Fujikavaning usuli ni olishning bir usuli chiral anomaliya yilda kvant maydon nazariyasi .
Aytaylik Dirak maydoni $ mathbb {r} $ ga qarab o'zgaradi vakillik ning ixcham Yolg'on guruhi G ; va bizda fon bor ulanish shakli qiymatlarini olish Yolg'on algebra g . { displaystyle { mathfrak {g}} ,.} Dirac operatori (ichida.) Feynman slash notation )
D. / = d e f ∂ / + men A / { displaystyle D ! ! ! ! / { stackrel { mathrm {def}} {=}} kısalt ! ! ! / + iA ! ! ! /} fermionik harakat esa tomonidan beriladi
∫ d d x ψ ¯ men D. / ψ { displaystyle int d ^ {d} x , { overline { psi}} iD ! ! ! ! / psi} The bo'lim funktsiyasi bu
Z [ A ] = ∫ D. ψ ¯ D. ψ e − ∫ d d x ψ ¯ men D. / ψ . { displaystyle Z [A] = int { mathcal {D}} { overline { psi}} { mathcal {D}} psi , e ^ {- int d ^ {d} x , { overline { psi}} iD ! ! ! / , psi}.} The eksenel simmetriya transformatsiya quyidagicha boradi
ψ → e men γ d + 1 a ( x ) ψ { displaystyle psi to e ^ {i gamma _ {d + 1} alpha (x)} psi ,} ψ ¯ → ψ ¯ e men γ d + 1 a ( x ) { displaystyle { overline { psi}} to { overline { psi}} e ^ {i gamma _ {d + 1} alpha (x)}} S → S + ∫ d d x a ( x ) ∂ m ( ψ ¯ γ m γ d + 1 ψ ) { displaystyle S to S + int d ^ {d} x , alfa (x) kısalt _ { mu} chap ({ overline { psi}} gamma ^ { mu} gamma _ {d + 1} psi o'ng)} Klassik ravishda, bu shiral oqimi, j d + 1 m ≡ ψ ¯ γ m γ d + 1 ψ { displaystyle j_ {d + 1} ^ { mu} equiv { overline { psi}} gamma ^ { mu} gamma _ {d + 1} psi} 0 = ∂ m j d + 1 m { displaystyle 0 = kısalt _ { mu} j_ {d + 1} ^ { mu}}
Kvant mexanik ravishda chiral oqimi saqlanib qolmaydi: Jekiv buni uchburchak diagrammasi yo'qolib qolmasligi tufayli topdi. Fujikava buni chiral transformatsiyasi ostida bo'linish funktsiyasi o'lchovining o'zgarishi sifatida qayta talqin qildi. Chiral transformatsiyadagi o'lchov o'zgarishini hisoblash uchun avval Dirak fermiyalarini o'z vektorlari asosida ko'rib chiqing. Dirac operatori :
ψ = ∑ men ψ men a men , { displaystyle psi = sum limitlar _ {i} psi _ {i} a ^ {i},} ψ ¯ = ∑ men ψ men b men , { displaystyle { overline { psi}} = sum limit _ {i} psi _ {i} b ^ {i},} qayerda { a men , b men } { displaystyle {a ^ {i}, b ^ {i} }} Grassmann baholangan koeffitsientlar va { ψ men } { displaystyle { psi _ {i} }} Dirac operatori :
D. / ψ men = − λ men ψ men . { displaystyle D ! ! ! ! / psi _ {i} = - lambda _ {i} psi _ {i}.} O'z funktsiyalari d-o'lchovli kosmosga integratsiyalashuvga nisbatan ortonormal deb qabul qilinadi,
δ men j = ∫ d d x ( 2 π ) d ψ † j ( x ) ψ men ( x ) . { displaystyle delta _ {i} ^ {j} = int { frac {d ^ {d} x} {(2 pi) ^ {d}}} psi ^ { xanjar j} (x) psi _ {i} (x).} Keyin yo'l integralining o'lchovi quyidagicha aniqlanadi:
D. ψ D. ψ ¯ = ∏ men d a men d b men { displaystyle { mathcal {D}} psi { mathcal {D}} { overline { psi}} = prod limits _ {i} da ^ {i} db ^ {i}} Cheksiz kichik chiral o'zgarishi ostida yozing
