(Tokyo Institute of Technology) Seminar at Ehime University ( ) 9 3 U(N C ), N F /2 BPS ( ) 12 5 (
|
|
- くにもと ねぎたや
- 5 years ago
- Views:
Transcription
1 (Tokyo Institute of Technology) Seminar at Ehime University ( ) 9 3 U(N C ), N F /2 BPS ( ) 12 5 ( ) 19 6 Conclusion 23
2 1 1.1 GeV SU(3) SU(2) U(1): W ±, Z 1. ( ) 2. W ±, Z 0 (1983), 3. LHC (2008 ) 2
3 トンネル周長 27km ( 参考 : 東京 JR 山手線の周長 34.5km) 4
4 ( ) 1. ( ) M GUT c GeV (1.1) 2. 3
5 ( ) M W c GeV M GUT c GeV ( ) M P c GeV M W 1. (Technocolor) TeV L. Susskind,Phys. Rev.D20 (1979) 2619; S. Weinberg, Phys. Rev.D19 (1979) 1277; D13 (1976) 974; S. Dimopoulos, and L. Susskind,Nucl. Phys. B155 (1979) 237; 4
6 2. 0 : m B m B = m F 1 2 : m F m F = 0 S.Dimopoulos, H.Georgi, Nucl.Phys.B193 (1981) 150; N.Sakai, Z.f.Phys.C11 (1981) 153; E.Witten, Nucl.Phys.B188 (1981) 513; (a) Lightest Supersymmetric Particle R (b) ( 1) (c) (h, H, A) (H +, H ) ( h ) (d) 150GeV 5
7 1: ( ) ( ) α i = gi 2 /4π, (i = 1, 2, 3) U(1), SU(2), SU(3) : (sequestering) 3. (Brane World)= 4 ( ) 6
8 y: 4 n 2πR V 4+n (r) = 1 m 1 m 2 M 2+n r 1 m 1 m 2 (n) 1+n M 2+n, for r R (1.2) (n) Rn r M 2+n (n) Rn = M P M (n) M P GeV P.Horava and E.Witten, Nucl.Phys.B475, 94 (1996); N.Arkani-Hamed, SDimopoulos, G.Dvali, Phys.Lett.B429 (1998) 263 ; I.Antoniadis, N.Arkani-Hamed, S.Dimopoulos, G.Dvali, Phys.Lett.B436 (1998) 257; Randall, Sundrum, Phys.Rev.Lett.83 (1999) 3370; 4690; = :,,... 7
9 LHC ( ) Landau-Ginzburg : U(1) φ L = 1 4e 2F µνf µν + D µ φ(d µ φ) λ ( φφ v 2) 2 (1.3) 4 : ( ) k = 1 d 2 x F 12 (1.4) 2π : = U(1) :, : λ < e 2 : (, ) λ > e 2 : 8
10 λ = e 2 : BPS BPS (SUSY) 2 ( ) ( ) : ( ) π 0 (M) φ (λ > 0) L = µ φ µ φ λ(φ 2 v 2 ) 2 (2.1) 9
11 : φ + v, φ v y = x 2 E = ( y φ) 2 + λ(φ 2 v 2 ) 2 = ( y φ + λ(φ 2 v 2 )) 2 + y [2 )] λ (v 2 φ φ3 3 [ dye 2 )] λ (v 2 φ φ3 3 Bogomol nyi-prasad-sommerfield (BPS) (2.2) Bogomol nyi, Sov.J.Nucl.Phys. 24 (1976) 449; Prasad and Sommerfield, Phys.Rev.Lett. 35 (1975) 760. BPS y φ + λ(φ 2 v 2 ) = 0 (2.3) : φ = v tanh( λvy) (2.4) References 10
12 3 U(N C ), N F L = 1 2g 2Tr(F MN(W )F MN (W )) + 1 g 2Tr(DM ΣD M Σ) +Tr [ D M H(D M H) ] V (3.1) V = g2 [ (HH 4 Tr ) 2 ] c1 NC + Tr [ (ΣH HM)(ΣH HM) ] D M H i = ( M + iw M )H i, D M Σ = M Σ + i[w M, Σ] F MN (W ) = M W N N W M + i[w M, W N ] W M, Σ (N C N C ) U(N C ) g : H ra H ra (N C N F matrix) (i = 1, 2 ; r = 1,, N C ; A = 1,, N F ), (M) A B m A δ A B : m A > m A+1 : U(1) N F 1 F : Σ 11
13 5 M, N, = 0, 1, 2, 3, 4 (8 SUSY) : A 1 A 2 A NC HH = c1 NC, ΣH HM = 0 (3.2) H ra = c δ A r A, Σ = diag(m A1,, m ANC ) (3.3) N : F! (N F N C )!N C! en F log(x x (1 x) (1 x)), x N C /N F : ( ),,, 4 1/2 BPS ( ) 1/2 BPS y x 4, 4 D W M y = 0 12
14 2: A 1 A 2 A NC B 1 B 2 B NC. E [ E = Tr D y H ra 2] + Tr [ ΣH HM 2] + 1 ( g 2Tr (D y Σ) 2) + g2 [ (HH 4 Tr ) 2 ] c1 NC = Tr D y H + ΣH HM ( g 2Tr D y Σ g2 ( c1nc HH )) 2 + c y TrΣ (4.1) 2 1/2 BPS : γ 4 ε i = i(σ 3 ) i jε j D y H = ΣH + HM, D y Σ = g 2 ( c1 NC HH ) /2 (4.2) 13
15 labeled by A 1 A 2 A NC B 1 B 2 B NC BPS + [ ] + Edy c Tr(Σ) = c N C m Ak k=1 N C k=1 m Bk (4.3) BPS Σ + iw y S 1 (y) y S(y) S(y) GL(N C, C) BPS : H(y) = S 1 (y)h 0 e My H 0 N C N F BPS : Ω SS ( y Ω 1 y Ω ) = g 2 c ( ) 1 C Ω 1 Ω 0, Ω0 c 1 H 0 e 2My H 0 H 0 Ω(y) S(y) Σ, W y, H 1 y = ±, U(1) H 0 14
16 V - : (S NC 2 ) (S, H 0 ) (S, H 0 ) H = S 1 H 0 e My (Σ, W y ) S S = V S, H 0 H 0 = V H 0, V GL(N C, C) 3: A B C ( ) A,B ( ) BPS : M = {H 0 H 0 V H 0, V GL(N C, C)} G NF,N C SU(N F ) SU(N C ) SU(N F N C ) U(1) (4.4) 15
