In quantum mechanics and statistical mechanics, parastatistics is a hypothetical alternative[1] to the established particle statistics models (Bose–Einstein statistics, Fermi–Dirac statistics and Maxwell–Boltzmann statistics). Other alternatives include anyonic statistics and braid statistics, both of these involving lower spacetime dimensions. Herbert S. Green[2] is credited with the creation of parastatistics in 1953.[3][4] The particles predicted by parastatistics have not been experimentally observed.
Consider the operator algebra of a system of N identical particles. This is a *-algebra. There is an SN group (symmetric group of order N) acting upon the operator algebra with the intended interpretation of permuting the N particles. Quantum mechanics requires focus on observables having a physical meaning, and the observables would have to be invariant under all possible permutations of the N particles. For example, in the case N = 2, R2 − R1 cannot be an observable because it changes sign if we switch the two particles, but the distance between the two particles : |R2 − R1| is a legitimate observable.
In other words, the observable algebra would have to be a *-subalgebra invariant under the action of SN (noting that this does not mean that every element of the operator algebra invariant under SN is an observable). This allows different superselection sectors, each parameterized by a Young diagram of SN.
In particular:
There are creation and annihilation operators satisfying the trilinear commutation relations[3]
[ a k , [ a l † † --> , a m ] ± ± --> ] − − --> = [ a k , a l † † --> ] ∓ ∓ --> a m ± ± --> a l † † --> [ a k , a m ] ∓ ∓ --> ± ± --> [ a k , a m ] ∓ ∓ --> a l † † --> + a m [ a k , a l † † --> ] ∓ ∓ --> = 2 δ δ --> k l a m {\displaystyle \left[a_{k},\left[a_{l}^{\dagger },a_{m}\right]_{\pm }\right]_{-}=[a_{k},a_{l}^{\dagger }]_{\mp }a_{m}\pm a_{l}^{\dagger }\left[a_{k},a_{m}\right]_{\mp }\pm [a_{k},a_{m}]_{\mp }a_{l}^{\dagger }+a_{m}\left[a_{k},a_{l}^{\dagger }\right]_{\mp }=2\delta _{kl}a_{m}}
[ a k , [ a l † † --> , a m † † --> ] ± ± --> ] − − --> = [ a k , a l † † --> ] ∓ ∓ --> a m † † --> ± ± --> a l † † --> [ a k , a m † † --> ] ∓ ∓ --> ± ± --> [ a k , a m † † --> ] ∓ ∓ --> a l † † --> + a m † † --> [ a k , a l † † --> ] ∓ ∓ --> = 2 δ δ --> k l a m † † --> ± ± --> 2 δ δ --> k m a l † † --> {\displaystyle \left[a_{k},\left[a_{l}^{\dagger },a_{m}^{\dagger }\right]_{\pm }\right]_{-}=\left[a_{k},a_{l}^{\dagger }\right]_{\mp }a_{m}^{\dagger }\pm a_{l}^{\dagger }\left[a_{k},a_{m}^{\dagger }\right]_{\mp }\pm \left[a_{k},a_{m}^{\dagger }\right]_{\mp }a_{l}^{\dagger }+a_{m}^{\dagger }\left[a_{k},a_{l}^{\dagger }\right]_{\mp }=2\delta _{kl}a_{m}^{\dagger }\pm 2\delta _{km}a_{l}^{\dagger }}
[ a k , [ a l , a m ] ± ± --> ] − − --> = [ a k , a l ] ∓ ∓ --> a m ± ± --> a l [ a k , a m ] ∓ ∓ --> ± ± --> [ a k , a m ] ∓ ∓ --> a l + a m [ a k , a l ] ∓ ∓ --> = 0 {\displaystyle \left[a_{k},\left[a_{l},a_{m}\right]_{\pm }\right]_{-}=[a_{k},a_{l}]_{\mp }a_{m}\pm a_{l}\left[a_{k},a_{m}\right]_{\mp }\pm [a_{k},a_{m}]_{\mp }a_{l}+a_{m}\left[a_{k},a_{l}\right]_{\mp }=0}
A paraboson field of order p, ϕ ϕ --> ( x ) = ∑ ∑ --> i = 1 p ϕ ϕ --> ( i ) ( x ) {\textstyle \phi (x)=\sum _{i=1}^{p}\phi ^{(i)}(x)} where if x and y are spacelike-separated points, [ ϕ ϕ --> ( i ) ( x ) , ϕ ϕ --> ( i ) ( y ) ] = 0 {\displaystyle [\phi ^{(i)}(x),\phi ^{(i)}(y)]=0} and { ϕ ϕ --> ( i ) ( x ) , ϕ ϕ --> ( j ) ( y ) } = 0 {\displaystyle \{\phi ^{(i)}(x),\phi ^{(j)}(y)\}=0} if i ≠ ≠ --> j {\displaystyle i\neq j} where [,] is the commutator and {,} is the anticommutator. Note that this disagrees with the spin-statistics theorem, which is for bosons and not parabosons. There might be a group such as the symmetric group Sp acting upon the φ(i)s. Observables would have to be operators which are invariant under the group in question. However, the existence of such a symmetry is not essential.
