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3 LGT U x, ˆ µ = U 11 L L M O M M L U Nc N c dµ(u) = e β UUU + U + x, ˆ µ det / D (U) + m x, ˆ µ ( ) N F du x, ˆ µ RMT H = H 11 L L M O M M L H NN dµ(h) = e tr H 2 dh

4 LGT RMT

5 Akemann-Damgaard-Magnea-SMN 97/98 SMN 98/99, Garcia-SMN-Verbaarschot 02 Nagao-SMN 01 Damgaard-SMN 98-01, Nagao-SMN 00/01 Dunne-SMN 03

7 s 1950s

9 H s P Wigner (s) = π 2 s e- π 4 s 2

11 3 3

12 n n 1 P Poisson (s) = e -s

13 V(x,y) = A x 4 + B y 4 + C x 2 y 2

14 2

15 ζ(1/2 + i x) x th ~ ( )-th zeroes 2 P(s)

16 H Prob(E,E ) ~ E -E β β=1,2,4 H

17 T = K C c.c. unitary T 2 = CC * =±1 symm. C = U T U antisymm. C = U T JU [H, T] = 0 H : R symm H : H selfdual L S [H, T] 0 H : C hermitian S B

18 SU(2)

20 # # L # # # H = # # # # # # # # M # O H 11 H 12 H 13 H 14 H 15 H 16 L H 21 H 22 H 23 H 24 H 25 H 26 L H 31 H 32 H 33 H 34 H 35 H 36 L H = H 41 H 42 H 43 H 44 H 45 H 46 L H 51 H 52 H 53 H 54 H 55 H 56 L H 61 H 62 H 63 H 64 H 65 H 66 L M M M M M M O

21 H = H + = (H ij ) R, C, H dµ(h) = exp(-tr H 2 ) Πd β H ij β = 1, 2, 4 dµ(h) = dµ(uhu + ) dµ({e}) = Π i de i exp(-tr E i2 ) Π i>j E i - E j β

22 N=2 N=

23 N=11, 21, 51 N= ( N=50 ) x 100

24 2 N=2 3λ 1 λ 2 3

25 N C =1, N F =0, 1-plaquette

27 2 β=0 β=1 β=2 β=4 exp(-s) ~ s β ~ exp(-c β s 2 )

28 s k E β (k ; s) GOE Poisson GUE

29 N P (s), ρ(λ), N= P (s), ρ(λ), exp(-tr H 2 ) exp(-tr V(H)) exp(-tr (H+A) 2 ) Anderson Gauss Anderson P (s)

30 Vector subgroup Sp(2n)

31 Riemann 1 U = V diag(e iθ 1,...,eiθ N ) V + Cartan Zirnbauer 96

32 λ µ µ U 1 nonchiral U/UxU U/O 1/2 nonchiral Sp/SpxSp U/Sp 2 nonchiral O/OxO U/UxU 1 Z+1/2 1/2 U O/OxO 1/2 Z/2 0 U/Sp Sp/SpxSp 2 2Z+3/2 3/2 U/O O 1 0, 1 0 O/U Sp Sp/U Sp/U 1/2 1/2 1/2 Sp O/U 2 1/2, 5/2 1/2 O Zirnbauer 96

34 N n N=n=1 any N, n U(1) dual C = U(2)/U(1)xU(1) Gauge U(N) dual U(2n)/U(n)xU(n) Meson Witten 79

35 det(z U) = du = du det(z U) exp( Ψ f f i (zδ ij U ij )Ψ ) j N N L LS [Z]

37 -

38 V x V(x) t t W x t W

39 t >> V 0 -V 1 t << V 0 -V 1 ψ x ψ x t > W Prob( ψ 0 ~ ψ n ) = 1 t < W Prob( ψ 0 ~ ψ n ) = (t/w) n 0

40 g(l) L L+dL g(l+dl) = F(g(L), (L+dL)/L) dlog g/dlog L = β(g) D>2 - β(g) g*. 0 g D>2 D=2 D<2 g(l) ~ exp(-l/ξ) β(g) ~ log g L g(l) ~ L D-1 /L β(g) ~ D-2

