Vasilev tenglamalari - Vasiliev equations

Bu maqola bu yetim, boshqa maqolalar singari unga havola. Iltimos havolalar bilan tanishtirish dan ushbu sahifaga tegishli maqolalar; harakat qilib ko'ring Havola vositasini toping takliflar uchun. (2016 yil iyun)

Vasilev tenglamalari bor rasmiy ravishda izchil o'lchovli o'zgaruvchan chiziqli bo'lmagan tenglamalar, ma'lum bir vakuum eritmasi bo'yicha chiziqlash erkin massasiz yuqori spinli maydonlarni tavsiflaydi anti-de Sitter maydoni. Vasilev tenglamalari klassik tenglamalar bo'lib, yo'q Lagrangian ma'lumki, kanonik ikki hosiladan boshlanadi Fronsdal Lagrangian va o'zaro ta'sir shartlari bilan yakunlanadi. Vasilev tenglamalarining uch, to'rtta va o'zboshimchalik bilan makon vaqt o'lchovlarida ishlaydigan bir qator o'zgarishlari mavjud. Vasilevning tenglamalari har qanday sonli simmetriya bilan super simmetrik kengaytmalarni qabul qiladi va ruxsat beradi Yang-Mills o'lchovlar. Vasilevning tenglamalari fonga bog'liq emas, eng sodda aniq echim anti-de-Sitter makonidir. Shuni ta'kidlash kerakki, lokalizatsiya to'g'ri bajarilmayapti va tenglamalar ma'lum rasmiy deformatsiya protsedurasining echimini beradi, bu esa maydon nazariyasi tilini xaritada ko'rsatishi qiyin. Yuqori aylanish AdS / CFT yozishmalar ko'rib chiqiladi Yuqori spin nazariyasi maqola.

Vasilev tenglamalari ma'lum yordamchi yo'nalishlarga qarab tartib bilan echilganda, fazoda vaqt ichida tenglamalar hosil qiladi va differentsial tenglamalarni hosil qiladi. Tenglamalar bir nechta tarkibiy qismlarga tayanadi: ochilmagan tenglamalar va yuqori spinli algebralar.

Quyidagi ekspozitsiya Vasilev tenglamalarini qurilish bloklariga bo'linadigan va keyin ularni birlashtiradigan tarzda tashkil etilgan. To'rt o'lchovli bosonik Vasilev tenglamalari misoli^[1] barcha boshqa o'lchovlar va o'ta nosimmetrik umumlashmalar ushbu asosiy misolning sodda modifikatsiyalari bo'lgani uchun uzoq vaqt davomida ko'rib chiqiladi.

• ning ta'rifi yuqori spinli algebra yuqori spinli nazariya tenglamalari yuqori spinli algebrada qiymatlarni oladigan ikkita maydon uchun tenglamalar bo'lib chiqqani uchun berilgan;
• Vasilev tenglamalariga kiradigan maydonlar qiymatlarni qabul qiladigan o'ziga xos yulduz mahsuloti aniqlanadi;
• Vasilev tenglamalarining bir qismi Harmonik osilatorning qiziqarli deformatsiyasi bilan bog'liq deformatsiyalangan osilatorlar ko'rib chiqilgan;
• The ochilgan yondashuv birinchi darajali shaklda differentsial tenglamalarni yozishning biroz rivojlangan shakli bo'lgan muhokama qilinadi;
• The Vasilev tenglamalari berilgan;
• Vasilyev tenglamalarini anti-de-Sitter fazosi bo'yicha chiziqli chiziqlash erkin massasiz yuqori spinli maydonlarni tasvirlashi isbotlangan.

Vasilev tenglamalarining uchta o'zgarishi ma'lum: to'rt o'lchovli,^[1] uch o'lchovli^[2][3] va d o'lchovli.^[4] Ular quyida muhokama qilinadigan yumshoq tafsilotlar bilan farq qiladi.

Yuqori spinli algebralar

Yuqori spinli algebralar^[5] yuqori spinli nazariya multipletining global simmetriyalari. Shu bilan birga ularni ba'zilarning global simmetriyalari sifatida aniqlash mumkin konformal maydon nazariyalari (CFT), bu kinematik qism asosida yotadi yuqori spinli AdS / CFT yozishmalari, bu alohida holat AdS / CFT. Boshqa bir ta'rif shundan iboratki, yuqori spinli algebralar universal qoplovchi algebra de-sitterga qarshi algebra ${ displaystyle shunday (d, 2)}$ ma'lum ikki tomonlama ideallar bilan. Yuqori spinli algebralarning bir nechta murakkab misollari mavjud, ammo ularning hammasini eng sodda yuqori spinli algebralarni matritsali algebralar bilan zichlash va keyinchalik cheklovlar qo'yish orqali olish mumkin. Yuqori spinli algebralar quyidagicha kelib chiqadi assotsiativ algebralar va Lie algebra kommutator orqali tuzilishi mumkin.

To'rt o'lchovli bosonik yuqori spinli nazariya uchun tegishli yuqori spinli algebra tufayli juda oddiy ${ textstyle so (3,2) sim sp (4, mathbb {R})}$ va ustiga qurilishi mumkin ikki o'lchovli kvant Harmonik osilator. Ikkinchi holatda yaratish / yo'q qilish operatorlarining ikki juftligi ${ textstyle a_ {1}, a_ {1} ^ { xanjar}, a_ {2}, a_ {2} ^ { xanjar}}$ kerak. Ular kvartetga joylashtirilishi mumkin ${ textstyle { hat {Y}} ^ {A}, A = 1, ..., 4}$ kommutatsiya kanonik munosabatlariga bo'ysunadigan operatorlarning

${ displaystyle [{ hat {Y}} ^ {A}, { hat {Y}} ^ {B}] = 2iC ^ {AB} ,,}$

qayerda ${ textstyle C ^ {AB} = - C ^ {BA}}$ bo'ladi ${ textstyle sp (4)}$ o'zgarmas tensor, ya'ni anti-nosimmetrikdir. Ma'lumki, bilinenlar osilatorni amalga oshirishni ta'minlaydi ${ textstyle sp (4)}$:

${ displaystyle T ^ {AB} = - { frac {i} {4}} {{ hat {Y}} ^ {A}, { hat {Y}} ^ {B} } ,, qquad [T ^ {AB}, T ^ {CD}] = T ^ {AD} C ^ {BC} + { text {3 more}} ,.}$

Yuqori spinli algebra barcha juft funktsiyalarning algebrasi sifatida aniqlanadi ${ textstyle f ({ hat {Y}}), f ({ hat {Y}}) = f (- { hat {Y}})}$ yilda ${ textstyle { hat {Y}} ^ {A}}$. Funktsiyalarning teng ekanligi yuqoriroq spin nazariyasining bosonik tarkibiga mos keladi ${ textstyle { hat {Y}} ^ {A}}$ kosmik vaqt nuqtai nazaridan va hatto kuchlaridan Majorana spinorlari bilan bog'liqligi ko'rsatiladi ${ textstyle { hat {Y}} ^ {A}}$ tensorlarga mos keladi. Bu assotsiativ algebra va mahsulot qulay tarzda amalga oshiriladi Moyal yulduz mahsuloti:

