접선벡터 좌표계를 사용한 열핵 전개

-44-

접선벡터 좌표계를 사용한 열핵 전개 II

이해원^*

충북대학교 물리학과, 청주 361-763 (2009년 12월 3일 받음)

d/2 개의 2계 타원 미분 연산자들의 곱으로 만들어진 d-계 타원연산자의 열핵(heat kernel) 전개를 연구하였다. 일반적인 차원 n을 갖는 굽어진 시공간에서 계산을 수행하였다. 열핵 전개의 처음 두 항들을 계산하였는데 그 결과는 각 2계 연산자의 경우에 얻어진 항들을 단순히 더한 것과 같지 않고 추가적인 항 들이 존재하게 된다. 이러한 계산을 위하여 좌표변환의 공변성을 유지하면서 접선벡터좌표계를 사용하는 방 법을 사용하였다.

핵심어: 열핵, 타원연산자, 접선벡터

_______________________________________________________________________________

Heat Kernel Expansions Using Tangent Vector Coordinate Systems II

Haewon LEE*

Department of Physics, Chungbuk National University, Cheongju 361-763 (Received 3 December 2009)

Heat kernel expansions of the d-th order elliptic operators obtained by using a product of d/2 second order elliptic operators in curved space-time with dimension n are investigated. We calculate the first two coefficient functions of the expansions, which are not the simple sum of the coefficients for each second order elliptic operator. For the calculations, we use a method with a tangent vector coordinate system, with which our calculatins are manifestly covariant under coordinate and gauge transformations.

Keywords: Heat Kernel, Elliptic Operator, Tangent Vector PACS numbers: 03.70.+k, 04.62.+v, 11.10.+z

* E-mail: hwlee@hep.chungbuk.ac.kr

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g_2r^{α1···αp;γ1···γq} ≡ g^α1α2g^α3α4· · · g^α2r−1α2rg^{α2r+1···γq} + g^α1α3g^α2α4· · · g^α2r−1α2rg^{α2r+1···γq}+ · · ·

¢

¸ôÇ S_2r^pq(i, j) H6£§õ °ú .

S_2r^pq(i, j) = i!

(i − p + r)!

2^p−r Cp+q

2 −r

×(−1)^i+j+q−p2 Γ(n+2i+2j+q−p

d )

(4π)^n2d₂ Γ(ⁿ₂) (26)

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Œl"f i ≥ p − r ≥ r s  âĺH 0_ °úכ`¦°úH.

S_2r^pq(i, j)\¦ iü< j \›'a> H†<Êú–Ð ‘:r€

X^α1···αp∂^γ1···γq ∼

[p/2]

X

r=0

S_2r^pqg_2r^{α1···αp;γ1···γq} (27)

ü

< °ú s jþt ú e”`¦ כ s. ôǼ# d” (26) \"f S<sup>pq</sup>2r = S<sub>2r−2s</sub><sup>(p−s)(q−s)</sup>e”`¦˜Ð{9 ú e”. ¢¸ôÇ ·ú¡Ü¼–ЍH S^pq≡ S^pq₀

–

Ð ³ðl½+É כ s.

d”

 (27)`¦6 x € >íߖõ #Q‹" J$™"f[þt_ ½+˓t p

o f”Œ•½+É Ãº e”. Óüt:ryŒ• J$™"f_ >úH S_2r^pq\ (i, j)

° ú

כ`¦ @/{9K ·ú˜ ú e”. >íߖ\ 1px©œ H#ŒQ €ªœ[þt

\

 @/K s\¦&h6 x #Œ ˜Ð€ 6£§õ °ú .

B ∼ −1 3S¹¹R G ∼ −2

3S⁰²R D ∼ −1

5(S²⁰+ S³¹)R_;β^β− 1

45(S³¹+ S²⁰)(R_αβR^αβ +3

2R_αβγδR^αβγδ) +1

4(S²⁰+ S₂²⁰)Ω_αβΩ^αβ B² ∼ 1

9S²²R²+2

9(S²²+ S¹¹)R_αβR^αβ GB ∼ 2

9S¹³(2R_αβR^αβ+ R²) +4

9S⁰²R_αβR^αβ BG ∼ 2

9S¹³(2R_αβR^αβ+ R²) G² ∼ 4

9S⁰⁴(2R_αβR^αβ+ R²) F_k ∼ 1

2(S²⁰+ S₂²⁰)C_k;β^β (28) 0

A_ d”[þt`¦6 x #Œ >ú a1, a2 Õªo“¦ a3 \¦½¨½+É Ã

º e”HX< õ\¦&ño € 6£§õ °ú .

a0 = Γ(ⁿ_d) (4π)^n2d₂ Γ(ⁿ₂) a1 = Γ(^n+d−2_d )

(4π)^n2d₂ Γ(ⁿ₂) X

k

( 2

3nR + Ck) a₂ = Γ(^n+d−4_d )

(4π)^n2d₂ Γ(ⁿ₂)[X

k

(n − 2)( 1

60R_;α^α+ 1 144R²

− 1

360R_αβR^αβ+ 1

360R_αβγδR^αβγδ+ 1

24Ω_αβΩ^αβ) +X

k

(1

6+(n − 4)(n + d)

48 )RCk+1 2

X

k⁰>k

[Ck⁰, Ck]

+1 2

X

k

C_k²+n − 4 2d (X

k

Ck)²

+X

k

(n − 2

6 −f (d, k) 12n )C_k;α^α] 0

A\"f f(d, k) = (d + 2)(d + 4 − 12k) + 24k² –Ð"f P

kf (d, k) = 0s.

IV. +sÇÂ]Ø

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 s âĺ\•¸ d = 2“ âĺ\H divergence †½Ó[þt`¦Ÿí

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[1] B. S. DeWitt, Dynamical Theory of Groups and Fields (Gordan and Breach, New York, 1965); Phys.

Rep. 19C, 295(1975).

[2] A. O. Barvinsky and G. A. Vilkovisky, Phys. Rep.

119, 1(1985); and references therein.

[3] Haewon Lee, Pong Youl Park and Hyun Kuk Shin, Phys. Rev. D 35, 2440 (1987).

[4] Haewon Lee and Sang Won Lee, Sae Mulli 58, 654 (2009).