42 δx 2 = D δτ >> L 2 /D ( ) 1 = h/ << E c hd / L 2

44 E ψ (x) 2 ξ >> L ξ ~ L ξ << L x

45 Σ ψ 2p ~ V -d p(p-1) Z(E-E ) = Σ ψ Ε 2 ψ Ε 2 ~ E E -(1-d 2) D=3 β=1 β=2 ( )

46 Σ ψ Ε 2 ψ Ε 2 ~ E E -(1-d 2) δn 2 ~ χ L, χ = (1-d 2 )/2 p(s) ~ exp(-κ s), κ = 1/(1-d 2 ) ( ) ( )

47 D=3 (β=1 ) ~ s 1 ~ exp(-s/2χ) ~ log L ~ χ L

48 L(Q) = 1/(V ) [D tr ( Q) 2 + δe tr ΛQ], Q(x) : NG E c 0 E c >> E c ~

50 ( ) H = (H ij ) = 1/(1+a 2 (i-j) 2 ) H UHU + d p ~ 1 - (a/2π 2 )p (g*>>1) d p ~ 1 - (1/g*)p 1 a

51 β=1 β=2 ~ s β ~ e -s/2χ β=4

52 D=3 Anderson

53 D=3 Anderson D=2 Anderson : Evangelou β=1 : β=4 ~ log L ~ χ L

54 [ ] [ ] [ ]

55 [ ] g*

56 D=4 vs Garcia-Verbaarschot β=2

57 D=3

58 g* L β=2 Σ 2 (L) ~ χ L, χ

59 g* g*>>1

60 : << Λ QCD QCD [π] : ε

initiated by Jac Verbaarschot Stony Brook + Akemann G, Altland A, Berbenni-Bitsch ME, Berg BA, Bietenholz W, Bittner E, Dalmazi D, Damgaard PH, Edwards RG, Farchioni F, de Forcrand F, Fyodorov YV,

62 initiated by Jac Verbaarschot Stony Brook + Akemann G, Altland A, Berbenni-Bitsch ME, Berg BA, Bietenholz W, Bittner E, Dalmazi D, Damgaard PH, Edwards RG, Farchioni F, de Forcrand F, Fyodorov YV, Garcia-Garcia AM, Giusti L, Gockeler M, Guhr T, Halasz MA, Hehl H, Heller UM, Hilmoine C, Hip I, Iida S, Jackson AD, Janik RA, Jansen K, Jurkiewicz J, Kaiser N, Kalkreuter T, Kanzieper E, Kiskis J, Klein B, Krasniz A, Lang CB, Luscher M, Lombardo M-P, Ma J-Z, Madsen T, Magnea U, Markum H, Meyer S, Nagao T, Narayanan R, Niclasen R, Nishigaki SM, Nowak MA, Osborn JC, Papp G, Pullirsch R, Rakow PEL, Rabitsch K, Rummukainen R, Schafer A, Schnabel M, Schwenk A, Seif B, Sener MK, Schlittgen B, Simons BD, Shcheredin S, Shrock RE, Shuryak EV, Smilga AV, Stephanov MA, Splittorff K, Toublan D, Takahashi K, Vanderheyden B, Weidenmuller HA, Weitz P, Wettig T, Wilke T, Wittig H, Wohlgenannt M, Zahed I, Zirnbauer MR, + many others RMT LGT

63 ψ D / ψ = ψ + L D µ σ µ ψ L +ψ + R D µ σ µ ψ R N F ψ L U L ψ L U L SU(N F ) L ψ R U R ψ R U R SU(N F ) R (N C, N F ) ψ ψ = ψ R + ψ L + ψ L + ψ R 0 ψ L Uψ L, ψ R Uψ R U SU(N F ) V m q 0

64 Banks Casher 80 d 4 x ψ (x)ψ(x) = tr 1 D / + m = 1 = iλ n + m 2m λ 2 n + m 2 λ n >0 V a = 1 2m dλ ρ(λ) 0 λ 2 + m 2 a 1 2m dλ ρ (λ) 0 λ 2 + m 2 π ρ (0) m 0

