( ) 1 (Quantum Field Theory) = Maxwell Dirac Quantum Electro-Dynamics (QED) 1
|
|
- あかり むらかわ
- 5 years ago
- Views:
Transcription
1 ( ) 1 (Quantum Field Theory) = Maxwell Dirac Quantum Electro-Dynamics (QED) 1
2 = 2
3 Key words = : = : = UV divergence = IR divergence massless ( photons) 3
4 QFT = : a = ( ) Λ = 1/a = = UV cut-off λ( a) QFT QFT Spin system on a lattice (microscopic scale = lattice spacing) Superstring theory (microscopic scale = Planck scale (10 34 cm)) 4
5 : QFT QFT : QFT Quantum Chromo Dynamics (Glashow-Weinberg-Salam theory) 5
6 QFT : QFT : scale-dependent (factorization) K.G. Wilson a( 1/Λ) Heisenberg H Λ = J S i S j <i,j> a L 6
7 L (mass scale µ = 1/L) L L/a re-scale = Scale µ : H µ = g n (µ) O }{{} n (µ), factorized form }{{} n UV IR O n (µ) S i S j, ( S i S j ) 2, etc. O n (µ) = local operators g n (µ) =. L 7
8 : ( ) QCD scale-independent Universality µ small g n (µ) g n (µ) O n (µ) = µ g n (µ) Hamiltonian scale independent 8
9 IR Universailty: (universality class) Universality class QFT g n (µ) 9
10 1.1 M.E. Peskin and D.V. Schroeder, An introduction to Quantum Field Theory, Perseus Books, 1995 S. Weinberg, The Quantum Theory of Fields I, II, III, Cambridge University press, 1995 M. Sredniki, Quantum Field Theory, Cambridge University press, 2007 J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 2002 (4th edition) C. Itzykson and J.B. Zuber, Quantum Field Theory, McGraw-Hill, 1980 J.D. Bjorken and S.D. Drell, Relativistic Quantum Fields, McGraw-Hill, 1965 M. Maggiore, A Modern Introduction to Quantum Field Theory, Oxford University press, 2005 G. Sterman, An Introduction to Quantum Field Theory, Cambridge University 10
11 press, 1993 F. Mandl and G. Shaw, Quantum Field Theory, John Wiley and Sons, 1984 P. Ramond, Field Theory, A Modern Primer, Addison-Wesley, 1990 I, II,, 1989,, 2002,,,, 1987,, 2001,
12 2 2.1 = phonon: magnon: plasmon: exciton: phonon qft1-2-1
13 2.1.1 (acoustic phonon) primitive cell n acoustic branch: 1 2 ω k 0 as k 0. optical branch: ( optical ). : ω k ω 0 0 as k 0. Acoustic + Optical qft1-2-2
14 1 + 1 ( ) relativistic massless field Massless phonon Nambu-Goldstone qft1-2-3
15 2.1.2 Discrete Elastic Line 1 : 1 2 N N + 1 Notation: m = (1) N = (2) a = = 1 Λ (3) L = line = Na (4) x n = na = n (5) ζ n (t) = n line (6) N n : N n N 2 (7) qft1-2-4
16 Lagrangian κ = a b = b a T 0 = κb Lagrangian Hamiltonian L = 1 2 m n ζ 2 n 1 2 κ n (ζ n+1 ζ n ) 2 (8) H = 1 π 2 n 2m 2 κ (ζ n+1 ζ n ) 2 n n (9) π n = m ζ n (10) m d2 ζ n dt 2 = κ [(ζ n+1 ζ n ) (ζ n ζ n 1 )] (11) qft1-2-5
17 Fourier : ζ n+n = ζ n ζ n (t) lattice Fourier series : ζ n (t) = 1 N k ζ k (t)e ikx n (12) ζ n = ζ k = ζ k (13) Periodicity k e ikna = 1 ka = 2πr N, r Z (14) x n = na e ikx n = e ikna = e i(k+(2π/a))na k 2π/a qft1-2-6
18 k first Brillouin zone π a < k π a (first Brillouin zone) (15) N r N 2 N First Brillouin zone a Fourier : ζ k = 1 N n ζ n e ikx n (16) e ika(m n) = Nδ mn (17) k k r qft1-2-7
19 (12) Lagrangian L = m 2 ζ k ζ k κ k (1 cos ka)ζ k ζ k (18) k : : ζ k + 2κ m (1 cos ka)ζ k }{{} ω 2 k = 0 (19) ζ k (t) = ζ k e iω kt ζ n (t) = 1 ζ k e ikx n iω k t N k (20) (21) qft1-2-8
20 v φ 2κ ω k = (1 cos ka) = m v φ ω k k 4κ m sin ka 2 (22) (23) k 0 : ω k k Hamiltonian: ζ k : π k = L ζ k = m ζ k (24) qft1-2-9
21 π n Fourier : π n = m ζ n = 1 N m ζk e ikx n = 1 N m ζ k e ikx n = 1 N πk e ikx n Exponent Hamiltonian: H = 1 2m π k π k + m 2 k ω 2 k ζ kζ k (25) k qft1-2-10
22 2.1.3 Continuum or low-energy limit: λ = 2π k a a 0 L k Infinite volume limit: L. k qft1-2-11
23 Continuum Limit: 1. L = Na a 0 N 2. ρ = m/a m 0 3. (Longitudinal) tension T l = κa κ renormalization cut-off Λ = 1/a a 0 : (11) a md 2 ζ n = κa [(ζ a dt 2 n+1 ζ n )/a (ζ n ζ n 1 )/a] 1 a ρ d2 ζ n dt 2 = T l [(ζ n+1 ζ n )/a (ζ n ζ n 1 )/a] 1 a qft (26)
24 : ζ n = ζ(x n, t) ζ n+1 ζ n a = 1 a (ζ(x n + a, t) ζ(x n, t)) = ζ (x n, t) + O(a) (27) ζ (x n, t) ζ (x n 1, t) = ζ (x n 1, t) + O(a) ζ (x n ) (28) a a 0 x n x 1 2 ζ v 2 t 2 v = = 2 ζ x 2 (29) T l ρ (30) qft1-2-13
25 { ( ζ ) 2 ( ) } ζ 2 L = 1 2 T l (vt) x (31) v ) π n = m ζ n = ρ ζ n a (32) a 0 π(x, t) qft1-2-14
26 π n π(x, t) = lim a 0 a = ρ ζ(x, t) (33) : κa ω(k) = k m/a a 0 k sin ka/2 ka/2 T l ρ = kv φ (34) v qft1-2-15
27 : x n = na 1 x n = a 1 ζ k (t) = ζ(x n, t)e ikx n N n = 1 1 ζ(x n, t)e ikx n x n a N n 1 a 1 L L/2 a 0 L/2 π k (t) = m ζ k (t) m a 1 L L/2 = a ρ L L/2 L/2 dxζ(x, t)e ikx (35) L/2 a 0. dx ζ(x, t)e ikx dx ζ(x, t)e ikx (36) qft1-2-16
