N=1 N=1 QCD N=1 non-abelian QCD X 0

Size: px
Start display at page:

Download "N=1 N=1 QCD N=1 non-abelian QCD X 0"

Transcription

1 N=1 N=1 QCD N=1 non-abelian QCD X 0

2 moduli Witten thooft QCD < N c = N c = N c > N c N c N c < < 3N c N c N c QCD+ X electric

3 5.2 magnetic A 62 B Kähler 63 C Wess-Zumino 64 2

4 1 Higgs SU(3) SU(2) U(1) Higgs Higgs Higgs Higgs (naturalness) log log SU(5) SU(3) SU(2) U(1) Gev [1] 3

5 QCD Λ ) 1/N Schwinger-Dyson 1/N [2] QCD QCD 1994 Seiberg [3] Seiberg [7][4][5][8][9] N=2 Seiberg Witten [6][10] 0 thooft 4

6 N=1 non- Abelian [7] non-abelian non-abelian Seiberg N=1 non-abelian Wilson Wilson 5

7 thooft [11] thooft Seiberg N=1 QCD [3][7] SU(N c ) < N c = N c = N c + 1 N c N c magnetic 3 2 N c < < 3N c 3N c 3 2 N c < < 3N c non-abelian Kutasov Schwimmer QCD [12][13][14] N=1 QCD non-abelian QCD non-abelian non-abelian QCD N=1 non-abelian SO(n) Sp(n) [15][16][17] 6

8 2 2.1 Poincaré [18][19] Lie Weyl spinor Q A α, Q βb {Q A α, Q βb } = 2σ m α β P mδb A {Q A α, Q Ḃ } β = { Q A α, Q βb } = 0 [ ] Pm, Q A α = [ P m, Q ] αa = 0 (1) Weyl spinor [20] P m Weyl spinor (A,B) N N N=1 N=2 N Appendix (2.5) 7

9 [22] [21] [1] 2.2 P m i / x m Q, Q {x m } {θ α, θ α } 8

10 F (x, θ, θ) = f(x) + θϕ(x) + θ χ(x) + θθm(x) + θ θn(x) + θσ m θvm (x) +θθ θ λ(x) + θ θθψ(x) + θθ θ θd(x) (2) θ θ Lorentz F (x, θ, θ) Q, Q Q α = θ α iσm α α θ α (3) x m Q α = θ + α iθα σα m α (4) x m P m = +i / x m (1) f(x) ϕ(x) δ ξ F (x, θ, θ) (ξq + ξ Q)F = δ ξ f(x) + θδ ξ ϕ(x) + (5) δ ξ f(x) = (ξq+ ξ Q)f(x) ξ, ξ Q, Q (1) P m = i x m (1) (2) 9

11 2.3 D α = θ + α iσm α α θ α (6) x m D α = θ α iθα σα m α (7) x m D α Φ = 0 (8) y m = x m + iθσ m θ Φ(y, θ, θ) = A(y) + 2θΨ(y) + θθf (y) (9) D α y m = D α θ β = 0 (10) D D A Weyl Ψ F D, D Q, Q (5) δ ξ A = 2ξΨ (11) δ ξ Ψ = i 2σ m ξ m A + 2ξF (12) δ ξ F = i 2 ξ σ m m Ψ (13) n Φ i L = d 2 θd 2 θk(φ [ i, Φ j ) + ] d 2 θw (Φ i ) + h.c. (14) h.c. d 2 θ θθ = 1 K, W K Kähler 10

12 W Appendix K = Φ i Φ i,w Wess-Zumino [23] Yukawa Wess-Zumino V = V (15) V = V V (x, θ, θ) = C(x) + iθχ(x) i θ χ(x) + i 2 θθ [M(x) + in(x)] i 2 θ θ [M(x) in(x)] ] θσ m θvm i (x) + iθθ θ [ λ(x) + 2 σm m χ(x) [ i θ θθ λ(x) + i ] 2 σm m χ(x) + 1 [D(x) 2 θθ θ θ + 1 ] 2 C(x) (16) v m (v m V a V = T a V a Λ a e V = e iλ e V e iλ (17) (T a ) ij Λ = T a Λ a T a Hausdorff V = V Hausdorff Λ V = V + i(λ Λ ) + (18) 11

13 QED Coulomb (16) C, χ, M, N λ, v m, D Wess-Zumino Wess-Zumino Wess-Zumino Appendix v m λ 1/2 Weyl D (TΦ) a ij (Φ) i = Φ i Φ = e iλ Φ (19) Φ W α = 1 4 D De V D α e V (20) W W α = e iλ W α e iλ (21) L = 1 16kg 2 Tr ( ) d 2 θ W α W α + h.c. + d 2 θd 2 θφ e V Φ (22) Tr(T a T b ) = kδ ab V 2gV L = 1 4 va mnv a mn Da D a D m A D m A i Ψ σ m D m Ψ + F F +i 2g ( A T a ΦΨλ a λ a T a ΦA Ψ ) + gd a A T a ΦA (23) [24][25] D m A = m A + igv a mt a ΦA (24) 12

14 D m Ψ = m Ψ + igvmt a ΦΨ a (25) D m λ a = m λ a gt abc vmλ b c (26) vmn a = m vn a n vm a gt abc vmv b n c (27) [T a, T b ] = it abc T c (28) Yukawa Kähler [26] L F = d 2 θ W (Φ i ) + h.c. (29) W Φ i F 2.4 moduli Nambu-Goldstone Higgs flat directions 13

15 moduli moduli moduli flat directions, moduli moduli moduli moduli D,F V (A) = 1 D a2 + F 2g 2 i 2 (30) a i where D a = (A i ) l (Ti a ) l k (A i ) k (31) i,k,l F i = W (A) A i (32) D a F i T a i i W(A) D a = 0 for all a (33) F i = 0 for all i (34) (35) D-flatness flat directions 14

16 flat directions U(1) 1-1 Q Q D = Q Q Q Q = 0 (36) flat directions < Q > = < Q > = a (37) < Q >=< Q >= a (38) a 0 Higgs ( Higgs Weyl Dirac Higgs Higgs U(1) a a 2.5 Witten spinor Majorana Q i = Q i (39) i=1,2,3,4 H = 1 4 (Q i ) 2 (40) 4 i=1 15

17 < 0 H 0 > 0 (41) H H = 0 H > 0 H Q Q 0 >= 0 N > 1 H = 1 4 i1 ) 4 i=1(q 2 = (Q i2 ) 2 = (42) i=1 Q i1 H > 0 N=0 H > 0 Witten [27] 16

18 Witten Tr( 1) F (43) F Tr ( 1) F = exp(2πij z ) (44) J z z 90 Tr Majorana H = Q 2 1 = Q 2 2 = Q 2 3 = Q 2 4 Q i Q j + Q j Q i = 0, for i j (45) Q = Q 1 Q b > = H f > Q f > = H b > (46) ( 1) F Tr ( 1) F Witten Witten Tr( 1) F = n H=0 B n H=0 F (47) 17

19 n H=0 B (n H=0 F ) Witten Tr( 1) F 0 (48) (47) Witten Tr( 1) F 0 n H=0 B nh=0 F 0 Witten Witten Witten Euler [27] Witten dual Coxeter [27] SU(N) N Witten Witten Witten 18

20 3 3.1 BCS Ginzburg-Landau QCD QCD QCD 3.2 d 2 θ 19

21 Yukawa Λ [28] Wilson log Legendre β [29] Wilson β [30] Wilson Wess-Zumino W tree = mϕ 2 + gϕ 3 (49) 20

22 ϕ m g Yukawa U(1) U(1) U(1) R ϕ 1 1 m 2 0 g 3 1 U(1) U(1) R (θ, θ) (-1,1) d 2 θw tree θ W eff = mϕ 2 f ( ) gϕ m (50) gϕ m f g/m g 0, m 0 g,m f(t) = 1 + t (51) W tree = mϕ 2 + gϕ 3 = W eff (52) [31] Kähler Wess- Zumino Wess-Zumino 21