ψ → ψ ′ = ( 1 + men a γ d + 1 ) ψ = ∑ men ψ men a ′ men , { displaystyle psi to psi ^ { prime} = (1 + i alfa gamma _ {d + 1}) psi = sum limitlar _ {i} psi _ {i} a ^ { prime i},} ψ ¯ → ψ ¯ ′ = ψ ¯ ( 1 + men a γ d + 1 ) = ∑ men ψ men b ′ men . { displaystyle { overline { psi}} to { overline { psi}} ^ { prime} = { overline { psi}} (1 + i alfa gamma _ {d + 1}) = sum limitlar _ {i} psi _ {i} b ^ { prime i}.} The Jacobian yordamida transformatsiyani hisoblash mumkin ortonormallik ning xususiy vektorlar
C j men ≡ ( δ a δ a ′ ) j men = ∫ d d x ψ † men ( x ) [ 1 − men a ( x ) γ d + 1 ] ψ j ( x ) = δ j men − men ∫ d d x a ( x ) ψ † men ( x ) γ d + 1 ψ j ( x ) . { displaystyle C_ {j} ^ {i} equiv left ({ frac { delta a} { delta a ^ { prime}}} right) _ {j} ^ {i} = int d ^ {d} x , psi ^ { xanjar i} (x) [1-i alfa (x) gamma _ {d + 1}] psi _ {j} (x) = delta _ { j} ^ {i} , - i int d ^ {d} x , alfa (x) psi ^ { xanjar i} (x) gamma _ {d + 1} psi _ {j} (x).} Koeffitsientlarning o'zgarishi { b men } { displaystyle {b_ {i} }}
D. ψ D. ψ ¯ = ∏ men d a men d b men = ∏ men d a ′ men d b ′ men det − 2 ( C j men ) , { displaystyle { mathcal {D}} psi { mathcal {D}} { overline { psi}} = prod limitler _ {i} da ^ {i} db ^ {i} = prod limitlar _ {i} da ^ { prime i} db ^ { prime i} { det} ^ {- 2} (C_ {j} ^ {i}),} qaerda Jacobian - bu determinantning o'zaro bog'liqligi, chunki integral o'zgaruvchilar Grassmannian, 2 esa a va b ning teng hissa qo'shganligi sababli paydo bo'ladi. Determinantni standart metodlar bo'yicha hisoblashimiz mumkin:
det − 2 ( C j men ) = tugatish [ − 2 t r ln ( δ j men − men ∫ d d x a ( x ) ψ † men ( x ) γ d + 1 ψ j ( x ) ) ] = tugatish [ 2 men ∫ d d x a ( x ) ψ † men ( x ) γ d + 1 ψ men ( x ) ] { displaystyle { begin {aligned} { det} ^ {- 2} (C_ {j} ^ {i}) & = exp left [-2 { rm {tr}} ln ( delta _ {j} ^ {i} -i int d ^ {d} x , alfa (x) psi ^ { xanjar i} (x) gamma _ {d + 1} psi _ {j} ( x)) right] & = exp left [2i int d ^ {d} x , alfa (x) psi ^ { xanjar i} (x) gamma _ {d + 1} psi _ {i} (x) right] end {aligned}}} a (x) da birinchi tartibga.
A doimiy bo'lgan holatga ixtisoslashgan, the Jacobian tartibga solinishi kerak, chunki integral yozilgan deb noto'g'ri aniqlangan. Fujikava ish bilan ta'minlangan issiqlik yadrosini tartibga solish , shu kabi
− 2 t r ln C j men = 2 men lim M → ∞ a ∫ d d x ψ † men ( x ) γ d + 1 e − λ men 2 / M 2 ψ men ( x ) = 2 men lim M → ∞ a ∫ d d x ψ † men ( x ) γ d + 1 e D. / 2 / M 2 ψ men ( x ) { displaystyle { begin {aligned} -2 { rm {tr}} ln C_ {j} ^ {i} & = 2i lim limit _ {M to infty} alpha int d ^ { d} x , psi ^ { xanjar i} (x) gamma _ {d + 1} e ^ {- lambda _ {i} ^ {2} / M ^ {2}} psi _ {i } (x) & = 2i lim limitlar _ {M to infty} alpha int d ^ {d} x , psi ^ { xanjar i} (x) gamma _ {d + 1} e ^ {{D ! ! ! / ,} ^ {2} / M ^ {2}} psi _ {i} (x) end {aligned}}} ( D. / 2 { displaystyle {D ! ! ! ! /} ^ {2}} D. 2 + 1 4 [ γ m , γ ν ] F m ν { displaystyle D ^ {2} + { tfrac {1} {4}} [ gamma ^ { mu}, gamma ^ { nu}] F _ { mu nu}}