A parafermion field ψ ψ --> ( x ) = ∑ ∑ --> i = 1 p ψ ψ --> ( i ) ( x ) {\textstyle \psi (x)=\sum _{i=1}^{p}\psi ^{(i)}(x)} of order p, where if x and y are spacelike-separated points, { ψ ψ --> ( i ) ( x ) , ψ ψ --> ( i ) ( y ) } = 0 {\displaystyle \{\psi ^{(i)}(x),\psi ^{(i)}(y)\}=0} and [ ψ ψ --> ( i ) ( x ) , ψ ψ --> ( j ) ( y ) ] = 0 {\displaystyle [\psi ^{(i)}(x),\psi ^{(j)}(y)]=0} if i ≠ ≠ --> j {\displaystyle i\neq j} . The same comment about observables would apply together with the requirement that they have even grading under the grading where the ψs have odd grading.
The parafermionic and parabosonic algebras are generated by elements that obey the commutation and anticommutation relations. They generalize the usual fermionic algebra and the bosonic algebra of quantum mechanics.[5] The Dirac algebra and the Duffin–Kemmer–Petiau algebra appear as special cases of the parafermionic algebra for order p = 1 and p = 2, respectively.[6]
Note that if x and y are spacelike-separated points, φ(x) and φ(y) neither commute nor anticommute unless p=1. The same comment applies to ψ(x) and ψ(y). So, if we have n spacelike separated points x1, ..., xn,
corresponds to creating n identical parabosons at x1,..., xn. Similarly,
corresponds to creating n identical parafermions. Because these fields neither commute nor anticommute
and
gives distinct states for each permutation π in Sn.
We can define a permutation operator E ( π π --> ) {\displaystyle {\mathcal {E}}(\pi )} by
respectively. This can be shown to be well-defined as long as E ( π π --> ) {\displaystyle {\mathcal {E}}(\pi )} is only restricted to states spanned by the vectors given above (essentially the states with n identical particles). It is also unitary. Moreover, E {\displaystyle {\mathcal {E}}} is an operator-valued representation of the symmetric group Sn and as such, we can interpret it as the action of Sn upon the n-particle Hilbert space itself, turning it into a unitary representation.
Плакат Якова Гуминера «Арифметика встречного промфинплана» (1931 год) Математический софизм (от греч. σόφισμα — уловка, хитрая выдумка, головоломка[1]) — ошибочное математическое утверждение, полученное с помощью рассуждений, которые кажутся правильными, но в действите…
لانسينجيرلاندLansingerland بلدية منظر فضائي لBerkel en Rodenrijs علم لانسينجيرلاندLansingerlandعلمشعار لانسينجيرلاندLansingerlandشعار Highlighted position of Lansingerland in a municipal map of South Hollandالموقع في جنوب هولندا الإحداثيات 51°59′N 4°30′E / 51.983°N 4.500°E / 51.983; 4.500 تأسست 1 يناير 2007[1] تقسيم إداري…
Hospital in Scotland, UKLauriston BuildingNHS LothianLauriston BuildingShown in EdinburghGeographyLocationEdinburgh, Scotland, UKCoordinates55°56′41″N 3°11′48″W / 55.9446°N 3.1968°W / 55.9446; -3.1968OrganisationCare systemNHS ScotlandTypeSpecialistServicesSpecialityOut-patient centreLinksWebsitewww.nhslothian.scot/GoingToHospital/Locations/Pages/LauristonBuilding.aspxOther linksList of hospitals in Scotland The Lauriston Building is an out-patient centre in L…
A90 kan verwijzen naar: Een verkeersweg: A90 (Duitsland), zie Bundesautobahn 90 A90 (Griekenland) A90 (Groot-Brittannië) A90 (Italië) Bekijk alle artikelen waarvan de titel begint met A90 of met A90 in de titel. Dit is een doorverwijspagina, bedoeld om de verschillen in betekenis of gebruik van A90 inzichtelijk te maken. Op deze pagina staat een uitleg van de verschillende betekenissen van A90 en verwijzingen daarnaartoe. Bent u hier via een pagina in Wikipedia terecht…
