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- ともみ ぜんじゅう
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2 [2, 3] : : : : : : : : : : : : : : : : 3 2, : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : , : : : : Borel. : : : : : : : : : : : : : : : : : : : :
3 1,..,.,,.,.,. t`hooft [2] SU(2). 1B =1L 6= 0 (1) (.) N. S. Manton [3] Sphaleron( 10Tev M W = weak ).. Sphaleron,., Tev. Shpaleron. H. Aoyama & H. Goldberg[4] Sphaleron. A. Ringwald [5]& o. Espinosa[6].! 2
4 . H. Aoyama & H. Kikuchi[7] Ringwald, Espinosa. V. V. Khose and A. Ringwald[8]. H. Aoyama & H. Kikuchi[9] )., H. Aoyama[10], H. Aoyama & A. Tamra[11], H. Aoyama & I. Okouchi[1]! Asympton 1.1 [2, 3] SU(2) W Sphaleron.,,,., ! 0 v! (r!1) (2),, 8! U 1 0 v! (r!1) (3) 3
5 . ( U.), A!0@ U 1 U 01 (r!1) (4)., U. 2 (SU(2)). 2 (SU(2)) = 1.,.,, ( ). (.),.. S 2 S 2 2 S 1., 2 (SU(2))! 3 (SU(2)) (5)., 3 (SU(2)) = Z.... (..) ( )... (..) Sphaleron. 4
6 Sphaleron,,SU(2) E Saddle Point (Sphaleron) Vacuum... Sphaleron.,Sphaleron.., a.,, Sphaleron.. =0; =0 = ( 5
7 J B J L = 3g Tr(FF3 ) (6) J B ;J L,.(.).., 1B =1L 6= 0,,B + L violation.(b 0 L.) q + q! 7q +3 l +(W;Z;H) (7) (W;Z.),., Sphaleron., Sphaleron M Sph O(M W = 10Tev),. 10Tev. 2,,.,.,. 6
8 2.1,. V(x) x., 9 L, 9 R.,,., H = E 0 E 0! (8) E 6 = E 0 6 (9) 9 + = 1 p2 (9 L +9 R ), 9 6 = 1 p ! (10) 7
9 x 9 0 = 1 p2 (9 L 0 9 R ), x,,..,.,.,. WKB. Z ' exp Pdx (11), P P = q 2V (x) (12). V (x) = 1 2 x2 (1 0 gx) 2 (13) 8
10 V(x) 1/32g 2 0 1/2g 1/g x., g 0. g! 0 V! 1=2x 2. V (x) V (x) = 1 2 x2 + gx g2 x 4 (14), g, g 2 g. P = x(1 0 gx) (15) Z ' exp 0 x(1 0 gx)dx (16) = exp 0 1! (17) 6g 2., g! 0 0. g n, 1 n! d d(g 2 )! n exp 0 1 6g 2! g=0 =( ) 2 exp 0 1 6g 2! g=0 (18) 9
11 exp (01=6g 2 ) 0. g 2..,. WKB,... x =0 x =1=g. hx =1=gj e 0iHt jx =0i = Z Dx e is[x] (19) x = 0 x = 1=g.,. hx =1=gj e 0H jx =0i = Z Dx e 0S E[x] (20) S E [x] = Z 1 dt 2 _x2 + V (x) (21), 0 +.., x = 0 x =1=g., 10
12 x τ,, E = 1 2 _x2 0 V (x) =0 (22) x() = 1 g e 0 (23) x I,.., h1=gje 0H j0i,.,, h1=gj e 0H j0i = X n = X n h1=gje n ihe n j e 0H j0i e 0En h1=gje n ihe n j0i E n..,...,, x() x I (). X x() =x I ()+ c n x (n) () (24) 11
13 x (n) ()., Z Dx e 0S E [x] = Z Y dcn e 0S E[x] (25) x I. P c n x (n) = x,, S E [x] = S E [x I ] Z Z d d 0 d S E x() x x=xi S E x() x( 0 ) x() x( 0 ) (26) x=xi S E =x()j x=xi x I 0. S E =x() x( 0 )j x=xi., x (n), n P n(1=2) n c 2 n., I,. S E (x I ), WKB Z I = e 0S E[x I ] Y P dcn e 0 1 n c 2 2 n (27) S E [x I ]= 1 6g 2 (28) R dc n 111 WKB,., 0., x I, S E x() =0 (29) x=xi Z d 0 S E x() x( 0 ) x=xi dx I d ( 0 )=0 (30) dx=d, 0.,,. 12
