communities that are not mainstream for high-powered computational chemistry—young researchers, researchers in industrial laboratories where they may be the only specialist, and researchers in developing countries.

The language used in Internet fora is neither written nor oral: it has been described as quasi-orality. In scientific mailing lists, it consists in a mix of scholar talk, informal talk and technical talk. In the CCL, this pidgin is perfectly suited to the diversity of topics of concerns to a professionally diverse community. It is notable that most conversations revolve around software as a topic, at the intersection of theories/modelling methods/publications/coding/software support and maintenance/licensing/hardware benchmarking/sales.

Topics
Some factors that have made the CCL so durable and resilient through changes are that the communication style is horizontal; discussions are frank and sometimes sharp but hardly ever hostile; the community is strongly self-regulated for productive discussions; the style and contents imprinted by the founder, Jan Labanowski; the perception of all members that the list provides value; the diversity, broad but not wild, of discussion subjects; and a level of tolerance by members to other members, especially to those less experienced who may inadvertently test established but unwritten rules, restart old discussions or thread well-trodden paths, or become shrill too fast.

The CCL is thus also an exceptional mailing list. Its transparency (the archive is open to all), inclusivity (a poster does not need to subscribe to send a message) and its ethos as designed by a mix of terms of service, moderation practices and moderator personality are key to its persistence. It is particularly important that its definition of topicality allows to blend the theoretical parts of daily practices with the technical parts, and even the commercial ones (something unique to the CCL). The CCL as a tool for the community is thus not only a way to “educate and get educated” (in Labanowski's words), but also an arena where a vast diversity of topics can be debated.

The “threaded conversation” structure (where the header of a first post defines the topic of a series of answers thus constituting a thread) is a typical and ubiquitous structure of discourse within lists and fora of the Internet. It is pivotal to the structure and topicality of debates within the CCL as an arena. The flame wars (as the liveliest episodes) give valuable and unique information to historians to comprehend what is at stake in the computational chemistry community.

Archive
The list also hosts many resources on computational chemistry. For example, it hosted a pre-publication version of Computational Chemistry by David Young.

Online social media not only comes to mind as an analogy for what the CCL achieves in its combination of mailing list, online forum, and portal, but also as a threat to the CCL in the long term. A number of groups exist on Facebook and LinkedIn, and increasingly also in the online platforms of learned societies like the Institute of Electrical and Electronics Engineers (IEEE) in which discussions on subjects similar to the CCL's take place. They may oppose less friction to membership, provide richer interactions, and supply software, datasets, literature, and other resources, in formats with which especially younger researchers and students are familiar. However, they have not “caught” enough to displace the CCL, in part because some of the most valued members of the CCL have not migrated to those platforms, either for professional purposes only, or at all. The most valuable resource that keeps the CCL together, in this view, is its own community.

It is a general trend in scholarly communities to give up on mailing lists (as a tool created, designed and maintained by scholars for scholars), like the CCL or the other chemistry related CHMINF-L list, and surrender to social media (as services to extract marketing value where scholars are dispossessed of their tools): the CCL is a place of resistance in this respect. There are a few groups and discussions on computational chemistry on Reddit and Facebook which we inspected for this paper but none achieve the function and reach of the CCL; the Reddit discussion thread points to the CCL.

Finally, from the historian's point of view, the issue of the preservation of CCL heritage (and scholar fora heritage in general) is essential. Not only the text of the corpus of messages has to be perennially archived, but also their related metadata, timestamps, headers that define topics, etc. Mailing lists archives are a unique opportunity for historians to explore interactions, debates, even tensions among scientists that reveal a lot about scientific communities: they constitute an important alternative to more official sources such as published papers.

title: Computational Chemistry List


text:
User:ConCuu

ConCuu: "a sheep"

User since: April 25th, 2005

Usually puts around in: Internet Phenomenon

Or related articles such as: Tricky the Clown (now deleted) or Dancing Banana

Also a big fan of HeroScape

Usually "anonymous" under: 1 2 3 4 5 6 7 (or others I'm too lazy to find)

title: User:ConCuu


text:
Membranelle

Membranelles (also membranellae) are structures found around the mouth, or cytostome, in ciliates. They are typically arranged in series, to form an "adoral zone of membranelles", or AZM, on the left side of the buccal cavity (peristome). The membranelles are made up of kinetosomes arranged in groups to make up polykinetids. The cilia which emerge from these structures appear to be fused and to function as a single membrane, which can be used to sweep particles of food into the cytostome, or for locomotion.

title: Membranelle


text:
User talk:Jwestbrook/archive1

 '''This is an archive of My Talk page. Archived November 18, 2005. Please direct any comments to the live talk page '''

Welcome to Wikipedia
Hello and welcome to Wikipedia. Thank you for your contributions. I hope you like the place and decide to stay. Here are a few good links for newcomers such as yourself:

 * Be Bold!
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 * Meet other new users
 * Learn from others
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 * Contribute, Contribute, Contribute!
 * Tell us about you

 * How to edit a page
 * Editing tutorial
 * Picture tutorial
 * How to write a great article
 * Naming conventions
 * Manual of Style

I hope you enjoy editing here and being a Wikipedian! By the way, you can sign your name on Talk and vote pages using three tildes, like this: ~~~. Four tildes (~~~~) produces your name and the current date. If you have any questions, see the help pages, add a question to the village pump or ask me on my talk page. Again, welcome! -- Spinboy 21:19, 14 October 2005 (UTC)

title: User talk:Jwestbrook/archive1


text:
User talk:Sauersauce24

Welcome
Welcome!

Hello, and welcome to Wikipedia! Thank you for your contributions. I hope you like the place and decide to stay. Here are a few good links for newcomers: I hope you enjoy editing here and being a Wikipedian! Please sign your name on talk pages using four tildes (~~); this will automatically produce your name and the date. If you have any questions, check out Where to ask a question or ask me on my talk page. Again, welcome! Friday (talk) 21:20, 14 October 2005 (UTC)
 * The five pillars of Wikipedia
 * How to edit a page
 * Help pages
 * Tutorial
 * How to write a great article
 * Manual of Style

Test
Thanks for experimenting with Wikipedia. Your test worked, and has been removed or reverted. Please use the sandbox for any other tests you want to do. Take a look at the welcome page if you would like to learn more about contributing to our encyclopedia. Thanks. Friday (talk) 21:20, 14 October 2005 (UTC)

title: User talk:Sauersauce24


text:
User:LonesomeDrifter

 * O Lord, grant me chastity and continence, but not yet.
 * —Saint Augustine

Welcome to my user page! I a study philosophy, sociology, and political science, and will make contributions accordingly.

title: User:LonesomeDrifter


text:
Talk:Burning Man/Archive 1

This archive page covers approximately the dates between Jul 15 2003 and August 2005.

Post replies to the main talk page, copying or summarizing the section you are replying to if necessary.

Please add new archivals to Talk:Burning Man/Archive03. (See How to archive a talk page.) Thank you. Kit 21:18, 22 October 2005 (UTC)

What do they do there?
What do they do there? --Menchi 23:26 15 Jul 2003 (UTC)

Well, having been there, the list given in the article is pretty good at showing the different things that go on at a burn. However, it does seem to lack a clear bit about "what goes on there". It is difficult to really say tho, unless you have been, and even then, its quite different for each participent. As there are no spectators, everyone is involved in the event in some way. The line "experiment in community, radical self-expression, and radical self-reliance." pretty much says it all - you come to burningman to participate. How and what kind of participation is up to the individual burner. Hyakugei 21:35, 19 Aug 2003 (UTC)

The writer(s) of

''The opposite of a burner is a "yahoo", the term used by burners to refer to individuals who come to Burning Man solely for the purpose of masturbatory self-gratification. Yahoos are individuals who express little sense of ownership or commitment to the community. They are boorish spectators and gawking tourists whose presence is wholly unwelcome.''

seem(s) to have a somewhat proprietary attitude towards the whole thing, although I wouldn't necessarily go so far as to call the passage POV. Maybe someone more familiar than I with burner culture could, er, depersonalize it a little? --Calieber 20:05, Oct 31, 2003 (UTC)

I'd have to agree, this does not follow POV rules, or even what "everyone" at BM thinks (and who the heck knows what that might be!). I've never even heard the term "yahoo", but i've not been to a burn in several years, so things change. Can we re-write it... maybe something like

"The opposite of a burner is a "yahoo". This term is applied to those that come to spectate rather then participate in the event. As the purpose of BM is to form community and radical self-expression, non-participants are generally frowned apon."

 * hyakugei 01:10, 12 Nov 2003 (UTC)

I've been to the last two burns and I have yet to hear anyone use the term "yahoo." On the other hand, "tourist" is pretty common. And there are quite a variety of burner attitudes towards both tourists and participation.

Gecko 19:03, 1 Mar 2004 (UTC)

We could fix the whole para by making some point about participants vs. non-participants, rather then about what they are called...

hyakugei 21:19, 2 Mar 2004 (UTC)

Whoa. Someone with a serioux axe to grind just demolished this page! There's a whole slew of very biased and inflammatory edits.

This appears to be the handiwork of 4.240.192.35, who appears to have never edited anything else. It looks like a smear job, so I'm reverting to the prior edit.

Gecko 00:29, 19 May 2004 (UTC)

 * Interesting. I only know about Burning Man from reading about it, but a decent portion of what I've read is from people who have become upset with the behavior of the organization.  (For example, here.)  Perhaps somebody should write a "Criticism" section to show the "pissed off at Burning Man" point of view, if it's common?

 * Surprisingly common. At the event itself, there is a "mainstream newspaper" called the Black Rock Gazette and an "alternative newspaper" called Pissing Clear.  IMHO, this article would benefit from presenting a section highlighting criticism of the event. // NetEsq 15:19, 17 Sep 2004 (UTC)

 * I agree that there needs to be a criticism section. Especially in regards to JWZ's view of the event's copyright policies  --69.237.198.2 16:42, 22 August 2005 (UTC)

 * The article should have NPOV. Noting that the event has its critics isn't a bad idea -- it's a fact. Trashing the event in the article isn't NPOV. Also... I don't understand why there's a crude survival guide in the middle of the article now. Much of it is of dubious quality. I recommend replacing it with an external link to the actual Survival Guide on the Burning Man website. // Gecko 23:54, 17 Sep 2004 (UTC)

Deleted reference to commissary as exception to no-commerce rule, as no commerce takes place there. Free meal passes are handed out to staff and volunteer workers beforehand.