28 Rescaling Q k (t) a ζ k (t), P k (t) 1 a π k (t) (37) rescaling : Fourier ζ(x, t) = 1 ζ k (t)e ikx = 1 1 a Q k (t)e ikx N N = 1 L Qk (t)e ikx (38) π n π(x, t) = lim a 0 a = 1 a 1 πk (t)e ikx = 1 Pk (t)e ikx N L (39) Fourier qft1-2-17
29 : H = 1 πk π k + 1 2m 2 m ω 2 k ζ kζ k = a Pk P k + m ω 2 k 2m 2a Q kq k = 1 Pk P k + ρ ω 2 k 2ρ 2 Q kq k (40) qft1-2-18
30 Infinite Volume (L ) Limit: Fourier k = 2π/L 0 (38) ζ(x, t) = 1 L Q k e ikx k L 2π k L dk Q k e ikx (41) 2π L Rescaling L : q k 2π Q k (42) ζ(x, t) = 1 dk q k e ikx (43) 2π P k : (39) Q k L : p k 2π P k (44) π(x, t) = 1 dk p k e ikx (45) 2π qft1-2-19
31 : a 0 ( ζ k π k ) rescaling H = 1 Pk P k + ρ ω 2 k 2ρ 2 Q kq k = 1 p k p k k + ρ ω 2 k 2ρ 2 q kq k k k = 1 dk p k p k + ρ dk ω 2 k 2ρ 2 q kq k (46) qft1-2-20
32 1. (k 0 ) 4κ ω k = m sin ka ( 4κ ka 2 = m 2 1 3! ( ) ) ka (47) : k irrelevant terms Universality ω k = v φ k qft1-2-21
33 I x-y a (a) (b) m T ±z ( a) Lagrangian (a 0 ) Lagrangian universality (a) (b) qft1-2-22
34 2. massless (a) Discrete translation in space x n x n + ma (48) a 0 (b) Continuous translation in time t t + t 0 (49) (c) global translation ( ) ζ n ζ n + c for all n (50) n ζ n = c 0 ( c 0 ) = c 0 ( c 0 = 0) massless (ω k k ) = Nambu-Goldstone qft1-2-23
35 : Global : T = 0 ( ) θ Ω 0 Ω θ { Ω θ } Ω θ = R Ω 0, [H, R] = 0, H Ω 0 = E Ω 0 H Ω θ = H(R Ω 0 ) = R(H Ω 0 ) = E(R Ω 0 ) = E Ω θ qft1-2-24
36 Local spin non-zero S( x) = 0 T T c ( ) S( x) 0: S( x) = order parameter Ω θ Massless (gapless) excitation = Nambu-Goldstone : NG : NG ( ) Massless qft1-2-25
37 2.2 : Phonon : [ζ m, π n ] = iδ mn, rest = 0 (51) [ζ k, π k ] = 1 [ζ m, π n ]e ikx m e ik x n N m,n = i e i(k k )am = iδ kk (52) N m qft1-2-26
38 : H = ( 1 2m π kπ k + m ) 2 ω2 k ζ kζ k k = ( 1 ω k 2 mω kζ k ζ k + 1 ) π k π k 2mω k k 1 a k (mω k ζ k + iπ k ) (54) 2mωk a 1 ) k (mω k ζ k iπ k (55) 2mωk (53) Heisenberg algebra [ ] ak, a k = δkk, rest = 0 (56) H = k ω k (a ka k ) (57) qft1-2-27
39 a k, a k (54) 1 ζ k = (a k + a k) (58) 2mωk mωk π k = i 2 (a k a k) (59) ζ n and π n ζ n = 1 N = k k 1 2mωk (a k + a k)e ikx n 1 2mNωk ( ak e ikx n + a ke ikx n ) (60) π n = 1 i mωk 2N ( ak e ikx n a ke ikx n ) (61) qft1-2-28
40 Fourier mode rescaling Q k = a ζ k, P k = π k a (62) Heisenberg algebra : 1 Q k = (a k + a k) (63) 2ρωk ρωk P k = i 2 (a k + a k) (64) ζ n : (60) ζ(x, t) = 1 ( ak e ikx + a ke ikx) (65) 2ρLωk k π n : π(x, t) = lim a 0 π n /a (61) π(x, t) = 1 i ρωk 2L ( ak e ikx a ke ikx) (66) qft1-2-29
41 ζ(x, t) π(y, t) : [ζ(x, t), π(y, t)] = lim [ζ m, π n /a] = i a 0 a δ mn (UV ) π(y, t) [ ] [ ζ(x, t), dyπ(y, t) = lim ζ m, ] (π n /a)a = i δ mn = i a 0 n n [ζ(x, t), π(y, t)] = = iδ(x y), rest = 0 (67) = : qft1-2-30
42 : a k (t) a k(t) t : (57) da k = i[h, a k ] = iω k a k (t) = a k e iω kt dt da k = i [ ] H, a k = iωk a k(t) = a ke iω kt dt ζ(x, t) ζ(x, t) = 1 ( ) a k e i(kx ωkt) + a ke i(kx ω kt) 2ρLωk k (68) (69) (70) qft1-2-31
43 Infinite Volume (L ) Limit ζ(x, t) ζ(x, t) = 1 L ( ) a k e i(kx ωkt) + a ke i(kx ω kt) k 2ρLωk 2π k 1 dk ( ) a(k)e i(kx ωkt) + a (k)e i(kx ω kt) 2πρ 2ωk k scaling L L a(k) 2π a k, a (k) 2π a k (72) scaling q k = L/2πQ k, p k = L/2πP k (71) qft1-2-32
44 L δ 1 = [ ] [ ] ak, a k = ak, a L k k 2π k k = dk [ a(k), a (k ) ] [ a(k), a (k ) ] = δ(k k ) (73) qft1-2-33
45 2.3 1 = = : ( ) Hilbert a n, a n Hilbert a(x), a(x) ζ(x), π(x) 1 Dirac (1927), Principles of Quantum Mechanics (1958) qft1-2-34
46 : atom ) N : U = Hilbert { φ n )}: U (CONS) 1 Hilbert : H N = U N H N Cn1,n 2,...,n N φ n1 ) φ n2 ) φ nn ) (74) qft1-2-35
47 N : Ψ n1...n N ) S( φ n1 ) φ n2 ) φ nn )) SU N (75) S = : a m, a n : V [ ] am, a n = δmn, rest = 0 (76) a m 0 = 0, 0 a m = 0 (77) qft1-2-36
48 Fock V : 0 a n N V N V N V = N V N (78) ψ n1...n N = a n 1 a n 2 a n N 0 V N (79) T : V N SU N T ( a n 1 a n 2 a n N 0 ) = S( φ n1 ) φ n2 ) φ nn )) (80) SU N V N qft1-2-37
49 : { f α )} = CONS: φ n ) = α f α )(f α φ n ) (81) a α a α = { f α )} 2 a n = α a α(f α φ n ) (82) 0 2 = Bogoliubov qft1-2-38
50 : { x)} a n a(x), a n a (x) a(x) and a (x) T ( ) a (x) = dka (k)(k x) = a(x) = dka(k)(x k) = T a (x) 0 = x) (83) 0 a(x)t 1 = (x (84) dk 2π a (k)e ikx (85) dk 2π a(k)e ikx (86) ζ(x) π(x) qft1-2-39
51 a(k), a (k) ζ(x) = 1 2 π(x) = 1 2 i dk 2π ( f(k)a(k)e ikx + f (k)a (k)e ikx) (87) dk 2π ( g(k)a(k)e ikx g (k)a (k)e ikx) (88) [ζ(x), π(y)] = iδ(x y) Remarks f(k)g (k) + f (k)g(k) = 2 (89) f(k) g(k) ( ) ζ(x), π(x) t t ( Schrödinger picture ) qft1-2-40