Barrera de hielo Larsen Larsen Ice Shelf Vista aérea de la barrera (NASA, 2004)Ubicación geográficaContinente AntártidaRegión Tierra de Graham, en la península AntárticaOcéano (o mar) Mar de Weddell (Antártico)Coordenadas 67°30′S 62°30′O / -67.5, -62.5Ubicación administrativaPaís Antártida AntártidaDivisión Antártida OccidentalCaracterísticasTipo Barrera de hieloLongitud kmAnchura kmSuperficie km²Mapa de localización Barrera de hielo Larsen Geolocaliz…
انتهاكات مسندة إلى صدام حسينالمعلومات الشخصيةاسم الميلادصدام حسين عبد المجيد الناصريالمهنةرئيس جمهورية العراقمواليد28 أبريل 1937العوجة، العراقالوفاة30 ديسمبر 2006 (69 سنة)الكاظمية، العراقسبب الوفاةالإعدام شنقاالعقوبةالإعدام شنقاالشريك القائمة .. علي حسن المجيد · عواد …
Municipio de Jackson Municipio Municipio de JacksonUbicación en el condado de Riley en Kansas Ubicación de Kansas en EE. UU.Coordenadas 39°27′26″N 96°45′19″O / 39.4572, -96.7553Entidad Municipio • País Estados Unidos • Estado Kansas • Condado RileySuperficie • Total 93.13 km² • Tierra 82.92 km² • Agua (10.96 %) 10.21 km²Altitud • Media 366 m s. n. m.Población (2010) …
Castiglion Fibocchi Gemeente in Italië Situering Regio Toscane (TOS) Provincie Arezzo (AR) Coördinaten 43° 32′ NB, 11° 46′ OL Algemeen Oppervlakte 25,7 km² Inwoners (1 januari 2018) 2.147[1] (81 inw./km²) Hoogte 300 m Overig Postcode 52029 Netnummer 0575 ISTAT-code 051011 Portaal Italië Castiglion Fibocchi is een gemeente in de Italiaanse provincie Arezzo (regio Toscane) en telt 2080 inwoners (31-12-2004). De oppervlakte bedraagt 25,7 km², de bevolkingsdi…
State highway in California, United States See also: U.S. Route 99 in California State Route 99SR 99 highlighted in redRoute informationMaintained by CaltransLength424.85 mi[1] (683.73 km)(includes overlap with US 50 and I-5 in Sacramento)Existed1926 as US 99, 1964 as SR 99–presentMajor junctionsSouth end I-5 near Wheeler RidgeMajor intersections SR 58 / SR 178 in Bakersfield SR 41 in Fresno SR 108 / SR 132 in Modesto SR&…
Celebration of a girl's 15th birthday For other uses, see Quinceañera (disambiguation). This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Quinceañera – news · newspapers · books · scholar · JSTOR (May 2020) (Learn how and when to remove this template message) Mexican American girls at a quinceañera celebration…
Church in SwedenAlva ChurchAlva kyrkaAlva Church, external viewAlva ChurchLocation on Gotland57°12′27″N 18°21′41″E / 57.20745°N 18.36141°E / 57.20745; 18.36141CountrySwedenDenominationChurch of SwedenPrevious denominationCatholicAdministrationDioceseVisby Alva Church (Swedish: Alva kyrka) is a medieval church in Alva on the Swedish island of Gotland. The oldest parts of the church date from the late 12th century; with the halted construction of the tower about…
For other uses, see Trespass (disambiguation). Not to be confused with Trespass (soundtrack). 1993 film score by Various artistsTrespassFilm score by Various artistsReleased1993 (1993)GenreJazz, Blues rock, Country and WesternLength39:11LabelSire / Warner Bros.ProducerRy CooderJim Keltner Professional ratingsReview scoresSourceRatingAllmusic[1] The film score to the 1992 film Trespass, was heavily influenced by experimental jazz. The performers include producers Ry Cooder an…
Unidentified ancient Greek vase painter Early Proto-Attic hydria by the Mesogeia Painter, note applied plastic snakes on lip, neck and handle, suggesting a use in funerary cult; neck depicts women dancing with a youth, belly a man behind two sphinxes, circa 700 BC, from Athens, now Antikensammlung, Berlin. The Mesogeia Painter, also Mesogaia Painter, was an Early Proto-Attic vase painter. His conventional name is derived from his name vases, several hydriai decorated by him and discovered in the…