14 ,, 0 1=g 0 x I (; 0 ). x, x 0,,,, 0., 0. 1= p 0, 0. Z I = e 0S E[x I ] Y P Z dc n e n1 nc 2 n n1 d 0 J (31) c 0, 0. J.. 0,, 0 1=g,. T, Z d 0 J = TJ (32) I = e S E[x I ] Y s 1 n1 n A TJ (33) T,. I, I = D 1=gj e 0HT j0 E (34) I = X n e 0EnT h1=gje n ihe n j0i (35) T E n e 0E 0T.. 0 1=g. 13
15 x τ0 τ 1:, 0 1=g, 0 1=g. x τ 2: 1=g 0.. AT = e S E[x I ] ( Q q n1 = n )JT ! A3 T 3 (36). 1=3!,.,,
16 AT + 1 3! A3 T ! A5 T = sinh AT = 1 e AT 0 e 0AT (37) 2 e AT e 0AT, E 0 A A, A. 1-, T,,,, + 0 e 0T e +T..,. 1-, =g,. 0 0, 0,,, cosh. cosh sinh h1=gje n i h0je n i. 2.2, Physics Today,,. 15
17 .,.,... V(x) x.., WKB,., -V(x) x,., x x ESC τ 16
18 x ESC. x ESC..,,,,.,,.,.,., 0. dx B =d ẋ τ, ẋ τ dx B =d dx I =d 0., 17
19 ) (0 d2 d + V 00 (x B ()) x (n) () = n x (n) () (38) 2 Schrodinger. Schrodinger 0, dx B =d.. 0 =0., Q n.. Colemann,..., e 0iE 0t 2 = e 00 0 t (39) E =2. 1=0 0.,,,., 0 0. WKB,.,.,,. 2.3,..,,.,.,,. 18
20 ,, Sphaleron. qq TeV,,.,.,..,,. x τ,. e 1=6g2,.,,.,.,,.,,,,. x 0, 1=g, h1=gje 0HT j0i T τ 19
21 ., h1=g 0 je 0HT j0+i.,,.,,,. 1 Z 3! d 1 Z d 2 Z d 3 = T 3 3! (40),, 1 ; 2 ; 3.,.,.,.. x τ δx T τ,,., 0. 0,,.,. 20
22 ,..,,,. T 3 =3!. T 3.,,.,,. V V () = (41) φ,.,.,,,.,,,.,., Colemann.,. 21
23 . V(x) x.,,,,,.,, H = E 0 E 1! (42),..,,.,,,., -V(x) x, x.,, x x,.,, 01,, 22
24 . Fake instability.,., (Complex time formalism),, (Valley method)., interacting instanton.,.,.,.,.,, , fake instability. fake 3. 4,.,,.. Euclidean,., Euclidean,.,. x(t),,,., 23
25 "ff" 3: "gg" 4: 24
26 saddle point 5:., resolvant. resolvant, G R (x i ;x f ; E) = x f 1 E + i 0 H x i = 0i Z 1 0 dt e i(e+i)t hx f je 0iHT jx i i (43) 2, x i x f T.,, G R (x i ;x f ; E) = X n 1 E + i 0 H hx fje n ihe n jx i i (44) resolvant, E. E,., T. T 0 1, saddle point 5.,.,.,.,T 0 d dt h e i(e+i)t hx f je 0iHT jx i i i =0 (45) 25
27 .,.., 0 h i e i(e+i)t hx f je 0iHT jx i i =0 (46) x resolvant T x, T 2,..,2 S x =0 (47) 1 T e i(e+i)t hx f je 0iHT jx i i. x i x f, T. x(t) x(0) = x i,x(t )=x f T ds dt, ds dt = = d dt Z T 0 dx dt 8 < dt : 1 2 dx dt! 2 0 V (x)3 d 2 x dt 2, dx dt. ( dx dt =! 2 = 0 V (x)9 ; 5x=xf + dv (x) + dx Z T 0 dt dx dt 0 t =0 t = T 0 dx dt 26 d 2 x dt dt 0 dv (x) dx dx dt! (48) = 0 (49) (50)
28 E x B x A : x i x f, 2 ds dt = ! 2 dx + V (x)3 5 dt t=t = 0E(T ) (51) T E = E(T ). resolvant resolvant E., resolvant E E.... x i x f 6. x i, x f,e 27
29 resolvant. T.,. Euclidean 1. T, T. resolvant,. x i x f 6 1.,Euclidean.., x i x 0 bounce x f 6 2,. 2. x i,x A 6 3..,. x x t..,x i x A x B, x A x f. T 0 2 T ,.,. G R.., 28
30 T : physical saddle point : unphysical saddle point T0 2T0 3T0 7:.,... (physical saddle point) (unphysical saddle point).. fake instability , 8. 29
31 T 8:.,,...,., WKB,...., T.,., T..resolvant T. 30