52 : Hamiltonian SU N V N 1 U 1 ô (1) ô (1) φ n ) = m φ m )o (1) mn (90) o (1) mn = (φ m ô (1) φ n ) (91) : T 1 (90) LHS = T 1 ô (1) φ n ) = (T 1 ô (1) T )T 1 φ n ) RHS = m = (T 1 ô (1) T )a n 0 (92) o (1) mn T 1 φ m ) = m o (1) mn a m 0 (93) qft1-2-41
53 Fock Ô (1) T 1 ô (1) T = m,n a mo (1) mn a n (94) 2 U U 2 ô (2) ô (2) S( φ m ) φ n )) = r,s S( φ r ) φ s ))o (2) rs,mn (95) T 1 LHS = (T 1 ô (2) T )T 1 S( φ m ) φ n )) = Ô (2) a ma n 0 RHS = = T 1 r,s S( φ r ) φ s ))o (2) rs,mn = r,s a ra so (2) rs,mn 0 qft1-2-42
54 Fock 2 Ô (2) = r,s,m,n 1 2 a ra so (2) rs,mn a ma n (96) : Ĥ n Ĥ = Ĥ (1) + Ĥ (2) + (97) Ĥ (1) = kinetic term hopping term Ĥ (2), Ĥ (3),... = Ĥ (1) : Ĥ (1) = dke(k)a (k)a(k) (98) [ a(k), a (k ) ] = δ(k k ) (99) qft1-2-43
55 dx a(k) = e ikx a(x) (100) 2π Ĥ (1) = dxa (x)e(k = i x )a(x) (101) Ĥ (2) : N 2 ĥ (2) = f( ˆx i ˆx j ) (102) 1 i j N 2 f( ˆx ŷ )S x) y) = f( x y )S x) y) (103) Ĥ (2) = 1 dxdya (x)a (y)f( x y )a(x)a(y) 2 = 1 dxdyf( x y )ρ(x)ρ(y) (104) 2 (105) qft1-2-44
56 ρ(x) = a (x)a(x) = [ρ(x), a(y)] = δ(x y)a(y), [ ρ(x), a (y) ] = δ(x y)a (y) (106) Schrödinger Heisenberg (Picture): Schrödinger t t-independent Schrödinger : i t ψ S (t) = H ψ S (t) (107) ψ S (t) = e iht ψ S (0) e iht ψ H (108) ψ H = Heisenberg qft1-2-45
57 Ô S : ψ 1 (t) Ô S ψ 2 (t) = ψ 1,H e iht Ô S e iht ψ 2,H ψ 1,H Ô H (t) ψ 2,H (109) Ô H (t) e iht Ô S e iht = (110) Ô H (t) dô H (t) dt [ ] = i H, Ô H (t) (111) Hisenberg ζ(x, t) : ζ(x, t) = e iht ζ(x)e iht (112) dζ(x, t) = i[h, ζ(x, t)] (113) dt qft1-2-46
58 a m a m Heisenberg da ( ) m = i[h, a m ] = i o (1) mn dt a n + a no (2) mn,rs a ra s + (114) da m = i [ ] ) H, a m = i (a no (1) nm dt + a ra so (2) rs,nm a n + (115) Remark: t m x ) Schrödinger qft1-2-47
59 : ζ(x) = dk 2π ζ(k)e ikx (116) ζ(k) = 1 ( f(k)a(k) + f ( k)a ( k) ) (117) 2 dk π(x) = π(k)e ikx (118) 2π π(k) = 1 2 i ( g( k)a( k) g (k)a (k) ) (119) [ζ(x), π(y)] = iδ(x y) f(k)g (k) + f (k)g(k) = 2 (120) qft1-2-48
60 (120) f(k) = f( k) = 1 g(k) = 1 g( k) = (121) 1 ζ(k) = (a(k) + a ( k)) (122) 2 g(k) π(k) = g(k) 2i (a( k) a (k)) (123) a(k), a (k) a(k) = 1 ( g(k)ζ(k) + 2 a (k) = 1 2 ( g(k)ζ( k) i ) g(k) π( k) i ) g(k) π(k) (124) (125) qft1-2-49
61 H = dke(k)a (k)a(k) = dk E(k) ( ) 1 2 g(k) 2π(k)π( k) + g(k)2 ζ(k)ζ( k) i dk E(k) [ζ(k), π(k)] } 2 {{} E 0 (126) g(k) 2 = E(k) (127) H = dk 1 2 ( π(k)π( k) + E(k) 2 ζ(k)ζ( k) ) E 0 (128) qft1-2-50
62 : E(k) = k 2 /2m ( 1 H = dx 2 π2 + 1 ) ζ) 2 E 8m 2( 2 0 (129) : E(k) = k 2 + m 2 H = dx 1 ( π 2 + ( ζ) 2 + m 2 ζ 2) E 0 (130) 2 : {b m, b n = { b m, n} b = 0 (131) { } bm, b n = 1 (132) qft1-2-51
63 2.4 2 k 2 /2m ζ(x, t) Ψ ζ c (x, t) Ψ ζ(x, t) Ψ (133) Ψ : qft1-2-52
64 Ψ 0 1. ζ c (x, t) 2. Ψ ζ(x, t) n Ψ n ζ c (x) n 2 : n = 2 Ψ ζ(x, t) 2 Ψ = Φ Ψ ζ(x, t) Φ Φ ζ(x, t) Ψ (134) Φ ζ c (x, t) 2 = Ψ ζ(x, t) Ψ Ψ ζ(x, t) Ψ qft1-2-53
65 : 1. ζ(x, t) ζ(x, t) = a(x, t) + a (x, t) [ a(x, t), a (y, t) ] O( ) (135) 0 2. Ψ z(x, t) a(x, t) a(x, t) Ψ = z(x, t) Ψ (136) a(x, t) n Ψ = z(x, t) n Ψ, Ψ a (x, t) n = Ψ z (x, t) n qft1-2-54
66 Ψ ζ(x, t) n Ψ = Ψ (a(x, t) + a (x, t)) n Ψ 2 = Ψ (z(x, t) + z (x, t)) n Ψ = ζ c (x, t) n (137) (136) = (coherent state) (1) (a, a ) (x, p) [p, x] = i p = i x pψ k (x) = kψ k (x) ψ k (x) = ce ikx (138) [ ] a, a = a = / a a z = z z, z = c(z)e za / 0 (139) z a = z z, z = 0 e z a/ c(z) (140) qft1-2-55
67 z = c(z)e za / 0 = c(z) ( 0 + z a ! ( ) ) z 2 a (141) = c(z) : w z = c(z)c(w) 0 e w a/ e za / 0 a/ = / a a e ω a/ f(a )e ω a/ = f(a + ω ) (142) w z = c(z)c(w) 0 e z(a +w )/ 0 = c(z)c(w) e w z/ (143) qft1-2-56
68 w = z c(z) convention z z = 1 c(z) = e z 2 /2 w z = e 1 2 ( z 2 + w 2 2w z) (144) (145) (2) ( ζ(x) ζ(x) = n (a n φ n (x) + a nφ (x)) (146) ζ(x) Ψ = e 1 2 Ψ ζ(x) Ψ = n n z n 2 e n z na n/ 0 (147) (z n φ n (x) + z n φ (x)) (148) qft1-2-57
69 : (point-splitting) Ψ ζ(x)ζ(y) Ψ = Ψ (a m φ m (x) + z m φ m (x))(z nφ n (y) + a nφ n (y)) Ψ = Ψ ζ(x) Ψ Ψ ζ(y) Ψ + φ n (x)φ n (y) n = Ψ ζ(x) Ψ Ψ ζ(y) Ψ + δ(x y) (149) aa = + a a ( δ(x y) = 0 ζ(x)ζ(y) 0 ) Coherent state Ψ qft1-2-58
70 2.5 massless 1 ( suppress ) Ψ(x) = {Ψ(x), Ψ(y)} = { Ψ (x), Ψ (y) } = 0 (150) { Ψ(x), Ψ (y) } = δ(x y) (151) H = dx Ψ (x) 1 ( ) 1 2 2m i x Ψ(x) (152) E = k2 2m (153) qft1-2-59
71 : E F E = E F, k = ±k F (left and right moving) (154) k F = 2mE F > 0 (155) k << k F E = k2 2m = (k F + k) 2 2m E F + k F m k = E F + v F k (156) E = E E F v F k (157) v F = massless fermion qft1-2-60