Grote Markt Geografische informatie Locatie Aalst Stadsdeel Centrum Algemene informatie Aangelegd in 12de eeuw Bestrating Kassei Opvallende gebouwen Belfort Aalst Portaal Steden Een ingekleurde ets met de grote markt van Aalst op afgebeeld in de 19de eeuw. De Grote Markt (Aalsters: Groeite Mert) is het grootste marktplein in Aalst en wordt beschouwd als het centrum van de stad. Het schepenhuis, het stadhuis, de Borse van Amsterdam en Herberg …
Japanese freestyle wrestler Medal record Women's freestyle wrestling Representing Japan Olympic Games 2004 Athens 48 kg 2008 Beijing 48 kg World Championships 2003 New York City 51 kg 2006 Guangzhou 48 kg 2007 Baku 48 kg Chiharu Icho (伊調 千春, Ichō Chiharu, born October 6, 1981 in Hachinohe, Aomori) is a Japanese wrestler who competed in the 48 kg weight class at the 2004 and 2008 Summer Olympics, winning the silver medal at both Games.[1] Her younger sister Kaori is al…
American rock bandThis article is about the rock band. For the X-Men villains, see U-Men (comics). The U-MenBackground informationOriginSeattle, Washington, United StatesGenres Punk rock post-punk garage rock noise rock rockabilly[1] Years active1980 (1980)–1989 (1989)Labels Bombshelter Homestead C/Z Sub Pop El Jefe Black Label Amphetamine Reptile Chuckie-Boy MembersJohn BigleyTom PriceJim TillmanCharlie RyanPast membersTom HazelmyerRobin BuchanTony Ransom The U-Men was an Am…
2022 South Korean Netflix TV series SomebodyPromotional posterHangul썸바디Revised RomanizationSseombadi Genre Thriller[1] Melodrama[1] Crime[1] Created byNetflix[2]Written byJung Ji-wooHan Ji-wan[2]Directed byJung Ji-woo[2]StarringKim Young-kwangKang Hae-rimKim Yong-jiKim Soo-yeonMusic byLee Ji-yeonCountry of originSouth KoreaOriginal languageKoreanNo. of episodes8ProductionProducersJung A-reumOk Kwang-heeRunning time50 minutesProduction company…
Ini adalah nama Maluku (Ambon), marganya adalah Matulessy Reza RahadianReza dalam ajang Indonesian Choice Awards 2017LahirReza Rahadian Matulessy5 Maret 1987 (umur 36)Bogor, Jawa Barat, IndonesiaPekerjaanPemeranmodelpenyanyisutradaraTahun aktif2004—sekarangTanda tangan Penghargaan Festival Film Indonesia Pemeran Pendukung Pria Terbaik 2009 Perempuan Berkalung Sorban Pemeran Utama Pria Terbaik 2010 3 Hati Dua Dunia Satu Cinta 2013 Habibie & Ainun 2016 My Stupid Boss Reza Rahadian …
Optical disk technology This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources in this section. Unsourced material may be challenged and removed. (February 2019) (Learn how and when to remove this template message) Logo of Compact Disc-ReWritable (CD-RW). CD-RW with distinctively darker data surface than a CD-R and a factory-pressed CD-ROM. Optical discs General Optical disc Optical disc drive Optical disc authoring …
Japanese media franchise Z/XCover of Z/X Zillions of enemy X volume 1 by Shueisha MangaZ/X Zillions of enemy XWritten byBroccoliIllustrated byKaregashi TsuchiyaPublished byShueishaMagazineV JumpDemographicShōnenOriginal runSeptember 2012 – October 2016Volumes6 GameZ/X Zillions of enemy X: Zekkai no CrusadeDeveloperBroccoliPublisherNippon Ichi SoftwareGenreCollectible card gamePlatformPlayStation 3ReleasedJP: May 23, 2013 Anime television seriesZ/X IgnitionDirected byYūji …
Lokasi Pengunjung: 13.58.40.171