= 2 men lim M → ∞ a ∫ d d x ∫ d d k ( 2 π ) d ∫ d d k ′ ( 2 π ) d ψ † men ( k ′ ) e men k ′ x γ d + 1 e − k 2 / M 2 + 1 / ( 4 M 2 ) [ γ m , γ ν ] F m ν e − men k x ψ men ( k ) { displaystyle = 2i lim limitlar _ {M to infty} alfa int d ^ {d} x int { frac {d ^ {d} k} {(2 pi) ^ {d} }} int { frac {d ^ {d} k ^ { prime}} {(2 pi) ^ {d}}} psi ^ { xanjar i} (k ^ { prime}) e ^ {ik ^ { prime} x} gamma _ {d + 1} e ^ {- k ^ {2} / M ^ {2} + 1 / (4M ^ {2}) [ gamma ^ { mu} , gamma ^ { nu}] F _ { mu nu}} e ^ {- ikx} psi _ {i} (k)} = − − 2 a ( 2 π ) d / 2 ( d 2 ) ! ( 1 2 F ) d / 2 , { displaystyle = - { frac {-2 alpha} {(2 pi) ^ {d / 2} ({ frac {d} {2}})!}} ({ tfrac {1} {2) }} F) ^ {d / 2},} o'z vektorlari uchun to'liqlik munosabatini qo'llaganidan so'ng, b-matritsalar ustida iz qoldirgan va M.dagi chegarani olganidan keyin natija maydon kuchi 2-shakl, F ≡ F m ν d x m ∧ d x ν . { displaystyle F equiv F _ { mu nu} , dx ^ { mu} wedge dx ^ { nu} ,.}
Ushbu natija tengdir ( d 2 ) t h { displaystyle ({ tfrac {d} {2}}) ^ { rm {th}}} Chern sinfi ning g { displaystyle { mathfrak {g}}} chiral anomaliya , chiral oqimining saqlanib qolmasligi uchun javobgardir.
Adabiyotlar
K. Fujikava va X. Suzuki (2004 yil may). Yo'l integrallari va kvant anomaliyalari . Clarendon Press. ISBN 0-19-852913-9 . S. Vaynberg (2001). Maydonlarning kvant nazariyasi . II jild: Zamonaviy dasturlar .. Kembrij universiteti matbuoti. ISBN 0-521-55002-5 . 
![{ displaystyle Z [A] = int { mathcal {D}} { overline { psi}} { mathcal {D}} psi , e ^ {- int d ^ {d} x , { overline { psi}} iD ! ! ! / , psi}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39689035711b3ff0b0547c3de59c52a2251defd5)
![C_ {j} ^ {i} equiv chap ({ frac { delta a} { delta a ^ { prime}}} o'ng) _ {j} ^ {i} = int d ^ {d } x , psi ^ {{ xanjar i}} (x) [1-i alfa (x) gamma _ {{d + 1}}] psi _ {j} (x) = delta _ {j} ^ {i} , - i int d ^ {d} x , alfa (x) psi ^ {{ xanjar i}} (x) gamma _ {{d + 1}}) psi _ {j} (x).](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ff4a9b91e03ec9e3a18e5f57eaadd03b10b1ec)
![{ begin {aligned} { det} ^ {{- 2}} (C_ {j} ^ {i}) & = exp left [-2 {{ rm {tr}}} ln ( delta _ {j} ^ {i} -i int d ^ {d} x , alfa (x) psi ^ {{ xanjar i}} (x) gamma _ {{d + 1}} psi _ {j} (x)) o'ng] & = exp chap [2i int d ^ {d} x , alfa (x) psi ^ {{ xanjar i}} (x) gamma _ {{d + 1}} psi _ {i} (x) right] end {hizalangan}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26a547accf4438f355faec9182c227ad3d642ad3)
![D ^ {2} + { tfrac {1} {4}} [ gamma ^ { mu}, gamma ^ { nu}] F _ {{ mu nu}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3b53ceb460bc4993a4bfd9e4e103daf9e2726f7)
![= 2i lim limitlar _ {{M to infty}} alfa int d ^ {d} x int { frac {d ^ {d} k} {(2 pi) ^ {d}} } int { frac {d ^ {d} k ^ { prime}} {(2 pi) ^ {d}}} psi ^ {{ xanjar i}} (k ^ { prime}) e ^ {{ik ^ { prime} x}} gamma _ {{d + 1}} e ^ {{- k ^ {2} / M ^ {2} + 1 / (4M ^ {2}) [ gamma ^ { mu}, gamma ^ { nu}] F _ {{ mu nu}}}} e ^ {{- ikx}} psi _ {i} (k)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7950abb404ba18c112133787793ba0fe6d8c870)