17 ( ) N C Ñ C N C (N F N C ) : 1,, N C Ñ C + 1,, N F dim R M 1,,N C Ñ C +1,,N F N F,N C = 2N wall = 2N C Ñ C (4.5) M = M 1/1 + M 1/2 = M 0 M 1 M N CÑC (4.6) H 0 : y 4:. U(1) : H 0 = (e r 1, e r 2,, e r N F ), H = S 1 H 0 e My = S 1 (e r 1+m 1 y,, e r N F +m NF y ) (4.7) i i + 1 Rer i + m i y Rer i+1 + m i+1 y Im(r i r i+1 ) : 16
18 U(N C ) : BPS Ñ C N F N C (T w : ) L = T w + d 4 θk(φ, φ ) + (4.8) [ K(φ, φ ) = dy c log detω + ctr ( Ω 0 Ω 1) + 1 ( 2g 2Tr Ω 1 y Ω ) ] 2 K (g ) : g 2 c/ m 1: Ω = Ω 0 c 1 H 0 e 2My H 0 (4.9) g 2 : (NLSM) 1/2 BPS N F = N C = N [ L 6 = Tr 1 2g 2F MNF MN + D M H(D M H) ] g2 4 Tr[( ) HH 2 ] c1 NC 17 Ω=Ω sol
19 BPS BPS 0 = D 1 H + id 2 H, 0 = F 12 + g2 2 (c1 N HH ) (4.10) H = S 1 H 0 (z), W 1 + iw 2 = i2s 1 z S, z x 1 + ix 2 (4.11) z (Ω 1 z Ω) = g2 4 (c1 N Ω 1 H 0 H 0 ) (4.12) k Z 0, (det(h 0 ) z k, z ) T c d 2 x TrF 12 = 2πck = i c dz log(deth 0 ) + c.c. 2 V - : H 0 V H 0, S V S, V = V (z) GL(N, C), det V = const. 0 : dim(m N,k ) = 2kN ( ) 1 N 1 R(z) k H 0 =, P (z) = (z z i ) (4.13) 0 P (z) i=1 18
20 z i C, R(z) k (Cosmic string) 5 ( ) : Tong, Phys.Rev.D (2004); Auzzi-Bolognesi-Evslin-Konishi, Nucl.Phys.B (2004); Shifman-Yung, Phys.Rev.D (2004); Auzzi-Bolognesi-Evslin, JHEP 0502 (2005) 046; 1/2 γ 123 ε i = ε i x 3 1/2SUSY : γ 12 (iσ 3 ) i jε j = ε i ( + ): 1/4 SUSY γ 3 (iσ 3 ) i jε j = ε i x 3 1/4 BPS D 3 Σ = g 2 ( c1 NC H 1 H 1 ) /2+F 12, D 3 H 1 = ΣH 1 + HM, 0 = D 1 H 1 + id 2 H 1, 0 = F 23 D 1 Σ, 0 = F 31 D 2 Σ 19
21 5: ( ) ( ) BPS E t w + t v + t m + m J m t w, t v, t m t w = c 3 Tr(Σ), t v = ctr(f 12 ), t m = 2 g 2 mtr( 1 2 ϵmnl F nl Σ) : [D 1 + id 2, D 3 + Σ] = 0 ( ) S(x m ) GL(N C, C) (D 3 + Σ)S 1 = 0 Σ + iw 3 S 1 3 S (5.1) (D 1 + id 2 )S 1 = 0 W 1 + iw 2 2iS 1 S (5.2) z x 1 + ix 2, and / z. BPS H 1 = S 1 (z, z, x 3 )H 0 (z)e Mx3 20
22 H 0 (z): z N C N F Ω SS (Ω 0 H 0 e 2My H 0 ) 4 (Ω 1 Ω) + 3 (Ω 1 3 Ω) = g 2 ( c Ω 1 Ω 0 ) (5.3) x x x 1 0 x x 3 a) x 3 0 b) : (t w + t v = 0.5c) a) : H 0 (z)e Mx3 = c(e x3, ze 4, e x3 ). z b) : H 0 (z)e Mx3 = c((z 4 2i)(z i)e 3/2x3, (z + 8 i)(z 7 + 6i)e 1/2x3 +15/2, z 2 e 1/2x3 +15/2, (z 6 5i)(z + 6 7i)e 3/2x3 ). (g 2 ) : U(1) (N C = 1) 21
23 H 0 (z) = c ( f 1 (z),..., f N F(z) ) : Ω = N F A=1 f A (z) 2 e 2m Ax 3 A x 3 A (z) = log f A+1(z) log f A (z) m A m A+1 f A (z) : f A (z) (z z A α )ka α : A z = z A α k A α x y 7: /4BPS 1/4BPS 22
24 6 Conclusion ( )U(N C ) N F BPS 3. ( ) H 0 M NF,N C {H 0 H 0 V H 0, V GL(N C, C)} SU(N F ) G NF,N C SU(N C ) SU(Ñ C ) U(1) (6.1) 4. (g 2 ) ( ) 5. BPS 6. 1/4 BPS ( ) 7. 1/2, 1/4 BPS 23
25 References of Solitons in 8 SUSY Theories Tokyo Tech Collaboration 1. Review Solitons in the Higgs Phase: Moduli matrix approach, hep-th/ , J. Phys. A 39 (2006) R , 2. Domain Walls in 5D Supersymmetric Theories Moduli space of BPS walls in supersymmetric gauge theories, hep-th/ , Commun. Math. Phys. 267 (2006) Global structure of moduli space for BPS walls, hep-th/ , Phys.Rev.D71 (2005) , D-brane Construction for Non-Abelian Walls, hep-th/ , Phys.Rev.D71, (2005), Non-Abelian Walls in Supersymmetric Gauge Theories, hep-th/ , Phys.Rev.D70 (2004) , Construction of Non-Abelian Walls and Their Complete Moduli Space, hep-th/ , Phys.Rev.Lett.93 (2004) , Exact Wall Solutions in 5-Dimensional SUSY QED, 24
26 hep-th/ , JHEP 11 (2003) 060, Massless Localized Vector Field on a Wall in Five Dimensions, hep-th/ , JHEP 11 (2003) 061, Vacua of Massive Hyper-Kähler Sigma Models with Non-Abelian Quotient, hep-th/ , Prog. Theor. Phys. 113 (2005) 657, Manifest Supersymmetry for BPS walls in N = 2 nonlinear sigma models,, hep-th/ , Nucl.Phys.B652 (2003) 35-71, BPS Wall in N = 2 SUSY Nonlinear Sigma Model with Eguchi-Hanson Manifold, hep-th/ , in Garden of Quanta - In honor of Hiroshi Ezawa, pages , 3. Vortex Statistical Mechanics of Vortices from D-branes and T-duality, hep-th/ , Phys.Rev.to appear Non-Abelian Vortices on Cylinder Duality between vortices and walls, hep-th/ , Phys.Rev.D73 (2006) , Moduli Space of Non-Abelian Vortices, hep-th/ , Phys.Rev.Lett.96 (2006) , Effective Theory on Non-Abelian Vortices in Six Dimensions, hep-th/ , Nucl.Phys.B701, 247 (2004), 4. 1/4 BPS states 25