The Getting Around in Black Rock section is written in the second-person whereas the rest of the article is written in the third-person. Does anyone mind if I fix this? Fallstorm 06:20, 17 July 2005 (UTC)

Black Rock City, LLC
I just noticed that one of my earlier edits was deleted by an IP registered to Black Rock City, LLC. I for one welcome our new NPOV Overlords.

Wikipedian Burners category
I thought there should be a category for Wikipedians who attend Burning Man, so I created Category: Wikipedian Burners. Hooray. Kit 01:41:13, 2005-08-02 (UTC)

the airport
I changed the section on flying -- it now links to the actual Burning Man page on the airport, and correctly reflects that flying is not an option just 'this year' but has been for several years. Kit 23:24:11, 2005-08-06 (UTC)
 * Added some more info on air travel. Kit 23:32:32, 2005-08-06 (UTC)

Vandalism
The other day I removed the following from the article, but figured I'd paste it here for posterity: one stab at why I go to Burning Man I like the art. I like the late nights. I like the fire, and the desert, and exhaustion, when others are exhausted, camping, funny names, gifts, gifting, not having to carry your wallet with you for a week, being naked, being dressed up, being encouraged, being mocked, working at the Post Office and commanding volunteers, making deliveries, being delivered, watching the sunrise in swivel chairs that swivel 360 degrees while playing trance through speakers by your ears (anyone else find that spot deep 	 playa, not far from the Temple?), making cool postcards to send to friends, sending friends art cards from the playa, playa stamps, riding on art cars with women dressed in pink or women and men in orange coveralls. the good music, collaborating with other musicians, being part of rock bands, losing my voice, the gifting economy, being away from the sstraight world for a while, making new friends, hugging old friends, trying new things, getting hurt, being thirsty, pissing clear, Piss Clear, the mountains, the drive there, the hotsprings on the way there and back, laughs, more laughs, crying wishing my family could be there with me, happy that my family is not there with me, talking to locals on the way there who sort of have an idea of what we're headed to, getting others excited about going, keeping in touch through the year, decompression, my Burning Man film collection, wearing pendants and necklaces, being recognized in a cafe cause I'm wearing a BRSD orange shirt, thinking about how this might be changing the world, noticing how it's changed me - caused me to get FreecyclePortland going, taught me about how to receive gifts and give them, taught me I don't need nearly as much stuff as I thought I did, figuring out how to live in the desert, sleeping in my tent, camping with my friends from SF, building the dome, visiting snuggletown,

Nutmeg Alfredo BRCPO.com - There is no Team in Fuck You!

Kit 22:56:55, 2005-08-21 (UTC)

title: Talk:Burning Man/Archive 1


text: User talk:81.154.134.54

title: User talk:81.154.134.54


text:
Covenant Code

The Covenant Code, or Book of the Covenant, is the name given by academics to a text appearing in the Torah, at Exodus –; or, more strictly, the term Covenant Code may be applied to Exodus 21:1–22:16. Biblically, the text is the second of the law codes said to have been given to Moses by God at Mount Sinai. This legal text provides a small but substantive proportion of the mitzvot (religious duties) within the Torah, and hence is a source of Jewish Law.

Academic context
The date that the Covenant Code was composed, and the details of how it found its way into the Bible, continue to be debated. Most proponents of the documentary hypothesis associate it with either the Elohist ("E") materials, or, less commonly, the Yahwist ("J") materials. (These are two of the four sources of the classic documentary hypothesis, the other two being the Deuteronomic ("D") material and Priestly ("P") material.) According to Joel Baden, "The Covenant Code is a part of E; the priestly laws [of Leviticus and Numbers] are part of P; and the deuteronomic laws [of Deuteronomy 12–26] stand at the center of D." Regardless of precise positions on the process, scholars agree that the Covenant Code was produced by a long process in which it changed over time.

A study of continuing importance is that of Albrecht Alt, who in 1934 published an analysis of the Covenant Code which hinges on the distinction between casuistic and apodictic law. The Covenant Code consists largely of case or casuistic law (often in the form of an "if-then" statement, in which specific situations are addressed), as for example Exodus 21:33–36. Apodictic laws (characterized by absolute or general commands or prohibitions, as in the Ten Commandments) on the other hand, also appear in the Covenant Code, for example in Exodus 21:17 ("Whoever curses father or mother shall be put to death"). Alt claimed, though some scholars disagree, that the apodictic laws were a feature only found in Israelite codes. Scholars do, however, agree that the contrast between the apodictic and casuistic forms is a clue to how multiple sources of law were edited together into the Covenant Code, although there remain disagreements over the precise details.

The form and content of the code is similar to many other codes from the Near East of the second millennium BC. It also resembles the Babylonian Code of Hammurabi. According to many scholars including Martin Noth and Albrecht Alt, the covenant code probably originated as a civil code with the Canaanites, and was altered to add Hebrew religious practices. Michael Coogan sees a noticeable difference between the Covenant Code and the non-biblical codes like the Code of Hammurabi. The Covenant Code, like other biblical codes, differs from these by including among the laws dealing with criminal and civil matters various regulations concerning worship. Both, however, set the laws in an explicitly religious context.

Relationship to the Ritual Decalogue
Some of the commandments in the Covenant Code overlap noticeably with the commandments in the Ritual Decalogue. Robert Pfeiffer suggested that the Covenant Code is an expansion of the Ritual Decalogue. Carol Meyers holds that the direction of influence is in the opposite direction: that Exodus 34 borrows material from within the Covenant Code.

Evident values
The Covenant Code portrays the values of the society in which it was produced, some of which are different from Western twentieth-century values.

With the ancient cultural view of women as property of men, the casuistic law regarding the seduced virgin in Exodus 22:16–17 portrays a woman who, as the property of her father, has had her value diminished by the loss of her virginity. However, this law still calls for restitution to be paid by the man who seduced her. A second example comes from Exodus 21:20–21, which describes the punishment required for a slave owner who strikes his slave with a rod. If the slave survives their injuries no punishment is required because they do not have the same rights as Israelite males.

In some instances, the values represented in the Covenant Code are more similar to present-day, Western values. Two examples include the placing of mothers on the same level as fathers in Exodus 21:15, 17, and providing for special care of members of lesser social classes, including converts, widows and orphans (Exodus 22:21–22).

title: Covenant Code


text:
Ducati Multistrada

First introduced in 2003, the Ducati Multistrada is a series of V-twin and V4 touring focused motorcycles. Essentially a hybrid of a supermoto and a sport-tourer, the Multistrada competes in the market with other dual-sport motorcycles such as the BMW GS. The first iteration of the Multistrada was, like the Yamaha TDM850, neither intended nor suitable for off-road use. Subsequent models were more suited to a proper dual-sport role.

Designed by Pierre Terblanche as an evolution to his previous design, the Cagiva Gran Canyon, the original Multistrada drew mixed reviews for its unconventional aesthetics. The word Multistrada is Italian for "many roads".

Reception
Motor Cycle News described the original Multistrada 1000 as "a truly versatile motorcycle", but let down by "a smallish tank and limited performance". Motorcycle Consumer News categorised the Multistrada as an "Open Standard".

Models
The Multistrada 1000DS was released in 2003. There were two basic Ducati Multistrada models in 2005, the 1000DS and the 620. There was also the 620 Dark entry model. For 2007, there were Multistrada 1100 and 1100S models.

Ducati Multistrada 1000DS
The 2005 Ducati Multistrada 1000DS has a 992 cc air cooled 90° V-twin engine with a 40 degree included valve angle, based on Ducati's existing engines, with twin-spark plug heads, pressure fed plain camshaft bearings, redesigned crankshaft, higher oil pressure and volume, and new alloy clutch basket, drive and driven plates. The 2006 1000DS uses Ducati's signature trellis frame with fully adjustable 165 mm travel Showa forks up front, and a single-sided swingarm, with a Showa fully adjustable rear shock coupled with a rising rate, height-adjustable suspension system at the rear. Brembo "Serie Oro" four piston brake callipers are used in the front and rear disc brakes. Front 320 mm discs, single 245 mm rear disc, and steel-braided brake lines are used throughout. The discs are now mounted directly to oversized hubs, eliminating the disc carriers. The wet weight is 504 –.

Ducati Multistrada 620
Entry level Multistradas feature a number of individual components, like the double sided "Monster" style swing arm, the integrated die cast front and rear footrests, and the three spoke alloy wheels. The cheapest Multistrada is the 620 Dark, with its matt black colour scheme and single 320 mm front disk. There are also parts that look the same, but aren't, like the fuel tank which although of identical size as the 1000s, due to design simplification will only hold 15 liters until the bike is ridden a short distance and refilled, to full 20 liter capacity and the simpler instrument panel without on-board computer functions. All body panels are identical to those of the 1000, and 1100: except the locking right front glove box which is not included on the 620 models (although the locking version can be retrofitted if desired).

The frame is as big as the Multistrada 1000 DS frame, but is different at the rear, where the single sided swing arm has been replaced by a double sided swing arm. Suspension features Marzocchi 43 mm upside down front forks and a Sachs rear monoshock with adjustable preload and rebound. The braking system has a rear disk and two 300 mm front disks with floating callipers operated by a new front master cylinder.

The 620 has an APTC wet clutch. According to Ducati, the wet clutch reduces lever operating effort, ensuring gradual engagement, and incorporates a racing derived "slipper" device to reduce the tendency for the rear wheel to lock up or chatter during aggressive downshifting.

2003–2004
The first version to be sold in 2003 had a 1,000 cc engine, but a smaller 620 cc version was introduced in 2005. The 2003 model Multistrada received most of its criticisms because of its seat, described as "a vinyl covered sheet of plywood" by Motorcyclist magazine, and windscreen that caused undue buffeting.

2005–2006
In 2005, the 1000DS Multistrada received an improved side stand (the 2003–2004 side stand was too short and made the bike prone to accidental tip-over), redesigned seats with softer padding and less angular shapes, improved mirrors with longer stems for better visibility and the optional touring screen from the 2003–2004 model year became standard. 2005 also saw the introduction of the 1000DS S model that came with Öhlins suspension, black wheels and carbon fiber belt covers and front fender. The S-model also came with an aluminum oversized handlebar that was said to reduce vibration and improve handling.

2007
In 2007, the 1,000 cc engine was replaced with a 1,100 cc engine that was claimed to produce an extra 5 ft.lbf of torque and the 620 model was dropped. Both the S and standard models came with black wheels but the suspension on the standard model was changed from the Showa components of previous years to a Marzocchi front fork and a Sachs rear shock. In order to make the S-model more appealing, it was fitted with fully adjustable Ohlin's suspension front and rear. Ducati also claimed to have improved fuel gauge accuracy, improved vibration isolated handlebars were introduced to reduce rider fatigue. A closed loop fuel injection system replaced the open loop FI system on the 1,000 cc models and the characteristic dry clutch was replaced with a conventional wet-clutch. Ducati also increased the valve adjust interval from 6000 to 7000 mi for a claimed 50% reduction in maintenance costs.