23 3.3 thooft Moduli thooft [11] Abel spectator Abel spectator Abel spectator spectator thooft thooft QCD 2 22

24 QCD 23

25 4 QCD N=1 QCD QCD SU(N c ) V a (a = 1, 2,, N c 2 1) SU(N c ) Q i r, Q s j (i, j = 1, 2,, ; r, s = 1, 2,, N c ) QCD D QCD 0 pure Yang-Mills QCD QCD Higgs [32] β β(g) = g3 16π 2 3N c + γ(g 2 ) 1 N c g 2 8π 2 γ(g 2 ) = g2 8π 2 N 2 c 1 N c + O(g 2 ) (53) [36] γ(g 2 ) 1-loop g 3 3N c 3N c < SU( ) L SU( ) R U(1) A U(1) B U(1) R (54) 0 QCD left-hand right-hand SU( ) U(1) 24

26 U(1) R R U(1) R W α (θ) e λ W α (e λ θ) Nc Q j λ N r(θ) e f Q j r (e λ θ) Nc Q s N j(θ) e λ Q s f j(e λ θ) (55) Q, Q U(1) A U(1) B ( Q, 1, 1, 1, N ) f N c ( Q 1,, 1, 1, N ) f N c (56) U(1) R (-1) U(1) R (+1) U(1) R U(1) A Λ SU( ) L SU( ) R U(1) B U(1) R (57) N c = 2 Q Q N c 2 N c = 2 QCD moduli U(1) D a = (Q ) r i (T a ) i jq j r ( Q ) i s (T a ) j i Q s j = 0 (58) 25

27 [33] a 1 Q = Q a 2 = a Nf (59) < N c a i Q = a 1 a 2 a Nc (60) Q = ã 1 ã 2 ã Nc (61) a i 2 ã i 2 i (62) N c a i Q, Q moduli M i j = Q i Q j (63) B i 1...i N c = Q i1 Q i Nc B i1...i N c = Q i1 Q in c (64) 26

28 i, j SU(N c ) SU(N c ) ϵ i 1...i Nc ϵi1...i N Bose c < N c Bose N c 4.1 < N c W eff = C Nc, ( Λ 3N c det QQ ) 1 Nc (65) C Nc, subtraction scheme 0 < N c determinant 0 flat directions (59) a Nf Higgs SU(N c ) Nf SU(N c 1) Nf 1 SU(N c ) Nf N c SU(N c 1) Λ L SU(N c ) Λ a Nf 27

29 Λ 3(N c 1) ( 1) L = Λ3N c a 2 (66) DR scheme subtraction scheme DR dimensional reduction Siegel[48] (65) a Nf C Nc, = C Nc SU(N c 1) Nf 1 (65) W tree = mm Q Nf, Q Nf SU(N c ) Nf 1 Λ 3N c ( 1) L = mλ 3N c (67) W exact = ( Λ 3N c det M ) 1 Nc ( ) f t = mm Λ 3N c 1 Nc (68) det M m f(t) = C Nc, + mm t W exact = C Nc, ( Λ 3N c det QQ ) 1 Nc + mm (69) m C Nc, 1 = (N c + 1) ( CNc, N c ) Nc Nc +1 (70) C Nc, = C Nc C Nc, = (N c ) const. = N c 1 (65) 1-instanton = N c 1 3Nc Nc N Λ f = Λ 3Nc (71) 28

30 Higgs instanton N c = 2, = 1 instanton DR scheme W eff = (N c ) ( Λ 3N c det QQ ) 1 Nc (72) [34] = N c 1 instanton < N c 1 (72) 0 well-defind [33] W tree = T r mm W full = W eff +W tree M < Mj i >= ( ) det mλ 3N 1 ( c Nc 1 i m) j (73) N c N c, N c SU(N c ) pure Yang-Mills Witten N c 4.2 = N c M i j B = 1 N c! ϵ i 1,i 2,...,i Nf Q i 1 1 Q i 2 2 Q i 29

31 B = 1 N c! ϵi 1,i 2,...,i Nf Q 1 i 1 Q 2 i 2 Q i Nf (74) moduli M i j B, B det M BB = 0 (75) M j i B, B moduli, xy = 0 x = 0 y = 0 rank(m) N c 2 (76) SU(N c rank(m)) 0 moduli 0 W tree = Tr mm QCD, N c < N c < Mj i >= ( ) det mλ 3Nc N 1 ( f Nc 1 i m) j (77) instanton [2] < BB >= 0 (78) [2] 0 M i j 0 QCD = N c m det < M >= Λ 2N c (79) moduli moduli M i j B, B det < M > < BB >= Λ 2Nc (80) 30

32 [3] 0 1-instanton moduli xy = Λ moduli Λ moduli moduli Higgs SU( ) L SU( ) R U(1) B U(1) R < M >=< B >=< B >= 0 moduli moduli M i j = Λδ i j B = B = 0 (81) SU( ) L SU( ) R U(1) B U(1) R SU( ) V U(1) B U(1) R (82) U(1) R thooft Q ( ) 1, 1 31

33 Q ( ) 1, 1 W (1) 0,1 (N 2 f 1) (83) Q, Q, W ( ) 1, 1 SU( ) V U(1) B U(1) R (-1) M j i B, B (81) (80) TrM BB M (Nf 2 1) 0, 1 1 B (1) Nf, 1 1 B (1) Nf, 1 1 (84) thooft SU( ) 2 V U(1) R d (2) ( ) d (2) ( ) = d (2) (Nf 2 1) U(1) 3 R 2Nf 2 ( 1) 3 + (Nf 2 1) = (Nf 2 1)( 1) 3 2 U(1) 2 BU(1) R 2Nf 2 = 2Nf 2 U(1) R 2Nf 2 + (Nf 2 1) = (Nf 2 1) 2 (85) d (2) (r) SU( ) V r Casimir d (2) ( ) = d (2) ( ) = 1, 2 d(2) (Nf 2 1) = (80) moduli Mj i = 0 B = B = Λ Nc (86) 32

34 SU( ) L SU( ) R U(1) B U(1) R SU( ) L SU( ) R U(1) R (87) thooft (80) Lagrangian Lagrange A W = A(detM BB Λ 2N c ) (88) Lagrange A < N c (72) pure Yang-Mills 4.3 = N c + 1 = N c + 1 = N c M i j B i = ϵ i j1 j 2 j N c Qj 1 Q j2 Q j Nc B i = ϵ i j 1j 2 j N c Qj1 Q j2 Q jn c (89) moduli 1, 2,, N c ( ) 1 j det M B B j i = 0 M i M i jb i = M i j B j = 0 (90) 33

35 ( ) 1 j det M B B j i = Λ 2Nc 1 m j i (91) M i (77) 0 0 moduli moduli = N c = N c + 1 m moduli moduli moduli 0 Higgs 0 0 M = B = B = 0 (92) SU( ) L SU( ) R U(1) B U(1) R (93) 0 thooft Q Q (, 1) 1, Nc (1, ) 1, Nc N c N c W (1, 1) 0,1 (N 2 c 1) (94) 34

36 Q, Q, W (, 1) 1, Nc SU( ) L SU( ) R 1 U(1) B U(1) R ( Nc ) M i j B i, B j M thooft (, ) 0,1 2Nc B (, 1) Nf 1, B (1, ) Nf +1, 1 1 (95) SU( ) 3 ( 1) d (3) ( ) SU( ) 2 U(1) R ( 1) 2 d (2) ( ) U(1) 3 R Nf Nf 2 U(1) 2 BU(1) R 2( 1) 2 SU( ) 2 U(1) B ( 1) d (2) ( ) U(1) 2 RU(1) B 0 U(1) R N 2 f (96) SU( ) SU( ) L SU( ) R d (3) ( ) SU( ) 3 Casimir d (3) ( ) = d (3) ( ) M = B = B = 0 35