27 Dynamics of Domain Wall Networks, arxiv: [hep-th], Phys.Rev.to appear. Effective Action of Domain Wall Networks, Phys.Rev.D75 (2007) , hepth/ , Non-Abelian Webs of Walls, Phys.Lett. B632 (2006) , hep-th/ , Webs of Domain Walls in Supersymmetric Gauge Theories, hep-th/ , Phys.Rev.D72 (2005) , Monopoles, Vortices, Domain Walls and D-branes: The rules of Interaction, hep-th/ , JHEP 03 (2005) 019, Instantons in the Higgs Phase, hep-th/ , Phys. Rev. D72, (2005), All Exact Solutions of a 1/4 Bogomol nyi-prasad-sommerfield Equation, hep-th/ , Phys.Rev.D71, (2005), Domain Wall Junction in N = 2 Supersymmetric QED in four dimensions, hep-th/ , Phys.Rev.D68 (2003) , BPS Lumps and Their Intersections in N = 2 SUSY Nonlinear Sigma Models, hep-th/ , Grav.Cosmol.8 (2002) , 5. Non-BPS Walls and Supersymmetry Breaking Non-BPS Walls and their stability in 5D Supersymmetric Theory, hep-th/ , Nucl.Phys.B696 (2004) 3-35, 6. Effective Lagrangian 26
28 Manifestly Supersymmetric Effective Lagrangians on BPS Solitons, hep-th/ , Phys.Rev.D73 (2006) , 7. Wall Solution in Supergravity Bogomol nyi-prasad-sommerfield Multiwalls in Five-Dimensional Supergravity, hep-th/ , Phys.Rev.D69 (2004) , Wall solution with weak gravity limit in five-dimensional Supergravity, hep-th/ , Phys.Lett.B556 (2003) , Solitons in 8 SUSY Theories 1. Review D. Tong, TASI lectures on solitons, arxiv:hep-th/ M. Shifman and A. Yung, Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories, arxiv:hep-th/ Early works M. Cvetic, F. Quevedo and S. J. Rey, Phys.Rev.Lett.67, 1836 (1991) E. Abraham and P. K. Townsend, Phys.Lett.B 291, 85 (1992) M. Cvetic, S. Griffies and S. J. Rey, Nucl.Phys.B 381, 301 (1992) M. Cvetic, S. Griffies and H. H. Soleng, Phys.Rev.D 48, 2613 (1993) 27
29 G. Dvali and M. Shifman, Phys.Lett.B396, 64 (1997) 3. Wall Solution in 8 SUSY Models J. P. Gauntlett, D. Tong, and P. K. Townsend, Phys.Rev.D63, (2001); J. P. Gauntlett, R. Portugues, D. Tong, P. K. Townsend, Phys.Rev.D63, (2001); J. P. Gauntlett, D. Tong, and P. K. Townsend, Phys.Rev.D64, (2001) R. Portugues, P. K. Townsend, JHEP , (2002) D. Tong, Phys.Rev.D66, (2002) The Moduli Space of BPS Domain Walls M. Shifman and A. Yung, Phys.Rev.D67, (2003) D. Tong, JHEP 0304, 031 (2003) Mirror Mirror on the Wall M. Arai, E. Ivanov and J. Niederle, Nucl.Phys.B680, 23 (2004) M. Shifman and A. Yung, Phys.Rev.D70, (2004) 4. Monopoles in Higgs Phase D. Tong, Phys.Rev.D69, (2004) R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and A. Yung, Nucl.Phys.B673, 187 (2003) R. Auzzi, S. Bolognesi, J. Evslin and K. Konishi, Nucl.Phys.B 686, 119 (2004) M. Shifman and A. Yung, Phys.Rev.D 70, (2004) R. Auzzi, M. Shifman and A. Yung, JHEP 0502 (2005)
30 5. Vortex A. Hanany and D. Tong, JHEP 0307, 037 (2003) R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and A. Yung, Nucl.Phys.B 673, 187 (2003) M. Shifman and A. Yung, Phys.Rev.D 70, (2004) A. Hanany and D. Tong, JHEP 0404, 066 (2004) A. Hanany and D. Tong, arxiv:hep-th/ ; M. A. C. Kneipp and P. Brockill, Phys.Rev.D 64, (2001) M. A. C. Kneipp, Phys.Rev.D 68, (2003) ; Phys.Rev.D 69, (2004) 6. Brane construction R. Auzzi, S. Bolognesi and J. Evslin, JHEP 0502 (2005) Index theorem K. S. M. Lee, Phys.Rev.D 67, (2003) introduction 29
( ) : (Technocolor)...