2008–2009
In 2008, the bike received minor tweaks to the decals changing to a font that closer resembles the graphics on the Ducati 1098. The closed loop FI systems also got some fine tuning to make it more refined. It also got a timer switch for the lights (the lights come on with the ignition and have no off switch) that shuts them off after 60 seconds if the engine has not been started, helping to increase battery life. The factory fitted immobiliser, for better security, remained standard, as it was since 2003 with the model's release. The 1100s model was fitted with Ohlins suspension.

2010
A new model called the Multistrada 1200 was presented at 2009 EICMA in Milan. It includes throttle by wire, a detuned version of the 1,200 cc liquid-cooled engine from the Ducati 1198 sportbike, and optional electronically adjustable suspension.

2015
With the third generation of Multistrada, Ducati focused particularly on the engine mechanics which includes a Testastretta DVT with 160 HP and 136 Nm of torque. It is equipped with Double Variable Timing - both the intake and exhaust valves can vary the timing. There is a later update with a new engine, the 1262 cc Testastretta 11° DVT that was used to power the Diavel.

2021
The 2021 range brings the fourth generation to its debut, featuring important innovations. The most obvious concerns the new 1158 cm³ and 170 HP at 10,500 rpm V4 Granturismo engine: it is a four-cylinder that abandons the historic desmodromic distribution for a more classic spring return of the valves. The V4 Granturismo engine adopts deactivation of the rear bank at idle so that there is no combustion in the cylinders, reducing consumption and improving thermal comfort thanks to lower temperatures. The frame was also revolutionized: from the steel tube trellis to the aluminum monocoque. As already seen on the previous Enduro and 950 models, the double sided swingarm and 19" front wheel are adopted across the entire range. On the V4 the suspension is fully adjustable Marzocchi mechanicals, while the V4 S is equipped with Ducati Skyhook semi-active Suspension Evolution with auto-leveling function for automatic height adjustment from the ground. An absolute first is the ARAS system featuring front and rear radar, through which it is possible to implement adaptive cruise control and blind spot detection.

With the 2022 model year, the Pikes Peak trim level returns, which is now clearly different from the rest of the range: the sporty characterization of the model includes the new single-sided swingarm (unpublished on the V4) and the 17" front wheel, combined with specific geometries for wheelbase, trail, steering head rake, raised footrests and narrower handlebars, and the "Race" configuration is making its debut on a Multistrada instead of the "Enduro" one, a mode recommended for circuit riding. the new Öhlins Smart EC 2.0 electronic suspension, the approved Akrapovic titanium and carbon silencer, the forged aluminum rims and various carbon racing details such as the mudguard and front part to lighten the bike. Also in 2022, the V4 version is presented a Rally version dedicated to travel, with semi-active suspension with increased travel to 200 mm, Power Mode dedicated to off-road use, Auto function levelling, which guarantees a constant set-up as the rider, passenger and luggage configuration changes and a new 30-litre aluminum tank.

2022
The Ducati Multistrada V2 replaced the Multistrada 950 for the 2022 model year. It has a 937 cc V-twin engine, and weighs as little as 489 lb, less than its predecessor.

title: Ducati Multistrada


text:
User talk:Betakate

Welcome
Hello, , and welcome to Wikipedia. Thank you for your contributions. I hope you like the place and decide to stay. Here are a few good links for newcomers: I hope you enjoy editing here and being a Wikipedian! By the way, you can sign your name on Talk and vote pages using three tildes, like this: ~~~. Four tildes (~~~~) produces your name and the current date. If you have any questions, see the help pages, add a question to the village pump or ask me on my talk page. Again, welcome! G Clark 21:21, 14 October 2005 (UTC)
 * The Five Pillars of Wikipedia
 * How to edit a page
 * Editing tutorial
 * Picture tutorial
 * How to write a great article
 * Naming conventions
 * Manual of Style

Basic taste
Hi Betakate! You delite "Styptic" taste without any motivation. But you really can separate "Styptic" taste to other "Basic taste"? If no, propose recover it in article. If yes - said what components (in %) formed "Styptic" taste :-) Alexandrov 12:27, 23 January 2006 (UTC)

 * Hi Alexandrov. I removed the heading "Styptic" in Basic taste because it contained no information in that section (diff). It would be great if you have sourced information on styptic taste you can share for that article!. -betakate 13:29, 23 January 2006 (UTC)

OAIS
Hello. OAIS was deleted because someone used Proposed deletion (PROD) on it, and there were no objections within 5 days. The deleting admin also presumably thought it should go. However, I've looked at the article and, whilst not outstanding it wasn't all that bad, and not apparently a copy-paste from anywhere. Part of the deal with PROD is that, if anyone later contests the deletion, it should be restored straightaway. I have therefore undeleted the article, at OAIS. Enjoy! -Splash - tk 16:15, 11 July 2006 (UTC)

DC Meetup Events: You're invited!
This has been an automated because you your name was on the invite list. BrownBot (talk) 16:20, 21 February 2009 (UTC)

D.C. Meetup, Saturday, June 6, 2009
The 7th DC Meetup dinner will be held this Saturday, June 6th, starting at 5 p.m. The event will be at Bertucci's, near George Washington University and the Foggy Bottom metro station. It will follow the Apps for Democracy open source event at GWU. For details or to RSVP if you haven't already, see Meetup/DC 7. (You have received this announcement because your user page indicates that you live in Maryland, Virginia, or DC.) Delivered by The  Helpful  Bot   at 19:37, 2 June 2009 (UTC) to report errors, please leave a note here.

Volunteer opportunity in Bethesda, Thursday, July 16
The Wikimedia Foundation will be conducting an all-day Academy at the National Institutes of Health, in Bethesda, Maryland, on Thursday, July 16. The team that will be teaching at the Academy, a mix of paid staff and volunteers, is looking for four more volunteers to be teaching assistants, providing one-to-one assistance in workshops whenever a workshop participant has a problem following the instructional directions. (We currently have two editors signed up as teaching assistants, and are looking for a total of six.)

The NIH editing workshops are only for two hours, but volunteers are asked to meet the Wikimedia Foundation team at the hotel in Bethesda at about 7:15 a.m. (time to be finalized shortly) and to stay for the entire day, which ends at 4:30 p.m. Lunch will be provided. (The full schedule can be found here.)

The team is not necessarily looking for expert editors (though they are welcome), just people who can help novices who might get stuck when trying to do some basic things. If you've been an editor for at least 3 months, and have done at least 500 edits, you probably qualify.

If you're interested, please send John Broughton an email. If you might be interested, but would like further information, please post a note on his user talk page, so that he can respond there, and others can see what was asked.

(You have received this posting because your user page indicates that you live in Maryland or DC. --EdwardsBot (talk) 04:11, 10 July 2009 (UTC))

D.C. Meetup, Saturday, September 26
The 8th DC Meetup dinner will be held this Saturday, September 26, starting at 6 p.m. The event will be at Burma Restaurant (740 6th St, NW near the Gallery Place-Chinatown Metro station). For details or to RSVP if you haven't already, see Meetup/DC 8. (You have received this announcement because your user page indicates that you live in Maryland, Virginia, or DC.) --EdwardsBot (talk) 06:49, 23 September 2009 (UTC)

Planning Discussions Now Finished Regarding DC Meetup #9
--NBahn (talk) 02:36, 10 December 2009 (UTC)
 * You are receiving this message either because you received a similar one before and didn't object, or you requested to receive a similar one in the future. If you don't wish to receive this message again, then please let me know either on my talk page or here.
 * Planning — for the most part, anyway — is now finished (see here) for DC Meetup #9.

Planning Discussions Now Underway Regarding DC Meetup #10

 * You are receiving this message either because you received a similar one before and didn't object, or you requested to receive a similar one in the future. If you don't wish to receive this message again, then please let me know either on my talk page or here.
 * Please be advised that planning is now underway (see here) for DC Meetup #10. --NBahn (talk) 15:16, 7 May 2010 (UTC)

Meetup/DC 11
Hey, just in case you missed it, there is an oppurtunity to get a free dinner this Tuesday August 11 and a chance to meet and hang out talk about WikiProject United States Public Policy and WP:GLAM/SI. Sorry that this is so late in the game, I was hoping the e-mail would be a better form of contact for active members (if you want to get on the e-mail list send me an User e-mail ). Hope that you can attend, User:Sadads (talk)11:40, 9 August 2010 (UTC)

DC Meetup #12
An off-wiki discussion is taking place concerning DC Meetup #12. Watch this page for announcements.

—NBahn (talk) 04:31, 9 September 2010 (UTC)

P.S. You are receiving this message either because you received a similar one before and didn't object, or you requested to receive a similar one in the future. If you don't wish to receive this message again, then please let me know either on my talk page or here.

title: User talk:Betakate


text: Category:Theatre in Ghana

title: Category:Theatre in Ghana


text:
Noth

Noth is a German surname. Notable people with the surname include:
 * Chris Noth (born 1954), American film and television actor
 * Martin Noth (1902–1968), German scholar of the Hebrew Bible
 * Winfried Nöth (born 1944), German linguist and semiotician

title: Noth


text:
User talk:209.68.147.232

April 2016
Hello. I wanted to let you know that I reverted one of your recent contributions —the one you made with this edit to Chicago— because it did not appear to be constructive. If you would like to experiment, please use the sandbox. If you think I made a mistake, or if you have any questions, you can leave me a message on my talk page. Thanks. Dcirovic (talk) 16:10, 29 April 2016 (UTC)
 * If this is a shared IP address, and you did not make the edits, consider creating an account for yourself so you can avoid further irrelevant notices.

January 2019
Hello, I'm Shellwood. I wanted to let you know that I reverted one of your recent contributions —specifically this edit to USS Coontz— because it did not appear constructive. If you would like to experiment, please use the sandbox. If you have any questions, you can ask for assistance at the Help Desk. Thanks. Shellwood (talk) 18:59, 17 January 2019 (UTC)

Please refrain from making unconstructive edits to Wikipedia, as you did with this edit to USS Coontz. Your edits appear to be vandalism and have been reverted or removed. If you would like to experiment, please use the sandbox. Repeated vandalism can result in the loss of editing privileges. Thank you. Shellwood (talk) 19:07, 17 January 2019 (UTC)

title: User talk:209.68.147.232


text:
Gamma matrices

In mathematical physics, the gamma matrices, $$\ \left\{ \gamma^0, \gamma^1, \gamma^2, \gamma^3 \right\}\ ,$$ also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra $$\ \mathrm{Cl}_{1,3}(\mathbb{R}) ~.$$ It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin $\tfrac{\ 1\ }{2}$ particles. Gamma matrices were introduced by Paul Dirac in 1928.

