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2 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
3 LGT RMT
4
5 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
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7 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 ψ R U R ψ R U L SU(N F ) L 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
8 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 a 0 π ρ (0) (continuum)
9 Σ ψ ψ = π ρ (0) V = π 0 Δ = O(V 1 ) = O(L d ) VΔ = 0 Δ = O(L 1 ) free Δ
10 V π ψ ψ SU(3), N F =0, staggered V=4 4 Gockeler et al 99
11 SU(2), N F =0, staggered V=10 4 Berbenni et al 97
12 SU(3), N F =0, staggered V=4 4 Damgaard et al 98 ρ(0) β SU, SO, Sp
13 probe fermionic & bosonic quarks f Z({m f },m m ) = [da] e S YM [ A] 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
14
15 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 σ L chpt = f π 2 tr µ U µ U + Σ Re tr MU +L Weinberg 67
16 σ spacetime G / H U(x)
17 σ spacetime G / H iϕ (x) U(x) U 0 e L >> m π 1 U 0 =1 vac. Z = DU(x) G / H e L kin [U ]+ L mass [U ] ( ) Dϕ(x) e ( ϕ )2 +m 2 ϕ 2 +λϕ 4 L ( )dx dx
18 σ spacetime ε G / H L << m π 1 U(x) U 0 G / H 0-mode only vac. Z = DU(x) G / H e L kin [U ]+ L mass [U ] du G / H 0 e V L mass [U 0 ] (N ) = lim Z chrmt N ( ) dx
19 " = # #m log Z m$% = & & quench L chpt = f π 2 tr µ U µ U + Σ Re tr MU +L f π 2 L 2 Σ m f π 2 L 2 >> Σ m Z chpt ε Leutwyler Smilga 92
20 1 L << m π Σ level spacing Thouless E hadron mass
21
22 s 1950s
23
24 H s P Wigner (s) = π 2 s e- π 4 s 2
25 H Prob(E,E ) ~ E -E β β=1,2,4 H
26 T = K C c.c. unitary T 2 = CC * =±1 symm. C = UT 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
27
28 # # 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
29 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
30
31 2 β=0 β=1 β=2 β=4 exp(-s) ~ s β ~ exp(-c β s 2 )
32 s k E β (k ; s) β=1 GOE β=2 GUE
33 N P (s), ρ(λ), N= P (s), ρ(λ), exp(-tr H 2 ) exp(-tr V(H)) exp(-tr (H+A) 2 ) Akemann Damgaard Magnea SN 97 P (s)
34
35 / 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
36 D / C R H SU Sp SO N F ν N L LS = Re tr MU
37 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 F Z = du e Re tr UM U: N F_ x _ N F
38 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 ) β(ν+1) 1 f λ } Π i λ 2 2 i λ β j i f i> j = Π{ dz i e z i Π( 2 z i + m ) β (ν+1)/2 1 f z } Π β i z i z j i f i> j z i 0
39 1~4 N F =0, C hermitian Damgaard SN 01
40 quark mass N F = 3 C hermitian Damgaard SN 98
41 N F = 1 R symmetric N F = 2 H selfdual Nagao SN 00
42
43 Quenched Dynamical quarks Topology Bulk N C U : = exp β plaquette U ij U jk U kl U li Finite density
44
45 SU(2), N F =4, staggered V=8 4 Berbenni et al 98, Akemann Kanzieper 00 µ m q ρ(0)/π
46 SU(2), N F =4, staggered, V=8 4 Berbenni et al 98
47 SU(2) SU(3) SU(3) adj ν=0 ν=1 N F =0, overlap V=4 4 Edwards et al 99
48 SU(3), N F =0, overlap V=10 4 Bietenholz et al 03 ψ ψ = (256 MeV) 3 (L =1.23 fm)
49 large physical size 1.23 fm 0.98 fm β=5.85 E c ~ 1.2 fm small physical size
50 : Damgaard-SN prediction 00 from chrmt SU(3), N F =0, overlap V=20 4 Giusti Luscher et al 03 (L =1.49 fm)
51 N F =0, overlap V=4 4 Edwards et al 99
52 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
53
54 µ ψ + ψ = µ ψ γ 0 ψ D / D / + µγ 0 det( / D + µγ 0 + m)
55 / D + µγ 0 SU(3), quenched β=5.2 (confine) V=6 3 x 4 Berg et al 00
56 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
57 SU(3), N F =0, staggered V=8 4 β=5.0 (confinement) Akemann Wettig 03
58
59 µ f Z = dh e tr H + H f m det f ih + µ f ih + + µ f m f L LS
60 G = SU, N F = 2 Klein Toublan Verbaarschot 03 1st order 2nd order
61 Ω LG ({σ};m,µ,t ) = L LS ({σ };m,µ) condensate modes + Tr log Δ({σ};m,µ) t [0, 1/T ] Splittorff Toublan Verbaarschot 02
62 QCD 3 G = Sp, N F = 2 Dunne SN 03 1st order 2nd order
63 chiral symmetry kinematics Dirac ψ ψ
QCD ( ), ( ) 8 23 Typeset by FoilTEX
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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
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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
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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 )
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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
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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
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72 Maxwell. Maxwell e r ( =,,N Maxwell rot E + B t = 0 rot H D t = j dv D = ρ dv B = 0 D = ɛ 0 E H = μ 0 B ρ( r = j( r = N e δ( r r = N e r δ( r r = : 2005 ( 2006.8.22 73 207 ρ t +dv j =0 r m m r = e E(
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