32 4.,.. B L. B L.. q, q l q + q! 7q +3 l; (52). G = h1jq 9 l3 j0i (53). j0i; j1i. (53) j0i j1i. Ringwald Espinosa. (53). j0i j1i, 1.., Z Z d 0 1 = 0 d 0 0 = 1 (54).,.. D 31
33 9: q 1 q 2! q 3 l q n ; q n. Dq n = n q n (55) n =0; 1; 2:::,. (55) Dq 0 = 0, S F = qdq q 0. G = h1j0i. (53)..,.. x I q 0 = AS F (x 0 x I ) (56).A., ( 9 (a))...,. g I q 9 l3, 32
34 σ E E 26 10: 1/E 2 E. g I,A,. g I. TeV. E 2 g 2 I E 26 (57). 3,,..., 1=E 2. (57) ( 10). BL, E 4, g I q 9 l 3 E 4, q; l E 3=2 (57). W,Z (57). 33
35 .,.. ( 9 (b))..,,., S =2S I. x I 0 x I.,..,W, 1.,.,.,,...,.,... R, R 11. R 2 34
36 S 2S E 0 R 11: R, R.,. R R,R.. 5., i. i,,.,, 35
37 S valley R 12:. S = S( i i S i... D j S i S: (58),D j S. (58), = 0, =0..,. D ij.. D ij. (59) D ij., 36
38 collective coordinate. collective coordinate,,..,.. i ( 1 2 (@ js) 2 0 S) =0: (60),., (@ js)2 (@ js)2 1 2 (@ js) 2 0 (S 0 c) (61) i.,,.. (62) ( 13.,..,. (58) (),,. i i.. ~ = 0 (), S() =S(()) i S(()) ~ i D ij(()) ~ i j ~ + 111: (63) 37
39 13: 2... R i d i =d.. (), h.. Fadeev-Popov. FP 1(()) R i i S= q (@S) (()) = = Z Z d( ~ i ()R i ()) d( ~ i R i ~! 01 i R i ~ i R i ~! 01 i R i : ~iri=0 1(()) = ~! i R i 0 i 38 R i + ~ jr i ~ i R i =0
40 fr i i R j j ~ g ~ i R i =0 : (65) Z = N Z Y j d j p 2 e 0S (66).N. (66) Z d( ~ i R i )1(()) = 1 (67) Z = N = N = N = N Z Z Z Z Z Y d j Z Y d j Z 1 dk d 01 2 d p j ( ~ i R i ) R i() d ~ j p 2 Z det D exp dk 2 eik ~ l R i 1 dq 2(RD 01 R) det e0s() R i() 0 2 RD01 Rk 2 i R i() e0s() R i() e0s(()) e0s(()) : (68) (58) D 01 R = R=, R (RD 01 R) det D = 1 det D det D (69). Z = N Z 1 dq 2 det R i() e0s(()) : (70) collective coordinate.,,. collective coordinate. 39
41 ,... d i d = f()@ is (71) f(). d i i S. FP.. D ij. (69),. (71),. (71),. 2,. S( 1 ; 2 )= 1 g 2 ( ) 2 +5( ) 2 (72) Green O(p; q) = Z 1 0 d 1 Z 1 01 d 2 p 1 q 2 e 0S( 1; 2 ) (73), y., 14, a; b; c; d (p; q) =(2; 2); (2; 12); (12; 2); (12; 12).. b; c. p; q y. 40
42 2 0.5 a b c d y x : (72).,. O(p; q). 1 = const, X Y..,,... 2.,,. (induced nucleation) z. S = Z d 4 x 1 2 (@ ) 2 + V () ; V () = (1 0 ) 2 0 ( ) (74). 16. collective coordinate. z, (induced nucleation). ( ),. 41
43 pnq : O(p; q) ( ) ( ) (g =0:2). V(φ) φ ε 15: =0:25 (74) 42
44 φ 1 λ3 λ4 λ5 λ2 0.5 λ ρ 16:. 1 =0:3, 2 = 00:2, 3 = 00:34, 4 = 00:25, 5 = 00:2.. collective coordinate,, collective coordinate,. 17, 18..,..,, S = Z 1 d 4 x 4 F 2 + jdj 2 + V () (75).,. S 1 ; S 2 ; S 3..,. 43
45 S λ2 λ1 λ φ λ4 17: jj collective coordinate , E λ1 λ2 λ φ -20 λ4 18: 44
46 (A(x); (x))! ~A(x) = 1 a A(x a ); ~ (x) =( x a ). ~ A(x); ~ (x), ~ S. ~ S S 1 ; S 2 ; S 3, (76) ~S = S 1 + a 2 S 2 + a 4 S 3 (77). S 1 ; S 2 ; S 3 S ~ a a =0. (76).. a.a(x); (x). (76) a. a =0,. I.Aeck 0. 0,. 0, 1,.,,.,.. 45