${ displaystyle (f star g) (Y) = f (Y) exp i left ({{ frac { overleftarrow { kısalt}} { qismli Y ^ {A}}} C ^ {AB} { frac { overrightarrow { qismli}} { qismli Y ^ {B}}}} o'ng) g (Y) ,,}$

operatorlar algebrasi degan ma'noni anglatadi ${ textstyle f ({ hat {Y}})}$ funktsiyasini algebra bilan almashtirish mumkin ${ textstyle f (Y)}$ oddiy qatnov o'zgaruvchilarida ${ textstyle {Y} ^ {A}}$ (shlyapalar yopiq) va mahsulot kommutativ bo'lmagan yulduz mahsulotiga almashtirilishi kerak. Masalan, kimdir topadi

${ displaystyle (Y ^ {A} yulduz g) (Y) = (Y ^ {A} + iC ^ {AB} qisman _ {B}) g (Y) ,, qquad (f yulduz Y ^ {B}) (Y) = (Y ^ {B} -iC ^ {BA} qisman _ {B}) f (Y) ,,}$

va shuning uchun ${ textstyle Y ^ {A} star Y ^ {B} -Y ^ {B} star Y ^ {A} = [Y ^ {A}, Y ^ {B}] _ { star} = 2iC ^ {AB}}$ operatorlar uchun bo'lgani kabi. Amalda bir xil yulduz mahsulotining yana bir vakili foydalidir:

${ displaystyle (f star g) (Y) = { frac {1} {(2 pi) ^ {4}}} int dUdVf (Y + U) g (Y + V) e ^ {iU_ { A} V_ {B} C ^ {AB}} ,.}$

Ko'rsatkichli formulani qismlar bo'yicha integratsiya qilish va chegara atamalarini tushirish orqali olish mumkin. Ta'minlash uchun prefaktor tanlangan ${ displaystyle 1 star 1 = 1}$. Lorents-kovariant bazasida biz bo'linishimiz mumkin ${ textstyle A = alfa, { nuqta { alfa}}; alfa = 1,2; { nuqta { alfa}} = 1,2}$ va biz ham bo'linib ketdik ${ displaystyle Y ^ {A} = y ^ { alfa}, y ^ { nuqta { alfa}}}$. Keyin Lorents generatorlari ${ textstyle L ^ { alpha beta} = T ^ { alpha beta}}$, ${ textstyle { bar {L}} ^ {{ nuqta { alfa}} { nuqta { beta}}} = T ^ {{ nuqta { alfa}} { nuqta { beta}}} }$ va tarjima generatorlari ${ textstyle P ^ { alpha { dot { beta}}} = T ^ {{ alpha} { dot { beta}}}}$. The ${ textstyle pi}$-avtomorfizm ikkita ekvivalent usulda amalga oshirilishi mumkin: yoki kabi ${ textstyle pi (y ^ { alfa}) = - y ^ { alfa}, pi (y ^ { nuqta { alfa}}) = y ^ { nuqta { alfa}}}$ yoki kabi ${ textstyle pi (y ^ { alfa}) = Y ^ { alfa}, pi (y ^ { nuqta { alfa}}) = - y ^ { nuqta { alfa}}}$. Ikkala holatda ham Lorents generatorlarini tegmasdan qoldiradi va tarjimalar belgisini aylantiradi.

Yuqorida qurilgan yuqori spinli algebra uch o'lchovli simmetriya algebrasi bo'lishi mumkin Klayn-Gordon tenglamasi ${ displaystyle square _ {3} phi (x) = 0}$. Ko'proq bepul CFTlarni hisobga olgan holda, masalan. bir qator skalar va bir qancha fermionlar, Maksvell maydoni va boshqalar, yuqori spinli algebralarning ko'proq misollarini yaratish mumkin.

Vasilev yulduz mahsuloti

Vasilev tenglamalari - bu echimini topadigan yordamchi yo'nalishlarga ega bo'lgan ma'lum kattaroq kosmosdagi tenglamalar. Qo'shimcha ko'rsatmalar dubllari bilan berilgan ${ textstyle {Y} ^ {A}}$, deb nomlangan ${ textstyle {Z} ^ {A}}$, ular Y bilan chalkashib ketgan. funktsiyalar algebrasidagi yulduzcha mahsulot ${ textstyle f (Y, Z)}$ yilda ${ textstyle {Y}, Z}$- o'zgaruvchilar

${ displaystyle F (Y, Z) yulduz G (Y, Z) = { frac {1} {(2 pi) ^ {4}}} int dU , dV , F (Y + U, Z + U) G (Y + V, ZV) exp {[iU_ {A} V_ {B} C ^ {AB}]} ,}$

Yuqorida keltirilgan ajralmas formula - bu Y va Z orasida kommentator uchun teskari belgilar bilan Veyl tartibiga mos keladigan ma'lum bir yulduz mahsuloti:

${ displaystyle [Y ^ {A}, Y ^ {B}] = 2iC ^ {AB} ,, qquad qquad [Z ^ {A}, Z ^ {B}] = - 2iC ^ {AB} ,.}$

Bundan tashqari, Y-Z yulduz mahsuloti, ko'rinib turibdiki, Y-Z va Y + Z ga nisbatan normal tartibda

${ displaystyle { begin {aligned} F (a, a ^ { xanjar}) yulduz G (a, a ^ { xanjar}) & = { frac {1} {(2 pi) ^ {4 }}} int dU , dV , F (a + 2U, a ^ { xanjar}) G (a, a ^ { xanjar} + 2V) exp {[iU_ {A} V_ {B} C ^ {AB}]} ,, & quad a = Y + Z ,, a ^ { xanjar} = YZ end {hizalanmış}}}$

Yuqori spinli algebra kengaytirilgan algebradagi assotsiativ subalgebra hisoblanadi. Bosonik proektsiyaga muvofiq tomonidan berilgan ${ displaystyle f (Y, Z) = f (-Y, -Z)}$.

Deformatsiyalangan osilatorlar

Vasilev tenglamalarining muhim qismi qiziqarli deformatsiyaga asoslanadi Kvantli harmonik osilator, deformatsiyalangan osilatorlar deb nomlanadi. Avvalo, odatdagi yaratish va yo'q qilish operatorlarini to'playlik ${ textstyle a ^ { xanjar}, a}$ dubletda ${ textstyle q _ { alfa} ,, alfa = 1,2}$. Kanonik kommutatsiya munosabatlari ( ${ textstyle 2i}$- Vasilev tenglamalari bilan taqqoslashni osonlashtiradigan omillar kiritildi)

${ displaystyle left [q _ { alpha}, q _ { beta} right] = - 2i epsilon _ { alpha beta} ,, qquad epsilon _ { alpha beta} = { begin {bmatrix} 0 & 1 - 1 & 0 end {bmatrix}} ,,}$

ning aniqlanganligini isbotlash uchun ishlatilishi mumkin ${ displaystyle q _ { alfa}}$ shakl ${ displaystyle sp (2) sim sl (2)}$ generatorlar

${ displaystyle { begin {aligned} T _ { alpha beta} & = { frac {i} {4}} {q _ { alpha}, q _ { beta} } ,, chap [T _ { alfa beta}, q _ { gamma} right] & = q _ { alpha} epsilon _ { beta gamma} + q _ { beta} epsilon _ { alpha gamma} , , chap [T _ { alpha beta}, T _ { gamma delta} right] & = T _ { alpha delta} epsilon _ { beta gamma} + T _ { beta delta} epsilon _ { alpha gamma} + T _ { alpha gamma} epsilon _ { beta delta} + T _ { beta gamma} epsilon _ { alpha delta} ,. end {hizalanmış }}}$