65 Σ ψ ψ = π ρ (0) V = π 0 = O(V 1 )=O(L d ) V = 0 = O(L 1 ) free

66 V π ψ ψ SU(3), N F =0, staggered V=4 4 Gockeler et al 99

67 SU(2), N F =0, staggered V=10 4 Berbenni et al 97 ρ s (ζ) = 1 ρ ζ

68 SU(3), N F =0, staggered V=4 4 Damgaard et al 98 ρ(0) β SU, SO, Sp

69 probe fermionic & bosonic quarks Z({m f },m m ) = [da] e S YM [ A] f det( / D + m f ) det( D / + m) det( D / + m ) m log Z({m },m m ) f m= m = tr 1 m + / D {m f } m iλ, Im ρ(λ) Z graded

70 1 Λ QCD << L π Z U =U R U L + : SU(N F ) L SU(N F ) R SU(N F ) V ψ + L Mψ R + c.c. + U u R Uu L + M u L Mu R L chpt = f π 2 tr µ U µ U + Σ Re tr MU +L Weinberg 67

71 Σ= m log Z m = ψ ψ quench L chpt = f π 2 tr µ U µ U + Σ Re tr MU +L f π 2 L 2 Σ m E C f 2 π >> m,λ Z 2 chpt ΣL ε Leutwyler Smilga 92

72 1 L << m π Σ level spacing Thouless E hadron mass

74 / D Z = dhdψ dψ exp tr H + H + ψ f f R ψ L f = dh e tr H + H f det m f ih + ih m f ( ) m f ih + ih m f ψ f L f ψ R H ψ L f, ψ L f ψ R f, ψ R f N x (N+v) N+v N N F

75 D / C SU R Sp H SO N F ν N L LS = Re tr MU

76 Z = dhdψ dψ exp H * f ij H ij + ψ,i f,i R ψ L f ( ) m f ih ji ih ij m f * ψ L f,i ψ R f,i f = dψ dψ exp (ψ,i R ψ g,i L )(ψ g,j f L ψ,j f R )+ m f (ψ,i f R ψ,i f L )+ (ψ,i f ( L ψ,i R )) f = dqdψ dψ exp Q * f fg Q fg + (iq fg + M fg )(ψ,i L ψ g,i R )+ (iq * f fg + M fg )(ψ,i f R ψ,i L ) = dq e -tr Q+Q det N (Q im )det N (Q + im ) { } Q: N F_ x _ N F N N 2 F Z = du e Re tr UM resc U: N F_ x _ N F

77 dµ(h ) = dh e tr H +H Π f det( H + 2 H + m ) f EV( / D ) =±i EV(H + H ) = { ±i λ 1,...,±i λ N,0,...,0} dµ(λ) =Π{ dλ i e λ 2 i Π( λ 2 2 i + m ) f λ } β(ν+1) 1 Π i λ 2 2 i λ β j i f i> j =Π{ dz i e z i Π( 2 z i + m ) f z } β (ν+1)/2 1 Π i i f i> j z i z j β z i 0 (βν/2) ν

78 1~4 N F =0, C hermitian Damgaard SN 01

79 quark mass N F = 3 C hermitian Damgaard SN 98

80 N F = 1 R symmetric N F = 2 H selfdual Nagao SN 00

82 2 SU(2), N F =0, staggered V=8 4 Berbenni et al 97

83 SU(2), N F =4, staggered V=8 4 Berbenni et al 98, Akemann Kanzieper 00 µ m q ρ(0)/π

84 SU(2), N F =4, staggered, V=8 4 Berbenni et al 98

85 SU(2) SU(3) SU(3) adj ν=0 ν=1 N F =0, overlap V=4 4 Edwards et al 99

86 SU(3), N F =0, overlap V=10 4 Bietenholz et al 03 ψ ψ = (256 MeV) 3 (L =1.23 fm)

87 large physical size 1.23 fm 0.98 fm β=5.85 E c ~ 1.2 fm small physical size

88 : Damgaard-SN prediction 00 from chrmt SU(3), N F =0, overlap V=20 4 Giusti Luscher et al 03 (L =1.49 fm)

89 N F =0, overlap V=4 4 Edwards et al 99

90 confine β =5.2 deconfine β =5.4 SU(3), N F =3, staggered V=6 3 x 4, ma=.05 Bittner et al 00 free β= confine β =0.9 Coulomb β =1.1 U(1), N F =0, staggered V=8 3 x 6 Bittner et al 00