37 (90) M = B = B = 0 0 M = B = B = 0 < N c + 1 QCD W eff = 1 Λ 2N c 1 ( M i j B i B j det M ) (97) Λ Q, Q = N c + 1 QCD M = B = B = 0 [3] Higgs QCD Higgs 4.4 > N c + 1 > N c + 1 = N c + 1 moduli moduli 0 thooft 36

38 > N c N c, N c QCD screening Coulomb non-abelian free electric [7] N c < < 3N c, N c QCD QCD Λ QCD [35] QCD β β(g) = g3 16π 2 3N c + γ(g 2 ) 1 N c g 2 8π 2 37

39 γ(g 2 ) = g2 8π 2 N 2 c 1 N c + O(g 2 ) (98) γ(g 2 ) β 1-loop 2-loop > 3N 3 c 2N 2 c 1 (99) β 0 g g = 0 N c g 2 N c N c, = 3 ϵ N c g 2 = 8π2 3 ϵ + O(ϵ2 ) (100) β 0 N c ϵ = 3 N c N c < < 3N c, N c [7] 3 2 N c < < 3N c 0 electric Abelian Coulomb V 1 R (101) non-abelian Coulomb [7] θ, θ (1,-1) R dilatation D 3 R (102) 2 [37] D = 3R, D = 3 R N=

40 O 1 (x)o 2 (0) R R = R(O 1 )+R(O 2 ) D D(O 1 )+D(O 2 ) O 1 (x)o 2 (0) x = 0 x 0 0 D = D(O 1 ) + D(O 2 ) R QCD U(1) R R SU( ) L SU( ) R U(1) B R Q Q D( QQ) = 3 2 R( QQ) = 3 N c D(B) = D( B) = 3N c( N c ) 2 (103) (98) β γ = 3 Nc + 1 QQ 2 D = γ + 2 = 3 N c [37] D 1 (104) D = 1 [38] D = N c < < 3N c g QQ D( QQ) = 3 N c < 3 2 N c D < 1 = 3 2 N c D = 1 39

41 QQ M = QQ < 3N 2 c 3N 2 c QCD non-abelian Coulomb N=1 QCD magnetic non-abelian [7] non-abelian non-abelian SU(N c ) QCD electric SU( N c ) magnetic q i, q i i QCD Mj i 3N 2 c < < 3N c non-abelian Coulomb 3N 2 c < < 3N c 3(N 2 f N c ) < < 3( N c ) Mj i W = 1 µ M i jq i q j (105) D(q q) = 3N c D(W ) = 1+ 3N c < 3 relevant magnetic magnetic electric 40

42 non-abelian quantum equivalence [40] N=4 Yang-Mills [41] N=2 QCD [6] N=1 non-abelian magnetic electric non-abelian Coulomb 0 magnetic electric electric SU( ) L SU( ) R U(1) B U(1) R (106) magnetic electric magnetic SU( ) L SU( ) R U(1) B U(1) R q in (, 1, N c N c, Nc ) q in (1,, Nc N c, Nc ) ( N c ) ( N c ) M in (,, 0, 2 N c ) 1 (107) M electric QQ U(1) B magnetic q, q q, q 41

43 Q, Q q( q) Q(Q) SU( ) U(1) R U(1) R U(1) R 1 thooft 42

44 SU( ) 3 N c d (3) ( ) SU( ) 2 U(1) R U(1) 3 R U(1) 2 BU(1) R N 2 c d (2) ( ) N 2 c 1 2N 4 c N 2 f 2N 2 c SU( ) 2 U(1) B N c d (2) ( ) U(1) 2 RU(1) B 0 U(1) R Nc 2 1 (108) electric magnetic thooft (105) electric magnetic electric M i j = QQ B i 1,...,i N c = Q i1 Q i Nc B i1,...,i N c = Q i1 Q in c magnetic M j i M i j = QQ magnetic b i1,...,t Nc = q i1 q inc b i 1,...,i Nc = q i1 q i Nc (109) N c = N c magnetic electric B i 1,...,i Nc B i1,...,i Nc = C ϵ i 1,...,i Nc,j 1,...,j Nc = C ϵ i1,...,i Nc,j 1,...,j Nc bj1,...,j Nc b j 1,...,j Nc (110) 43

45 C = ( µ) N c Λ 3N c C magnetic electric magnetic magnetic q i q j (105) M (105) magnetic electric magnetic (105) 1 µ µ electric M i j M i j = Q i Q j (103) magnetic M m 1 µ M = µm m µ (105) M m µ magnetic Λ electric Λ M m M µ Λ 3Nc Λ 3( N c) = ( 1) N c µ (111) [39] scale matching relation scale matching relation ( 1) N c scale matching relation electric magnetic g 1 g 44

46 electric log Λ magnetic scale matching relation log Λ W 2 α = W 2 α (112) W α magnetic F 2 = F 2 E 2 B 2 = (Ẽ2 B 2 ) Wα 2 = W α 2 λλ = λ λ scale matching relation magnetic SU( ( N c )) = SU(N c ) M j i N j i = q i q j d i, d j scale matching relation Λ 3N c Nf Λ 3( N c ) = ( 1) ( N c) µ = ( 1) N c µ Λ = Λ µ = µ (113) W = 1 µ N j i d i d j + 1 µ M i jn j i = 1 µ N j i ( d i d j + M i j) (114) magnetic magnetic M, N N j i = 0 M i j = d i d j W = 0 d i, d j Q i, Q j electric N c N c non-abelian free electric 3N c N c 3 2 N c < < 3N c non- Abelian Coulomb N c N c magnetic 45

47 magnetic non-abelian Coulomb 3( N c ), electric non-abelian free electric magnetic magnetic irrelevant q, q, M electric electric N c + 2 3N 2 c magnetic magnetic electric non-abelian free magnetic = N c + 1 magnetic SU( N c ) = SU(1) flat directios electric W tree = mm SU(N c ) Nf 1 QCD Λ L SU(N c ) Nf Λ Λ 3N c ( 1) L = mλ 3N c SU(N c ) Nf 1 SU(N c ) Nf 46

48 magnetic W tree = mm W = 1 µ M j i q j q i + mm (115) M, M j, M i, q Nf, q q Nf q = µm, q i q = q Nf q i = 0 (116) M = M j = M i = 0 (117) (116) magnetic Higgs SU( N c 1) Nf 1 W = 1 M j i i ˆq jˆ q µ (118) M, ˆq, ˆ q SU(Nf N c 1) Nf 1 1 i, ĩ = 1,..., 1 electric SU(N c ) Nf 1 SU( N c 1) Nf 1 = SU(( 1) N c ) Nf 1 magnetic Λ 3(( N c ) 1) ( 1) L = Λ (3 Nc) <q Nf q Nf > = Λ (3Nf Nc) Nf µm magnetic scale matching relations = N c + 2 magnetic Higgs instanton W inst = Λ 6 N c+2 L det( 1 µ M) q Nf q = det M Λ 3N c (N c +1) L (119) 47

49 W = 1 Λ 2Nc 1 L ( M i jq i q j det M) (120) q, q electric = N c + 1 B i = q i B j = q j (120) = N c + 2 electric magnetic magnetic electric QCD (120) electric Λ L magnetic loop instanton electric flat directions < Q >=< Q Nf > < Q N c >=< Q Nc > electric SU(N c ) Nf Higgs SU(N c 1) Nf 1 Λ 3(Nc 1) ( 1) L = Λ3N c <Q QNf > electric magnetic < M > q Nf, q magnetic SU( N c ) Nf 1 electric Λ 3( N c) ( 1) L = 1 µ < M > Λ 3( N c) magnetic 48