72 Ψ(x) Ψ(x) = ψ L (x) }{{} slow e ik F x }{{} fast + ψ R (x) }{{} slow e ik F x }{{} fast (158) ψ L (x) ψ R (x) k( k F ) e ±2ik F x x 2ψ xψ k/k F k/m H = iv F ˆN = dx (ψ L xψ L ψ R xψ R ) + E F ˆN }{{} f ermi energy (159) dx (ψ L ψ L + ψ R ψ R) = number operator (160) qft1-2-61
73 : H eff = iv F dx (ψ L xψ L ψ R xψ R ) (161) 1+1 masssless fermion (with c = 1) S = dtdx ψiγ µ µ ψ = i dtdx ψ(γ 0 t + γ 1 x )ψ = i dtdxψ ( t + γ 5 x )ψ (162) ( ) ψ = ψ R ψ L, ψ = ψ γ 0 (163) γ 0 = σ 1, γ 1 = iσ 2 (164) γ 5 γ 0 γ 1 = σ 3 (165) qft1-2-62
74 : π = iψ (166) H D = dxπ ψ { } L = dx iψ ψ (iψ ψ + iψ γ 5 x ψ) = i dxψ γ 5 x ψ = i dx(ψ L xψ L ψ R xψ R ) (167) v F = c H eff Massless fermion : Ψ Ψ + θ(θ ) E F ˆN Nambu-Goldstone fermion qft1-2-63
75 3 : Lorentz 3.1 = ( ) qft1-3-1
76 3.1.1 : SU(2) SL(2, C) SU(2): U G = SU(2) (special unitary) { U = 2 2 U U = 1 unitary, det U = 1 special (1) U i SU(2) = U 1 U 2 SU(2) U 1 = U (U 1 U 2 )U 3 = U 1 (U 2 U 3 ) associativity (2) (i) (U 1 U 2 ) (U 1 U 2 ) = U 2 U 1 U 1U 2 = 1 (ii) det(u 1 U 2 ) = det U 1 det U 2 = 1 qft1-3-2
77 SU(2) ( ) a b U = c d det U = 1 ad bc = 1 (3) U U = 1 a 2 + c 2 = b 2 + d 2 = 1 (4) 0 = a b + c d (5) (3) 2 (4) 2 (5) = 3 U = ( a b b a ), a 2 + b 2 = 1 (6) qft1-3-3
78 su(2): SU(2) Lie : U = e ix Unitary U = 1 + ix + O(X 2 ) (7) U U (1 ix )(1 + ix) 1 + i(x X ) = 1 X = X X (8) det U = 1 : Y det e Y = e TrY (9) qft1-3-4
79 : Y y i det e Y = i e y i = e i y i = e TrY // (10) det U = det e ix = e itrx = 1 TrX = 0 (11) e ix SU(2) X = X, TrX = 0 (12) qft1-3-5
80 X ( ) a b (i) X = c d = X = a = a, d = d real, c = b (13) (ii) traceless TrX = a + d = 0 d = a (14) ( a b c d ) X = ( a b b a ) (15) qft1-3-6
81 Pauli θ a (a = 1 3) =3 X = 1 ( ) θ 3 θ 1 iθ 2 = θ a s a (16) 2 θ 1 + iθ 2 θ 3 a s a = 1 2 σ a ( ) 0 1 σ 1 = 1 0, σ 2 = ( 0 i i 0 ), σ 3 = ( ) (17) s a traceless s a [s a, s b ] = iɛ abc s c (18) SU(2) Lie su(2) qft1-3-7
82 3.1 1 U SU(2) e ix, X =traceless hermitian e i a θ aσ a /2 = cos θ 2 + iˆθ σ sin θ 2 (19) θ θ, ˆθ θ θ (20) SL(2, C) Lie : U G = SL(2, C) (speical linear) { U = 2 2 det U = 1 (21) det U = = 6 3 qft1-3-8
83 SL(2, C) Lie e ix SL(2, C) TrX = 0 (22) 3 X = φ a s a, s a = 1 2 σ a (23) a=1 φ a = (24) sl(2, C) 3 s a. su(2) SL(2, C) SU(2) U = e ix, X = c a s a, c a = SL(2, R) { θ a for SU(2) φ a for SL(2, C) (25) U = e X, X = c a s a, c a = (26) qft1-3-9
84 3.1.2 x µ : convention 1 η µν = diag (1, 1, 1, 1) time-favored (27) µ, ν = 0, 1, 2, 3 (28) x µ : x µ = Λ µ νx ν (29) x µ x µ = x µ x ν η µν = (ct) 2 x x x µ x ν η µν = Λ µ ρλ ν σx ρ x σ η µν = x ρ x σ η ρσ (30) Λ µ ρλ ν ση µν = η ρσ Λ T ηλ = η (31) 1 Time-favored convention Space-favored η µν = diag( 1, 1, 1, 1) qft1-3-10
85 Λ = e ξ ξ (32) (31) Λ = e ξ 1 + ξ (32) x µ x µ + ξ µ νx ν (33) ξ T η + ηξ = 0 (ηξ) T = ηξ (34) ηξ : ηξ 6 L ρσ (L ρσ ) µν = i(η ρµ η σν η σµ η ρν ) (35) L T ρσ = L ρσ ηξ α ρσ qft1-3-11
86 ηξ = α ρσ L ρσ (ηξ) µν = η µρ ξ ρ ν = ξ µν = α ρσ (L ρσ ) µν = iα ρσ (η ρµ η σν η σµ η ρν ) = 2iα µν α µν = i 2 ξ µν (36) ξ µ ν ξ µ ν = i 2 ξρσ (L ρσ ) µ ν (37) (L ρσ ) µ ν = i(δ µ ρ η σν δ µ σ η ρν) (38) L ρσ = Lorentz qft1-3-12
87 : [L µν, L ρσ ] = 1 i (η µρl νσ η νρ L µσ + η νσ L µρ η µσ L νρ ) (39) 3.2 (39) L µν : L µν (i = 1, 2, 3): 3 rotations I i 1 2 ɛ ijkl jk (40) 3 boosts K i L i0 (41) qft1-3-13
88 3.3 I 3 = L 12 z δ I3 x i = iθ(l 12 ) i jx j (42) 3.4 K 1 = L (x 0 ± x 1 ) 3.5 [I i, I j ] = iɛ ijk I k (43) [I i, K j ] = iɛ ijk K k (K i 3 ) (44) [K i, K j ] = iɛ ijk I k (45) qft1-3-14
89 3.1.3 Lorentz SO(1, 3) J (±) i 2 J (±) i 1 2 (I i ik i ) (46) I i K i : J (±) (43) (45) J (±) k k J ( ) k = J (+) k (47) [ ] J (±) i, J (±) j [ ] J (+) i, J ( ) j = iɛ ijk J (±) k (48) = 0 (49) 2 qft1-3-15
90 SU(2) k : 1 2 ξρσ L ρσ = ξ i0 L i ξij L ij = ξ i0 K i ξij ɛ ijk I k ( ) ( ) 1 1 = 2 ξij ɛ ijk + iξ k0 J (+) k + 2 ξij ɛ ijk iξ k0 J (±) J ( ) k θ k J (+) k + θ k J ( ) k (50) θ k 1 2 ξij ɛ ijk + iξ k0 = 3 complex parameters (51) J ( ) k = J (+) k i 2 ξρσ L ρσ = iθ k J (+) k iθ k J ( ) k = ( iθ k J (+) k) + ( iθ k J (+) k) (52) qft1-3-16
91 3 + 1 SO(1, 3) = SL(2, C) SL(2, C) ( 6 ) Lorentz SL(2, C) SL(2, C) G : n n ρ : U G ρ(u) = n n (53) ρ(u 1 U 2 ) = ρ(u 1 )ρ(u 2 ) (54) qft1-3-17
92 = (defining) : M = ( a b c d ), det M = ad bc = 1 (55) u α = M α β u β SL(2, C) spinor (56) : u α ɛ αβ v β = SL(2, C), ɛ 12 1 (57) u αɛ αβ v β = ɛ αβ M α α M β β u α v β (58) ɛ αβ α M β α M β = ɛ α β ɛ αβ (59) cɛ α β c = 1 α = 1, β = 2 ɛ αβ M α 1 M β 2 = det M = 1 = cɛ 12 = c (60) qft1-3-18