In Dirac representation, the four contravariant gamma matrices are

\begin{align} \gamma^0 &= \begin{pmatrix} 1 & 0 & 0 & 0 \\  0 & 1 &  0 & 0 \\   0 & 0 &  -1 & 0 \\  0 & 0 &  0 & -1 \end{pmatrix}, & \gamma^1 &= \begin{pmatrix} 0 & 0 & 0 & 1 \\   0 &  0 & 1 & 0 \\   0 & -1 & 0 & 0 \\  -1 &  0 & 0 & 0 \end{pmatrix}, \\ \\ \gamma^2 &= \begin{pmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{pmatrix}, & \gamma^3 &= \begin{pmatrix} 0 & 0 & 1 & 0 \\   0 & 0 & 0 & -1 \\  -1 & 0 & 0 &  0 \\   0 & 1 & 0 &  0 \end{pmatrix} ~. \end{align} $$

$$\gamma^0$$ is the time-like, Hermitian matrix. The other three are space-like, anti-Hermitian matrices. More compactly, $$\ \gamma^0 = \sigma^3 \otimes I_2\ ,$$ and $$\ \gamma^j = i \sigma^2 \otimes \sigma^j \ ,$$ where $$\ \otimes\ $$ denotes the Kronecker product and the $$\ \sigma^j\ $$ (for $j$ = 1, 2, 3) denote the Pauli matrices.

In addition, for discussions of group theory the identity matrix ($I$) is sometimes included with the four gamma matricies, and there is an auxiliary, "fifth" traceless matrix used in conjunction with the regular gamma matrices
 * $$\begin{align}

\ I_4 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\  0 & 0 & 1 & 0 \\  0 & 0 & 0 & 1 \end{pmatrix} \, \qquad \gamma^5 \equiv i\gamma^0\gamma^1\gamma^2\gamma^3 = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\  1 & 0 & 0 & 0 \\  0 & 1 & 0 & 0 \end{pmatrix} ~. \end{align} $$ The "fifth matrix" $$\ \gamma^5\ $$ is not a proper member of the main set of four; it is used for separating nominal left and right chiral representations.

The gamma matrices have a group structure, the gamma group, that is shared by all matrix representations of the group, in any dimension, for any signature of the metric. For example, the 2×2 Pauli matrices are a set of "gamma" matrices in three dimensional space with metric of Euclidean signature (3, 0). In five spacetime dimensions, the four gammas, above, together with the fifth gamma-matrix to be presented below generate the Clifford algebra.

Mathematical structure
The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation
 * $$\left\{ \gamma^\mu, \gamma^\nu \right\} = \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^{\mu \nu} I_4\ ,$$

where the curly brackets $$\ \{, \}\ $$ represent the anticommutator, $$\ \eta_{\mu \nu}\ $$ is the Minkowski metric with signature (+ − − −), and $$I_4$$ is the 4 × 4 identity matrix.

This defining property is more fundamental than the numerical values used in the specific representation of the gamma matrices. Covariant gamma matrices are defined by
 * $$\ \gamma_\mu = \eta_{\mu \nu} \gamma^\nu = \left\{\gamma^0, -\gamma^1, -\gamma^2, -\gamma^3 \right\}\ ,$$

and Einstein notation is assumed.

Note that the other sign convention for the metric, (− + + +) necessitates either a change in the defining equation:
 * $$\ \left\{ \gamma^\mu, \gamma^\nu \right\} = -2 \eta^{\mu \nu} I_4\ $$

or a multiplication of all gamma matrices by $$i$$, which of course changes their hermiticity properties detailed below. Under the alternative sign convention for the metric the covariant gamma matrices are then defined by
 * $$\ \gamma_\mu = \eta_{\mu \nu} \gamma^\nu = \left\{-\gamma^0, +\gamma^1, +\gamma^2, +\gamma^3 \right\} ~.$$

Physical structure
The Clifford algebra $$\ \mathrm{Cl}_{1,3}(\mathbb{R})\ $$ over spacetime $V$ can be regarded as the set of real linear operators from $V$ to itself, $End(V)$, or more generally, when complexified to $$\ \mathrm{Cl}_{1,3}(\mathbb{R})_\mathbb{C}\ ,$$ as the set of linear operators from any four-dimensional complex vector space to itself. More simply, given a basis for $V$, $$\ \mathrm{Cl}_{1,3}(\mathbb{R})_\mathbb{C}\ $$ is just the set of all $4×4$ complex matrices, but endowed with a Clifford algebra structure. Spacetime is assumed to be endowed with the Minkowski metric $ημν$. A space of bispinors, $Ux$, is also assumed at every point in spacetime, endowed with the bispinor representation of the Lorentz group. The bispinor fields $Ψ$ of the Dirac equations, evaluated at any point $x$ in spacetime, are elements of $Ux$ (see below). The Clifford algebra is assumed to act on $Ux$ as well (by matrix multiplication with column vectors $Ψ(x)$ in $Ux$ for all $x$). This will be the primary view of elements of $$\ \mathrm{Cl}_{1,3}(\mathbb{R})_\mathbb{C}\ $$ in this section.

For each linear transformation $S$ of $Ux$, there is a transformation of $End(Ux)$ given by $S E S−1$ for $E$ in $$\ \mathrm{Cl}_{1,3}(\mathbb{R})_\mathbb{C} \approx \operatorname{End}(U_x) ~.$$ If $S$ belongs to a representation of the Lorentz group, then the induced action $E ↦ S E S−1$ will also belong to a representation of the Lorentz group, see Representation theory of the Lorentz group.

If $S(Λ)$ is the bispinor representation acting on $Ux$ of an arbitrary Lorentz transformation $Λ$ in the standard (4 vector) representation acting on $V$, then there is a corresponding operator on $$\ \operatorname{End}\left( U_x \right) = \mathrm{Cl}_{1,3}\left( \mathbb{R} \right)_\mathbb{C}\ $$ given by equation:

 * $$\ \gamma^\mu \ \mapsto \ S(\Lambda)\ \gamma^\mu\ {S(\Lambda)}^{-1} = {\left( \Lambda^{-1} \right)^\mu}_\nu\ \gamma^\nu = {\Lambda_\nu}^\mu\ \gamma^\nu \ ,$$

showing that the quantity of $γμ$ can be viewed as a basis of a representation space of the 4 vector representation of the Lorentz group sitting inside the Clifford algebra. The last identity can be recognized as the defining relationship for matrices belonging to an indefinite orthogonal group, which is $$\ \eta\Lambda^\textsf{T}\eta = \Lambda^{-1}\ ,$$ written in indexed notation. This means that quantities of the form

 * $$a\!\!\!/ \equiv a_\mu\gamma^\mu$$

should be treated as 4 vectors in manipulations. It also means that indices can be raised and lowered on the $γ$ using the metric $ημν$ as with any 4 vector. The notation is called the Feynman slash notation. The slash operation maps the basis $eμ$ of $V$, or any 4 dimensional vector space, to basis vectors $γμ$. The transformation rule for slashed quantities is simply

 * $${a\!\!\!/}^\mu \mapsto {\Lambda^\mu}_\nu {a\!\!\!/}^\nu ~.$$

One should note that this is different from the transformation rule for the $γμ$, which are now treated as (fixed) basis vectors. The designation of the 4 tuple $$\left( \gamma^\mu \right)_{\mu=0}^{3} = \left(\gamma^0, \gamma^1, \gamma^2, \gamma^3 \right)$$ as a 4 vector sometimes found in the literature is thus a slight misnomer. The latter transformation corresponds to an active transformation of the components of a slashed quantity in terms of the basis $γμ$, and the former to a passive transformation of the basis $γμ$ itself.

The elements $$\ \sigma^{\mu \nu} = \gamma^\mu \gamma^\nu - \gamma^\nu \gamma^\mu\ $$ form a representation of the Lie algebra of the Lorentz group. This is a spin representation. When these matrices, and linear combinations of them, are exponentiated, they are bispinor representations of the Lorentz group, e.g., the $S(Λ)$ of above are of this form. The 6 dimensional space the $σ^{μν}$ span is the representation space of a tensor representation of the Lorentz group. For the higher order elements of the Clifford algebra in general and their transformation rules, see the article Dirac algebra. The spin representation of the Lorentz group is encoded in the spin group Spin(1, 3) (for real, uncharged spinors) and in the complexified spin group Spin(1, 3) for charged (Dirac) spinors.

Expressing the Dirac equation
In natural units, the Dirac equation may be written as
 * $$\ \left(i \gamma^\mu \partial_\mu - m\right) \psi = 0\ $$

where $$\ \psi\ $$ is a Dirac spinor.