50 Λ 3(Nc 1) ( 1) L Λ 3( N c) ( 1) L = 1 µ Λ3Nc N Λ f 3( N c) = ( 1) N c µ 1 scale matching relations flat directions electric magnetic electric magnetic moduli electric rank < M > N c magnetic M magnetic M N j i = q i q j < N j i >= 0 rank < M > magnetic rank < M >< N c rank <M > < N c < N j i >= 0 magnetic rank < M > N c rank < M >= N c magnetic 0 rank < M >= N c = Ñc detn b b = Λ 2Ñ c L (121) Λ L Λ 2Ñ c L = det < 1 µ M > Λ 3Ñ c det 0 (121) b, b B, B < N j i >= 0 scale matching relations < B B >= det < M > (122) electric 49

51 non-abelian, N c QCD 3N c non Abelianfree electric electric magnetic N c + 2 3N 2 c 3 2 c < < 3N c non AbelianCoulomb electric magnetic N c + 2 3N 2 c non Abelianfree magnetic magnetic electric 3N c = N c + 1 Higgs moduli = moduli = N c Higgs moduli moduli < N c instanton W eff = 0 N c pure Y amg Mills = 2N c (123) N=2 QCD N=1 non-abelian N=2 QCD [42] 50

52 5 QCD+ X QCD X j i (TrX = 0) > 0 QX Q N=2 N=2 Seiberg Witten N=1 Kähler N=1 W = s 0 k + 1 TrXk+1 (124) Kutasov Schwimmer [12][13] QCD 5.1 electric electric SU(N c ) Q i α, Q β j ; α, β = 1,, N c ; i, j = 1,, X β α; α, β = 1,, N c ; TrX = 0 W = s 0 k + 1 TrXk+1 (125) k > 2 irrelevant dangerously irrelevant relevant 51

53 SU( ) L SU( ) R U(1) B U(1) R (126) U(1) R R Q (, 1, 1, 1 2 N c ) k + 1 Q (1,, 1, 1 2 N c ) k X (1, 1, 0, k + 1 ) (127) 1-loop β (2N c ) 2N c < (M j ) ĩ i = QĩX j 1 Q i ; j = 1, 2,, k (128) B (n 1,n 2,,n k ) = Q n 1 (1) Qn 2 (2) Qn k (k) k n l = N c (129) l=1 ϵ 1,2,,N c Q (l) Q (l) = X l 1 Q; l = 1,, k (130) TrX j ; 2 j k (131) j (125) 52

54 5.2 magnetic ppp 2N c > > 3 N c non-abelian Coulomb 2 k non-abelian magnetic magnetic SU( N c ); Nc = k N c q α i, q j β ; α, β = 1,, N c ; i, j = 1,, Y β α ; α, β = 1,, N c ; TrY = 0 (M j ) ĩ i = QĩX j 1 Q i ; j = 1, 2,, k (132) electric magnetic electric (126) q Y q M j ( (, 1, N c k N c, 1 2 k + 1 ( N c 1,,, 1 2 k N c k + 1 ( ) 1, 1, 0, 2 k + 1,, 0, 2 4 ) k N c k N c ) N c + 2 ) (j 1) k + 1 k + 1 (133) electric magnetic thooft SU( ) 3 N c d (3) ( ) SU( ) 2 U(1) R 2 Nc 2 d (2) ( ) k + 1 SU( ) 2 U(1) B N c d (2) ( ) U(1) R 2 k + 1 (N c 2 1) ( ) U(1) 3 2 R ( k + 1 1)3 + 1 (Nc 2 1) 16 Nc 4 (k + 1) 3 Nf 2 53

55 U(1) 2 BU(1) R 4 k + 1 N 2 c (134) magnetic W mag = s 0 k + 1 TrY k+1 + s 0 µ 2 k M j qy k j q (135) j=1 s 0 Y s 0 = s 0 [14] electric magnetic TrY j = TrX j ; j = 2,, k 1 TrY k = N c N c TrX k (M j ) ĩ i = QĩX j 1 Q i ; j = 1, 2,, k B (n 1,n 2,,n k ) el B (m1,m2,,m k) mag ; m l = n k+1 i ; l = 1, 2,, k (136) B mag magnetic electric (125) k l=1 n l = N c, kl=1 m l = N c Q (l) 0 n l, 0 m l k l=1 m l = N c k l=1 n l = N c k k k m l = ( n k+1 i ) = k n k+1 i = k N c = N c (137) l=1 l=1 l=1 TrY j [14] 54

56 k=1 electric X QCD QCD k electric electric W el = s 0 Tr X k+1 + m Q Nf Q (138) (N c, ) (N c, 1) (k N c k, 1) magnetic W mag = s 0 k + 1 TrY k+1 + s 0 µ 2 k j=1 M j qy k j q + m(m 1 ) (139) q Nf Y l 1 q = δ l,k m; l = 1, 2,, k (140) q α = δ α,1 q α = δ α,k Y α β = δ α β+1 β = 1,, k 1 0 (141) Higgs (k N c, ) (k N c k, 1) electric 55

57 W el = k i=0 s i k + 1 i Tr Xk+1 i (142) s k TrX s k TrX = 0 k=2 s i ( 1 W el = Tr 3 X3 + m ) 2 X2 + λx (143) X X x 2 + mx + λ = 0 (144) V = W (X) 2 x +, x X x + r = 0, 1,, N c N c + 1 x N c r r N c r Λ TrX = rx + (Λ) + (N c r)x (Λ) = 0 r Higgs SU(N c ) SU(r) SU(N c r) U(1) (145) X QCD magnetic electric (2 N c ) + 1 Higgs SU(2 N c ) SU(l) SU(2 N c r) U(1) (146) electric magnetic QCD N c > l, 2 N c l 56

58 l = N c, N c 1,, (147) magnetic electric QCD, l = r magnetic M 1, M 2 QCD k > 2 k=2 k W (X) = 0 X k x l ; l = 1,, k X x i i l k l=1 i l = N c SU(N c ) SU(i 1 ) SU(i 2 ) SU(i k ) U(1) k 1 (148) magnetic k l=1 j l = k N c k j l ; l = 1,, k SU(k N c ) SU(i 1 ) SU(i 2 ) SU(i k ) U(1) k 1 (149) k=2 QCD j l = i l < N c k X W = s o k+1 TrXk+1 < N c k k > 2 s l W (X) = 0 W (x) = i (x a i ) n i ; n i = k (150) 57

59 n i 1 r i X a i SU(N c ) i SU(r i ) U(1) k 1 ; ri = N c (151) SU(r i ) X i 0 W L = i W (n i+1) (a i )TrX (n i+1) i + X i (152) W (n i+1) (a i ) W(x) n i x=a i magnetic SU(k N c ) i SU( r i ) U(1) k 1 ; ri = k N c (153) W = s o k+1 TrXk+1 < N c k r i = n i r i SU(N c ) SO(N), Sp(N) X QCD non-abelian [43][44] 58

60 6 Seiberg QCD Kutasov-Schwimmer non-abelian Coulomb non- Abelian Moduli N=1 thooft N=1 QCD QCD 3N 2 c < < 3N c non-abelian QCD Kutasov-Schwimmer non-abelian Kutasov-Schwimmer 59

61 non-abelian non-abelian non-abelian [45] N=2 non-abelian electric magnetic electric magnetic [46] N=1 non-abelian 60