93 (59) M T ɛm = ɛ ɛmɛ T = M T 1 (61) (contragredient) M T 1 : M 1 M 2 = M 3 M T 2 M T 1 = M T 3 M T 1 1 M T 1 2 = M T 1 3 (62) u = Mu ɛ (61) u = Mu (ɛu ) = (ɛmɛ T )(ɛu) = M T 1 (ɛu) (63) ɛu u α u α ɛ αβ u β (64) qft1-3-19
94 ɛ γα u α = ɛ γα ɛ αβ }{{} δ β γ u β = u γ u α = ɛ αβ u β = u β ɛ βα (65) u α ɛ αβ v β = u α v α = u β v β (66) : M : M 1 M 2 = M 3 M 1 M 2 = M 3 (67) (dotted spinor) u α (u α ) (68) u α = M α u β (69) qft1-3-20
95 3.1.5 J (±) k T Lorentz SO(1, 3) = SL(2, C) SL(2, C) J (±) k J (+) 1 = i 0 0 i 0, J (+) 2 = i i 0 0 etc. : ( x y) im x i y m A B (A B)( x y) (A x) (B y) (A B) im;jn = A ij B mn (70) qft1-3-21
96 (A B)(C D) = (AC) (BD) (71) 3.6 σ k 2 1 = 1 2 ( ) (σ k ) 11 1 (σ k ) 12 1 (σ k ) 21 1 (σ k ) 22 1, 1 σ k 2 = 1 2 ( σ k 0 0 σ k ) A B A B J (+) k J ( ) k T T J (+) i T 1 = J (+) i T J ( ) i T 1 = J ( ) k Σ(+) i 2 1 (72) 1 Σ( ) i 2 (73) 1 2 Σ(±) i SL(2, C) SL(2, C) 2 2 J ± i qft1-3-22
97 T T : SO(1, 3) SL(2, C) SL(2, C) x µ u α u α (74) T : x µ u α u α T T α α,µ T µ : T T α α,µ (T µ ) α α (75) [(72), (73)] T 2T J (±) i = 2J (±) i T (76) (+) 2(T µ ) α α (J (+) i ) µ ν = 2(J (+) β i ) β α α (T ν ) β β = Σ(+) i α β δ β α (T ν ) β β = Σ (+) i α β (T ν ) β α = (Σ (+) i T ν ) α α (77) qft1-3-23
98 ( ) 2 2 2T µ J (+) µ i ν = Σ (+) i T ν (78) 2T µ J ( ) µ i ν = T ν Σ ( ) T i (79) J (±) i (46) 2J (±) µ i ν = iɛ ijk η µj δ k ν ± (δµ i η ν0 δ µ 0 η νi) (80) (78) (79) iɛ ijk T j δ k ν + T iη ν0 T 0 η νi = Σ (+) i T ν (81) iɛ ijk T j δ k ν T iη ν0 + T 0 η νi = T ν Σ ( ) T i (82) qft1-3-24
99 ν : (+) case: ν = 0 T i = Σ (+) i T 0 ν = i T 0 = Σ (+) i T i ν = k i iɛ ijk T j = Σ (+) ( ) case: ν = 0 T i = T 0 Σ ( ) ν = i ν = k i i i T k T T T 0 = T i Σ ( ) i iɛ ijk T j = T k Σ ( ) i = σ i Σ (+) i (+) ( ) Σ ( ) i T 0 = 1, T i = σ i (83) Σ ( ) i = σ T i = σ i (84) T qft1-3-25
100 T µ σ µ (σ µ ) α α = T α α,µ = (1, σ) (85) T = i i , T 1 = i i = 1 2 T (86) T 1, σ 1, σ 2, σ 3 ( ) 2 T 1 T = δ ν µ. T 1 1, σ1, σ 2, σ 3 T 1 σ µ : T : T α β,µ ( (α β) ) T 1 : (T 1 µ,α β ) qft1-3-26
101 2 2 : T α α,µ (σ µ ) α α (87) (T 1 ) µ,α β 1 2 ( σµ ) βα (88) α β α β σ µ (T 1 ) µ,α β T α β,µ = 1 2 ( σµ ) βα (σ µ ) α β (89) ( σ µ ) αβ = (1, σ) αβ (90) ( σ µ ) αβ = (1, σ) αβ (91) qft1-3-27
102 σ µ σ µ : (i) T T 1 = 1 (σ µ ) α β( σ µ ) γδ = 2δ δ α δ γ β (92) (ii) (iii) T 1 T = Tr( σµ σ ν ) = δ µ ν (93) bilinear : σ µ σ ν + σ ν σ µ = 2η µν (94) σ µ σ ν + σ ν σ µ = 2η µν (95) σ µ = η 0µ η µi σ i, σ ν = η 0ν + η νj σ j (96) qft1-3-28
103 (iv) σ µ σ µ Pauli ɛσ i ɛ T = σ i, σt i = σ i (97) ɛσ T µ ɛt = ɛ(1, σ T )ɛ T = (1, σ) = σ µ (98) SO(1, 3) SL(2, C) SL(2, C) x µ x = Λx = e iθ kj (+) k e iθ k J( ) k x (99) qft1-3-29
104 SL(2, C) SL(2, C) T T x = T e iθ kj (+) k T 1 T e iθ k J( ) k T 1 T x = e iθ kj (+) k e iθ k J ( ) k T x = (M M ) T x (100) M e iθ kσ k /2 SL(2, C) (101) M = e iθ k σ k /2 SL(2, C) (102) σ ν (100) = (T x ) α α = (σ µ ) α α x µ = M β α M α β (σ ν ) β β xν = (Mσ ν M ) α α x ν (103) qft1-3-30
105 SO(1, 3) SL(2, C) SL(2, C) σ µ x µ = M σ ν x ν M (104) σ µ Λ µ ν = M σ ν M (105) Λ = e iθ kj (+) k e iθ k J( ) k (106) M = e iθ kσ k /2 SL(2, C) (107) SL(2, C) SL(2, C) SU(2) SL(2, C) ( ) qft1-3-31
106 u α =SL(2, C) = 1/2 building block SL(2, C) u α = M α β u β, M = u 1 = au 1 + bu 2 u 2 = cu 1 + du 2 ( a b c d ), ad bc = 1 (108) J 3 = 1 2 σ 3 ( ) ( ) u 1 u 1 u = (1 + θj 3 ) 2 u 2 ( u 1 = ) 2 θ u 1, j 3 = 1 (109) ( 2 u 2 = 1 1 ) 2 θ u 2, j 3 = 1 (110) 2 qft1-3-32
107 n + 1 u 1 u 2 n n + 1 ζ k u n k 1 u k 2 SL(2, C), k = 0, 1, 2,..., n = integer (111) ζ k = u n k 1 u k 2 = (Mu)n k 1 (Mu) k 2 = (au 1 + bu 2 ) n k (cu 1 + du 2 ) k D kl (M)ζ l (112) n + 1 ζ k (n + 1) D(M) 3.7 D(M M) = D(M )D(M) SL(2, C) n + 1 qft1-3-33
108 : D(M ) kl D(M) lm ζ m = D(M ) kl (au 1 + bu 2 ) n l (cu 1 + du 2 ) l = (a (au 1 + bu 2 ) + b (cu 1 + du 2 )) n k (c (au 1 + bu 2 ) + d (cu 1 + du 2 )) k = ((a a + b c)u 1 + (a b + b d)u 2 ) n k + ((c a + d c)u 1 + (c b + d d)u 2 ) k = ((M M) 11 u 1 + (M M) 12 u 2 ) n k + ((M M) 21 u 1 + (M M) 22 u 2 ) k = D(M M) kl ζ l (113) : ζ 0 = u n θj 3 ζ 0 = (u 1 )n = ( ) n θ u (1 n1 + n2 ) θ ζ 0 (114) j = n/2 n + 1 = 2j + 1 (2j + 1) D j (M) qft1-3-34
109 : D 0 (M) = 1. D 1/2 (M) = M. Clebsh-Gordan : CG : D j 1 D j 2 = D j 1+j 2 D j 1+j 2 1 D j 1 j 2 (115) D 1/2 D 1/2 = D 1 D 0 (116) qft1-3-35
110 3.1.8 SL(2, C) SL(2, C) SL(2, C) SL(2, C) ζ kk = (u 2j k 1 u k k 2 )(u2j u k 1 2 ) (117) 0 k 2j, 0 k 2j (118) ζ kk = D jj (M, M ) kk ;ll ζ ll (119) (2j + 1)(2j + 1) ( ) : D 00 (M, M ) = 1 D (M, M ) = M D 01 2(M, M ) = M D (M, M ) = M M Lorentz (cf (100)) qft1-3-36