Switching to Feynman notation, the Dirac equation is
 * $$\ (i {\partial\!\!\!/} - m) \psi = 0 ~.$$

The fifth "gamma" matrix, 5
It is useful to define a product of the four gamma matrices as $$\gamma ^5 = \sigma_1\otimes I  $$, so that
 * $$\ \gamma^5 \equiv i\gamma^0\gamma^1\gamma^2\gamma^3 =

\begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\  1 & 0 & 0 & 0 \\  0 & 1 & 0 & 0 \end{pmatrix} \qquad $$ (in the Dirac basis). Although $$\ \gamma^5\ $$ uses the letter gamma, it is not one of the gamma matrices of $$\ \mathrm{Cl}_{1,3}(\mathbb{R}) ~.$$ The index number 5 is a relic of old notation: $$\ \gamma^0\ $$ used to be called "$$\gamma^4$$".

$$\ \gamma^5\ $$ has also an alternative form:
 * $$\ \gamma^5 = \tfrac{i}{4!} \varepsilon^{\mu \nu \alpha \beta} \gamma_{\mu} \gamma_{\nu} \gamma_{\alpha} \gamma_{\beta}\ $$

using the convention $$\varepsilon_{0 1 2 3} = 1\ ,$$ or
 * $$\ \gamma^5 = -\tfrac{i}{4!} \varepsilon^{\mu \nu \alpha \beta} \gamma_{\mu} \gamma_{\nu} \gamma_{\alpha} \gamma_{\beta}\ $$

using the convention $$\varepsilon^{0 1 2 3} = 1 ~.$$ Proof:

This can be seen by exploiting the fact that all the four gamma matrices anticommute, so
 * $$ \gamma^0 \gamma^1 \gamma^2 \gamma^3 = \gamma^{[0} \gamma^1 \gamma^2 \gamma^{3]} = \tfrac{1}{4!} \delta^{0123}_{\mu\nu\varrho\sigma} \gamma^\mu \gamma^\nu \gamma^\varrho \gamma^\sigma\ ,$$

where $$\delta^{\alpha\beta\gamma\delta}_{\mu\nu\varrho\sigma}$$ is the type (4,4) generalized Kronecker delta in 4 dimensions, in full antisymmetrization. If $$\ \varepsilon_{\alpha \dots \beta}\ $$ denotes the Levi-Civita symbol in $n$ dimensions, we can use the identity $$ \delta^{\alpha\beta\gamma\delta}_{\mu\nu\varrho \sigma} = \varepsilon^{\alpha\beta\gamma\delta} \varepsilon_{\mu\nu\varrho\sigma} $$. Then we get, using the convention $$\ \varepsilon^{0123} = 1\ ,$$
 * $$\ \gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3 = \frac{i}{4!} \varepsilon^{0123}\varepsilon_{\mu\nu\varrho\sigma} \,\gamma^\mu\gamma^\nu\gamma^\varrho \gamma^\sigma = \tfrac{i}{4!} \varepsilon_{\mu\nu\varrho\sigma} \,\gamma^\mu\gamma^\nu\gamma^\varrho \gamma^\sigma = -\tfrac{i}{4!} \varepsilon^{\mu\nu\varrho\sigma} \,\gamma_\mu\gamma_\nu\gamma_\varrho \gamma_\sigma$$

This matrix is useful in discussions of quantum mechanical chirality. For example, a Dirac field can be projected onto its left-handed and right-handed components by:
 * $$\ \psi_{\mathrm L} = \frac{\ I - \gamma^5\ }{2}\ \psi, \qquad \psi_{\mathrm R} = \frac{\ I + \gamma^5\ }{2}\ \psi ~.$$

Some properties are:
 * It is Hermitian:
 * $$\left(\gamma^5\right)^\dagger = \gamma^5 ~.$$
 * Its eigenvalues are ±1, because:
 * $$\left(\gamma^5\right)^2 = I_4 ~.$$
 * It anticommutes with the four gamma matrices:
 * $$\left\{ \gamma^5,\gamma^\mu \right\} = \gamma^5 \gamma^\mu + \gamma^\mu \gamma^5 = 0 ~.$$

In fact, $$\ \psi_{\mathrm L}\ $$ and $$\ \psi_{\mathrm R}\ $$ are eigenvectors of $$\ \gamma^5\ $$ since
 * $$\gamma^5 \psi_{\mathrm L} = \frac{\ \gamma^5 - \left(\gamma^5\right)^2\ }{2} \psi = - \psi_{\mathrm L}\ ,$$ and $$ \gamma^5 \psi_{\mathrm R} = \frac{\ \gamma^5 + \left(\gamma^5\right)^2\ }{2} \psi = \psi_{\mathrm R} ~.$$

Five dimensions
The Clifford algebra in odd dimensions behaves like two copies of the Clifford algebra of one less dimension, a left copy and a right copy. Thus, one can employ a bit of a trick to repurpose $i γ5$ as one of the generators of the Clifford algebra in five dimensions. In this case, the set ${γ0, γ1, γ2, γ3, i γ5 }$ therefore, by the last two properties (keeping in mind that $i2 ≡ −1$) and those of the ‘old’ gammas, forms the basis of the Clifford algebra in $5$ spacetime dimensions for the metric signature $(1,4)$. . In metric signature $(Γ^{a}) = (γμ, i γ5 )$, the set $a = (0, 1, 2, 3, 4)$ is used, where the ${Γa, Γb}$ are the appropriate ones for the $(4,1)$ signature. This pattern is repeated for spacetime dimension ${γ^{ 0}, γ^{ 1}, γ^{ 2}, γ^{ 3}, γ^{ 5} }$ even and the next odd dimension $γ^{ μ}$ for all $(3,1)$. For more detail, see higher-dimensional gamma matrices.

Identities
The following identities follow from the fundamental anticommutation relation, so they hold in any basis (although the last one depends on the sign choice for $$\gamma^5$$).

Miscellaneous identities
1. $$\gamma^\mu\gamma_\mu = 4 I_4$$

2. $$\gamma^\mu\gamma^\nu\gamma_\mu = -2\gamma^\nu$$

3. $$\gamma^\mu\gamma^\nu\gamma^\rho\gamma_\mu = 4\eta^{\nu\rho} I_4$$

4. $$\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma\gamma_\mu = -2\gamma^\sigma\gamma^\rho\gamma^\nu$$

5. $$\gamma^\mu\gamma^\nu\gamma^\rho = \eta^{\mu\nu}\gamma^\rho + \eta^{\nu\rho}\gamma^\mu - \eta^{\mu\rho}\gamma^\nu - i\epsilon^{\sigma\mu\nu\rho}\gamma_\sigma\gamma^5$$

6. $$\gamma^5\sigma^{\nu\rho} = \tfrac{i}{2} \epsilon^{\sigma\mu\nu\rho} \sigma_{\sigma\mu}\ ,$$ where $$\ \sigma_{\mu\nu} = \tfrac{i}{2} [\gamma_{\mu},\gamma_{\nu}] =  \tfrac{i}{2}(\gamma_{\mu}\gamma_{\nu}-\gamma_{\nu}\gamma_{\mu}) \ $$

Trace identities
The gamma matrices obey the following trace identities: 1. $\operatorname{tr} \left(\gamma^\mu\right) = 0$

2. Trace of any product of an odd number of $\gamma^\mu$ is zero

3. Trace of $\gamma^5$ times a product of an odd number of $\gamma^\mu$ is still zero

4. $\operatorname{tr} \left(\gamma^\mu\gamma^\nu\right) = 4\eta^{\mu\nu}$

5. $\operatorname{tr} \left(\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma\right) = 4\left(\eta^{\mu\nu}\eta^{\rho\sigma} - \eta^{\mu\rho}\eta^{\nu\sigma} + \eta^{\mu\sigma}\eta^{\nu\rho}\right)$

6. $\operatorname{tr} \left(\gamma^5\right) = \operatorname{tr} \left(\gamma^\mu\gamma^\nu\gamma^5\right) = 0$

7. $\operatorname{tr} \left(\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma\gamma^5\right) = -4i\epsilon^{\mu\nu\rho\sigma}$

8. $\operatorname{tr} \left(\gamma^{\mu_1}\dots\gamma^{\mu_n}\right) = \operatorname{tr} \left(\gamma^{\mu_n}\dots\gamma^{\mu_1}\right)$

Proving the above involves the use of three main properties of the trace operator:
 * tr(A + B) = tr(A) + tr(B)
 * tr(rA) = r tr(A)
 * tr(ABC) = tr(CAB) = tr(BCA)

This implies $$\operatorname{tr}(\gamma^\nu) = 0$$
 * }

This can only be fulfilled if
 * $$\operatorname{tr} \left(\gamma^\mu \gamma^\nu \gamma^\rho \right) = 0 $$

The extension to 2n + 1 (n integer) gamma matrices, is found by placing two gamma-5s after (say) the 2n-th gamma-matrix in the trace, commuting one out to the right (giving a minus sign) and commuting the other gamma-5 2n steps out to the left [with sign change (-1)^2n = 1]. Then we use cyclic identity to get the two gamma-5s together, and hence they square to identity, leaving us with the trace equalling minus itself, i.e. 0.
 * }

 * }

For the term on the right, we'll continue the pattern of swapping $$\gamma^\sigma $$ with its neighbor to the left,

 * $$\operatorname{tr} \left( \gamma^\mu \gamma^\nu \gamma^\sigma \gamma^\rho \right) $$
 * $$= \operatorname{tr} \left(\gamma^\mu \left(2 \eta^{\nu \sigma} - \gamma^\sigma \gamma^\nu\right) \gamma^\rho \right) $$
 * $$= 2 \eta^{\nu \sigma} \operatorname{tr} \left(\gamma^\mu \gamma^\rho \right) - \operatorname{tr} \left(\gamma^\mu \gamma^\sigma \gamma^\nu \gamma^\rho \right) \quad \quad (2) $$
 * }
 * $$= 2 \eta^{\nu \sigma} \operatorname{tr} \left(\gamma^\mu \gamma^\rho \right) - \operatorname{tr} \left(\gamma^\mu \gamma^\sigma \gamma^\nu \gamma^\rho \right) \quad \quad (2) $$
 * }

Again, for the term on the right swap $$\gamma^\sigma$$ with its neighbor to the left,

 * $$\operatorname{tr} \left( \gamma^\mu \gamma^\sigma \gamma^\nu \gamma^\rho \right)$$
 * $$ = \operatorname{tr} \left( \left(2 \eta^{\mu \sigma} - \gamma^\sigma \gamma^\mu\right) \gamma^\nu \gamma^\rho \right) $$
 * $$= 2 \eta^{\mu \sigma} \operatorname{tr} \left( \gamma^\nu \gamma^\rho \right) - \operatorname{tr} \left( \gamma^\sigma \gamma^\mu \gamma^\nu \gamma^\rho \right)\quad \quad (3) $$
 * }
 * $$= 2 \eta^{\mu \sigma} \operatorname{tr} \left( \gamma^\nu \gamma^\rho \right) - \operatorname{tr} \left( \gamma^\sigma \gamma^\mu \gamma^\nu \gamma^\rho \right)\quad \quad (3) $$
 * }

Eq (3) is the term on the right of eq (2), and eq (2) is the term on the right of eq (1). We'll also use identity number 3 to simplify terms like so:
 * $$2 \eta^{\rho \sigma} \operatorname{tr} \left(\gamma^\mu \gamma^\nu \right) = 2 \eta^{\rho \sigma} \left(4 \eta^{\mu \nu}\right) = 8 \eta^{\rho \sigma} \eta^{\mu \nu} .$$

So finally Eq (1), when you plug all this information in gives
 * $$\operatorname{tr} \left(\gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma\right) = 8 \eta^{\rho \sigma} \eta^{\mu \nu} - 8 \eta^{\nu \sigma} \eta^{\mu \rho} + 8 \eta^{\mu \sigma} \eta^{\nu \rho} $$
 * $$ - \ \operatorname{tr} \left( \gamma^\sigma \gamma^\mu \gamma^\nu \gamma^\rho \right) \quad \quad \quad \quad \quad \quad (4) $$

The terms inside the trace can be cycled, so
 * $$\operatorname{tr} \left( \gamma^\sigma \gamma^\mu \gamma^\nu \gamma^\rho \right) = \operatorname{tr} \left(\gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma\right). $$

So really (4) is
 * $$2 \ \operatorname{tr} \left(\gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma\right) = 8 \eta^{\rho \sigma} \eta^{\mu \nu} - 8 \eta^{\nu \sigma} \eta^{\mu \rho} + 8 \eta^{\mu \sigma} \eta^{\nu \rho} $$

or
 * $$ \operatorname{tr} \left(\gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma\right) = 4 \left( \eta^{\rho \sigma} \eta^{\mu \nu} - \eta^{\nu \sigma} \eta^{\mu \rho} + \eta^{\mu \sigma} \eta^{\nu \rho} \right) $$

 * }

Add $$\operatorname{tr}\left(\gamma^5\right)$$ to both sides of the above to see
 * $$2\operatorname{tr}\left(\gamma^5\right) = 0 $$.

