Feynman parametrlanishi baholash texnikasi halqa integrallari kelib chiqadi Feynman diagrammalari bir yoki bir nechta ilmoq bilan. Biroq, ba'zida bu sohalarda birlashishda foydalidir sof matematika shuningdek.
Formulalar
Richard Feynman quyidagilarni kuzatdi:
![{ frac {1} {AB}} = int _ {0} ^ {1} { frac {du} { left [uA + (1-u) B right] ^ {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba6124fb1e6f1989accc58fa8f8fdefeb8f767bf)
har qanday murakkab sonlar uchun amal qiladi A va B 0 chiziqli segmentga ulanmasa A va B. Formula quyidagi kabi integrallarni baholashga yordam beradi:
![int { frac {dp} {A (p) B (p)}} = int dp int _ {0} ^ {1} { frac {du} { left [uA (p) + (1 -u) B (p) o'ng] ^ {2}}} = int _ {0} ^ {1} du int { frac {dp} { chap [uA (p) + (1-u) B (p) o'ng] ^ {2}}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd7ec9daf0ae058280778e79ab74831a23848444)
Agar A (p) va B (p) ning chiziqli funktsiyalari p, keyin oxirgi integralni almashtirish yordamida baholash mumkin.
Umuman olganda Dirac delta funktsiyasi
:[1]
![{ displaystyle { begin {aligned} { frac {1} {A_ {1} cdots A_ {n}}} & = (n-1)! int _ {0} ^ {1} du_ {1} cdots int _ {0} ^ {1} du_ {n} { frac { delta (1- sum _ {k = 1} ^ {n} u_ {k}) ;} { left ( sum _ {k = 1} ^ {n} u_ {k} A_ {k} right) ^ {n}}} & = (n-1)! int _ {0} ^ {1} du_ { 1} int _ {0} ^ {u_ {1}} du_ {2} cdots int _ {0} ^ {u_ {n-2}} du_ {n-1} { frac {1} { chap [A_ {1} + u_ {1} (A_ {2} -A_ {1}) + nuqta + u_ {n-1} (A_ {n} -A_ {n-1}) o'ng] ^ { n}}}. end {hizalangan}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a049f04b1de5dae0e89f3be60db0369592b58769)
Ushbu formula har qanday murakkab sonlar uchun amal qiladi A1,...,An 0 ularning tarkibida bo'lmasa qavariq korpus.
Umuman olganda, bu shart bilan
Barcha uchun
:
![{ displaystyle { frac {1} {A_ {1} ^ { alpha _ {1}} cdots A_ {n} ^ { alpha _ {n}}}} = { frac { Gamma ( alpha _ {1} + nuqta + alfa _ {n})} { Gamma ( alfa _ {1}) cdots Gamma ( alfa _ {n})}} int _ {0} ^ {1 } du_ {1} cdots int _ {0} ^ {1} du_ {n} { frac { delta (1- sum _ {k = 1} ^ {n} u_ {k}) ; u_ {1} ^ { alfa _ {1} -1} cdots u_ {n} ^ { alfa _ {n} -1}} { left ( sum _ {k = 1} ^ {n} u_ { k} A_ {k} o'ng) ^ { sum _ {k = 1} ^ {n} alfa _ {k}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f73bbf734bd1234c270cf0e5486f568e8543c1e)
qaerda Gamma funktsiyasi
ishlatilgan.[2]
Hosil qilish
![{ frac {1} {AB}} = { frac {1} {AB}} chap ({ frac {1} {B}} - { frac {1} {A}} right) = { frac {1} {AB}} int _ {B} ^ {A} { frac {dz} {z ^ {2}}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4743252ca77e44d06fc812c80cf26fd6af6d357e)
Endi o'rnini bosuvchi yordamida integralni chiziqli ravishda o'zgartiring,
olib keladi
shunday ![z = uA + (1-u) B](https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb2c60830cf00b2de1f286b5086889325081c73)
va biz kerakli natijani olamiz:
![{ frac {1} {AB}} = int _ {0} ^ {1} { frac {du} { left [uA + (1-u) B right] ^ {2}}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c889abc83102e38acad92e417ca8271f2b7dbf2)
Ko'proq umumiy holatlarda, lotinlarni juda samarali ravishda bajarish mumkin Shvinger parametrlari. Masalan, Feynmanning parametrlangan shaklini olish uchun
, biz birinchi navbatda maxrajdagi barcha omillarni Shviner parametrlangan shaklida ifodalaymiz:
![{ displaystyle { frac {1} {A_ {i}}} = int _ {0} ^ { infty} ds_ {i} , e ^ {- s_ {i} A_ {i}} { text {for}} i = 1, ldots, n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1edfa49cc66e4dffd2239ac37e2e0146637a723)
va qayta yozing,
![{ displaystyle { frac {1} {A_ {1} cdots A_ {n}}} = int _ {0} ^ { infty} ds_ {1} cdots int _ {0} ^ { infty } ds_ {n} exp chap (- chap (s_ {1} A_ {1} + cdots + s_ {n} A_ {n} o'ng) o'ng).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/783107414dd7997127c95d749be0d782b7d13155)
Keyin integral o'zgaruvchilarning quyidagi o'zgarishini amalga oshiramiz,
![alfa = s_1 + ... + s_n,](https://wikimedia.org/api/rest_v1/media/math/render/svg/524d8dd63ee1f9d69791e56ebdfabc757ef4e9a6)
![{ displaystyle alpha _ {i} = { frac {s_ {i}} {s_ {1} + cdots + s_ {n}}}; i = 1, ldots, n-1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b77d09b4a08126830ea3dbae7566882c316227a3)
olish,
![{ displaystyle { frac {1} {A_ {1} cdots A_ {n}}} = int _ {0} ^ {1} d alpha _ {1} cdots d alpha _ {n-1 } int _ {0} ^ { infty} d alfa alfa ^ {n-1} exp left (- alfa left { alpha _ {1} A_ {1} + cdots + alfa _ {n-1} A_ {n-1} + chap (1- alfa _ {1} - cdots - alfa _ {n-1} o'ng) A_ {n} o'ng } o'ng).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a4e468d97de7bcb2cb2b10e1d286b4085761cf8)
qayerda
mintaqa bo'yicha integratsiyani bildiradi
bilan
.