( ) 2007.5.14 1 3 1.1............................. 3 1.2 :........... 5 1.3........................ 7 1.4................. 8 2 11 2.1 (Technocolor)................ 11 2.2............................. 12
More informationNorisuke Sakai (Tokyo Institute of Technology) In collaboration with M. Eto, T. Fujimori, Y. Isozumi, T. Nagashima, M. Nitta, K. Ohashi, K. Ohta, Y. T
Norisuke Sakai (Tokyo Institute of Technology) In collaboration with M. Eto, T. Fujimori, Y. Isozumi, T. Nagashima, M. Nitta, K. Ohashi, K. Ohta, Y. Tachikawa, D. Tong, M. Yamazaki, and Y. Yang 2008.3.21-26,
More informationKaluza-Klein(KK) SO(11) KK 1 2 1
Maskawa Institute, Kyoto Sangyo University Naoki Yamatsu 2016 4 12 ( ) @ Kaluza-Klein(KK) SO(11) KK 1 2 1 1. 2. 3. 4. 2 1. 標準理論 物質場 ( フェルミオン ) スカラー ゲージ場 クォーク ヒッグス u d s b ν c レプトン ν t ν e μ τ e μ τ e h
More informationSeiberg Witten 1994 N = 2 SU(2) Yang-Mills 1 1 3 2 5 2.1..................... 5 2.2.............. 8 2.3................................. 9 3 N = 2 Yang-Mills 11 3.1............................... 11 3.2
More informationSUSY DWs
@ 2013 1 25 Supersymmetric Domain Walls Eric A. Bergshoeff, Axel Kleinschmidt, and Fabio Riccioni Phys. Rev. D86 (2012) 085043 (arxiv:1206.5697) ( ) Contents 1 2 SUSY Domain Walls Wess-Zumino Embedding
More information3 exotica
( / ) 2013 2 23 embedding tensors (non)geometric fluxes exotic branes + D U-duality G 0 R-symmetry H dim(g 0 /H) T-duality 11 1 1 0 1 IIA R + 1 1 1 IIB SL(2, R) SO(2) 2 1 9 GL(2, R) SO(2) 3 SO(1, 1) 8
More information中央大学セミナー.ppt
String Gas Cosmology References Brandenberger & Vafa, Superstrings in the early universe, Nucl.Phys.B316(1988) 391. Tseytlin & Vafa, Elements of string cosmology, Nucl.Phys.B372 (1992) 443. Brandenberger,
More informationq quark L left-handed lepton. λ Gell-Mann SU(3), a = 8 σ Pauli, i =, 2, 3 U() T a T i 2 Ỹ = 60 traceless tr Ỹ 2 = 2 notation. 2 off-diagonal matrices
Grand Unification M.Dine, Supersymmetry And String Theory: Beyond the Standard Model 6 2009 2 24 by Standard Model Coupling constant θ-parameter 8 Charge quantization. hypercharge charge Gauge group. simple
More informationYITP50.dvi
1 70 80 90 50 2 3 3 84 first revolution 4 94 second revolution 5 6 2 1: 1 3 consistent 1-loop Feynman 1-loop Feynman loop loop loop Feynman 2 3 2: 1-loop Feynman loop 3 cycle 4 = 3: 4: 4 cycle loop Feynman
More informationD-brane K 1, 2 ( ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane
D-brane K 1, 2 E-mail: sugimoto@yukawa.kyoto-u.ac.jp (2004 12 16 ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane D-brane RR D-brane K D-brane K D-brane K K [2, 3]
More information1/2 ( ) 1 * 1 2/3 *2 up charm top -1/3 down strange bottom 6 (ν e, ν µ, ν τ ) -1 (e) (µ) (τ) 6 ( 2 ) 6 6 I II III u d ν e e c s ν µ µ t b ν τ τ (2a) (
August 26, 2005 1 1 1.1...................................... 1 1.2......................... 4 1.3....................... 5 1.4.............. 7 1.5.................... 8 1.6 GIM..........................
More informationTOP URL 1
TOP URL http://amonphys.web.fc.com/ 1 19 3 19.1................... 3 19.............................. 4 19.3............................... 6 19.4.............................. 8 19.5.............................
More informationQCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1
QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 (vierbein) QCD QCD 1 1: QCD QCD Γ ρ µν A µ R σ µνρ F µν g µν A µ Lagrangian gr TrFµν F µν No. Yes. Yes. No. No! Yes! [1] Nash & Sen [2] Riemann
More informationChern-Simons Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q
Chern-Simons E-mail: fuji@th.phys.nagoya-u.ac.jp Jones 3 Chern-Simons - Chern-Simons - Jones J(K; q) []Jones q J (K + ; q) qj (K ; q) = (q /2 q /2 )J (K 0 ; q), () J( ; q) =. (2) K Figure : K +, K, K 0
More informationTOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
More information·«¤ê¤³¤ß·²¤È¥ß¥ì¥Ë¥¢¥àÌäÂê
.. 1 10-11 Nov., 2016 1 email:keiichi.r.ito@gmail.com, ito@kurims.kyoto-u.ac.jp ( ) 10-11 Nov., 2016 1 / 45 Clay Institute.1 Construction of 4D YM Field Theory (Jaffe, Witten) Jaffe, Balaban (1980).2 Solution
More informationConfinement dual Meissener effect dual Meissener effect
BASED ON WORK WITH KENICHI KONISHI (UNIV. OF PISA) [0909.3781 TO APPEAR IN NPB] Confinement dual Meissener effect dual Meissener effect 1) Perturbed SU(N) Seiberg WiRen theory : 2) SU(N) with Flavors at
More information0. Intro ( K CohFT etc CohFT 5.IKKT 6.
E-mail: sako@math.keio.ac.jp 0. Intro ( K 1. 2. CohFT etc 3. 4. CohFT 5.IKKT 6. 1 µ, ν : d (x 0,x 1,,x d 1 ) t = x 0 ( t τ ) x i i, j, :, α, β, SO(D) ( x µ g µν x µ µ g µν x ν (1) g µν g µν vector x µ,y
More informationCKY CKY CKY 4 Kerr CKY
( ) 1. (I) Hidden Symmetry and Exact Solutions in Einstein Gravity Houri-Y.Y: Progress Supplement (2011) (II) Generalized Hidden Symmetries and Kerr-Sen Black Hole Houri-Kubiznak-Warnick-Y.Y: JHEP (2010)
More information1 (Contents) (1) Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji
8 4 2018 6 2018 6 7 1 (Contents) 1. 2 2. (1) 22 3. 31 1. Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji SETO 22 3. Editorial Comments Tadashi
More information20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................