47 . (60) S = (@ is) 2 (78),. (78)., Borel....,,.,.., Z,.,, : Z = 1X n=0 c n g 2n : (79) c n,. c n n., 46
48 ,. : c n (01) n n! (n!1): (80), J.Zinn-Justin L.N.Lipatov.,. (80).. Borel.. Z : Z = 1X n=0 0(z) =, 0(n +1)=n! c n g 2n 1 n! Z 1 0 Z 1 0 dt t n e 0t : (81) dt t z01 e 0t (82) 1= 1 n! Z 1 0 dt t n e 0t (83) (79)., t, n., Z e Z : ez = Z 1 0 dt 1X n=0 1X n=0 c n n! (tg2 ) n e 0t : (84) c n n! (tg2 ) n (85). f(tg 2 ), t e Z. Z, Z 1, Z = e Z (86) 47
49 ., Z e Z ; e Z Z Borel.. Borel, Borel.. c n =(01) n n! (87). Z. e Z. f(tg 2 ) f(tg 2 ) = = 1X (01) n n! (tg 2 ) n n! n=0 1 (88) 1+tg 2 e Z., Z e Z. n (80) c n Z e. (80), Borel. Borel.. Borel. (87) c n = n! (89). f(tg 2 ), f(tg 2 )= tg 2 (90). f(tg 2 ) t =1=g 2 (88). Borel t f(tg 2 ) 48
50 ,..,. Borel singularity. n, Borel singularity., Borel singularity Borel., Borel Pade. (80),.,, QED. c n n! (n!1) (91). Borel singularity,.,.. Borel singularity,,,. Borel Borel singularity,.,.,,.. 49
51 n!, (01) n n!. asympton. Zinn-Justin, Lipatov,,. n!. asympton. asympoton,,. expert x ,.,,. (.) 2. ( 19..).,,.,..,., Euclid., x, D2... address dai@phys.h.kyotou.ac.jp. 50
52 g 1 x 19: 2. x =1=g 0.. ( 1 )., x =0,.,.. ( 21.),., x 1=g,. 0 1=g,,,. 1=g 0. 1=g. 51
53 S E bounce valley line 20:.. x g 1 21:.. 52
54 ,,.,.,,,., x =0 x =0, x =1=g.,..,. x 1=g.., 0..,.,. x = 0 x = 1=g,.,, 0 1=g, 22,,.,..,. 0 1=g,.,. 53
55 g 1 x 22: x =0 x =1=g.,. 0.,.,., S =0 (92) xi 2 S dx I x I x I d =0 (93).... X 2 @ i (94) 54
56 X j F j 2 j F j = F i (96)., x, i j, (95), (96) 0@ 2 x + V 0 (x) =F (97) 0@ 2 + V 00 (x) F = F (98)., F =0, (97), (98) 0=0,. F (97). F.,. ( ( 19).), F (98)., (97) F : Z d F 0@ 2 x + V x = Z F@ Fd: (99) Z d x (100)., 0. F, F, F 0. Z d d d F 2 (101)
57 F 23: F.. Z d x =0 (102). 0, 0,. x 0 1=g. F ( x F. =0.,.,. 1 q 5 0 n n (103) 56
58 ., Coleman \The uses of instantons"[12].., Faddeev-Popov.,,., (97) (98). =0 2 x + V 0 (x) = F (104) 0@ 2 + V 00 (x) F = 0 (105)..,.. 2 V (x) = 1 2 x2 (1 0 gx 2 ) (106), (106) x. V (x) = 1 2 x2 (1 0 gx 2 ) 0 gx (107) g g 1.. x =0 x =1=g V (0) = V (1=g) =0, V (1=g) =0. x =1=g,.,.. x 0 () = 1 g e 0 (108)
59 . =0.. ( g 2.) x = x 0 + g 2 (109) F = 06 g 2 _x 0 +18(g 2 ) 2 _x 0 (110) (_x 0 x 0.)... (110) < 1 g 2 (111) > 1 (112) x F x,f., g 2 (111) (112)., (109) (110),..,., 2. 58
60 ,,, 0,.....,,,.!...,, N L (113) L = N L ! 2NL+ (01) N L+1 0(0N L 0 ) (114) g 2 N L! E (N:P:).,.. (114),..,, ( ), exp 0 ( ),., N:P:. 59
61 (114) 0 2 g 2! 2NL+ (115).,. g 2, ImE (N:P:) L sin (116).... ImE (P ) L = 0ImE (N:P:) L : (117) (E (P ) L Borel.,.. E (P ) L g 2.. g 2, E (P ) L g 2 Cauchy. E (P ) (g 2 )= 1 2i I dt E(P ) t 0 g 2 (118). E (P ), E (P ).. (118) E (P ) L (g 2 )= 1 Z dt ImE(P ) L (t) (119) t 0 g 2