Keyingi qadam
integratsiya.
![{ displaystyle int _ {0} ^ { infty} d alfa alfa ^ {n-1} exp (- alfa x) = { frac { qismli ^ {n-1}} { qisman (-x) ^ {n-1}}} chap ( int _ {0} ^ { infty} d alfa exp (- alfa x) o'ng) = { frac { chap (n -1 o'ng)!} {X ^ {n}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/415c7c775445e9e28dd6949f4dac1a83caa307a2)
biz aniqlagan joyda ![{ displaystyle x = alfa _ {1} A_ {1} + cdots + alfa _ {n-1} A_ {n-1} + chap (1- alfa _ {1} - cdots - alfa _ {n-1} o'ng) A_ {n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21c7cf5c04a1fd6fe7721c48d604e1d6effd3610)
Ushbu natijani o'rnini egallab, oldingi shaklga o'tamiz,
![{ displaystyle { frac {1} {A_ {1} cdots A_ {n}}} = chap (n-1 o'ng)! int _ {0} ^ {1} d alfa _ {1} cdots d alpha _ {n-1} { frac {1} {[ alpha _ {1} A_ {1} + cdots + alpha _ {n-1} A_ {n-1} + chap (1- alfa _ {1} - cdots - alfa _ {n-1} o'ng) A_ {n}] ^ {n}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f292fd8ef3883120a5bf7f93df4cabf998a83cc)
va qo'shimcha integralni kiritgandan so'ng, biz Feynman parametrlashining yakuniy shakliga kelamiz, ya'ni
![{ displaystyle { frac {1} {A_ {1} cdots A_ {n}}} = chap (n-1 o'ng)! int _ {0} ^ {1} d alfa _ {1} cdots int _ {0} ^ {1} d alfa _ {n} { frac { delta left (1- alpha _ {1} - cdots - alpha _ {n} right)} {[ alpha _ {1} A_ {1} + cdots + alpha _ {n} A_ {n}] ^ {n}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b11d20312b5ef0eb58f51c9cb4b61c2cc8b95f71)
Xuddi shu tarzda, Feynman parametrlash shaklini eng umumiy holatni olish uchun:
Bu maxrajdagi omillarning mos keladigan turli xil Shvinger parametrlash shakli bilan boshlanishi mumkin, ya'ni
![frac {1} {A_1 ^ { alpha_1}} = frac {1} { left ( alpha_1-1 right)!} int ^ infty_0 ds_1 , s_1 ^ { alpha_1-1} e ^ {-s_1 A_1} = frac {1} { Gamma ( alpha_1)} frac { kısmi ^ { alfa_1-1}} { qismli (-A_1) ^ { alpha_1-1}} chap ( int_ {0} ^ { infty} ds_1 e ^ {- s_1 A_1} o'ng)](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa325ae2b9d6f17b2c7a24fde357527e32d480bc)
va keyin oldingi holat bo'yicha aniq davom eting.
Muqobil shakl
Parametrlashning ba'zan foydali bo'lgan muqobil shakli
![frac {1} {AB} = int_ {0} ^ { infty} frac {d lambda} { left [ lambda A + B right] ^ 2}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/96b103991b478aa4c7f21ffc0d51b9c8e7711846)
Ushbu shakl o'zgaruvchilar o'zgarishi yordamida olinishi mumkin
.Bizdan foydalanishimiz mumkin mahsulot qoidasi buni ko'rsatish uchun
, keyin
![start {align}
frac {1} {AB} & = int ^ 1_0 frac {du} { left [uA + (1-u) B right] ^ 2}
& = int ^ 1_0 frac {du} {(1-u) ^ {2}} frac {1} { left [ frac {u} {1-u} A + B right] ^ 2}
& = int_ {0} ^ { infty} frac {d lambda} { left [ lambda A + B right] ^ 2}
end {align}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53d4182ef7a1d46a52f9f40f0c9cea3988f7c028)
Umuman olganda bizda
![frac {1} {A ^ {m} B ^ {n}} = frac { Gamma (m + n)} { Gamma (m) Gamma (n)} int_ {0} ^ { infty } frac { lambda ^ {m-1} d lambda} { left [ lambda A + B right] ^ {n + m}},](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b986497e8262710b61788c4b56e6b6753d55a3e)
qayerda
bo'ladi gamma funktsiyasi.
Ushbu shakl chiziqli maxrajni birlashtirganda foydali bo'lishi mumkin
kvadrat maxraj bilan
kabi og'ir kvark samarali nazariyasi (HQET).
Nosimmetrik shakl
Parametrlashning nosimmetrik shakli vaqti-vaqti bilan ishlatiladi, bu erda integral uning o'rniga intervalda bajariladi
, olib boradi:
![{ frac {1} {AB}} = 2 int _ {{- 1}} ^ {1} { frac {du} { left [(1 + u) A + (1-u) B right] ^ {2}}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/869668a62d68d9029027a8be5272a4ec432944ef)
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