111 3.1.9 : (σ µ ) α β : SO(3, 1) SL(2, C) SL(2, C) D V α β (σ µ ) α β V µ (120) : ( σ µ ) βα V α β ( σµ ) βα = (σ ν ) α β ( σµ ) βα V ν = Tr(σ ν σ µ )V ν = 2V µ qft1-3-37
112 V µ = 1 2 Tr(V σµ ) (121) V µ U µ : bispinor bispinor : U α β = ɛ αγ ɛ β δu γ δ = ɛ αγ ɛ β δ(σ µ ) γ δu µ V α β = (ɛσ µ ɛ T ) α βu µ = (ɛσ T µ ɛt ) βα U µ = ( σ µ ) βα U µ (122) V α βu α β = (σ µ ) α β( σ ν ) βα V µ U ν = Tr(σ µ σ ν )V µ U ν = 2V µ U µ (123) qft1-3-38
113 : µ Lorentz bi-spinor α β = (σ µ) α β µ = T α β,µ µ (124) : = i i } {{ } T = i 2 1 i β α β γ = (σ µ ) α β ( σν ) βγ µ ν = 1 2 (σµ σ ν + σ ν σ µ ) α γ µ ν = δ γ α µ µ = δ γ α 2 (125) qft1-3-39
114 α β α γ = (σ µ ) α β( σ ν ) γα µ ν = 1 2 ( σν σ µ + σ µ σ ν ) γ β µ ν β γ = δ γ β µ µ = γ β 2 (126) α β α β = 2 2 (127) T µν = T α β;γ δ = (σ µ ) α β(σ ν ) γ δt µν (128) qft1-3-40
115 D D SL(2, C) SL(2, C) Clebsh-Gordan = 1 s 0 a, a = antisymmetric, s = symmetric D D 2 2 = D 11 (9) }{{} sym. traceless D 10 (3) }{{} SD D 01 (3) }{{} ASD D 00 (1) }{{} scalar (129) ( T µν = T (µν) 1 ) 4 ηµν T ρ ρ + T [µν] ηµν T ρ ρ = (10 1) 6 1 (130) T (µν) = 1 2 (T µν + T νµ ), T [µν] = 1 2 (T µν T νµ ) (131) qft1-3-41
116 T [µν] = T [µν] SD + T [µν] ASD D 10 (3) D 01 (3) (132) (Anti-)Self-Dual : F µν = : (dual) F µν F µν i 2 ɛ µνρσf ρσ (133) ɛ , ɛ 0123 = 1 (134) i F µν = F µν (135) qft1-3-42
117 F µν = i 2 ɛ F µνρσ ρσ = i 2 ɛ i µνρσ 2 ɛρσλτ F λτ = 1 4 ɛ µνρσɛ λτ ρσ F λτ = 1 2 (δλ µ δτ ν δτ µ δλ ν )F λτ = F µν (136) Self-dual (SD) anti-self-dual(asd) : (SD) F (+) µν = F (+) µν (137) (ASD) F ( ) µν = F ( ) µν (138) SD ASD F (+) µν F µν = F (+) µν + F ( ) µν (139) F (+) µν = 1 2 (F µν + F µν ), F ( ) µν = 1 2 (F µν F µν ) (140) F ( ) µν F ( ) µν = F (+) µν qft (141)
118 (A)SD (i.e. det Λ = +1) (A)SD 3.8 F µν SD F µν = Λ µ ρλ ν σf ρσ SD : F µν = i 2 ɛ µνρσf ρσ = i 2 ɛ µνρσλ ρ τλ σ λf τ λ = i 2 ɛ µνρσλ ρ τλ σ i λ 2 ɛτ λαβ F αβ (142) Λ µ ν Λ ρ ν Λ ν σ = δ ρ σ (143) F αβ = Λ γ αλ δ βf γδ (144) qft1-3-44
119 ɛ 0123 = 1 F µν = 1 4 ɛ µνρσλ ρ τλ σ λɛ τ λαβ Λ γ αλ δ βf γδ = 1 4 ɛ µνρσɛ ρσγδ F γδ det Λ = det Λ F µν = F µν // (145) (A)SD : SD D 10 T (αγ) = 1 2 (T αγ + T γα ) (146) T αγ ɛ β δt α β;γ δ = ɛ β δ(σ µ ) α β(σ ν ) γ δt µν = (σ µ ɛσ T ν ) αγt µν = (σ µ σ ν ɛ) αγ T µν (147) σ µ σ ν ɛ (σ µ σ ν ɛ) T = ɛ T σ T ν σt µ = (ɛt σ ν ɛ) T (ɛ T σ T µ ɛ)ɛt = σ ν σ µ ɛ T = σ ν σ µ ɛ (148) qft1-3-45
120 T (αγ) = 1 2 [(σ µ σ ν σ ν σ µ )ɛ] αγ T µν = 1 2 [(σ µ σ ν σ ν σ µ )ɛ] αγ T [µν] = i(σ µν ɛ) αγ T [µν] (149) (σ µν ) α β i 2 (σ µ σ ν σ ν σ µ ) α β (150) σ µν SD σ µν = i 2 ɛ µνρσσ ρσ (151) 3.9 σ µ, σ µ qft1-3-46
121 T (αγ) = i i 2 ɛ µνρσ(σ ρσ ɛ) αγ T [µν] = i(σ ρσ ɛ) αγ T [ρσ] = i(σ ρσ ɛ) αγ T [ρσ] (152) (149) ASD D 01 T (αβ) = i(σ µν ɛ) αβ T (+)µν (153) ( σ µν ) α β i 2 ( σ µσ ν σ ν σ µ ) α β (154) σ µν = σ µν (155) T ( α β) = ɛαγ T α β;γ δ = i(ɛt σ µν ) α β T ( )µν (156) 3.10 qft1-3-47
122 3.2 = SL(2, C) SL(2, C) Klein-Gordon(scalar) φ D 00 α βφ = 0 (= (σ µ ) α β µ φ) (157) µ φ = 0 φ = constant: α β α β = 2 2 qft1-3-48
123 φ 1 2 α β α βφ = 2 φ = m 2 φ ( 2 + m 2 )φ = 0 Klein-Gordon (158) = E 2 = p 2 + m 2 (159) Weyl Weyl ξ α D : α β ξ α = ( σ µ ) βα µ ξ α = 0 Weyl (160) σ µ = (1, σ) (µ ) qft1-3-49
124 m 2 = 0 KG : 0 = γ β α β ξ α = δ α γ 2 ξ α = 2 ξ γ (161) ( t + σ i i )ξ = 0 E p σ = 0 (162) Helicity 1 massless Weyl η β D 01 2 helicity h = p σ p = 1 (163) α βη β = (σ µ ) α β µ η β = 0 (164) ( t σ i i )η. = 0 σ µ = (1, σ) (165) h = 1 (166) Helicity 1 massless qft1-3-50
125 3.2.3 Dirac Weyl massless fermion Massive fermion ξ α D η β D 01 2 ( ) α β ξ α = aη β D D 0 2 = D 11 2 D 01 2 (167) ( ) α β η β = bξ α D D = D D (168) a, b KG γ β 2 γ β α βξ α = δ α γ 2 ξ α = m 2 ξ γ = a γ βη β = abξ γ ab = m 2 (169) rescale a = b a = b = im qft1-3-51
126 α βξ α = imη β (170) α βη β = imξ α (171) Dirac spinor Dirac ξ α η β 4 Dirac ψ = ( ξ α η β ) (172) ( ) ( ) m 0 0 i ψ = α β 0 m α β ψ 0 = ( 0 (σ µ ) α β µ ( σ µ ) βα µ 0 ) ψ (173) qft1-3-52
127 γ µ ( ) {( ) ( γ µ 0 σ µ σ σ µ =, σ 0 ψ Dirac : )} (174) (iγ µ µ m)ψ = 0 (175) σ µ σ µ biliear γ µ Clifford : σ µ σ ν + σ ν σ µ = 2η µν (176) σ µ σ ν + σ ν σ µ = 2η µν (177) {γ µ, γ ν } = 2η µν (178) qft1-3-53
128 Chiral(Weyl) : Dirac D D 01 2 = chiral ( Weyl) γ 5 ( ) γ 5 iγ 0 γ 1 γ 2 γ = 0 1 (179) γ 2 5 = 1 (180) γ 5 ( ξ α η α ) = ( ξ α η α γ 5 =chirality ) (181) P ± 1 2 (1 ± γ 5), P 2 ± = P ±, P + P = 0 (182) P + ψ = ξ D P ψ = η D 01 2 qft1-3-54