Jumladan, ${ displaystyle T _ { alpha beta}}$ aylantiradi ${ displaystyle q _ { alfa}}$ sifatida ${ displaystyle sp (2)}$-vektor bilan ${ displaystyle epsilon _ { alpha beta}}$ rolini o'ynash ${ displaystyle sp (2)}$-variant metrikasi. Deformatsiyalangan osilatorlar aniqlanadi^[6] generatorlar to'plamini qo'shimcha ishlab chiqaruvchi element bilan qo'shib ${ displaystyle Q}$ va postulat

${ displaystyle {q _ { alfa}, Q } = 0 ,, qquad chap [q _ { alfa}, q _ { beta} o'ng] = - 2i epsilon _ { alpha beta} (1 + Q) ,.}$

Shunga qaramay, buni ko'rish mumkin ${ displaystyle T _ { alpha beta}}$, yuqorida ta'riflanganidek, shakl ${ displaystyle sp (2)}$- generatorlar va to'g'ri aylantirish ${ displaystyle q _ { alfa}}$. Da ${ displaystyle Q = 0}$ biz deformatsiz osilatorlarga qaytamiz. Aslini olib qaraganda, ${ displaystyle q _ { alfa}}$ va ${ displaystyle T _ { alpha beta}}$ ning generatorlarini hosil qiling Yolg'on superalgebra ${ displaystyle osp (1 | 2)}$, qayerda ${ displaystyle q _ { alfa}}$ g'alati generatorlar sifatida qaralishi kerak. Keyin, ${ displaystyle {q _ { alfa}, q _ { beta} } = - 4iT _ { alpha beta}}$ ning belgilaydigan munosabatlar qismidir ${ displaystyle osp (1 | 2)}$.Deformatsiyalangan osilator munosabatlarining bitta (yoki ikkita) nusxasi generatorlar maydonlar bilan almashtiriladigan va kommutatsiya munosabatlari maydon tenglamalari sifatida o'rnatiladigan Vasilev tenglamalarining bir qismini tashkil etadi.

Katlanmagan tenglamalar

Yuqori spinli maydonlar uchun tenglamalar ochilmagan shaklda Vasilev tenglamalaridan kelib chiqadi va har qanday differentsial tenglamalar to'plamini lotinlarni belgilash uchun yordamchi maydonlarni kiritish orqali birinchi tartibda qo'yish mumkin. Katlanmagan yondashuv^[7] o'lchov simmetriyalari va diffeomorfizmlarni hisobga oladigan ushbu g'oyaning rivojlangan islohidir. Faqat o'rniga ${ textstyle kısalt _ { mu} phi ^ {i} (x) = f _ { mu} ^ {i} ( phi)}$ ochilmagan tenglamalar differentsial shakllar tilida shunday yoziladi

${ displaystyle dW ^ {A} = F ^ {A} (W) ,,}$

bu erda o'zgaruvchilar mavjud differentsial shakllar ${ textstyle W ^ {A} = W _ { mu _ {1} ... mu _ {q}} ^ {A} (x) , dx ^ { mu _ {1}} wedge .. . wedge dx ^ { mu _ {q}}}$ mavhum indeks bilan sanab o'tilgan har xil darajadagi ${ textstyle A}$; ${ textstyle d}$ bo'ladi tashqi hosila ${ textstyle d = dx ^ { mu} kısalt _ { mu}}$. Tuzilishi funktsiyasi ${ textstyle F ^ {A} (W)}$ kabi tashqi mahsulot Teylor seriyasida kengaytirilishi mumkin deb taxmin qilinadi

${ displaystyle F ^ {A} (W) = sum _ {q_ {1} + ... + q_ {n} = q + 1} F_ {B_ {1} ... B_ {n}} ^ { A} W ^ {B_ {1}} wedge ... wedge W ^ {B_ {n}} ,,}$

qayerda ${ textstyle W ^ {A}}$ shakl darajasiga ega ${ textstyle q}$ va yig'indisi shakli darajalari qo'shiladigan barcha shakllar ustidan ${ textstyle q + 1}$. Katlanmagan tenglamalarning eng oddiy misoli bu nol egrilik tenglamalari ${ textstyle d omega = { tfrac {1} {2}} [ omega, omega]}$ bir shaklli ulanish uchun ${ textstyle omega}$ har qanday Lie algebra ${ textstyle { mathfrak {g}}}$. Bu yerda ${ textstyle A}$ Lie algebra asosini bosib o'tadi va tuzilish vazifasini bajaradi ${ textstyle F ^ {A} ( omega) = f_ {BC} ^ {A} , omega ^ {A} wedge omega ^ {B}}$ Lie algebrasining tuzilish konstantalarini kodlaydi.

Beri ${ textstyle dd equiv 0}$ ochilmagan tenglamalarning izchilligini talab qiladi

${ displaystyle 0 equiv ddW ^ {A} = dF ^ {A} (W) = dW ^ {B} { frac { qismli} { qisman W ^ {B}}} F ^ {A} (W ) = F ^ {B} { frac { qismli} { qismli W ^ {B}}} F ^ {A} (W) qquad longleftrightarrow qquad F ^ {B} { frac { qismli} { qisman W ^ {B}}} F ^ {A} (W) = 0 ,,}$

qaysi Frobenius integralligi sharti. Nolinchi egrilik tenglamasi holatida bu faqat Jakobining o'ziga xosligi. Tizim birlashtirilgandan so'ng, u aniq o'lchov simmetriyasiga ega ekanligini ko'rsatishi mumkin. Har bir soha ${ textstyle W ^ {A}}$ bu nolga teng bo'lmagan darajaning shakli ${ textstyle q}$ o'lchov parametriga ega ${ textstyle xi ^ {A}}$ bu daraja shakli ${ textstyle q-1}$ va o'lchov o'zgarishlari

${ displaystyle delta W ^ {A} = d xi ^ {A} + xi ^ {B} { frac { qismli} { qisman W ^ {B}}} F ^ {A} (W) }$

Vasilev tenglamalari bir shakldan iborat bo'lgan aniq maydon mazmuni uchun ochilmagan tenglamalarni hosil qiladi ${ textstyle omega}$ va nol shakli ${ textstyle C}$, ikkalasi ham qiymatlarni qabul qiladi yuqori spinli algebra. Shuning uchun, ${ displaystyle W ^ {A} = ( omega, C)}$ va ${ displaystyle omega = omega _ { mu} (Y | x) dx ^ { mu}, omega (Y | x) = omega (-Y | x)}$, ${ displaystyle C = C (Y | x), C (Y | x) = C (-Y | x)}$. Yuqori spinli maydonlarning o'zaro ta'sirini tavsiflovchi ochilmagan tenglamalar

${ displaystyle { begin {aligned} d omega & = omega star omega + { mathcal {V}} ( omega, omega, C) + { mathcal {V}} ( omega, omega, C, C) + ... ,, dC & = omega star CC star pi ( omega) + { mathcal {V}} ( omega, C, C) + ... ,, end {hizalangan}}}$

qayerda ${ textstyle { mathcal {V}} ( omega, ..., C)}$ ichida yuqori va yuqori darajadagi o'zaro ta'sir tepaliklari ${ textstyle C}$- maydon. Yuqori spinli algebrada hosila bilan belgilanadi ${ displaystyle star}$. Tepaliklarning aniq shakli Vasilev tenglamalaridan chiqarilishi mumkin. Dalalarda bilinear bo'lgan tepaliklar yuqori spinli algebra bilan belgilanadi. Automorfizm ${ textstyle pi}$ ning avtomorfizmi tomonidan vujudga keladi anti-de Sitter Agar algebra tarjimalari belgisini silkitsa, quyida ko'rib chiqing ${ textstyle C}$- kengayish, tenglamalar shunchaki ulanish uchun nol egrilik shartidir ${ textstyle omega}$ yuqori spinli algebra va nol shakli uchun kovariant doimiylik tenglamasi ${ textstyle C}$ bu o'ralgan qo'shma tasvirda qiymatlarni oladi^[8] (burilish avtomorfizm tomonidan amalga oshiriladi ${ textstyle pi}$).