62 7 61

63 { Q L α, QᾱM } = 2σ m αᾱ p m δ L M A Poincarè [P m, P n ] = 0 [ Pm, Qα] L = [ ] P m, QᾱL = 0 [P m, B l ] = [ P m, X ] LM = 0 { Q L α, Q βm } = ϵαβ X LM { QᾱL, Q βm } = ϵᾱ βx LM [ X LM, QᾱK ] = [ X LM, Q K α ] = 0 [ X LM, X KN ] = [ X LM, B l ] = 0 [B l, B m ] = ic k lmb k [ Q L α, B l ] [ QᾱL, B l ] = S L l MQ M α = S l M L QᾱM X LM = a l,lm B l (154) X LM a l,lm L, M B l c k lm XLM Sl L M Jacobi B l Jacobi S M l Ka k,kl = a k,mk S l L K (155) a S ( S ) intertwiner Weyl spinorq Q 1 N=1 ) Lorentz 62

64 B Kähler n Φ i L = d 2 θd 2 θk(φ [ i, Φ j ) + ] d 2 θw (Φ i ) + h.c. (156) A i χ i L = g ij m A i m A j ig ij χ j σ m D m χ i R ij kl χi χ k χ j χ l 1 2 D id j W χ i χ j 1 2 D i D j W χ i χ j g ij D i W D j W (157) D i W = A i W D i D j W = 2 A i A W j Γk ij A W (158) k g ij = K(A) A i A j g ij,k = A g k ij = g mj Γm ik g ij,k = A g k ij = g im Γm i k (159) g ij K Kähler Kähler Γ m ik Kähler [47] 63

65 C Wess-Zumino Wess-Zumino v m v m + m χ Wess-Zumino Wess-Zumino Wess-Zumino δ ξ A = 2ξΨ δ ξ Ψ = i 2σ m ξdm A + 2ξF δ ξ F = i 2 ξ σ m D m Ψ + i2gt (a) A ξ λ (a) δ ξ v (a) m = i λ (a) σ m ξ + i ξ σ m λ (a) δ ξ λ (a) = σ mn ξv (a) mn + iξd (a) δ ξ D (a) = ξσ m D m λ(a) D m λ (a) σ m ξ (160) 64

66 [1] S.Dimopoulos,S.Raby and F.Wilczek, Phys. Rev. D24 (1981) [2] D.Amati, K.Konishi, Y.Meurice, G.C.Rossi and G.Veneziano, Phys. Rep. 162 (1988) 169 and references therein. [3] N.Seiberg, Phys. Rev. D49 (1994)6857. [4] K.Intriligator and N.Seiberg, Nucl. Phys. B431 (1994) 551. [5] K.Intriligator and N.Seiberg, Nucl. Phys. B444 (1995) 125. [6] N.Seiberg and E.Witten, Nucl. Phys. B431 (1994) 484. [7] N.Seiberg, Nucl. Phys. B435 (1995) 129. [8] K.Intriligator and R.G.Leigh and N.Seiberg, Phys. Rev. D50 (1994) [9] K.Intriligator, Phys. Lett. 336B (1994) 409. [10] N.Seiberg and E.Witten, Nucl. Phys. B426 (1994) 19. [11] G. thooft: in Recentdevelopments in Gauge Theoriees,eds. G. thooft et al.(plenum Press, New York,1980). [12] D.Kutasov, Phys. Lett. 351B (1995) 230. [13] D.Kutasov and A.Schwimmer, Phys. Lett. 354B (1995) 315. [14] D.Kutasov and A.Schwimmer and N.Seiberg, hep-th/ [15] K.Intriligator and R.G.Leigh and M.Strassler, hep-th/ [16] M.Berkooz, Nucl.Phys. B452 (1995)

67 [17] P.Pouliot, Phys.Lett. 359B (1995) 108 ;P.Pouliot, hep-th/ ; P. Pouliot and M.J. Strassler, hep-th/ [18] S.Coleman and J.Mandula, Phys. Rev. 159 (1967) [19] R.Haag, J.Lopuszanski and M.Sohnius, Nucl. Phys. B88 (1975) 257. [20] J.Wess and J.Bagger, Supersymmetry And Supergravity,Princeton University Press(second edition,1992). [21] E.Witten, Nucl. Phys. B185 (1981) 513. [22] H.P.Nilles, Phys. Rep. 110 (1984) 1 and references therein. [23] J.Wess and B.Zumino, Nucl. Phys. B70 (1974) 39. [24] J.Wess and B.Zumino, Nucl. Phys. B78 (1974) 1. [25] S.Ferrara and B.Zumino, Nucl. Phys. B79 (1974) 413. [26] J.Bagger and E.Witten, Phys.Lett. B118 (1982) 103. [27] E.Witten, Nucl. Phys. B202 (1982) 253. [28] N.Seiberg, Phys. Lett. 318B (1993) 469. [29] M.A.Shifman and A.I.Vainshtein, Nucl. Phys. B277 (1986) 456. [30] M.A.Shifman and A.I.Vainshtein, Nucl. Phys. B359 (1991) 571. [31] M.T.Grisaru, M.Roček and W.Siegel, Nucl. Phys. B159 (1979) 429. [32] T.Banks and E.Rabinovici, Nucl. Phys. B160 (1979) 349; E.Fradkin and S.Shenker, Phys. Rev. D19 (1979)

68 [33] I.Affleck, M.DIne, N.Seiberg, Nucl. Phys. B241 (1984) 493; Nucl. Phys. B256 (1985) 557. [34] D.Finnell and P.Pouliot, Nucl. Phys. B453 (1995) 225. [35] T.Banks and A.Zaks, Nucl. Phys. B196 (1982) 189. [36] V.Novikov, M.Shifman, A.Vainshtein and V.Zakharov, Nucl. Phys. B229 (1983) 381. [37] M.Flato and C.Fronsdal, Lett. Math. Phys. 8 (1984) 159; V.K.Dobrev and V.B.Petkova, Phys. Lett. 162B (1985) 127. [38] G.Mack, Comm. Math. Phys. 55 (1977) 1. [39] K. Intriligator and N. Seiberg, hep-th/ [40] P.Goddard and D.Olive, Int. J. Mod. Phys. A1 (1986) 303 [41] H.Osborn, Phys. Lett. 83B (1979) 321; A.Sen, Int. J. Mod. Phys. A9 (1994) 3707; Phys. Lett. 329B (1994) 217; C.Vafa and E.Witten Nucl. Phys. B431 (1994) 3. [42] R.G. Leigh and M.J. Strassler, Nucl. Phys. B447 (1995) 95. [43] R.G. Leigh and M.J. Strassler Phys. Lett. 356B (1995) 492. [44] K. Intriligator and P.Pouliot, Phys. Lett. 353B (1995) 471. [45] M.Berkooz, hep-th/ [46] J.L.F.Barbón and S.Ramgoolam, hep-th/ [47] B.Zumino, Phys. Lett. 87B (1979) 203. [48] W.Siegel, Phys. Lett. 84B (1979)

2017 II 1 Schwinger Yang-Mills 5. Higgs 1

2017 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 information

SO(3) 49 u = Ru (6.9), i u iv i = i u iv i (C ) π π : G Hom(V, V ) : g D(g). π : R 3 V : i 1. : u u = u 1 u 2 u 3 (6.10) 6.2 i R α (1) = 0 cos α

SO(3) 49 u = Ru (6.9), i u iv i = i u iv i (C ) π π : G Hom(V, V ) : g D(g). π : R 3 V : i 1. : u u = u 1 u 2 u 3 (6.10) 6.2 i R α (1) = 0 cos α SO(3) 48 6 SO(3) t 6.1 u, v u = u 1 1 + u 2 2 + u 3 3 = u 1 e 1 + u 2 e 2 + u 3 e 3, v = v 1 1 + v 2 2 + v 3 3 = v 1 e 1 + v 2 e 2 + v 3 e 3 (6.1) i (e i ) e i e j = i j = δ ij (6.2) ( u, v ) = u v = ij