129 3.2.4 V µ : Proca Maxwell m 0 Massive : Proca = Maxwell massive : : V µ ξ α β = (σ µ ) α βv µ D ν α β = (σ ν ) α β ν D ν V µ α βξ γ δ D D = D 11 D 10 D 01 D 00 (183) (1: 0: ) (16 ) qft1-3-55
130 1. SL(2) ( ɛ αγ ) D D D 10 D 01 χ D 10 η D χ χ D D 10 = D D D D ξ 5. η D D 01 = D D ξ ξ D 1 2 2, 1 χ D 10, η D 01 qft1-3-56
131 3 (i) α( βξ γ) α = a 1 η ( β γ) D 01 (ASD) (ii) α β η( β γ) γ = a 2 ξ α D (vector) (iii) (α γ ξ γ β) = a 3 χ (αβ) D 10 (SD) (iv) α βχ (αγ) = a 4 ξ β γ D (vector) : Dirac-Fierz-Pauli : 4(ξα γ ) + 3(η β γ ) + 3(χ αβ ) = 10 3 ( ) 3 ( ) qft1-3-57
132 Dirac-Fierz-Paluli Proca : KG a i α β = (σµ ) α β µ (184) α β = ( σ µ ) βα µ (185) ξ α β = (σµ ) α β V µ ξ β α = ɛ β γ ξ α γ = (σ µ ɛ T ) β α V µ (186) ξ β γ = ɛ β α ξ Ṫ αγ = (ɛσµt ) β γ V µ = ( σ µ ɛ) β γ V µ (187) χ αβ = i 2 (σµν ɛ) αβ F (+) µν (188) η α β = i 2 (ɛt σ µν ) α βf ( ) µν η α β = i 2 ( σµν ɛ T α ) βf ( ) µν (189) convention χ αβ χ αβ = i(σ µν ɛ) αβ χ (+) µν qft1-3-58
133 χ (+) µν = 1 2 F µν (+) η( ) µν = 1 2 F ( ) µν (i) (iii) (i) α( β ξ α γ) = a 1 η ( β γ) (190) Eq.(i) LHS = 1 { } ( σ µ ) βα µ (σ ν ɛ T ) γ α V ν + ( β γ) 2 = 1 { } ( σ µ σ ν ɛ T ) β γ µ V ν + ( β γ) 2 ( σ µ σ ν ɛ T ) T = σ ν σ µ ɛ T LHS = 1 2 ( ( σ µ σ ν σ ν σ µ )ɛ T ) β γ µ V ν = 1 2i ( σµν ɛ T ) β γ V µν (191) V µν µ V ν ν V µ (192) qft1-3-59
134 σ µν ASD Eq.(i) Eq.(iii) a 1 F ( ) µν = V ( ) µν (193) a 3 F (+) µν = V (+) µν (194) (ii) (iv) (ii) α βη ( β γ) = a 2 ξ α γ (195) LHS = (σ µ 1 ) α β µ 2i ( σνρ ɛ T ) β γ F ( ) νρ = 1 2i (σµ σ νρ ɛ T ) α γ µ F ( ) νρ (196) qft1-3-60
135 σ µ σ νρ : D D 01 = D D D σ µ σ νρ D σ µ σ µ σ νρ = b(η µν σ ρ η µρ σ ν ) + cɛ µνρλ σ λ (197) b, c µ, ν, ρ σ µ σ νρ = i(η µν σ ρ η µρ σ ν ) + ɛ µνρλ σ λ (198) σ µ σ νρ = i(η µν σ ρ η µρ σ ν ) + ɛ µνρλ σ λ (199) Eq. (ii) (iv) : qft1-3-61
136 µ F ( ) µν = a 2 V ν (200) µ F (+) µν = a 4 V ν (201) a 2 0 a 4 0 µ V µ = 0 µ (204) µ F ( ) µν = a 2 V ν (202) µ F (+) µν = a 4 V ν (203) a 1 F ( ) µν = V ( ) µν (204) a 3 F (+) µν = V (+) µν (205) a 1 µ F ( ) µν = µ V ( ) µν = 1 2 µ (V µν Ṽ µν ) = 1 2 ( 2 V ν ν ( V )) µ Ṽ µν = 0 (206) qft1-3-62
137 (202) (202) (203) a 1 a 2 V ν = 1 2 ( 2 V ν ν ( V )) (207) a 3 a 4 V ν = 1 2 ( 2 V ν ν ( V )) (208) 4 V ν m KG a 1 a 2 = a 3 a 4 = m2 a 1 = 1, a 3 = 1 F µν (±) = V µν (±) (204), (205) a 2 = a 4 = m 2 /2 2 (209) Proca 4 V = 0 qft1-3-63
138 F µν = µ V ν ν V µ (210) µ F µν = m 2 V ν ( j ν ) (211) µ V µ = 0 (212) m 2 0 µ V µ = 0 2 Proca 3 m 2 = 0 massless Maxwell µ V µ = 0 Proca ( ) qft1-3-64
139 Proca London : Proca London 5 V µ effective Maxwell A µ Proca : j µ = m 2 A µ London ( ) (213) (i.e. A 0 = 0) ( t 2 A = massive KG ( 2 + m 2 ) A = 0 (214) curl A = B 2 B = 1 λ 2 L B, λ L = 1 m = London penetration depth Meissner : B( x) = B 0 e 1 λ L ˆn x, ˆn 2 = 1 (215) 5 F. London and H. London, Proc. Roy. Soc. (London) A149, (1935) 72. qft1-3-65
140 3.11 3/2 massive Rarita-Schwinger ψ µα ξ (αβ) γ D 11 2, χα ( β γ) D (216) : ψ µα 16 ξ χ 12 ψ µα 1/2 4 qft1-3-66
141 Locality: 2. Reality or Hermiticity: = ( ) 3. Lorentz Invariance: : O = v = λ Ov = λv (v, Ov) = λ(v, v) = (Ov, v) = (v, Ov) = λ (v, v) (λ λ )(v, v) = 0 (217) λ (v, v) 0 v λ qft1-3-67
142 3.3.1 : = c = 1 S px Mcx ML 1 L 1 M (218) O mass [O] Klein-Gordon (scalar) : ( 2 + m 2 )φ = 0 (219) S = d 4 x 1 2 φ( 2 + m 2 )φ (220) qft1-3-68
143 φ : Weyl : 0 = [φ] [φ] = 1 (221) ξ α D : α βξ α = 0 (222) D 01 2 ξ α (ξ α ) = ξ α Reality I ξ = ξ β α β ξ α = ξ β ( σµ ) βα µ ξ α = ξ σ µ µ ξ (223) (222) ξ α (c#) qft1-3-69
144 ξ α odd element ( = c#) α, β αβ = βα, (αβ) β α (224) qft1-3-70
145 I ξ : ( σ µ ) = (ɛσ µt ɛ T ) = ɛσ µ ɛ T (223) I ξ = (ξ σ µ µ ξ) = µ ξ α (( σµ ) βα ) ξ β = µ ξ α (ɛσµ ɛ T ) β α ξ β = µ ξ α (ɛσµt ɛ T ) αβ ξ β = µ ξ α ( σµ ) αβ ξ β = µ ξ σ µ ξ (225) I ξ + Iξ! : L ξ = i 2 ( ξ σ µ µ ξ µ ξ σ µ ξ ) = i 2 ξ σ µ µ ξ (226) δ d 4 xl ξ = 0 σ µ µ ξ = 0: η α D 01 2 : L η = i 2 η σ µ µ η (227) Weyl ( Dirac) : 4 = 1 + 2[ξ] [ξ] = 3/2 qft1-3-71
146 Dirac : Dirac : L kin ψ = L ξ + L η = i 2 ξ σ µ µ ξ + i 2 η σ µ µ η = 2(η i (, ξ ) 0 σ µ ) ( ) µ ξ σ µ µ 0 η = 2(ξ i ( ) ( ), η ) 0 1 γ µ ξ µ 1 0 η = i 2 ψ γ 0 γ µ µ ψ = i 2 ψγ µ µ ψ (228) ψ ψ γ 0 = Dirac conjugate (229) qft1-3-72
147 : ξ η ( L m ψ = mη ξ + ξ η = m (ξ, η ) ) ( = m ψψ (230) ξ η ) L ψ = L kin ψ + Lm ψ = i 2 ψγ µ µ ψ m ψψ (231) d 4 x i 2 ψγ µ µ ψ d 4 x i ψγ µ µ ψ 2 d 4 x ψ(iγ µ µ m)ψ (232) S ψ = qft1-3-73
148 Proca : Proca F µν = µ A ν ν A µ µ F µν = m 2 A ν (233) µ A µ = 0 S A = d 4 x ( 14 ) F µν [A]F µν [A] + m2 2 Aµ A µ (234) A µ 1 qft1-3-74