Dala tarkibi

Vasilev tenglamalarining maydon tarkibi uchta maydon tomonidan berilgan va ularning barchasi Y va Z funktsiyalarining kengaytirilgan algebraidagi qiymatlarni oladi:

• o'lchov aloqasi ${ displaystyle W = W _ { mu} (Y, Z | x) , dx ^ { mu}}$, uning qiymati Z = 0 yuqori spinli algebra aloqasini beradi ${ displaystyle omega = omega _ { mu} (Y | x) , dx ^ { mu}}$. Bosonik proektsiyani nazarda tutadi ${ displaystyle W (Y, Z | x) = W (-Y, -Z | x)}$;
• nol shakl ${ displaystyle B = B (Y, Z | x)}$, uning qiymati Z = 0 yuqori spinli algebraning nol shaklini beradi ${ displaystyle C = C (Y | x)}$. Bosonik proektsiyani nazarda tutadi ${ displaystyle B (Y, Z | x) = B (-Y, -Z | x)}$;
• yordamchi maydon ${ displaystyle S = S_ {A} (Y, Z | x) , dZ ^ {A}}$, ba'zida uni yordamchi Z-kosmosda bitta shakl sifatida ko'rish foydali bo'ladi, shuning uchun farqlar: ${ displaystyle dZ ^ {A} wedge dZ ^ {B} = - dZ ^ {B} wedge dZ ^ {A} ,.}$

Ushbu maydon Z ga bog'liqlikni echishda yo'q qilinishi mumkin. Uchun bosonik proektsiya ${ displaystyle S}$- maydon ${ displaystyle S_ {A} (Y, Z | x) = - S_ {A} (- Y, -Z | x)}$ qo'shimcha indeks tufayli ${ displaystyle A}$ oxir-oqibat Y, Z tomonidan olib boriladi.

Z-kosmosdagi differentsial shakllar natijasida yuzaga keladigan chalkashliklarni oldini olish va ga bo'lgan munosabatni ochish uchun deformatsiyalangan osilatorlar Vasilev tenglamalari quyida komponent shaklida yozilgan. Vasilev tenglamalarini ikki qismga bo'lish mumkin. Birinchi qism faqat nol egrilik yoki kovariant doimiylik tenglamalarini o'z ichiga oladi:

${ displaystyle { begin {aligned} dW & = W star W ,, dB & = W star BB star pi (W) ,, dS_ {A} & = W star S_ {A } -S_ {A} star W ,, end {hizalanmış}}}$

bu erda yuqori spinli algebra avtomorfizmi ${ displaystyle pi}$ kabi to'liq algebraga kengaytirilgan

${ displaystyle { begin {aligned} pi (W) (y _ { alpha}, y _ { dot { alfa}}, z _ { alpha}, z _ { dot { alfa}}) & = { W} (- y _ { alfa}, y _ { nuqta { alfa}}, - z _ { alfa}, z _ { nuqta { alfa}}) = {W} (y _ { alfa}, - y_ { nuqta { alfa}}, z _ { alfa}, - z _ { nuqta { alfa}}) ,, end {aligned}}}$

bosonik proektsiya tufayli so'nggi ikki shakl ekvivalentdir ${ displaystyle W (Y, Z | X)}$.

Shuning uchun, tenglamalarning birinchi qismi shundan beri x fazoda noan'anaviy egrilik yo'qligini anglatadi ${ displaystyle W}$ tekis. Ikkinchi qism tizimni norivial qiladi va yordamchi ulanishning egriligini aniqlaydi ${ displaystyle S}$:

${ displaystyle { begin {aligned} left [S_ {A}, S_ {B} right] _ { star} & = - 2i { begin {bmatrix} epsilon _ { alpha beta} (1 + B star varkappa) & 0 0 & epsilon _ {{ dot { alpha}} { dot { beta}}} (1 + B star { bar { varkappa}}) end { bmatrix}} ,, {B star varkappa, S _ { alpha} } _ { star} & = 0 ,, {B star { bar { varkappa}}, S _ { dot { alpha}} } _ { star} & = 0 ,, end {aligned}}}$

bu erda ikkita Klein operatorlari tanishtirildi

${ displaystyle varkappa = exp {iy _ { alpha} z ^ { alpha}} ,, qquad { bar { varkappa}} = exp {iy _ { dot { alpha}} z ^ { nuqta { alfa}}}}$

Klein operatorlarining mavjudligi tizim uchun juda muhimdir. Ular buni tushunadilar ${ displaystyle pi}$ avtomorfizm ichki sifatida

${ displaystyle { begin {aligned} varkappa star f (y _ { alpha}, y _ { dot { alpha}}, z _ { alpha}, z _ { dot { alpha}}) star varkappa & = f (-y _ { alpha}, y _ { nuqta { alfa}}, - z _ { alfa}, z _ { nuqta { alfa}}) ,, qquad && varkappa star varkappa = 1 ,, { bar { varkappa}} yulduz f (y _ { alfa}, y _ { nuqta { alfa}}, z _ { alfa}, z _ { nuqta { alfa} }) star { bar { varkappa}} & = f (y _ { alpha}, - y _ { nuqta { alfa}}, z _ { alfa}, - z _ { nuqta { alfa}}) ,, qquad && { bar { varkappa}} star { bar { varkappa}} = 1 ,. end {aligned}}}$

Boshqacha qilib aytganda, Klein operatori ${ displaystyle varkappa}$ kabi harakat qilish ${ displaystyle (-1) ^ {N_ {y} + N_ {z}}}$, ya'ni y, z dagi juft funktsiyalarga o'tishga va juft funktsiyalarga o'tishga qarshi.

Ushbu 3 + 2 tenglamalar Vasilev tenglamalari^[1] to'rt o'lchovli bosonik yuqori spinli nazariya uchun. Bir nechta sharhlar tartibda.