More information

量子力学 問題

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

Seiberg 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 information

SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ

SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ SO(3) 71 5.7 5.7.1 1 ħ L k l k l k = iϵ kij x i j (5.117) l k SO(3) l z l ± = l 1 ± il = i(y z z y ) ± (z x x z ) = ( x iy) z ± z( x ± i y ) = X ± z ± z (5.118) l z = i(x y y x ) = 1 [(x + iy)( x i y )

More information

q 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

q 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 information

susy.dvi

susy.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

0406_total.pdf

0406_total.pdf 59 7 7.1 σ-ω σ-ω σ ω σ = σ(r), ω µ = δ µ,0 ω(r) (6-4) (iγ µ µ m U(r) γ 0 V (r))ψ(x) = 0 (7-1) U(r) = g σ σ(r), V (r) = g ω ω(r) σ(r) ω(r) (6-3) ( 2 + m 2 σ)σ(r) = g σ ψψ (7-2) ( 2 + m 2 ω)ω(r) = g ω ψγ

More information

Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 = ( p µ γ µ + m)(p ν γ ν + m) (5.1) γ = p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 = 1 2 p µp ν {γ µ, γ ν } + m

Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 = ( p µ γ µ + m)(p ν γ ν + m) (5.1) γ = p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 = 1 2 p µp ν {γ µ, γ ν } + m Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 p µ γ µ + mp ν γ ν + m 5.1 γ p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 1 2 p µp ν {γ µ, γ ν } + m 2 5.2 p m p p µ γ µ {, } 10 γ {γ µ, γ ν } 2η µν 5.3 p µ γ µ + mp

More information

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5. A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c

More information

,,..,. 1

,,..,. 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 information

6 6.1 L r p hl = r p (6.1) 1, 2, 3 r =(x, y, z )=(r 1,r 2,r 3 ), p =(p x,p y,p z )=(p 1,p 2,p 3 ) (6.2) hl i = jk ɛ ijk r j p k (6.3) ɛ ijk Levi Civit

6 6.1 L r p hl = r p (6.1) 1, 2, 3 r =(x, y, z )=(r 1,r 2,r 3 ), p =(p x,p y,p z )=(p 1,p 2,p 3 ) (6.2) hl i = jk ɛ ijk r j p k (6.3) ɛ ijk Levi Civit 6 6.1 L r p hl = r p (6.1) 1, 2, 3 r =(x, y, z )=(r 1,r 2,r 3 ), p =(p x,p y,p z )=(p 1,p 2,p 3 ) (6.2) hl i = jk ɛ ijk r j p k (6.3) ɛ ijk Levi Civita ɛ 123 =1 0 r p = 2 2 = (6.4) Planck h L p = h ( h

More information

( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes )

( ) ) ) ) 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 information

( 3) b 1 b : b b f : a b 1 b f = f (2.7) g : b c g 1 b = g (2.8) 1 b b (identity arrow) id b f a b g f 1 b b c g (2.9) 3 C C C a, b a b Hom C (a, b) h

( 3) b 1 b : b b f : a b 1 b f = f (2.7) g : b c g 1 b = g (2.8) 1 b b (identity arrow) id b f a b g f 1 b b c g (2.9) 3 C C C a, b a b Hom C (a, b) h 2011 9 5 1 Lie 1 2 2.1 (category) (object) a, b, c, a b (arrow, morphism) f : a b (2.1) f a b (2.2) ( 1) f : a b g : b c (composite) g f : a c ( 2) f f a b g f g c g h (2.3) a b c d (2.4) h (g f) = (h

More information

( ) ( 40 )+( 60 ) Schrödinger 3. (a) (b) (c) yoshioka/education-09.html pdf 1

( ) ( 40 )+( 60 ) Schrödinger 3. (a) (b) (c)   yoshioka/education-09.html pdf 1 2009 1 ( ) ( 40 )+( 60 ) 1 1. 2. Schrödinger 3. (a) (b) (c) http://goofy.phys.nara-wu.ac.jp/ yoshioka/education-09.html pdf 1 1. ( photon) ν λ = c ν (c = 3.0 108 /m : ) ɛ = hν (1) p = hν/c = h/λ (2) h

More information

QCD 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 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 information

第10章 アイソパラメトリック要素

第10章 アイソパラメトリック要素 June 5, 2019 1 / 26 10.1 ( ) 2 / 26 10.2 8 2 3 4 3 4 6 10.1 4 2 3 4 3 (a) 4 (b) 2 3 (c) 2 4 10.1: 3 / 26 8.3 3 5.1 4 10.4 Gauss 10.1 Ω i 2 3 4 Ξ 3 4 6 Ξ ( ) Ξ 5.1 Gauss ˆx : Ξ Ω i ˆx h u 4 / 26 10.2.1

More information

ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx

ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx i B5 7.8. p89 4. ψ x, tψx, t = ψ R x, t iψ I x, t ψ R x, t + iψ I x, t = ψ R x, t + ψ I x, t p 5.8 π π π F e ix + F e ix + F 3 e 3ix F e ix + F e ix + F 3 e 3ix dx πψ x πψx p39 7. AX = X A [ a b c d x

More information

0. Intro ( K CohFT etc CohFT 5.IKKT 6.

0. 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 information

1 (Contents) (1) Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji

1 (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 information

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

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 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 information

φ 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 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 information

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345

More information

newmain.dvi

newmain.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 information

Λ(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 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 information

) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4

) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4 1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev

More information

N cos s s cos ψ e e e e 3 3 e e 3 e 3 e

N 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

II 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.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 information

2016 ǯ¥Î¡¼¥Ù¥ëʪÍý³Ø¾Þ²òÀ⥻¥ß¥Ê¡¼ Kosterlitz-Thouless ž°Ü¤È Haldane ͽÁÛ

2016 ǯ¥Î¡¼¥Ù¥ëʪÍý³Ø¾Þ²òÀ⥻¥ß¥Ê¡¼  Kosterlitz-Thouless ž°Ü¤È Haldane ͽÁÛ 2016 Kosterlitz-Thouless Haldane Dept. of Phys., Kyushu Univ. 2016 11 29 2016 Figure: D.J.Thouless F D.M.Haldane J.M.Kosterlitz TOPOLOGICAL PHASE TRANSITIONS AND TOPOLOGICAL PHASES OF MATTER ( ) ( ) (Dirac,

More information

·«¤ê¤³¤ß·²¤È¥ß¥ì¥Ë¥¢¥àÌäÂê

·«¤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 information

: , 2.0, 3.0, 2.0, (%) ( 2.

: , 2.0, 3.0, 2.0, (%) ( 2. 2017 1 2 1.1...................................... 2 1.2......................................... 4 1.3........................................... 10 1.4................................. 14 1.5..........................................

More information

v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i

v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i 1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1 1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [

More information

Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ

Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ Galois ) 0 1 1 2 2 4 3 10 4 12 5 14 16 0 Galois Galois Galois TaylorWiles Fermat [W][TW] Galois Galois Galois 1 Noether 2 1 Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R

More information

数学Ⅱ演習(足助・09夏)

数学Ⅱ演習(足助・09夏) II I 9/4/4 9/4/2 z C z z z z, z 2 z, w C zw z w 3 z, w C z + w z + w 4 t R t C t t t t t z z z 2 z C re z z + z z z, im z 2 2 3 z C e z + z + 2 z2 + 3! z3 + z!, I 4 x R e x cos x + sin x 2 z, w C e z+w

More information

30

30 3 ............................................2 2...........................................2....................................2.2...................................2.3..............................

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

.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 information

untitled

untitled 0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.