More informationcm λ λ = h/p p ( ) λ = cm E pc [ev] 2.2 quark lepton u d c s t b e 1 3e electric charge e color charge red blue green qq
2007 2007 7 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 2007 2 4 5 6 6 2 2.1 1: KEK Web page 1 1 1 10 16 cm λ λ = h/p p ( ) λ = 10 16 cm E pc [ev] 2.2 quark lepton 2 2.2.1 u d c s t b + 2 3 e 1 3e electric charge
More information20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33
More information1. Introduction Palatini formalism vierbein e a µ spin connection ω ab µ Lgrav = e (R + Λ). 16πG R µνab µ ω νab ν ω µab ω µac ω νcb + ω νac ω µcb, e =
Chiral Fermion in AdS(dS) Gravity Fermions in (Anti) de Sitter Gravity in Four Dimensions, N.I, Takeshi Fukuyama, arxiv:0904.1936. Prog. Theor. Phys. 122 (2009) 339-353. 1. Introduction Palatini formalism
More information『共形場理論』
T (z) SL(2, C) T (z) SU(2) S 1 /Z 2 SU(2) (ŜU(2) k ŜU(2) 1)/ŜU(2) k+1 ŜU(2)/Û(1) G H N =1 N =1 N =1 N =1 N =2 N =2 N =2 N =2 ĉ>1 N =2 N =2 N =4 N =4 1 2 2 z=x 1 +ix 2 z f(z) f(z) 1 1 4 4 N =4 1 = = 1.3
More information(τ τ ) τ, σ ( ) w = τ iσ, w = τ + iσ (w ) w, w ( ) τ, σ τ = (w + w), σ = i (w w) w, w w = τ w τ + σ w σ = τ + i σ w = τ w τ + σ w σ = τ i σ g ab w, w
S = 4π dτ dσ gg ij i X µ j X ν η µν η µν g ij g ij = g ij = ( 0 0 ) τ, σ (+, +) τ τ = iτ ds ds = dτ + dσ ds = dτ + dσ δ ij ( ) a =, a = τ b = σ g ij δ ab g g ( +, +,... ) S = 4π S = 4π ( i) = i 4π dτ dσ
More information' , 24 :,,,,, ( ) Cech Index theorem 22 5 Stability 44 6 compact 49 7 Donaldson 58 8 Symplectic structure 63 9 Wall crossing 66 1
1998 1998 7 20 26, 44. 400,,., (KEK), ( ) ( )..,.,,,. 1998 1 '98 7 23, 24 :,,,,, ( ) 1 3 2 Cech 6 3 13 4 Index theorem 22 5 Stability 44 6 compact 49 7 Donaldson 58 8 Symplectic structure 63 9 Wall crossing
More informationSO(2)
TOP URL http://amonphys.web.fc2.com/ 1 12 3 12.1.................................. 3 12.2.......................... 4 12.3............................. 5 12.4 SO(2).................................. 6
More information反D中間子と核子のエキゾチックな 束縛状態と散乱状態の解析
.... D 1 in collaboration with 1, 2, 1 RCNP 1, KEK 2 . Exotic hadron qqq q q Θ + Λ(1405) etc. uudd s? KN quasi-bound state? . D(B)-N bound state { { D D0 ( cu) B = D ( cd), B = + ( bu) B 0 ( bd) D(B)-N
More information( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes )
( 3 7 4 ) 2 2 ) 8 2 954 2) 955 3) 5) J = σe 2 6) 955 7) 9) 955 Statistical-Mechanical Theory of Irreversible Processes 957 ) 3 4 2 A B H (t) = Ae iωt B(t) = B(ω)e iωt B(ω) = [ Φ R (ω) Φ R () ] iω Φ R (t)
More informationmain.dvi
SGC - 48 208X Y Z Z 2006 1930 β Z 2006! 1 2 3 Z 1930 SGC -12, 2001 5 6 http://www.saiensu.co.jp/support.htm http://www.shinshu-u.ac.jp/ haru/ xy.z :-P 3 4 2006 3 ii 1 1 1.1... 1 1.2 1930... 1 1.3 1930...
More informationIA hara@math.kyushu-u.ac.jp Last updated: January,......................................................................................................................................................................................
More information余剰次元のモデルとLHC
余剰次元のモデルと LHC 松本重貴 ( 東北大学 ) 1.TeraScale の物理と余剰次元のモデル.LHC における ( 各 ) 余剰次元モデル の典型的なシグナルについて TeraScale の物理と余剰次元のモデル Standard Model ほとんどの実験結果を説明可能な模型 でも問題点もある ( Hierarchy problem, neutrino mass, CKM matrix,
More informationx (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s
... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z
More informationZ: Q: R: C: sin 6 5 ζ a, b
Z: Q: R: C: 3 3 7 4 sin 6 5 ζ 9 6 6............................... 6............................... 6.3......................... 4 7 6 8 8 9 3 33 a, b a bc c b a a b 5 3 5 3 5 5 3 a a a a p > p p p, 3,
More informationEuler, Yang-Mills Clebsch variable Helicity ( Tosiaki Kori ) School of Sciences and Technology, Waseda Uiversity (i) Yang-Mills 3 A T (T A) Poisson Ha
Euler, Yang-ills Clebsch variable Helicity Tosiaki Kori ) School of Sciences and Technology, Waseda Uiversity i) Yang-ills 3 A T T A) Poisson Hamilton ii) Clebsch parametrization iii) Y- Y-iv) Euler,v)
More informationpptx
Based on N. Nagata, S. Shirai, JHEP 1403 (2014) 049. Ø Ø Y. Okada, M. Yamaguchi, T. Yanagida (1991), H. E. Haber, R. Hempfling (1991) J. R. Ellis, G. Ridolfi, F. Zwirner (1991) Scalar Par cles Gravi no
More informationIntroduction SFT Tachyon condensation in SFT SFT ( ) at 1 / 38
( ) 2011 5 14 at 1 / 38 Introduction? = String Field Theory = SFT 2 / 38 String Field : ϕ(x, t) x ϕ x / ( ) X ( σ) (string field): Φ[X(σ), t] X(σ) Φ (Φ X(σ) ) X(σ) & / 3 / 38 SFT with Lorentz & Gauge Invariance
More informationW u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)