Now, this pattern can also be used to show
 * $$\operatorname{tr}\left(\gamma^\mu \gamma^\nu \gamma^5\right) = 0$$.

Simply add two factors of $$\gamma^\alpha$$, with $$\alpha$$ different from $$\mu$$ and $$\nu$$. Anticommute three times instead of once, picking up three minus signs, and cycle using the cyclic property of the trace.

So,
 * $$\operatorname{tr}\left(\gamma^\mu \gamma^\nu \gamma^5\right) = 0$$.

 * }

Conjugating with $$\gamma^0$$ one more time to get rid of the two $$\gamma^0$$s that are there, we see that $$\gamma^0 \Gamma^\dagger \gamma^0$$ is the reverse of $$\Gamma$$. Now,

 * $$\operatorname{tr} \left(\gamma^0 \Gamma^\dagger \gamma^0\right)$$
 * $$= \operatorname{tr} \left(\Gamma^\dagger\right)$$
 * (since trace is invariant under similarity transformations)
 * $$= \operatorname{tr} \left(\Gamma^*\right)$$
 * (since trace is invariant under transposition)
 * $$= \operatorname{tr} \left(\Gamma\right)$$
 * (since the trace of a product of gamma matrices is real)
 * }
 * }
 * $$= \operatorname{tr} \left(\Gamma\right)$$
 * (since the trace of a product of gamma matrices is real)
 * }
 * }

Normalization
The gamma matrices can be chosen with extra hermiticity conditions which are restricted by the above anticommutation relations however. We can impose
 * $$\left( \gamma^0 \right)^\dagger = \gamma^0 $$, compatible with $$\left( \gamma^0 \right)^2 = I_4$$

and for the other gamma matrices (for k = 1, 2, 3)
 * $$\left( \gamma^k \right)^\dagger = -\gamma^k $$, compatible with $$\left( \gamma^k \right)^2 = -I_4.$$

One checks immediately that these hermiticity relations hold for the Dirac representation.

The above conditions can be combined in the relation
 * $$\left( \gamma^\mu \right)^\dagger = \gamma^0 \gamma^\mu \gamma^0. $$

The hermiticity conditions are not invariant under the action $$\gamma^\mu \to S(\Lambda) \gamma^\mu {S(\Lambda)}^{-1}$$ of a Lorentz transformation $$\Lambda$$ because $$S(\Lambda)$$ is not necessarily a unitary transformation due to the non-compactness of the Lorentz group.

Charge conjugation
The charge conjugation operator, in any basis, may be defined as
 * $$C\gamma_\mu C^{-1} = -(\gamma_\mu)^\textsf{T}$$

where $$(\cdot)^\textsf{T}$$ denotes the matrix transpose. The explicit form that $$C$$ takes is dependent on the specific representation chosen for the gamma matrices, up to an arbitrary phase factor. This is because although charge conjugation is an automorphism of the gamma group, it is not an inner automorphism (of the group). Conjugating matrices can be found, but they are representation-dependent.

Representation-independent identities include:
 * $$\begin{align}

C\gamma_5 C^{-1} &= +(\gamma_5)^\textsf{T} \\ C\sigma_{\mu\nu} C^{-1} &= -(\sigma_{\mu\nu})^\textsf{T} \\ C\gamma_5\gamma_\mu C^{-1} &= +(\gamma_5\gamma_\mu)^\textsf{T} \\ \end{align}$$ The charge conjugation operator is also unitary $$C^{-1}=C^\dagger$$, while for $$\mathrm{Cl}_{1,3}(\mathbb{R})$$ it also holds that $$C^T = -C$$ for any representation. Given a representation of gamma matrices, the arbitrary phase factor for the charge conjugation operator can also be chosen such that $$ C^\dagger = -C$$, as is the case for the four representations given below (Dirac, Majorana and both chiral variants).

Feynman slash notation
The Feynman slash notation is defined by
 * $${a\!\!\!/} := \gamma^\mu a_\mu $$

for any 4-vector $$a$$.

Here are some similar identities to the ones above, but involving slash notation:
 * $${a\!\!\!/}{b\!\!\!/} = \left[a \cdot b - i a_\mu \sigma^{\mu\nu} b_\nu \right] I_4 $$
 * $${a\!\!\!/}{a\!\!\!/} = \left[ a^\mu a^\nu \gamma_\mu \gamma_\nu \right] I_4 = \left[\tfrac{1}{2} a^\mu a^\nu \left(\gamma_\mu \gamma_\nu + \gamma_\nu \gamma_\mu\right) \right] I_4 = \left[ \eta_{\mu\nu} a^\mu a^\nu \right] I_4 = a^2I_4$$
 * $$\operatorname{tr}\left({a\!\!\!/}{b\!\!\!/}\right) = 4 (a \cdot b)$$
 * $$\operatorname{tr}\left({a\!\!\!/}{b\!\!\!/}{c\!\!\!/}{d\!\!\!/}\right) = 4 \left[(a \cdot b)(c \cdot d) - (a \cdot c)(b \cdot d) + (a \cdot d)(b \cdot c) \right]$$
 * $$\operatorname{tr}\left(\gamma_5 {a\!\!\!/}{b\!\!\!/}\right) = 0$$
 * $$\operatorname{tr}\left(\gamma_5 {a\!\!\!/}{b\!\!\!/}{c\!\!\!/}{d\!\!\!/}\right) = -4 i \epsilon_{\mu \nu \rho \sigma} a^\mu b^\nu c^\rho d^\sigma$$
 * $$\gamma_\mu {a\!\!\!/} \gamma^\mu = -2 {a\!\!\!/} $$
 * $$\gamma_\mu {a\!\!\!/} {b\!\!\!/} \gamma^\mu = 4 (a \cdot b) I_4 $$
 * $$\gamma_\mu {a\!\!\!/} {b\!\!\!/} {c\!\!\!/} \gamma^\mu = -2 {c\!\!\!/} {b\!\!\!/} {a\!\!\!/}$$
 * where $$\epsilon_{\mu \nu \rho \sigma}$$ is the Levi-Civita symbol and $$\sigma^{\mu\nu} = \tfrac{i}{2} \left[\gamma^\mu, \gamma^\nu\right] ~.$$ Actually traces of products of odd number of $$\ \gamma\ $$ is zero and thus
 * $$\operatorname{tr}(a_1\!\!\!\!\!\!/ \,\,\, a_2\!\!\!\!\!\!/ \,\,\,\cdots a_n\!\!\!\!\!\!/\,\,\,) = 0\ $$ for $ab$ odd.

Many follow directly from expanding out the slash notation and contracting expressions of the form $$\ a_\mu b_\nu c_\rho\ \ldots\ $$ with the appropriate identity in terms of gamma matrices.

Other representations
The matrices are also sometimes written using the 2×2 identity matrix, $$I_2$$, and
 * $$ \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix} $$

where k runs from 1 to 3 and the σk are Pauli matrices.

Dirac basis
The gamma matrices we have written so far are appropriate for acting on Dirac spinors written in the Dirac basis; in fact, the Dirac basis is defined by these matrices. To summarize, in the Dirac basis:

 * $$\gamma^0 = \begin{pmatrix} I_2 & 0 \\ 0 & -I_2 \end{pmatrix},\quad \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix},\quad \gamma^5 = \begin{pmatrix} 0 & I_2 \\ I_2 & 0 \end{pmatrix} ~.$$

In the Dirac basis, the charge conjugation operator is real antisymmetric,

C = i\gamma^2\gamma^0 = \begin{pmatrix} 0 & -i\sigma^2 \\ -i\sigma^2 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 &  -1 \\      0 & 0 &  1 & 0 \\      0 &  -1 &  0 & 0 \\      1 & 0 &  0 & 0      \end{pmatrix} ~. $$

Weyl (chiral) basis
Another common choice is the Weyl or chiral basis, in which $$\gamma^k$$ remains the same but $$\gamma^0$$ is different, and so $$\gamma^5$$ is also different, and diagonal,
 * $$\gamma^0 = \begin{pmatrix} 0 & I_2 \\ I_2 & 0 \end{pmatrix},\quad \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix},\quad \gamma^5 = \begin{pmatrix} -I_2 & 0 \\ 0 & I_2 \end{pmatrix},$$

or in more compact notation:

\gamma^\mu = \begin{pmatrix} 0 & \sigma^\mu \\ \overline{\sigma}^\mu & 0 \end{pmatrix}, \quad \sigma^\mu \equiv (1, \sigma^i), \quad \overline{\sigma}^\mu \equiv \left(1, -\sigma^i\right). $$

The Weyl basis has the advantage that its chiral projections take a simple form,

\psi_{\mathrm L} = \tfrac{1}{2}\left(1 - \gamma^5\right)\psi = \begin{pmatrix} I_2 & 0 \\ 0 & 0 \end{pmatrix}\psi, \quad \psi_{\mathrm R} = \tfrac{1}{2}\left(1 + \gamma^5\right)\psi = \begin{pmatrix} 0 & 0 \\ 0 & I_2 \end{pmatrix}\psi ~. $$

The idempotence of the chiral projections is manifest.

By slightly abusing the notation and reusing the symbols $$\psi_{\mathrm L/R}$$ we can then identify
 * $$\psi = \begin{pmatrix} \psi_{\mathrm L} \\ \psi_{\mathrm R} \end{pmatrix},$$

where now $$\psi_{\mathrm L}$$ and $$\psi_{\mathrm R}$$ are left-handed and right-handed two-component Weyl spinors.