• Komponentlarga bo'linishda tizimning algebraik qismi ${ displaystyle A = alfa, { nuqta { alfa}}; alfa = 1,2; { nuqta { alfa}} = 1,2}$ tanloviga muvofiq ${ displaystyle sp (4)}$-metrik

${ displaystyle C_ {AB} = { begin {bmatrix} epsilon _ { alpha beta} & 0 0 & epsilon _ {{ dot { alpha}} { dot { beta}}}} end {bmatrix}}}$

o'zaro harakatlanadigan deformatsiyalangan osilatorlarning ikki nusxasiga teng bo'ladi:

${ displaystyle { begin {array} {lll} left [S _ { alpha}, S _ { beta} right] = - 2i epsilon _ { alpha beta} (1 + B star varkappa) & chap [S _ { alpha}, S _ { dot { beta}} right] = 0 & {B star varkappa, S _ { alpha} } _ { star} = 0 chap [S _ { nuqta { alfa}}, S _ { beta} o'ng] = 0 va chap [S _ { nuqta { alfa}}, S _ { nuqta { beta}} o'ng] = - 2i epsilon _ {{ dot { alpha}} { dot { beta}}} (1 + B star { bar { varkappa}}) & {B star { bar { varkappa}}, S _ { dot { alpha}} } _ { star} = 0 end {array}}}$

Shuning uchun oxirgi ikkita tenglama ikkita nusxaning aniqlanish munosabatlariga tengdir ${ displaystyle osp (1 | 2)}$ bilan ${ displaystyle S _ { alpha}}$ va ${ displaystyle S _ { dot { alfa}}}$ g'alati generatorlar rolini o'ynash va ${ displaystyle B star varkappa}$ va ${ displaystyle B star { bar { varkappa}}}$ deformatsiyalar rolini o'ynash. Beri ${ displaystyle B}$ ikki nusxada bir xil, ular mustaqil emas, bu esa izchillikni buzmaydi.

• Tizim izchil. Dastlabki uchta tenglamaning izchilligi aniq, chunki ular nol egrilik / kovariant-barqarorlik tenglamalari. So'nggi ikkita tenglamaning mustahkamligi deformatsiyalangan osilatorlar tufayli. Tenglamalarning ikkala qismining o'zaro mutanosibligi shundan iboratki, burama kovariant barqarorligi ${ displaystyle B}$-field ikkalasining odatiy kovariant barqarorligiga tengdir ${ displaystyle B star varkappa}$ yoki ${ displaystyle B star { bar { varkappa}}}$. Haqiqatdan ham,

${ displaystyle d (B star varkappa) - [W, (B star varkappa)] = (dB-W star B + B star varkappa star W star varkappa) star varkappa = (dB-W star B + B star pi (W)) star varkappa = 0}$

biz qayerda foydalanganmiz ${ displaystyle varkappa star varkappa = 1}$ va uning bilan bog'liqligi ${ displaystyle pi}$-avtomorfizm. Keyin, ${ displaystyle varkappa}$ bekor qilinishi mumkinligi sababli bekor qilinishi mumkin;

• Tenglamalar o'lchov o'zgarmasdir. O'lchov simmetriyasining o'zgarishi ${ displaystyle xi = xi (Y, Z | x)}$ ular:

${ displaystyle { begin {aligned} delta W & = d xi - [W, xi] _ { star} delta B & = xi star BB star pi ( xi) delta S_ {A} & = xi star S_ {A} -S_ {A} star pi ( xi) end {hizalanmış}}}$

• Tenglamalar fonga bog'liq emas va chiziqli echimning izohini berish uchun ba'zi vakuumni ko'rsatish kerak
• Eng sodda aniq echim - bu anti-de Sitter maydoni:

${ displaystyle W = Omega = { frac {1} {2}} varpi ^ { alpha alpha} L _ { alpha alpha} + h ^ { alpha { dot { alfa}}} P_ { alfa { nuqta { alfa}}} + { frac {1} {2}} varpi ^ {{ nuqta { alfa}} { nuqta { alfa}}} { bar {L} } _ {{ nuqta { alfa}} { nuqta { alfa}}} ,, qquad B = 0 ,, qquad S_ {A} = Z_ {A} ,,}$

qayerda ${ displaystyle Omega}$ tekis ulanishdir ${ displaystyle d Omega = Omega star Omega}$ anti-de-Sitter algebrasi va Lorents va uning tarjimalari bo'yicha komponentlar spin-ulanishga mos keladi ${ Displaystyle varpi ^ { alfa alfa}, varpi ^ {{ nuqta { alfa}} { nuqta { alfa}}}}$ va vierbein ${ displaystyle h ^ {{ alpha} { dot { alpha}}}}$navbati bilan. Bu muhim ${ displaystyle S}$maydonida noan'anaviy vakuum qiymati mavjud, bu tufayli echim bo'ladi ${ displaystyle [Z_ {A}, Z_ {B}] _ { star} = - 2iC_ {AB}}$ va haqiqat ${ displaystyle B = 0}$.

• Sitterga qarshi vakuum ustida chiziqli Vasilev tenglamalari spinning s = 0,1,2,3, ... barcha bo'sh massasiz maydonlarini tavsiflaydi, bu ba'zi bir hisoblashlarni talab qiladi va quyida keltirilgan.

Lineerizatsiya

Lineerlashtirilgan Vasilev tenglamalari erkin massasiz yuqori spinli maydonlarni tavsiflashini isbotlash uchun anti-de-Sitter vakuumidagi chiziqli dalgalanmaları ko'rib chiqishimiz kerak. Avvalo biz aniq echimni qaerdan olamiz ${ displaystyle W = Omega}$ anti-de Sitter algebrasining tekis aloqasi, ${ displaystyle B = 0}$ va ${ displaystyle S_ {A} = Z_ {A}}$ va dalgalanmalar qo'shing

${ displaystyle W = Omega + w ,, qquad B = 0 + 2ib ,, qquad S_ {A} = Z_ {A} + 2is_ {A} ,.}$

Keyin, biz Vasilev tenglamalarini chiziqli qilib qo'yamiz

${ displaystyle { begin {aligned} dw- Omega star ww star Omega & = 0 ,, db- Omega star b + b star pi ( Omega) & = 0 , , ds_ {A} - Omega star s_ {A} + s_ {A} star Omega & = kısalt _ {A} w ,, qisman _ {A} b & = 0 , , kısalt _ { alfa} s _ { beta} - qisman _ { beta} s _ { alfa} & = epsilon _ { alfa beta} b star varkappa ,, qisman _ { nuqta { alfa}} s _ { nuqta { beta}} - qisman _ { nuqta { beta}} s _ { nuqta { alfa}} & = epsilon _ {{ dot { alpha}} { dot { beta}}} b star { bar { varkappa}} ,, kısalt _ { alpha} s _ { dot { beta}} - qismli _ { dot { beta}} s _ { alpha} & = 0 ,, end {aligned}}}$

Yuqorida u bir necha marta ishlatilgan ${ textstyle [Z ^ {A}, f (Z)] _ { star} = - 2i qisman _ {A} f (Z), qismli _ {A} equiv { frac { qismli} { qisman Z ^ {A}}}}$, ya'ni S-maydonning vakuum qiymati komutator ostida hosila vazifasini bajaradi. To'rt komponentli Y, Z ni ikki komponentli o'zgaruvchilarga quyidagicha ajratish qulay ${ displaystyle Y ^ {A} = (y ^ { alfa}, y ^ { nuqta { alfa}}), Z ^ {A} = (z ^ { alfa}, z ^ { dot { alfa}})}$. To'rtinchi tenglamada ishlatilgan yana bir hiyla-nayrang Klein operatorlarining o'zgaruvchanligi:

${ displaystyle {b star varkappa, z _ { alpha} } = b star varkappa star z _ { alpha} + z _ { alpha} star b star varkappa = (b star varkappa yulduz z _ { alfa} star varkappa + z _ { alfa} yulduz b) yulduz varkappa = [z _ { alfa}, b] yulduz varkappa ,.}$

Vasilev tenglamalarining beshinchisi endi yuqoridagi so'nggi uchta tenglamaga bo'lingan.