91 µ ψ + ψ = µ ψ γ 0 ψ D / D / + µγ 0 det( / D + µγ 0 + m)

92 / D + µγ 0 SU(3), quenched β=5.2 (confine) V=6 3 x 4 Berg et al 00

93 Hanano Nelson 96 ih ih + ih + µ ih + + µ Z = dh e tr H + H f m det f ih + + µ ih + µ m f Stephanov 96 Splittorff Verbaarschot 03

94 µ=0.53 Halasz et al 97

95 SU(3), N F =0, staggered V=8 4 β=5.0 (confinement) Akemann Wettig 03

97 µ µ T

98 q i q j 0 ee 0

99 µ ψ + ψ = µ ψ γ 0 ψ D / D / + µγ 0 Boltzmann det( D / + µγ 0 + m) µ

100 Sp(N C ), SO(N C ) [ D / + µγ 0 + m,t ] = 0 ( / D + µγ 0 + m) ab det( / D + µγ 0 + m) R R H gauge spinor c.c. T = JCγ 5K, T 2 =±1 Cγ 5 K Jσ 2 ψ L * σ 2 ψ L * ψ R SU(N F ) L x SU(N F ) R SU(2N F )

101 qq SU(N C 3) det( / D + µγ 0 + m) R : µ Sp(N C ), SO(N C ) det( D / + µγ 0 + m)det( D / + µγ 0 + m) * det( D / + µγ 0 + m) R qq : µ Kogut Stephanov Toublan 99~01

102 µ f Z = dh e tr H + H f m det f ih + µ f ih + + µ f m f L LS

103 G = SU, N F = 2 Klein Toublan Verbaarschot 03 1st order 2nd order

104 2N F {ψ f, χ f } L mass = m ( f ψ + f ψ f χ + f χ ) f f det( D / 2 + m 2 f ) 0 P x Z 2 (ψ, χ )- m f m 0 f ( ψ + f ψ f χ + f χ ) f SU(2N F ) SU(N F ) x SU(N F )

105 Sp Jσ 2 ψ T ψ gauge spinorflavor ψ χ L kin = 1 2 ΨT (Jσ 2 D / I)Ψ, Ψ= Jσ 2 ψ T Jσ 2 χ T SU(2N F ) Sp(4N F ) SO J 1, Sp(4N F ) O(4N F )

106 Sp 1 L mass = 1 2 ΨT (Jσ 2 M ˆ )Ψ, M ˆ = 1 1 Sp(4N F ) Sp(2N F ) x Sp(2N F ) 1 SO J 1, Sp O, ˆ M

107 Sp(N C )- Dunne-SMN 03 L = 1 2 ΨT (Jσ 2 D / I)Ψ+ m 2 ΨT (Jσ 2 M ˆ )Ψ µ 2 ΨT (ijσ 1 C ˆ )Ψ 1 1 C ˆ = 1 1 Sp(N C ), O(N C )-, O (N C )-

108 1 Λ QCD << L π Sp(4N Σ : F ) Sp(2N F ) Sp(2N F ) m 0, µ 0

109

110

111 L LS

112 L st =L LS cosα = min 1, 4µ2 2 m π

113 L st [Σ]

114

115 Ω LG ({σ}; m,µ,t ) = L LS ({σ };m,µ) condensate modes + Tr log ({σ};m,µ) t [0, 1/T ] Splittorff Toublan Verbaarschot 02

116 QCD 3 G = Sp, N F = 2 Dunne SMN 03 tricr 1st 2nd QCD 3 QCD 4 µ m π 2 ~ T logt ~ T 3/2

117 L eff [Σ;µ,m] F π mass ~ exp(- F π /T) may be generated

118 chiral symmetry kinematics Dirac ψ ψ

°ø±â¾Ð±â±â

°ø±â¾Ð±â±â 20, 30, 40 20, 30, 40 1 2 3 4 5 6 7 8 9 10 3.1 6.3 9.4 12.6 15.7 18.8 22.0 25.1 28.3 31.4 2.4 4.7 7.1 9.4 11.8 14.1 16.5 18.8 21.2 23.6 7.1 14.1 21.2 28.3 35.3 42.4 49.5 56.5 63.6 70.7 5.9 11.9 17.8 23.7