3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)
More information,,..,. 1
016 9 3 6 0 016 1 0 1 10 1 1 17 1..,,..,. 1 1 c = h = G = ε 0 = 1. 1.1 L L T V 1.1. T, V. d dt L q i L q i = 0 1.. q i t L q i, q i, t L ϕ, ϕ, x µ x µ 1.3. ϕ x µ, L. S, L, L S = Ld 4 x 1.4 = Ld 3 xdt 1.5
More information211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
More informationIII 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F F 1 F 2 F, (3) F λ F λ F λ F. 3., A λ λ A λ. B λ λ
More information163 KdV KP Lax pair L, B L L L 1/2 W 1 LW = ( / x W t 1, t 2, t 3, ψ t n ψ/ t n = B nψ (KdV B n = L n/2 KP B n = L n KdV KP Lax W Lax τ KP L ψ τ τ Cha
63 KdV KP Lax pair L, B L L L / W LW / x W t, t, t 3, ψ t n / B nψ KdV B n L n/ KP B n L n KdV KP Lax W Lax τ KP L ψ τ τ Chapter 7 An Introduction to the Sato Theory Masayui OIKAWA, Faculty of Engneering,
More informationuntitled
2 : n =1, 2,, 10000 0.5125 0.51 0.5075 0.505 0.5025 0.5 0.4975 0.495 0 2000 4000 6000 8000 10000 2 weak law of large numbers 1. X 1,X 2,,X n 2. µ = E(X i ),i=1, 2,,n 3. σi 2 = V (X i ) σ 2,i=1, 2,,n ɛ>0
More information2 (March 13, 2010) N Λ a = i,j=1 x i ( d (a) i,j x j ), Λ h = N i,j=1 x i ( d (h) i,j x j ) B a B h B a = N i,j=1 ν i d (a) i,j, B h = x j N i,j=1 ν i
1. A. M. Turing [18] 60 Turing A. Gierer H. Meinhardt [1] : (GM) ) a t = D a a xx µa + ρ (c a2 h + ρ 0 (0 < x < l, t > 0) h t = D h h xx νh + c ρ a 2 (0 < x < l, t > 0) a x = h x = 0 (x = 0, l) a = a(x,
More informationX G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
More informationD = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j
6 6.. [, b] [, d] ij P ij ξ ij, η ij f Sf,, {P ij } Sf,, {P ij } k m i j m fξ ij, η ij i i j j i j i m i j k i i j j m i i j j k i i j j kb d {P ij } lim Sf,, {P ij} kb d f, k [, b] [, d] f, d kb d 6..
More informationDonaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib
( ) Donaldson Seiberg-Witten Witten Göttsche [GNY] L. Göttsche, H. Nakajima and K. Yoshioka, Donaldson = Seiberg-Witten from Mochizuki s formula and instanton counting, Publ. of RIMS, to appear Donaldson
More informationx V x x V x, x V x = x + = x +(x+x )=(x +x)+x = +x = x x = x x = x =x =(+)x =x +x = x +x x = x ( )x = x =x =(+( ))x =x +( )x = x +( )x ( )x = x x x R
V (I) () (4) (II) () (4) V K vector space V vector K scalor K C K R (I) x, y V x + y V () (x + y)+z = x +(y + z) (2) x + y = y + x (3) V x V x + = x (4) x V x + x = x V x x (II) x V, α K αx V () (α + β)x
More informationII No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
More informationΛ(1405) supported by Global Center of Excellence Program Nanoscience and Quantum Physics 2009, Aug. 5th 1
Λ(1405) supported by Global Center of Excellence Program Nanoscience and Quantum Physics 2009, Aug. 5th 1 S KN Λ(1405) Λ(1405) CDD Nc 2 L = q(i/ m)q P L = 1 2 (1 γ 5), P R = 1 2 (1 + γ 5), q L P L q, q
More informationmain.dvi
Ver. 1.50 2001 ( ) 1 4 2 Effective Theory 5 2.1 Effective theory... 5 2.2 massless 2-flavor QCD... 5 2.3..................... 9 2.4 Standard model... 10 2.5... 11 2.6... 13 3 Supersymmetry 15 3.1 Supersymmetry...
More informationPart () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
More informationIntroduction 2 / 43
Batalin-Vilkoviski ( ) 2016 2 22 at SFT16 based on arxiv:1511.04187 BV Analysis of Tachyon Fluctuation around Multi-brane Solutions in Cubic String Field Theory 1 / 43 Introduction 2 / 43 in Cubic open
More information橋本研
-Goldstone ( ) 2011 4-2012 3 -Goldstone UV IR SSB+ : Chiral Lagrangian :,... -Goldstone Nambu( 60), Goldstone(61), Nambu, Jona-Lasinio( 61), Goldstone, Salam, Weinberg( 62) Lorentz ( ) = NG 2002~2004?
More information1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
More information( ) ) AGD 2) 7) 1
( 9 5 6 ) ) AGD ) 7) S. ψ (r, t) ψ(r, t) (r, t) Ĥ ψ(r, t) = e iĥt/ħ ψ(r, )e iĥt/ħ ˆn(r, t) = ψ (r, t)ψ(r, t) () : ψ(r, t)ψ (r, t) ψ (r, t)ψ(r, t) = δ(r r ) () ψ(r, t)ψ(r, t) ψ(r, t)ψ(r, t) = (3) ψ (r,
More informationYang-Mills Yang-Mills Yang-Mills 50 operator formalism operator formalism 1 I The Dawning of Gauge T
Yang-Mills 50 E-mail: kugo@yukawa.kyoto-u.ac.jp 2004 Yang-Mills 50 2004 Yang-Mills 50 operator formalism operator formalism 1 I The Dawning of Gauge Theory O Raifeartaigh [1] I, II, III O Raifeartaigh
More informationN=1 N=1 QCD N=1 non-abelian QCD X 0
2 945207 18 9 22 N=1 N=1 QCD N=1 non-abelian QCD X 0 1 3 2 7 2.1..................................... 7 2.2.................................. 8 2.3............ 10 2.4 moduli...............................