Chiziqli dalgalanmalar tahlili tenglamalarni to'g'ri tartibda birma-bir echishda. Eslatib o'tamiz, ikkita maydon uchun ochilmagan tenglamalarni topishni kutmoqdamiz: bitta shakl ${ displaystyle omega = omega _ { mu} (Y | x) dx ^ { mu}}$ va nol shakl ${ displaystyle C = C (Y | x)}$. To'rtinchi tenglamadan shunday xulosa kelib chiqadi ${ displaystyle b}$ yordamchi Z yo'nalishiga bog'liq emas. Shuning uchun, kimdir aniqlay oladi ${ displaystyle b = C (Y | x)}$. Keyin ikkinchi tenglama darhol olib keladi

${ displaystyle nabla C + ih ^ { alfa { dot { alpha}}} chap (y _ { alpha} y _ { dot { alfa}} - { frac { qismli ^ {2}} { qisman y ^ { alfa} qisman y ^ { nuqta { alfa}}}} o'ng) C = 0 ,,}$

qayerda ${ displaystyle nabla}$ Lorentsning kovariant hosilasi

${ displaystyle nabla = d- varpi ^ { alpha beta} chap (y _ { alpha} { frac { qismli} { qisman y ^ { beta}}} + y _ { beta} { frac { qismli} { qismli y ^ { alfa}}} o'ng) -...}$

bu erda ... bilan atamani belgilang ${ displaystyle varpi ^ {{ dot { alpha}} { dot { beta}}}}$ bu birinchisiga o'xshash. Lorents kovariant hosilasi spin-ulanish qismining odatiy kommutator harakatidan kelib chiqadi ${ displaystyle Omega}$. Vierbein bilan atama ${ displaystyle pi}$- AdS-tarjimalari belgisini silkitadigan va piyodalarga-kommutator ishlab chiqaradigan avomorfizm ${ displaystyle h ^ { alpha { nuqta { alfa}}} chap {P _ { alfa { nuqta { alfa}}}, bullet right } _ { star}}$.

S-tenglamaning mazmunini o'qish uchun uni Yda kengaytirish va C-tenglama komponentini tahlil qilish kerak

${ displaystyle C = sum _ {k + m = { text {even}}} C _ { alpha _ {1} ... alfa _ {k}, { dot { alpha}} _ {1 } ... { nuqta { alfa}} _ {m}} (X) , y ^ { alfa _ {1}} ... y ^ { alfa _ {k}} , y ^ { { nuqta { alfa}} _ {1}} ... y ^ {{ nuqta { alfa}} _ {m}}}$

Keyin turli xil tarkibiy qismlarni quyidagicha talqin qilish mumkin:

• Birinchi komponent ${ displaystyle C (x)}$ skalar maydoni. Uning yonidagi, ${ displaystyle C _ { alpha { dot { alpha}}}}$ skalar hosilasi sifatida C tenglamasi fazilati bilan ifodalanadi. Komponentli tenglamalardan biri Klayn-Gordon tenglamasini o'rnatadi ${ displaystyle ( square -2) C (x) = 0}$, bu erda kosmologik doimiy birga o'rnatiladi. Nuqta va belgisiz indekslarning teng soniga ega komponentlar skalyarning qobiqdagi hosilalari sifatida ifodalanadi

${ displaystyle C _ { alpha _ {1} ... alfa _ {k}, { nuqta { alfa}} _ {1} ... { dot { alfa}} _ {k}} ( x) , h _ { mu _ {1}} ^ { alfa _ {1} { nuqta { alfa}} _ {1}} ... h _ { mu _ {k}} ^ { alfa _ {k} { dot { alfa}} _ {k}} sim nabla _ { mu _ {1}} ... nabla _ { mu _ {k}} C (x)}$

• ${ displaystyle C _ { alpha beta}, C _ {{ dot { alpha}} { dot { beta}}}}$ ning o'z-o'zini o'zi boshqaradigan va o'z-o'ziga qarshi bo'lgan komponentlari Maksvell tensori ${ displaystyle F _ { mu nu}}$. C-tenglama Maksvell tenglamalarini o'rnatadi. K + 2 = m va k = m + 2 bo'lgan komponentlar Maksvell tenzorining qobiqdagi hosilalari;
• ${ displaystyle C _ { alpha beta gamma delta}, C _ {{ dot { alpha}} { dot { beta}} { dot { gamma}} { dot { delta}}} }$ ning o'z-o'zini dual va anti-dual-komponentlari Veyl tensori ${ displaystyle C _ { mu nu, lambda rho}}$. C-tenglama Veyl tenzori uchun Byanki identifikatorlarini o'rnatadi. K + 4 = m va k = m + 4 bo'lgan komponentlar Veyl tensorining qobiqdagi hosilalari;
• ${ displaystyle C _ { alpha _ {1} ... alpha _ {2s}}, C _ {{ dot { alpha}} _ {1} ... { dot { alpha}} _ {2s }}}$ Veyl tenzorining yuqori spinli umumlashmasining o'z-o'ziga xos va o'z-o'ziga qarshi komponentlari. C-tenglama Byanki identifikatorlarini o'rnatadi va k + 2s = m va k = m + 2s bo'lgan komponentlar yuqori spinli Veyl tensorining qobiqdagi hosilalari hisoblanadi;

Oxirgi uchta tenglama shaklning tenglamalari deb tan olinishi mumkin ${ displaystyle mathrm {d} mu = nu, mathrm {d} nu = 0}$ qayerda ${ displaystyle mathrm {d}}$ Z fazosidagi differentsial shakllar makonining tashqi hosilasi. Bunday tenglamalarni. Yordamida echish mumkin Poincare Lemma. Bundan tashqari, yulduz mahsuloti uchun ajralmas formuladan kelib chiqadigan Klein operatori tomonidan o'ngdan qanday ko'paytirilishini bilish kerak:

${ displaystyle f (y _ { alpha}, z _ { beta}) star varkappa = f (-z _ { alpha}, - y _ { beta}) varkappa ,.}$

Ya'ni. natija Y va Z o'zgaruvchilarining yarmini almashtirish va belgini aylantirishdir. Oxirgi uchta tenglamaning echimi quyidagicha yozilishi mumkin

${ displaystyle s _ { alpha} = z _ { alpha} int _ {0} ^ {1} t , dt , C (-zt, { bar {y}}) e ^ {ity ^ { alfa} z _ { alfa}} + qisman _ { alfa} epsilon ,,}$

shunga o'xshash formula mavjud bo'lgan joyda ${ displaystyle s _ { dot { alfa}}}$Bu erda oxirgi muddat o'lchov noaniqligi, ya'ni Z maydoniga aniq shakllarni qo'shish erkinligi va ${ displaystyle epsilon = epsilon (Y, Z | x)}$.Uni tuzatish uchun o'lchash mumkin ${ displaystyle kısalt _ { alpha} epsilon = 0}$. Keyin, echimni uchinchi tenglamaga qo'shadi, qaysi biri bir xil bo'lsa, ya'ni Z fazosidagi birinchi darajadagi differentsial tenglama. Uning umumiy echimini yana Puankare Lemma beradi