More information

ʪ¼Á¤Î¥È¥Ý¥í¥¸¥«¥ë¸½¾Ý (2016ǯ¥Î¡¼¥Ù¥ë¾Þ¤Ë´ØÏ¢¤·¤Æ)

ʪ¼Á¤Î¥È¥Ý¥í¥¸¥«¥ë¸½¾Ý  (2016ǯ¥Î¡¼¥Ù¥ë¾Þ¤Ë´ØÏ¢¤·¤Æ) (2016 ) Dept. of Phys., Kyushu Univ. 2017 8 10 1 / 59 2016 Figure: D.J.Thouless F D.M.Haldane J.M.Kosterlitz TOPOLOGICAL PHASE TRANSITIONS AND TOPOLOGICAL PHASES OF MATTER 2 / 59 ( ) ( ) (Dirac, t Hooft-Polyakov)

More information

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google

I 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 information

2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................

More information

1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x

1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x . P (, (0, 0 R {(,, R}, R P (, O (0, 0 OP OP, v v P (, ( (, (, { R, R} v (, (, (,, z 3 w z R 3,, z R z n R n.,..., n R n n w, t w ( z z Ke Words:. A P 3 0 B P 0 a. A P b B P 3. A π/90 B a + b c π/ 3. +

More information

SO(2)

SO(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

DVIOUT-fujin

DVIOUT-fujin 2005 Limit Distribution of Quantum Walks and Weyl Equation 2006 3 2 1 2 2 4 2.1...................... 4 2.2......................... 5 2.3..................... 6 3 8 3.1........... 8 3.2..........................

More information

D 24 D D D

D 24 D D D 5 Paper I.R. 2001 5 Paper HP Paper 5 3 5.1................................................... 3 5.2.................................................... 4 5.3.......................................... 6

More information

A 2 3. m S m = {x R m+1 x = 1} U + k = {x S m x k > 0}, U k = {x S m x k < 0}, ϕ ± k (x) = (x 0,..., ˆx k,... x m ) 1. {(U ± k, ϕ± k ) 0 k m} S m 1.2.

A 2 3. m S m = {x R m+1 x = 1} U + k = {x S m x k > 0}, U k = {x S m x k < 0}, ϕ ± k (x) = (x 0,..., ˆx k,... x m ) 1. {(U ± k, ϕ± k ) 0 k m} S m 1.2. A A 1 A 5 A 6 1 2 3 4 5 6 7 1 1.1 1.1 (). Hausdorff M R m M M {U α } U α R m E α ϕ α : U α E α U α U β = ϕ α (ϕ β ϕβ (U α U β )) 1 : ϕ β (U α U β ) ϕ α (U α U β ) C M a m dim M a U α ϕ α {x i, 1 i m} {U,

More information

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

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 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 information

[1] convention Minkovski i Polchinski [2] 1 Clifford Spin 1 2 Euclid Clifford 2 3 Euclid Spin 6 4 Euclid Pin Clifford Spin 10 A 12 B 17 1 Cliffo

[1] convention Minkovski i Polchinski [2] 1 Clifford Spin 1 2 Euclid Clifford 2 3 Euclid Spin 6 4 Euclid Pin Clifford Spin 10 A 12 B 17 1 Cliffo [1] convention Minkovski i Polchinski [2] 1 Clifford Spin 1 2 Euclid Clifford 2 3 Euclid Spin 6 4 Euclid Pin + 8 5 Clifford Spin 10 A 12 B 17 1 Clifford Spin D Euclid Clifford Γ µ, µ = 1,, D {Γ µ, Γ ν

More information

( )

( ) 7..-8..8.......................................................................... 4.................................... 3...................................... 3..3.................................. 4.3....................................

More information

W 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)

W 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

7 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 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

I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )

I A A441 : April 15, 2013 Version : 1.1 I   Kawahira, Tomoki TA (Shigehiro, Yoshida ) I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17

More information

Donaldson 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 [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 information

SAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T

SAMA- 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

all.dvi

all.dvi 29 4 Green-Lagrange,,.,,,,,,.,,,,,,,,,, E, σ, ε σ = Eε,,.. 4.1? l, l 1 (l 1 l) ε ε = l 1 l l (4.1) F l l 1 F 30 4 Green-Lagrange Δz Δδ γ = Δδ (4.2) Δz π/2 φ γ = π 2 φ (4.3) γ tan γ γ,sin γ γ ( π ) γ tan

More information

2 (2016 3Q N) c = o (11) Ax = b A x = c A n I n n n 2n (A I n ) (I n X) A A X A n A A A (1) (2) c 0 c (3) c A A i j n 1 ( 1) i+j A (i, j) A (i, j) ã i

2 (2016 3Q N) c = o (11) Ax = b A x = c A n I n n n 2n (A I n ) (I n X) A A X A n A A A (1) (2) c 0 c (3) c A A i j n 1 ( 1) i+j A (i, j) A (i, j) ã i [ ] (2016 3Q N) a 11 a 1n m n A A = a m1 a mn A a 1 A A = a n (1) A (a i a j, i j ) (2) A (a i ca i, c 0, i ) (3) A (a i a i + ca j, j i, i ) A 1 A 11 0 A 12 0 0 A 1k 0 1 A 22 0 0 A 2k 0 1 0 A 3k 1 A rk

More information

2016 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 16 2 1 () X O 3 (O1) X O, O (O2) O O (O3) O O O X (X, O) O X X (O1), (O2), (O3) (O2) (O3) n (O2) U 1,..., U n O U k O k=1 (O3) U λ O( λ Λ) λ Λ U λ O 0 X 0 (O2) n =

More information

X 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 information

Norisuke 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. 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 information

meiji_resume_1.PDF

meiji_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

20 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 information

d ϕ i) t d )t0 d ϕi) ϕ i) t x j t d ) ϕ t0 t α dx j d ) ϕ i) t dx t0 j x j d ϕ i) ) t x j dx t0 j f i x j ξ j dx i + ξ i x j dx j f i ξ i x j dx j d )

d ϕ i) t d )t0 d ϕi) ϕ i) t x j t d ) ϕ t0 t α dx j d ) ϕ i) t dx t0 j x j d ϕ i) ) t x j dx t0 j f i x j ξ j dx i + ξ i x j dx j f i ξ i x j dx j d ) 23 M R M ϕ : R M M ϕt, x) ϕ t x) ϕ s ϕ t ϕ s+t, ϕ 0 id M M ϕ t M ξ ξ ϕ t d ϕ tx) ξϕ t x)) U, x 1,...,x n )) ϕ t x) ϕ 1) t x),...,ϕ n) t x)), ξx) ξ i x) d ϕi) t x) ξ i ϕ t x)) M f ϕ t f)x) f ϕ t )x) fϕ

More information

Part () () Γ Part ,

Part () () 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 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 (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 information

9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ0 ψ n (r) ω n Schrödinger Ĥ 0 ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ0 + Ĥint (

9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ0 ψ n (r) ω n Schrödinger Ĥ 0 ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ0 + Ĥint ( 9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) 2. 2.1 Ĥ ψ n (r) ω n Schrödinger Ĥ ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ + Ĥint (t)] ψ (r, t), (2) Ĥ int (t) = eˆxe cos ωt ˆdE cos ωt, (3)

More information

24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x

24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 24 I 1.1.. ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 1 (t), x 2 (t),, x n (t)) ( ) ( ), γ : (i) x 1 (t),

More information

IA

IA IA 31 4 11 1 1 4 1.1 Planck.............................. 4 1. Bohr.................................... 5 1.3..................................... 6 8.1................................... 8....................................