The charge conjugation operator in this basis is real antisymmetric,
 * $$C = i\gamma^2\gamma^0 = \begin{pmatrix} i\sigma^2 & 0 \\ 0 & -i\sigma^2 \end{pmatrix}$$

The Dirac basis can be obtained from the Weyl basis as
 * $$\gamma^\mu_{\mathrm W} = U \gamma^\mu_{\mathrm D} U^\dagger, \quad \psi_{\mathrm W} = U \psi_{\mathrm D}$$

via the unitary transform
 * $$U = \tfrac{1}{\sqrt{2\ }\ } \left(1 + \gamma^5 \gamma^0 \right) = \tfrac{1}{\sqrt{2\ }\ } \begin{pmatrix}I_2 & -I_2 \\I_2 & I_2\end{pmatrix}.$$

Weyl (chiral) basis (alternate form)
Another possible choice of the Weyl basis has
 * $$\gamma^0 = \begin{pmatrix} 0 & -I_2 \\ -I_2 & 0 \end{pmatrix},\quad \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix},\quad \gamma^5 = \begin{pmatrix} I_2 & 0 \\ 0 & -I_2 \end{pmatrix}.$$

The chiral projections take a slightly different form from the other Weyl choice,
 * $$\psi_{\mathrm R} = \begin{pmatrix} I_2 & 0 \\0 & 0 \end{pmatrix}\psi,\quad \psi_{\mathrm L} = \begin{pmatrix} 0 & 0 \\0 & I_2 \end{pmatrix}\psi.$$

In other words,
 * $$\psi = \begin{pmatrix} \psi_{\mathrm R} \\\psi_{\mathrm L} \end{pmatrix},$$

where $$\psi_{\mathrm L}$$ and $$\psi_{\mathrm R}$$ are the left-handed and right-handed two-component Weyl spinors, as before.

The charge conjugation operator in this basis is
 * $$C = i\gamma^2\gamma^0

= \begin{pmatrix} -i\sigma^2 & 0 \\ 0 & i\sigma^2 \end{pmatrix} = \begin{pmatrix} 0 & -1 &  0 & 0  \\      1 & 0 &  0 & 0  \\      0 & 0 &  0 & 1  \\      0 & 0 &   -1 & 0  \\     \end{pmatrix} ~ = -i\sigma^3\otimes\sigma^2. $$

This basis can be obtained from the Dirac basis above as $$\gamma^\mu_{\mathrm W} = U \gamma^\mu_{\mathrm D} U^\dagger, \psi_{\mathrm W} = U \psi_{\mathrm D}$$ via the unitary transform
 * $$U = \tfrac{1}{\sqrt{2\ }\ }\left(1 - \gamma^5 \gamma^0\right)

= \tfrac{1}{\sqrt{2\ }\ } \begin{pmatrix} I_2 & I_2 \\ -I_2 & I_2 \end{pmatrix} ~.$$

Majorana basis
There is also the Majorana basis, in which all of the Dirac matrices are imaginary, and the spinors and Dirac equation are real. Regarding the Pauli matrices, the basis can be written as
 * $$\begin{align}

\gamma^0 &= \begin{pmatrix} 0 & \sigma^2 \\ \sigma^2 & 0 \end{pmatrix}\, ~& \gamma^1 &= \begin{pmatrix} i\sigma^3 & 0 \\ 0 & i\sigma^3 \end{pmatrix}\, ~& \gamma^2 &= \begin{pmatrix} 0 & -\sigma^2 \\ \sigma^2 & 0 \end{pmatrix},\\ \gamma^3 &= \begin{pmatrix} -i\sigma^1 & 0 \\ 0 & -i\sigma^1 \end{pmatrix}\, ~& \gamma^5 &= \begin{pmatrix} \sigma^2 & 0 \\ 0 & -\sigma^2 \end{pmatrix}\, ~& C &= \begin{pmatrix} 0 & -i\sigma^2 \\ -i\sigma^2 & 0 \end{pmatrix}\ , \end{align}$$ where $$C$$ is the charge conjugation matrix, which matches the Dirac version defined above.

The reason for making all gamma matrices imaginary is solely to obtain the particle physics metric (+, −, −, −), in which squared masses are positive. The Majorana representation, however, is real. One can factor out the $$\ i\ $$ to obtain a different representation with four component real spinors and real gamma matrices. The consequence of removing the $$\ i\ $$ is that the only possible metric with real gamma matrices is (−, +, +, +).

The Majorana basis can be obtained from the Dirac basis above as $$\gamma^\mu_{\mathrm M} = U \gamma^\mu_{\mathrm D} U^\dagger, \psi_{\mathrm M} = U \psi_{\mathrm D}$$ via the unitary transform
 * $$U = U^\dagger = \tfrac{1}{\sqrt{2\ }\ } \begin{pmatrix}I_2 & \sigma^2 \\\sigma^2 & -I_2\end{pmatrix} ~.$$

Cl1,3(C) and Cl1,3(R)
The Dirac algebra can be regarded as a complexification of the real algebra Cl1,3($$\mathbb{R}$$), called the space time algebra:
 * $$ \mathrm{Cl}_{1,3}(\mathbb{C}) = \mathrm{Cl}_{1,3}(\mathbb{R}) \otimes \mathbb{C} $$

Cl1,3($$\mathbb{R}$$) differs from Cl1,3($$\mathbb{C}$$): in Cl1,3($$\mathbb{R}$$) only real linear combinations of the gamma matrices and their products are allowed.

Two things deserve to be pointed out. As Clifford algebras, Cl1,3($$\mathbb{C}$$) and Cl4($$\mathbb{C}$$) are isomorphic, see classification of Clifford algebras. The reason is that the underlying signature of the spacetime metric loses its signature (1,3) upon passing to the complexification. However, the transformation required to bring the bilinear form to the complex canonical form is not a Lorentz transformation and hence not "permissible" (at the very least impractical) since all physics is tightly knit to the Lorentz symmetry and it is preferable to keep it manifest.

Proponents of geometric algebra strive to work with real algebras wherever that is possible. They argue that it is generally possible (and usually enlightening) to identify the presence of an imaginary unit in a physical equation. Such units arise from one of the many quantities in a real Clifford algebra that square to −1, and these have geometric significance because of the properties of the algebra and the interaction of its various subspaces. Some of these proponents also question whether it is necessary or even useful to introduce an additional imaginary unit in the context of the Dirac equation.

In the mathematics of Riemannian geometry, it is conventional to define the Clifford algebra Clp,q($$\mathbb{R}$$) for arbitrary dimensions $2n$. The Weyl spinors transform under the action of the spin group $$\mathrm{Spin}(n)$$. The complexification of the spin group, called the spinc group $$\mathrm{Spin}^\mathbb{C}(n)$$, is a product $$\mathrm{Spin}(n)\times_{\mathbb{Z}_2} S^1$$ of the spin group with the circle $$S^1 \cong U(1).$$ The product $$\times_{\mathbb{Z}_2}$$ just a notational device to identify $$(a,u)\in \mathrm{Spin}(n)\times S^1$$ with $$(-a, -u).$$ The geometric point of this is that it disentangles the real spinor, which is covariant under Lorentz transformations, from the $$U(1)$$ component, which can be identified with the $$\mathrm{U}(1)$$ fiber of the electromagnetic interaction. The $$\times_{\mathbb{Z}_2}$$ is entangling parity and charge conjugation in a manner suitable for relating the Dirac particle/anti-particle states (equivalently, the chiral states in the Weyl basis). The bispinor, insofar as it has linearly independent left and right components, can interact with the electromagnetic field. This is in contrast to the Majorana spinor and the ELKO spinor (Eigenspinoren des Ladungskonjugationsoperators), which cannot (i.e. they are electrically neutral), as they explicitly constrain the spinor so as to not interact with the $$S^1$$ part coming from the complexification. The ELKO spinor is a Lounesto class 5 spinor.

However, in contemporary practice in physics, the Dirac algebra rather than the space-time algebra continues to be the standard environment the spinors of the Dirac equation "live" in.

Other representation-free properties
The gamma matrices are diagonalizable with eigenvalues $$\pm 1$$ for $$\gamma^0$$, and eigenvalues $$\pm i$$ for $$\gamma^i$$.

In particular, this implies that $$\gamma^0$$ is simultaneously Hermitian and unitary, while the $$\gamma^i$$ are simultaneously anti–Hermitian and unitary.

Further, the multiplicity of each eigenvalue is two.

More generally, if $$\ \gamma^\mu X_\mu\ $$ is not null, a similar result holds. For concreteness, we restrict to the positive norm case $$\ \gamma^\mu p_\mu = p\!\!\! / \ $$ with $$\ p \cdot p = m^2 > 0 ~.$$ The negative case follows similarly.

It follows that the solution space to $$\ p\!\!\! / - m = 0\ $$ (that is, the kernel of the left-hand side) has dimension 2. This means the solution space for plane wave solutions to Dirac's equation has dimension 2.

This result still holds for the massless Dirac equation. In other words, if $$p_\mu$$ null, then $$p\!\!\! /$$ has nullity 2.

Euclidean Dirac matrices
In quantum field theory one can Wick rotate the time axis to transit from Minkowski space to Euclidean space. This is particularly useful in some renormalization procedures as well as lattice gauge theory. In Euclidean space, there are two commonly used representations of Dirac matrices:

Chiral representation

 * $$\gamma^{1, 2, 3} = \begin{pmatrix}

0 & i\sigma^{1, 2, 3} \\ -i\sigma^{1, 2, 3} & 0 \end{pmatrix}, \quad \gamma^4 = \begin{pmatrix} 0 & I_2 \\ I_2 & 0 \end{pmatrix} $$

Notice that the factors of $$i$$ have been inserted in the spatial gamma matrices so that the Euclidean Clifford algebra
 * $$\left\{\gamma^\mu, \gamma^\nu \right\} = 2\delta^{\mu\nu} I_4$$

will emerge. It is also worth noting that there are variants of this which insert instead $$-i$$ on one of the matrices, such as in lattice QCD codes which use the chiral basis.