${ displaystyle w = omega + Z ^ {A} int _ {0} ^ {1} dt , f_ {A} (zt) ,, qquad f_ {A} = dS_ {A} - [ Omega, S_ {A}] _ { star} ,,}$

qayerda ${ displaystyle omega = omega (Y | x)}$ - bu Z fazosidagi integral sobit, ya'ni de-Rham kohomologiyasi. Aynan shu integralning doimiysi bitta shakl bilan aniqlanishi kerak ${ displaystyle omega (Y | x)}$ nomidan ko'rinib turibdiki. Biroz algebra topilgandan keyin

${ displaystyle w = omega + int _ {0} ^ {1} dt (1-t) left (t varpi ^ { alpha beta} z _ { alpha} z _ { beta} -ih ^ { alfa { nuqta { alfa}}} z _ { alfa} { frac { qismli} { qismli y ^ { nuqta { alfa}}}} o'ng) C (-z _ { gamma} t, y _ { dot { gamma}}) e ^ {ity ^ { delta} z _ { delta}} + ...}$

bu erda biz yana nuqta va belgisiz indekslar almashtirilgan muddatni tashladik. Oxirgi qadam, echimni topish uchun birinchi tenglamaga qo'shishdir

${ displaystyle nabla omega -h ^ { alpha { nuqta { alfa}}} chap (y _ { alfa} { frac { qismli} { qisman y ^ { nuqta { alfa}} }} + y _ { nuqta { alfa}} { frac { qismli} { qismli y ^ { alfa}}} o'ng) omega = - { frac {1} {2}} h _ { gamma} {} ^ { dot { alpha}} wedge h ^ { gamma { dot { beta}}} { frac { qismli ^ {2}} { qisman y ^ { dot { alfa}} kısmi y ^ { nuqta { beta}}}} C (0, y _ { dot { delta}}) + ...}$

va yana o'ngdagi ikkinchi atama qoldiriladi. Bu muhim ${ displaystyle omega}$ tekis ulanish emas, ammo ${ displaystyle w}$ tekis ulanishdir. Tahlil qilish uchun ${ displaystyle omega}$-tenglamalarni kengaytirish foydalidir ${ displaystyle omega}$ Yda

${ displaystyle omega = sum _ {k + m = { text {even}}} omega _ { alpha _ {1} ... alpha _ {k}, { dot { alpha}} _ {1} ... { nuqta { alfa}} _ {m}} (X) , y ^ { alfa _ {1}} ... y ^ { alfa _ {k}} , y ^ {{ nuqta { alfa}} _ {1}} ... y ^ {{ nuqta { alfa}} _ {m}}}$

Ning mazmuni ${ displaystyle omega}$-tenglama quyidagicha:

• K = m bo'lgan diagonal komponentlar yuqori spinli vierbeinlar bo'lib, ularning to'liq nosimmetrik komponentlarini Fronsdal maydoni kabi

${ displaystyle h _ { mu _ {1}} ^ { alpha _ {1} { dot { alpha}} _ {1}} ... h _ { mu _ {s-1}} ^ { alfa _ {s-1} { nuqta { alfa}} _ {s-1}} , omega _ { mu _ {s} | alfa _ {1} ... alfa _ {s- 1}, { dot { alpha}} _ {1} ... { dot { alpha}} _ {s-1}} sim Phi _ { mu _ {1} ... mu _ {s}}}$

chapda simmetrizatsiya nazarda tutilgan joyda;

• The ${ displaystyle omega}$s = 2,3,4, ... uchun Fronsdal tenglamalarini o'rnatish uchun tenglamani ko'rsatish mumkin. Multipletning s = 1 va s = 0 komponentlari uchun Maksvell tenglamalari va Klayn-Gordon tenglamalari C-tenglamada;
• Boshqa komponentlar Fronsdal maydonining qobiqdagi hosilalari sifatida ifodalanadi;
• Fronsdal maydonining order-s hosilasi yuqori spinli Veyl tensorining simmetriyasiga ega bo'lib, S-maydonning tegishli komponentini ${ displaystyle omega}$-tenglama.

Xulosa qilish kerakki, anti-de-Sitter maydoni - bu Vasilev tenglamalarining aniq echimi va uning ustida chiziqlash natijasida s = 0,1,2,3, ... bo'lgan maydonlar uchun Fronsdal tenglamalariga teng bo'lgan ochilmagan tenglamalar topiladi.

Boshqa o'lchamlar, kengaytmalar va umumlashmalar

• tenglikni buzish bilan bog'liq bo'lgan to'rt o'lchovli tenglamalarda erkin parametrni kiritishning muhim varianti mavjud. Faqatgina o'zgartirishlar kerak

${ displaystyle { begin {array} {ll} left [S _ { alpha}, S _ { beta} right] = - 2i epsilon _ { alpha beta} (1 + e ^ {i theta } B star varkappa) & left [S _ { dot { alpha}}, S _ { dot { beta}} right] = - 2i epsilon _ {{ dot { alpha}} { nuqta { beta}}} (1 + e ^ {- i theta} B star { bar { varkappa}}) end {array}}}$

Ushbu bepul parametr muhim rol o'ynaydi yuqori spinli AdS / CFT yozishmalari. Nazariyasi ${ displaystyle theta = 0, pi / 2}$ paritet o'zgarmasdir;

Biri ham olishi mumkin ${ displaystyle theta}$ har qanday teng funktsiya bo'lish ${ displaystyle theta (x) = theta (-x)}$ ning ${ displaystyle B star varkappa}$ yuqoridagi va ning birinchi tenglamasida ${ displaystyle B star { bar { varkappa}}}$ ikkinchisida, bu tenglamalarning izchilligini buzmaydi.

• Yang-Mills guruhlarini tanishtirish mumkin^[9] by letting the fields take values in the tensor product of the Y-Z algebra with the matrix algebra and then imposing truncations as to get ${displaystyle o(N),u(N),usp(N)}$;
• the four-dimensional equations reviewed above can be extended with super-symmetries.^[9] One needs to extend the Y-Z algebra with additional Clifford-like elements

${displaystyle left{xi _{i},xi _{j} ight}=2delta _{ij}}$

so that the fields are now function of ${displaystyle Y,Z,xi }$ and space-time coordinates. The components of the fields are required to have the right spin-statistic. The equations need to be slightly modified.^[10]

There also exist Vasiliev's equations in other dimensions:

• in three dimensions there is the minimal higher-spin theory^[2] and its development, known as Prokushkin-Vasiliev theory,^[3] that is based on a one-parameter family of higher-spin algebras (usually the family is denoted as ${displaystyle hs(lambda )}$) and also allows for super-symmetric extensions;
• there exist Vasiliev equations that operate in any space-time dimension.^[4] The spectrum of the theory consists of all the fields with integer (or even only) spins.

The equations are very similar to the four-dimensional ones, but there are some important modifications in the definition of the algebra that the fields take values in and there are further constraints in the d-dimensional case.

Discrepancies between Vasiliev equations and Higher Spin Theories

There is a number of flaws/features of the Vasiliev equations that have been revealed over the last years. First of all, classical equations of motion, e.g. the Vasiliev equations, do not allow one to address the problems that require an action, the most basic one being quantization. Secondly, there are discrepancies between the results obtained from the Vasiliev equations and those from the other formulations of higher spin theories, from the AdS / CFT yozishmalari or from general field theory perspective. Most of the discrepancies can be attributed to the assumptions used in the derivation of the equations: gauge invariance is manifest, but locality was not properly imposed and the Vasiliev equations are a solution of a certain formal deformation problem. Practically speaking, it is not known in general how to extract the interaction vertices of the higher spin theory out of the equations.