More information量子力学 問題
3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,
More information“‡”�„³…u…›…b…N…z†[…‰
2009 8 31 / 4 : : Outline 4 G MN T t g x mu... g µν = G MN X M x µ X N x ν X G MN c.f., : X 5, X 6,... g µν : : g µν, A µ : R µν 1 2 Rg µν = 8πGT µν : G MN : R MN 1 2 RG MN = 0 ( ) Kaluza-Klein G MN =
More information. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n
003...............................3 Debye................. 3.4................ 3 3 3 3. Larmor Cyclotron... 3 3................ 4 3.3.......... 4 3.3............ 4 3.3...... 4 3.3.3............ 5 3.4.........
More informationsusy.dvi
1 Chapter 1 Why supper symmetry? 2 Chapter 2 Representaions of the supersymmetry algebra SUSY Q a d 3 xj 0 α J x µjµ = 0 µ SUSY ( {Q A α,q βb } = 2σ µ α β P µδ A B (2.1 {Q A α,q βb } = {Q αa,q βb } = 0
More information.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T
NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977
More informationI, II 1, A = A 4 : 6 = max{ A, } A A 10 10%
1 2006.4.17. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 1. 1. 2. 3. 4. 5. 2. ɛ-δ 1. ɛ-n
More informationA
A 2563 15 4 21 1 3 1.1................................................ 3 1.2............................................. 3 2 3 2.1......................................... 3 2.2............................................
More information( ),.,,., C A (2008, ). 1,, (M, g) (Riemannian symmetric space), : p M, s p : M M :.,.,.,, (, ).,, (M, g) p M, s p : M M p, : (1) p s p, (
( ),.,,., C A (2008, ). 1,,. 1.1. (M, g) (Riemannian symmetric space), : p M, s p : M M :.,.,.,, (, ).,,. 1.2. (M, g) p M, s p : M M p, : (1) p s p, (2) s 2 p = id ( id ), (3) s p ( )., p ( s p (p) = p),,
More information2017 II 1 Schwinger Yang-Mills 5. Higgs 1
2017 II 1 Schwinger 2 3 4. Yang-Mills 5. Higgs 1 1 Schwinger Schwinger φ 4 L J 1 2 µφ(x) µ φ(x) 1 2 m2 φ 2 (x) λφ 4 (x) + φ(x)j(x) (1.1) J(x) Schwinger source term) c J(x) x S φ d 4 xl J (1.2) φ(x) m 2
More information4/15 No.
4/15 No. 1 4/15 No. 4/15 No. 3 Particle of mass m moving in a potential V(r) V(r) m i ψ t = m ψ(r,t)+v(r)ψ(r,t) ψ(r,t) = ϕ(r)e iωt ψ(r,t) Wave function steady state m ϕ(r)+v(r)ϕ(r) = εϕ(r) Eigenvalue problem
More informationD 2009 A * 1 ( ) *1 ( ) 0 1 1 6 2 32 2.1............................................. 32 2.2.................................. 41 2.3...................................... 47 3 65 3.1..............................................
More informationl µ l µ l 0 (1, x r, y r, z r ) 1 r (1, x r, y r, z r ) l µ g µν η µν 2ml µ l ν 1 2m r 2mx r 2 2my r 2 2mz r 2 2mx r 2 1 2mx2 2mxy 2mxz 2my r 2mz 2 r
2 1 (7a)(7b) λ i( w w ) + [ w + w ] 1 + w w l 2 0 Re(γ) α (7a)(7b) 2 γ 0, ( w) 2 1, w 1 γ (1) l µ, λ j γ l 2 0 Re(γ) α, λ w + w i( w w ) 1 + w w γ γ 1 w 1 r [x2 + y 2 + z 2 ] 1/2 ( w) 2 x2 + y 2 + z 2
More information7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±
7 7. ( ) SU() SU() 9 ( MeV) p 98.8 π + π 0 n 99.57 9.57 97.4 497.70 δm m 0.4%.% 0.% 0.8% π 9.57 4.96 Σ + Σ 0 Σ 89.6 9.46 K + K 0 49.67 (7.) p p = αp + βn, n n = γp + δn (7.a) [ ] p ψ ψ = Uψ, U = n [ α
More information(1) (2) (3) (4) 1
8 3 4 3.................................... 3........................ 6.3 B [, ].......................... 8.4........................... 9........................................... 9.................................
More information²ÄÀÑʬΥ»¶ÈóÀþ·¿¥·¥å¥ì¡¼¥Ç¥£¥ó¥¬¡¼ÊýÄø¼°¤ÎÁ²¶á²òÀÏ Asymptotic analysis for the integrable discrete nonlinear Schrödinger equation
Asymptotic analysis for the integrable discrete nonlinear Schrödinger equation ( ) ( ) 2016 12 17 1. Schrödinger focusing NLS iu t + u xx +2 u 2 u = 0 u(x, t) =2ηe 2iξx 4i(ξ2 η 2 )t+i(ψ 0 +π/2) sech(2ηx
More informationTeV b,c,τ KEK/ ) ICEPP
TeV b,c,τ KEK/ ) ICEPP 2 TeV TeV ~1930 ~1970 ~2010 LHC TeV LHC TeV LHC TeV CKM K FCNC K CP violation c b, τ B-B t B CP violation interplay 6 Super B Factory Super KEKB LoI (hep-ex/0406071) SLAC Super B
More informationmeiji_resume_1.PDF
β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
More information, 1.,,,.,., (Lin, 1955).,.,.,.,. f, 2,. main.tex 2011/08/13( )
81 4 2 4.1, 1.,,,.,., (Lin, 1955).,.,.,.,. f, 2,. 82 4.2. ζ t + V (ζ + βy) = 0 (4.2.1), V = 0 (4.2.2). (4.2.1), (3.3.66) R 1 Φ / Z, Γ., F 1 ( 3.2 ). 7,., ( )., (4.2.1) 500 hpa., 500 hpa (4.2.1) 1949,.,
More information( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) =
1 9 8 1 1 1 ; 1 11 16 C. H. Scholz, The Mechanics of Earthquakes and Faulting 1. 1.1 1.1.1 : - σ = σ t sin πr a λ dσ dr a = E a = π λ σ πr a t cos λ 1 r a/λ 1 cos 1 E: σ t = Eλ πa a λ E/π γ : λ/ 3 γ =
More informationI A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google
I4 - : April, 4 Version :. Kwhir, Tomoki TA (Kondo, Hirotk) Google http://www.mth.ngoy-u.c.jp/~kwhir/courses/4s-biseki.html pdf 4 4 4 4 8 e 5 5 9 etc. 5 6 6 6 9 n etc. 6 6 6 3 6 3 7 7 etc 7 4 7 7 8 5 59
More informationMilnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, P
Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, PC ( 4 5 )., 5, Milnor Milnor., ( 6 )., (I) Z modulo
More informationO x y z O ( O ) O (O ) 3 x y z O O x v t = t = 0 ( 1 ) O t = 0 c t r = ct P (x, y, z) r 2 = x 2 + y 2 + z 2 (t, x, y, z) (ct) 2 x 2 y 2 z 2 = 0
9 O y O ( O ) O (O ) 3 y O O v t = t = 0 ( ) O t = 0 t r = t P (, y, ) r = + y + (t,, y, ) (t) y = 0 () ( )O O t (t ) y = 0 () (t) y = (t ) y = 0 (3) O O v O O v O O O y y O O v P(, y,, t) t (, y,, t )
More information1 1, 2016 D B. 1.1,.,,. (1). (2). (3) Milnor., (1) (2)., (3). 1.2,.,, ( )..,.,,. 1.3, webpage,.,,.