More information

main.dvi

main.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 information

H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [

H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [ 3 3. 3.. H H = H + V (t), V (t) = gµ B α B e e iωt i t Ψ(t) = [H + V (t)]ψ(t) Φ(t) Ψ(t) = e iht Φ(t) H e iht Φ(t) + ie iht t Φ(t) = [H + V (t)]e iht Φ(t) Φ(t) i t Φ(t) = V H(t)Φ(t), V H (t) = e iht V (t)e

More information

linearal1.dvi

linearal1.dvi 19 4 30 I 1 1 11 1 12 2 13 3 131 3 132 4 133 5 134 6 14 7 2 9 21 9 211 9 212 10 213 13 214 14 22 15 221 15 222 16 223 17 224 20 3 21 31 21 32 21 33 22 34 23 341 23 342 24 343 27 344 29 35 31 351 31 352

More information

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ = 1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A

More information

1 filename=mathformula tex 1 ax 2 + bx + c = 0, x = b ± b 2 4ac, (1.1) 2a x 1 + x 2 = b a, x 1x 2 = c a, (1.2) ax 2 + 2b x + c = 0, x = b ± b 2

1 filename=mathformula tex 1 ax 2 + bx + c = 0, x = b ± b 2 4ac, (1.1) 2a x 1 + x 2 = b a, x 1x 2 = c a, (1.2) ax 2 + 2b x + c = 0, x = b ± b 2 filename=mathformula58.tex ax + bx + c =, x = b ± b 4ac, (.) a x + x = b a, x x = c a, (.) ax + b x + c =, x = b ± b ac. a (.3). sin(a ± B) = sin A cos B ± cos A sin B, (.) cos(a ± B) = cos A cos B sin

More information

2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =

2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) = 1 1 1.1 I R 1.1.1 c : I R 2 (i) c C (ii) t I c (t) (0, 0) c (t) c(i) c c(t) 1.1.2 (1) (2) (3) (1) r > 0 c : R R 2 : t (r cos t, r sin t) (2) C f : I R c : I R 2 : t (t, f(t)) (3) y = x c : R R 2 : t (t,

More information

DaisukeSatow.key

DaisukeSatow.key Nambu-Goldstone Fermion in Quark-Gluon Plasma and Bose-Fermi Cold Atom System ( /BNL! ECT* ") : Jean-Paul Blaizot (Saclay CEA #) ( ) (SUSY) = b f b f 2 (SUSY) Q: supercharge b f b f SUSY: [Q, H]=0 Supercharge

More information

1. 1.1....................... 1.2............................ 1.3.................... 1.4.................. 2. 2.1.................... 2.2..................... 2.3.................... 3. 3.1.....................

More information

25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3

More information

prime number theorem

prime number theorem For Tutor MeBio ζ Eite by kamei MeBio 7.8.3 : Bernoulli Bernoulli 4 Bernoulli....................................................................................... 4 Bernoulli............................................................................

More information

1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc

1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc 013 6 30 BCS 1 1.1........................ 1................................ 3 1.3............................ 3 1.4............................... 5 1.5.................................... 5 6 3 7 4 8

More information

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi) 0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()

More information

SUSY DWs

SUSY 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 information

ver Web

ver Web ver201723 Web 1 4 11 4 12 5 13 7 2 9 21 9 22 10 23 10 24 11 3 13 31 n 13 32 15 33 21 34 25 35 (1) 27 4 30 41 30 42 32 43 36 44 (2) 38 45 45 46 45 5 46 51 46 52 48 53 49 54 51 55 54 56 58 57 (3) 61 2 3

More information

1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3

1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3 1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A 2 1 2 1 2 3 α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3 4 P, Q R n = {(x 1, x 2,, x n ) ; x 1, x 2,, x n R}

More information

TOP URL 1

TOP 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 information

20 4 20 i 1 1 1.1............................ 1 1.2............................ 4 2 11 2.1................... 11 2.2......................... 11 2.3....................... 19 3 25 3.1.............................

More information

S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d

S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5....

More information

構造と連続体の力学基礎

構造と連続体の力学基礎 II 37 Wabash Avenue Bridge, Illinois 州 Winnipeg にある歩道橋 Esplanade Riel 橋6 6 斜張橋である必要は多分無いと思われる すぐ横に道路用桁橋有り しかも塔基部のレストランは 8 年には営業していなかった 9 9. 9.. () 97 [3] [5] k 9. m w(t) f (t) = f (t) + mg k w(t) Newton

More information

t χ 2 F Q t χ 2 F 1 2 µ, σ 2 N(µ, σ 2 ) f(x µ, σ 2 ) = 1 ( exp (x ) µ)2 2πσ 2 2σ 2 0, N(0, 1) (100 α) z(α) t χ 2 *1 2.1 t (i)x N(µ, σ 2 ) x µ σ N(0, 1

t χ 2 F Q t χ 2 F 1 2 µ, σ 2 N(µ, σ 2 ) f(x µ, σ 2 ) = 1 ( exp (x ) µ)2 2πσ 2 2σ 2 0, N(0, 1) (100 α) z(α) t χ 2 *1 2.1 t (i)x N(µ, σ 2 ) x µ σ N(0, 1 t χ F Q t χ F µ, σ N(µ, σ ) f(x µ, σ ) = ( exp (x ) µ) πσ σ 0, N(0, ) (00 α) z(α) t χ *. t (i)x N(µ, σ ) x µ σ N(0, ) (ii)x,, x N(µ, σ ) x = x+ +x N(µ, σ ) (iii) (i),(ii) z = x µ N(0, ) σ N(0, ) ( 9 97.

More information

YITP50.dvi

YITP50.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 information

Chern-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   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 information

Macdonald, ,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdona

Macdonald, ,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdona Macdonald, 2015.9.1 9.2.,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdonald,, q., Heckman Opdam q,, Macdonald., 1 ,,. Macdonald,

More information

1 (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

1 (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

Kaluza-Klein(KK) SO(11) KK 1 2 1

Kaluza-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 information

Dynkin Serre Weyl

Dynkin Serre Weyl Dynkin Naoya Enomoto 2003.3. paper Dynkin Introduction Dynkin Lie Lie paper 1 0 Introduction 3 I ( ) Lie Dynkin 4 1 ( ) Lie 4 1.1 Lie ( )................................ 4 1.2 Killing form...........................................

More information

201711grade1ouyou.pdf

201711grade1ouyou.pdf 2017 11 26 1 2 52 3 12 13 22 23 32 33 42 3 5 3 4 90 5 6 A 1 2 Web Web 3 4 1 2... 5 6 7 7 44 8 9 1 2 3 1 p p >2 2 A 1 2 0.6 0.4 0.52... (a) 0.6 0.4...... B 1 2 0.8-0.2 0.52..... (b) 0.6 0.52.... 1 A B 2

More information

2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n

2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n . X {x, x 2, x 3,... x n } X X {, 2, 3, 4, 5, 6} X x i P i. 0 P i 2. n P i = 3. P (i ω) = i ω P i P 3 {x, x 2, x 3,... x n } ω P i = 6 X f(x) f(x) X n n f(x i )P i n x n i P i X n 2 G(k) e ikx = (ik) n

More information

2011de.dvi

2011de.dvi 211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37

More information

006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................

More information

http://www.ike-dyn.ritsumei.ac.jp/ hyoo/wave.html 1 1, 5 3 1.1 1..................................... 3 1.2 5.1................................... 4 1.3.......................... 5 1.4 5.2, 5.3....................

More information

ii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,.

ii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,. 24(2012) (1 C106) 4 11 (2 C206) 4 12 http://www.math.is.tohoku.ac.jp/~obata,.,,,.. 1. 2. 3. 4. 5. 6. 7.,,. 1., 2007 (). 2. P. G. Hoel, 1995. 3... 1... 2.,,. ii 3.,. 4. F. (),.. 5... 6.. 7.,,. 8.,. 1. (75%)

More information

20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................

More information