Most of the studies concern with the four-dimensional Vasiliev equations. The correction to the free spin-2 equations due to the scalar field stress-tensor was extracted out of the four-dimensional Vasiliev equations and found to be^[11]

${displaystyle G_{mu u }+Lambda g_{mu u }=Re(b_{1}^{2})left[sum _{k}left(xi _{k}g_{mu u } abla _{ ho (k+1)}phi abla ^{ ho (k+1)}phi +eta _{k} abla _{mu ho (k)}phi abla ^{ ho (k)}{}_{ u }phi +zeta _{k} abla _{mu u ho (k)}phi abla ^{ ho (k)}phi ight)-{frac {4}{9}}g_{mu u }phi ^{2} ight]}$

qayerda ${ extstyle abla _{ ho (k)}phi =' abla _{ ho _{1}}... abla _{ ho _{k}}phi +{ ext{symmetrization}}-{ ext{traces}}'}$ are symmetrized derivatives with traces subtracted. The most important information is in the coefficients ${ extstyle xi _{k},eta _{k},zeta _{k}}$ and in the prefactor ${ extstyle Re(b_{1}^{2})}$, qayerda ${ extstyle b_{1}=exp[i heta ]}$ is a free parameter that the equations have, see Other dimensions, extensions, and generalisations. It is important to note that the usual stress-tensor has no more than two derivative and the terms ${ displaystyle k> 0}$ are not independent (for example, they contribute to the same ${ extstyle langle T_{ab}j_{0}j_{0} angle }$ AdS/CFT three-point function). This is a general property of field theories that one can perform nonlinear (and also higher derivative) field redefinitions and therefore there exist infinitely many ways to write the same interaction vertex at the classical level. The canonical stress-tensor has two derivatives and the terms with contracted derivatives can be related to it via such redefinitions.

A surprising fact that had been noticed^[11][12] before its inconsistency with the AdS/CFT was realized is that the stress-tensor can change sign and, in particular, vanishes for ${ extstyle heta =pi /4}$. This would imply that the corresponding correlation function in the Chern-Simons matter theories vanishes, ${ extstyle langle T_{ab}j_{0}j_{0} angle =0}$, which is not the case.

The most important and detailed tests were performed much later. It was first shown^[13] that some of the three-point AdS/CFT functions, as obtained from the Vasiliev equations, turn out to be infinite or inconsistent with AdS/CFT, while some other do agree. Those that agree, in the language of Unfolded equations mos keladi ${displaystyle omega star C-Cstar pi (omega )}$ and the infinities/inconsistencies resulted from ${displaystyle {mathcal {V}}(omega ,C,C)}$. The terms of the first type are local and are fixed by the higher spin algebra. The terms of the second type can be non-local (when solved perturbatively the master field ${ displaystyle C}$ is a generating functions of infinitely many derivatives of higher spin fields). These non-localities are not present in higher spin theories as can be seen from the explicit cubic action^[14].

Further infinities, non-localities or missing structures were observed^{[15][16][17][18][19]}. Some of these tests explore the extension of the Klebanov-Polyakov gumoni to Chern-Simons matter theories where the structure of correlation functions is more intricate and certain parity-odd terms are present. Some of these structures were not reproduced by the Vasiliev equations. General analysis of the Vasiliev equations at the second order^[20] showed that for any three fixed spins the interaction term is an infinite series in derivatives (similar to ${ displaystyle k}$-sum above); all of the terms in the series contribute to the same AdS/CFT three-point function and the contribution is infinite. All the problems can be attributed to the assumptions used in the derivation of the Vasiliev equations: restrictions on the number of derivatives in the interaction vertices or, more generally, locality was not imposed, which is important for getting meaningful interaction vertices, see e.g. Noether protsedurasi. The problem how to impose locality and extract interaction vertices out of the equations is now under active investigation^[21].

As is briefly mentioned in Other dimensions, extensions, and generalisations there is an option to introduce infinitely many additional coupling constants that enter via phase factor ${displaystyle heta (x)= heta _{0}+ heta _{2}x^{2}+...}$. As was noted^[22], the second such coefficient ${ displaystyle theta _ {2}}$ will affect five-point AdS/CFT correlation functions, but not the three-point ones, which seems to be in tension with the results obtained directly from imposing higher spin symmetry on the correlation functions. Later, it was shown^[20] that the terms in the equations that result from ${displaystyle heta _{2,4,...}}$ are too non-local and lead to an infinite result for the AdS/CFT correlation functions.

In three dimensions the Prokushkin-Vasiliev equations, which are supposed to describe interactions of matter fields with higher spin fields in three dimensions, are also affected by the aforementioned locality problem. For example, the perturbative corrections at the second order to the stress-tensors of the matter fields lead to infinite correlation functions^[23]. There is, however, another discrepancy: the spectrum of the Prokushkin-Vasiliev equations has, in addition to the matter fields (scalar and spinor) and higher spin fields, a set of unphysical fields that do not have any field theory interpretation, but interact with the physical fields.

Aniq echimlar

Since the Vasiliev equations are quite complicated there are few exact solutions known

• as it was already shown, there is an important solution --- empty anti-de Sitter space, whose existence allows to interpret the linearized fluctuations as massless fields of all spins;
• in three dimensions to find anti-de Sitter space as an exact solution for all values of the parameter ${ displaystyle lambda}$ turns out to be a nontrivial problem, but it is known;^[3]
• there is a domain-wall type solution of the four-dimensional equations;^[24]
• there is a family of the solutions to the four-dimensional equations that are interpreted as black holes, although the metric transforms under the higher-spin transformations and for that reason it is difficult to rely on the usual definition of the horizon etc.;^[25][26][27]
• in the case of three-dimensions there is a consistent truncation that decouples the scalar field from the higher-spin fields, the latter being described by the Chern–Simons theory. In this case any flat connection of the higher-spin algebra is an exact solution and there has been a lot of works on this subclass;

Shuningdek qarang

Izohlar

Adabiyotlar

Sharhlar

• Vasiliev, M.A. (1996). "Higher-spin gauge theories in four, three and two dimensions". Xalqaro zamonaviy fizika jurnali D. 05 (6): 763–797. arXiv:hep-th/9611024. Bibcode:1996IJMPD...5..763V. doi:10.1142/S0218271896000473. S2CID  119436825.
• Vasiliev, M.A. (2004). "Higher spin gauge theories in various dimensions". Fortschritte der Physik. 52 (67): 702–717. arXiv:hep-th/0401177. Bibcode:2004ForPh..52..702V. doi:10.1002/prop.200410167.
• Vasiliev, Mikhail (2000). "Spin o'lchovining yuqori nazariyalari: Yulduzli mahsulot va AdS maydoni". Super Dunyoning Ko'p Yuzlari. 533-610 betlar. arXiv:hep-th / 9910096. Bibcode:2000mfsw.book..533V. doi:10.1142/9789812793850_0030. ISBN  978-981-02-4206-0. S2CID  15804505. Yo'qolgan yoki bo'sh sarlavha = (Yordam bering)
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