1 1, 2016 D B. 1.1,.,,. (1). (2). (3) Milnor., (1) (2)., (3). 1.2,.,, ( )..,.,,. 1.3, 2015. webpage,.,,. 2 1 (1),, ( ). (2),,. (3),.,, : Hashinaga, T., Tamaru, H.: Three-dimensional solvsolitons and the
More informationb.dvi
, 0 1 2 1.1 [2, 3] : : : : : : : : : : : : : : : : 3 2, 6 2.1 : : : : : : : : : : : : : : : 7 2.2 : : : : : : : : : : : : : : : : : : : 15 2.3, : : : : 18 3 23 4 31 5 35 6 46 6.1 Borel. : : : : : : : :
More informationφ 4 Minimal subtraction scheme 2-loop ε 2008 (University of Tokyo) (Atsuo Kuniba) version 21/Apr/ Formulas Γ( n + ɛ) = ( 1)n (1 n! ɛ + ψ(n + 1)
φ 4 Minimal subtraction scheme 2-loop ε 28 University of Tokyo Atsuo Kuniba version 2/Apr/28 Formulas Γ n + ɛ = n n! ɛ + ψn + + Oɛ n =,, 2, ψn + = + 2 + + γ, 2 n ψ = γ =.5772... Euler const, log + ax x
More informationDecember 28, 2018
e-mail : kigami@i.kyoto-u.ac.jp December 28, 28 Contents 2............................. 3.2......................... 7.3..................... 9.4................ 4.5............. 2.6.... 22 2 36 2..........................
More information2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c
GL 2 1 Lie SL(2, R) GL(2, A) Gelbart [Ge] 1 3 [Ge] Jacquet-Langlands [JL] Bump [Bu] Borel([Bo]) ([Ko]) ([Mo]) [Mo] 2 2.1 H = {z C Im(z) > 0} Γ SL(2, Z) Γ N N Γ (N) = {γ SL(2, Z) γ = 1 2 mod N} g SL(2,
More information1 : ( ) ( ) ( ) ( ) ( ) etc (SCA)
START: 17th Symp. Auto. Decentr. Sys., Jan. 28, 2005 Symplectic cellular automata as a test-bed for research on the emergence of natural systems 1 : ( ) ( ) ( ) ( ) ( ) etc (SCA) 2 SCA 2.0 CA ( ) E.g.
More informationi
i 3 4 4 7 5 6 3 ( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n,
More informationChebyshev Schrödinger Heisenberg H = 1 2m p2 + V (x), m = 1, h = 1 1/36 1 V (x) = { 0 (0 < x < L) (otherwise) ψ n (x) = 2 L sin (n + 1)π x L, n = 0, 1, 2,... Feynman K (a, b; T ) = e i EnT/ h ψ n (a)ψ
More informationLebesgue Fubini L p Banach, Hilbert Höld
II (Analysis II) Lebesgue (Applications of Lebesgue Integral Theory) 1 (Seiji HIABA) 1 ( ),,, ( ) 1 1 1.1 1 Lebesgue........................ 1 1.2 2 Fubini...................... 2 2 L p 5 2.1 Banach, Hilbert..............................
More informationA = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B
9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A
More information30
3 ............................................2 2...........................................2....................................2.2...................................2.3..............................
More information橡超弦理論はブラックホールの謎を解けるか?
1999 3 (Can String Theory Solve the Puzzles of Black Holes?) 305-0801 1-1 makoto.natsuume@kek.jp D-brane 1 Schwarzschild 60 80 2 [1] 1 1 1 2 2 [2] 25 2.2 2 2.1 [7,8] Schwarzschild 2GM/c 2 Schwarzschild
More informationSAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T
SAMA- SUKU-RU Contents 1. 1 2. 7.1. p-adic families of Eisenstein series 3 2.1. modular form Hecke 3 2.2. Eisenstein 5 2.3. Eisenstein p 7 3. 7.2. The projection to the ordinary part 9 3.1. The ordinary
More information.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,
[ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b
More information( ) Note (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e
( ) Note 3 19 12 13 8 8.1 (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e µ τ, e R, µ R, τ R (1a) L ( ) ) * 3) W Z 1/2 ( - )
More informationnewmain.dvi
数論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published
More informationarxiv: v1(astro-ph.co)
arxiv:1311.0281v1(astro-ph.co) R µν 1 2 Rg µν + Λg µν = 8πG c 4 T µν Λ f(r) R f(r) Galileon φ(t) Massive Gravity etc... Action S = d 4 x g (L GG + L m ) L GG = K(φ,X) G 3 (φ,x)φ + G 4 (φ,x)r + G 4X (φ)
More informationEinstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x
7 7.1 7.1.1 Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x 3 )=(x 0, x )=(ct, x ) (7.3) E/c ct K = E mc 2 (7.4)
More informationN cos s s cos ψ e e e e 3 3 e e 3 e 3 e
3 3 5 5 5 3 3 7 5 33 5 33 9 5 8 > e > f U f U u u > u ue u e u ue u ue u e u e u u e u u e u N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 > A A > A E A f A A f A [ ] f A A e > > A e[ ] > f A E A < < f ; >
More information