62 .,.. E (P ) L. g 2. E (P ) L, E (P ) L (g 2 )= E (P ) L;n = 1 Z 1 0 1X n=0 E (P ) L;n g 2n (120) dt ImE(P ) L (t) (121) t n+1. E (P ) L;n. n,. ( g.) t E (P ) L;n., n t E (P ) L;n. : E n = 3 6 0( +1) 0(n + +1) 1+O. 1 n ; (n!1): (122) g 2..., n O(1=n). n,.., (117),, (122). (122). 200 ( g 400 ). 61
63 F 24: (122) 3 n n!. 3 n n!. (122) ,. 62
64 24,., Borel., Borel. Borell.,.,,. 0 0 Borel =1.,, Borel. 1 Borel. meta 63
65 SUSY Borel,.,, Zinn-Justin, Lipatov., ,,. transparent...,... 64
66 .,..,,.,. stream line method.,..,...,,..,.,,.. Lagrange..,.,. Q.? A. Bohor-Sommerfeld,,.,.,,.. 65
67 ,. [1] H. Aoyama, H. Kikuchi, T. Harano, I. Okouchi, M. Sato and S. Wada, Prog. Theor. Phys. Supplement 127 (191997) 1 and references contained therein. [2] t'hooft Phys. Rev. Lett. 37 (191976) 8;PR D14 (191976) [3] N. S. Manton, Phys. Rev. D28 (191983) F. R. Klinkhamar and N. S. Manton, Phys. Rev. D30 (191984) [4] H. Aoyama and H. Goldberg, Phys. Rev. Lett. 188B (191987) 506. [5] A. Ringwald. [6] O. Espinosa, Nucl. Phys. B343 (191990) 310. [7] H. Aoyama and H. Kikuchi, Phys. Lett. B247 (1990) 75; Phys. Rev. D43 (1991) 1999; Int. Mod. Phys. A7 (1992) [8] V. V. Khoze and A. Ringwald, Nucl. Phys. B259 (1991) 106. [9] H. Aoyama and H. Kikuchi, Nucl. Phys. B369 (191992) 219. [10] H. Aoyama,Mod. Phys. Lett. A7 (191992) [11] H. Aoyama, A. M. Tamra, Nucl. Phys. B384 (1992) 229. [12] S. Coleman, Aspects of Symmetry "The uses of instantons" (Cambridge,1985). [13] H. Aoyama, H. Kikuchi, I. Okouchi, M. Sato and S. Wada, quant-ph/ , to be published Phys. Lett. B. 66
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
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72 9 Hooke,,,. Hooke. 9.1 Hooke 1 Hooke. 1, 1 Hooke. σ, ε, Young. σ ε (9.1), Young. τ γ G τ Gγ (9.2) X 1, X 2. Poisson, Poisson ν. ν ε 22 (9.) ε 11 F F X 2 X 1 9.1: Poisson 9.1. Hooke 7 Young Poisson G
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IA hara@math.kyushu-u.ac.jp Last updated: January,......................................................................................................................................................................................
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[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin
2 2.1 F (t) 2.1.1 mẍ + kx = F (t). m ẍ + ω 2 x = F (t)/m ω = k/m. 1 : (ẋ, x) x = A sin ωt, ẋ = Aω cos ωt 1 2-1 x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ
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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
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