In Euclidean space,
 * $$\gamma_{\mathrm M}^5 = i \left(\gamma^0\gamma^1\gamma^2\gamma^3\right)_{\mathrm M} = \tfrac{1}{i^2} \left(\gamma^4\gamma^1\gamma^2\gamma^3 \right)_{\mathrm E} = \left(\gamma^1\gamma^2\gamma^3\gamma^4\right)_{\mathrm E} = \gamma^5_{\mathrm E} ~.$$

Using the anti-commutator and noting that in Euclidean space $$\left(\gamma^\mu\right)^\dagger = \gamma^\mu$$, one shows that
 * $$\left( \gamma^5 \right)^\dagger = \gamma^5$$

In chiral basis in Euclidean space,
 * $$\gamma^5 = \begin{pmatrix} -I_2 & 0\\ 0 & I_2 \end{pmatrix}$$

which is unchanged from its Minkowski version.

Non-relativistic representation

 * $$\gamma^{1,2,3} = \begin{pmatrix} 0 & -i \sigma^{1,2,3} \\ i \sigma^{1,2,3} & 0 \end{pmatrix}\, \quad

\gamma^4 = \begin{pmatrix} I_2 & 0 \\ 0 & -I_2 \end{pmatrix}, \quad \gamma^5 = \begin{pmatrix} 0 & -I_2 \\ -I_2 & 0 \end{pmatrix} $$

title: Gamma matrices


text:
Rainhard Fendrich

Rainhard Jürgen Fendrich (born 27 February 1955) is an Austrian singer, composer, entertainer, and actor. He is one of the most successful Austropop musicians, and his songs are written in Viennese German. He is very popular in Austria and in Bavaria, but less so in other German-speaking countries and regions. In non-German-speaking countries he is little known. His song from 1990, "I Am from Austria" (its lyrics, except for the title, are in Viennese German) is still popular in Austria and considered to be the "secret national anthem".

Early life and education
Fendrich's mother was a Sudeten German while his father's family originated from Serbia. His father was a mechanical engineer, his mother a model. He has a brother, Harald Fendrich, who is six years younger than him and also a musician who played the bass guitar in his band and is now part of WIR4 (Ulli Bäer, Gary Lux, Harry Stampfer). At the age of ten, Fendrich was sent to a Catholic boarding school, where he lived until the age of 17. At the boarding school he was an altar boy and sang in the choir. His piano lessons were cancelled because he was too bad at mathematics. According to his own statements, he thought he was "chubby" and not very attractive as a child. When he got a guitar at the age of 15, he taught himself the chords and also began to write lyrics. Frustrating experiences with girls at that time find their expression later, for example in the songs Cyrano (1991) or Frieda (2001). He dropped out of law school in order to finance acting and singing lessons by working various jobs.

Beginning of career and commercial breakthrough (1980-1986)
Fendrich appeared at the Theater an der Wien from 1980 (in Die Gräfin vom Naschmarkt and Chicago) and played Judas in the musical Jesus Christ Superstar there in 1982. In 1980 he was also engaged by Hans Gratzer for a performance series of Hamlet at the Schauspielhaus, made one of his first television appearances as a singer on the ORF program Wir-extra for the benefit of children in the Third World, and received his first recording contract.

Another appearance followed in Tritsch Tratsch in 1981 with the song Zweierbeziehung (Two-way relationship of a man with his car). The debut album Ich wollte nie einer von denen sein (I never wanted to be one of them) was released in May of the same year, but initially failed to achieve high sales figures.

In August 1981 he landed the Austrian summer hit of the year with Strada del sole (in a similar style, about a "vacation relationship"). The single sold 99,000 copies in Austria, which would correspond to today's status of triple platinum. Moreover, in 2020 the song was voted 14th among the "100 most important Austrian pop songs" by the pop culture magazine The Gap as part of the AustroTOP ranking. Equally successful was the second album Und alles ist ganz anders word’n (And everything has become completely different). The next number one hits followed in 1982 with Schickeria and Oben Ohne (Topless). At that time Fendrich was already celebrated as a shooting star of Austropop.

In 1983 Fendrich presented his first compilation called A winzig klaner Tropfen Zeit (A tiny drop of time). In the same year he also performed with Wolfgang Ambros at the School Closing Open Air in Vienna's Gerhard-Hanappi Stadium. This concert was released on a live compilation with the album Open Air.

Fendrich achieved his next success in 1983 with Auf und davon (Up and away). Although the album couldn't quite meet the commercial expectations of the previous album and the singles didn't manage that either, Fendrich performed at several sold-out concerts on his tour. This led to the song Weus'd a Herz hast wia a Bergwerk (Cause you have a heart like a mine) being available twice as a single; in a studio version and in a live version.

After a short retreat from the public eye, the albums Wien bei Nacht (Vienna by night) and Kein schöner Land (No country more beautiful) were released in 1985 and 1986. He also released another live compilation.

Breakthrough in Germany (1988–1997)
With Macho, Macho, sung largely in Viennese, another of his singles reached the top position in 1988. In addition, this was followed by the big breakthrough in Germany with number 2. Fendrich also achieved a top 3 position in Switzerland. He wrote the song in five minutes, it was inspired by an article in Brigitte magazine.

Equally well known in Austria and Germany is his appearance with Reinhard Mey, with Ein Loch in der Kanne (A hole in the pot) in the television program Was wäre wenn (What if) in 1988.

In 1989 the album Von Zeit zu Zeit (From time to time) was released, which could not meet Fendrich's commercial expectations. The singles Von Zeit zu Zeit and I Am from Austria were also not successful at first. Later, however, the title I Am from Austria developed into the "secret federal anthem of Austria" and was voted the "greatest Austropop hit of all time" in 2011 as part of the ORF program Österreich wählt (Austria votes).

In 1991 the album Nix is fix (Nothing is fixed), produced by Tato Gomez at BMG Ariola, was released, which could repeat the success of earlier times. The album stayed in the Austrian charts for 32 weeks and was number 1 for five weeks.

A year later, Fendrich performed with the Vienna Symphony Orchestra at the opening of the Vienna Festival. Under the project "Vienna Symphonic Orchestra Project" (VSOP), by and with Christian Kolonovits, who conducted the concert and in which the Arnold Schoenberg Choir also participated. This concert was released as an album and after some time made it to the first place in the Austrian music charts.

Fendrich took over as host of the ARD program Herzblatt (Sweetheart) in 1993 after Rudi Carrell, which he presented on German television until 1997. In the same year, he launched the comedy show Nix is fix (ORF/ARD). The cabaret and media-critical format was not continued despite good reviews.

A special postage stamp was issued by the Austrian Post Office in 1993.

Along the way, Fendrich recorded the album Brüder (Brothers), which also managed to place in Germany after a long time. The singles Midlife Crisis and Brüder, which were rather unpopular in Austria, made it into the German charts. The album was, despite the great commercial success, rather negatively received by the media in Fendrich's home country. The reason for this was that Fendrich slowly turned away from the Viennese dialect with this album. As a promotion for Herzblatt, the ballads album Lieder mit Gefühl (Songs with feeling) was also presented.

In the mid-1990s, Fendrich did not release any music. He made his comeback with the album Blond (Blonde). The title song became his last major number one hit to date. The single Blond sold 10,000 copies in Austria and was awarded gold. In addition, after the programming reform of the Austrian station Ö3, the song was one of the few Austrian songs played on the radio.

Return to Austria (from 1997)
In 1997, Fendrich initiated a benefit concert for the homeless, which was intended as a one-off event, at which he performed together with Wolfgang Ambros and Georg Danzer. The overwhelming success led to this team performing as Austria 3 until 2006, creating joint live albums and compilations. The band announced their split on 10 June 2006. Their last concert took place on 24 July 2006 in Altusried, Swabia. On 16 April 2007 the circle of friends reunited at Georg Danzer's comeback concert in the Viennese City Hall to perform three of their old songs; afterwards, several parties planned a possible comeback, but Georg Danzer died of lung cancer on 21 June 2007.

Further artistic achievements
Also in 1997, Fendrich had success as a film actor in the lead role of Fröhlich geschieden (Happily divorced) (ZDF) alongside Christina Plate and Helmut Fischer. In 1998 he played Billy Flynn in the musical Chicago at the Theater an der Wien, and in 2002 and 2003 he appeared as Jeff Zodiak in the musical WakeUp, written by him and arranged by Christian Kolonovits and Harold Faltermeyer, with less success at the Raimund Theater.

Fendrich received the Austrian Golden Romy award for most popular presenter in TV entertainment three times. In 1994, 1995 and 2000, he was nominated for the Amadeus Austrian Music Award four times before winning it in 2002. Austrian NEWS magazine readers voted him "Best Entertainer of the Decade" in same year.

As an entertainer, he was the first host of Die Millionenshow, the Austrian version of Who Wants to Be a Millionaire. He also hosted the game show Deal or No Deal in 2005. Both shows aired on ORF. From 2008 he hosted the show Sing and Win! (a variation of the US format Don't forget the lyrics) on the private TV station ATV.

Rest of career
He has appeared in numerous Austrian and German movies.

The concert on Donauinselfest which Fendrich gave "instead and for Georg Danzer" who a few weeks before the event had to refuse because of his rapidly progressing disease, attracted 200.000 fans on 23 June 2007.

Private life
Fendrich was married to Andrea Sator from 1984 to 2003, with whom he has two sons. Their daughter died of a viral disease in 1989 at the age of 17 months.

When Fendrich was found to have bought cocaine in early April 2006 in the course of police surveillance of a dealer ring, he confessed and claimed to have been using it for 15 years. Immediately afterwards, the artist went through voluntary withdrawal and has since participated in anti-drug campaigns, but was sentenced to an unconditional fine of 37,500 euros in May for possession of cocaine and the (later largely abandoned) "passing on of drugs".

On 4 December 2010 Fendrich married his long-time girlfriend Ina Nadine Wagler in Berlin. Their son was born on 10 March 2011. After the couple separated in 2012, the divorce took place in November 2016.

Austria 3

 * 1998: Austria3 - Live (with Wolfgang Ambros and Georg Danzer)
 * 1998: Austria3 - Live Vol. 2 (with Wolfgang Ambros and Georg Danzer)
 * 2000: Austria3 - Die Dritte (with Wolfgang Ambros and Georg Danzer)

DVDs

 * 2002: Ein Saitensprung
 * 2004: Jetzt

Singles

 * "Little Drummer Boy" with Eric Minsk, and "Stille Nacht" with Andrew Edge.

title: Rainhard Fendrich


text:
Wikipedia:Articles for deletion/Rough-Rider

I can't really tell what the user tried to do when creating the page. It appears that 62.6.139.11 tried to create a user page. Or in another mannerism, 62.6.139.11 was trying to create a personal page, and personal pages are not allowed on Wikipedia. If the latter category, then this page is not allowed. If the first category, then the thing to do is to Create an account. Rhetoricalwater 21:21, 14 October 2005 (UTC)