149 3.3.2 Scaling : : L int = i g i O i, [g i ] = 4 [O i ] (235) g i (mass) mass scale M L int i : L int = i g (0) g (0) i M [O i] 4 O i (236) M µ g (0) i M [O i] 4 µ[oi] = ( ) µ [Oi g (0) ] 4 i µ 4 (237) M i i qft1-3-75
150 4 4 O i = irrelevant operator M Λ i g (0) i M [O i] 4 Λ[O i] = i ( Λ g (0) i Λ 4 M ) [Oi ] 4 (238) 4 irrelevant operators irrelevant qft1-3-76
151 : 0, 1 2, 1 ( ) 6 [O] = 2: [O] = 3: [O] = 4: φ 2, A 2 (239) φ 3, φ A, ψψ, ψγ 5 ψ, (240) φ 4, φ φ, (A 2 ) 2, φ 2 A 2, φ 2 A, A 2 A ( A) 2, ψψφ, ψγ 5 ψφ, ψγ µ ψa µ, ψγ µ γ 5 ψa µ (241) 6 3/2 2 qft1-3-77
152 : φ log(λ/m) φ 8 φ 8 10 qft1-3-78
153 4 4.1 Noether 4.2 Schwinger 4.1 Noether ( ) : Emmy Noether (1918) : : ( ) qft1-4-1
154 : φ(x) : ( ) 1 S = [dx]l(φ(x), µ φ(x)) (1) [dx] = Ω = Ω x µ y µ = x µ + x µ, Ω Ω (2) φ(x) φ (y) = φ(x) + φ(x) (3) φ total variation ( φ(x) φ = 0 : x µ = 0 ) 1 µ ν φ qft1-4-2
155 Lie = x : δφ(x) = φ (x) φ(x) (3) : φ (x + x) φ (x) + x µ µ φ(x) δφ(x) = φ (x) φ(x) = φ(x) x µ µ φ(x) (4) Lie φ(x) = δφ(x) + x µ µ φ(x) (5) Lagrange : Lagrange δl/δφ δl δφ L φ L µ µ φ (6) φ Lagrange qft1-4-3
156 Noether : 1. : current j µ S S = j µ = Ω [dx] ( µ j µ + δl ) δφ δφ (7) L δφ + x µ L (8) µ φ 2. : Ω µ j µ + δl δφ = 0 (9) δφ qft1-4-4
157 µ j µ = 0 space-like surface Σ charge Σ2 0 = µ j µ = j µ dσ µ j µ dσ µ (10) Σ 1 Σ 2 Σ 1 Q(Σ 1 ) = Q(Σ 2 ), Q(Σ) j µ dσ µ (11) Σ Σ t = dσ µ (d 3 x, 0, 0, 0) Q = d 3 xj 0 j µ parameter qft1-4-5
158 : S = [dy]l(φ (y), µ φ (y)) (12) Ω y Lie : φ (y) = φ(y) + δφ(y) ( L S = [dy]l(φ(y), µ φ(y)) + [dx] δφ + L ) Ω Ω φ µ φ µ δφ (13) Ω x : [dy]l[y] = y Ω Ω x [dx]l[x + x] = y x [dx] (L[x] + xµ µ L) (14) Ω ( µ L L x ) qft1-4-6
159 : M M µ ν µ x ν (= ) y x = det 1 + M = exp Tr ln(1 + M) exp TrM 1 + µ x µ [dy] = [dx](1 + µ x µ ) (15) ( S = [dx](1 + ρ x ρ ) L + x µ µ L + L δφ + L ) Ω φ µ φ µ δφ [ = [dx] L + µ ( x µ L) + δl ( )] L δφ + µ δφ Ω δφ µ φ [ = [dx] L + δl ( )] L δφ + µ δφ + x µ L (16) δφ µ φ Ω S (7) // qft1-4-7
160 Remark 1: (13) S [dy]l[y] [dx]l[x] (17) Ω Ω Ω = Ω Ω x µ dσ µ Stokes dσ µ x µ L[x] = [dx] µ ( x µ L) Ω Ω (18) x µ dσ µ Ω = x µ dσ µ x µ qft1-4-8
161 Remark 2: S : space-like Σ 1, Σ 2 S = G[Σ] = Σ2 1 Σ [dx] µ j µ = G[Σ 2 ] G[Σ 1 ] (19) j µ dσ µ = Charge (20) Σ Schwinger 1: 1 : Lagrangian L t ( L = L(x(t), ẋ(t))) t t = t + ɛ (21) x(t) x = 0 qft1-4-9
162 Lie δx = 0 ɛ t x = ɛẋ (22) j = L δx + ɛl = p( ɛẋ) ɛl = ɛ(l pẋ) = ɛh (23) ẋ ɛ H 2: U(1) : φ(x) = L = 1 2 µφ µ φ m 2 φ 2 λ 4 φ 4 (24) L global U(1) φ (x) = e ieλ φ(x) (25) qft1-4-10
163 Λ φ(x) = δφ(x) = ieλφ(x) (26) φ(x) = δφ(x) = ieλφ(x) (27) j µ = L δφ + µ φ L µ φ δφ = 1 2 µ φ ( ieλφ) µ φ(ieλφ ) = ie 2 φ µ φλ (28) Λ J µ = ie 2 φ µ φ (29) qft1-4-11
164 Noether : G dim G = { finite : : global sym local or gauge sym (30) Noether = G Lie X a : x µ G T a : φ G ɛ a : global parameter x µ = ɛ a X a x µ = ɛ a (ξ ν a (x) ν)x µ = ξ µ a (x)ɛa (31) φ = ɛ a T a φ (32) φ T a qft1-4-12
165 Lie δφ = φ x µ µ φ = (T a φ ξ µ a µφ)ɛ a (33) ɛ a Noether µ j µ a = δl δφ (T aφ ξ µ a µφ) (34) j µ a = L µ φ (T aφ ξ µ a µφ) + ξ µ a L (35) currents j a µ qft1-4-13
166 Remark 1: L = Lagrange K = Lagrange = L + K δ(l + K) δφ = 0 δl δφ = δk δφ L Noether (34) (36) µ j µ a = δl δφ (T aφ ξ µ a µφ) = δk δφ (T aφ ξ µ a µφ) 0 (37) current partial conservation qft1-4-14
167 Remark 2: : x µ = 0 j µ = µ φ δφ. Lagrangian : δl = L δφ + L φ µ φ µ δφ = δl ( ) L δφ + µ δφ δφ µ φ j µ = δl δφ + µ j µ = 0 (38) δφ x µ 0 Lagrangian Lie variation x µ L δl = δl ( L δφ + µ δφ δφ µ φ ) = δl δφ + µ j µ µ ( x µ L) = µ ( x µ L) (39) δφ j µ ( L/ µ φ) δφ + x µ L L qft1-4-15
168 Remark 3: ɛ a ɛ a (x) local 1. : Lagrangian µ φ ɛ a local T a φ µ ɛ a δl = L µ φ T aφ µ ɛ a = j µ a µɛ a (40) µ ɛ a j a µ 2. x µ = ξ a µ ɛa 0 : (39) µ j µ µ ɛ a µ j µ = µ j µ a ɛa + j µ a µɛ a (41) j a µ (35) j µ a = L µ φ (T aφ ξ ν a νφ) + ξ µ a L (42) qft1-4-16
169 (δl/δφ) δφ + µ j a µ ɛa = 0 (39) δl = δl δφ + µ j µ δφ }{{} µ ( x µ L) µ j a µ ɛ a +j a µ µ ɛ a = j µ a µɛ a µ ( x }{{ µ } L) ξ a µ ɛ a (x) = L µ φ (T aφ ξ ν a νφ) µ ɛ a + ξ µ a L µɛ a µ (ξ µ a ɛa L) = L µ φ (T aφ ξ ν a νφ) µ ɛ a µ (ξ µ a L)ɛa = j µ,0 a µɛ a µ j µ,1 a ɛa (43) j a µ δl µ ɛ a ɛ a ( ) qft1-4-17
170 δl = j µ,0 a µɛ a µ j µ,1 a ɛa (44) j µ a = jµ,0 a + j µ,1 a (45) : 1 : L = 1 2ẋ2 V (x), δx = ɛẋ (46) ɛ ɛ(t) δl δẋ + L δx = ẋ d ( ɛẋ) + ɛẋdv x dt dx = ɛẋ2 ɛẋẍ + ɛẋ dv ( ) dx = ɛ( ẋ }{{} 2 d 1 ) ɛ (47) dt j 0 δl = L ẋ 2ẋ2 V }{{} j 1 j 0 + j 1 = 1 2ẋ2 V = H qft1-4-18
171 Noether : : ɛ a (x) total variation φ = A a (x, φ, φ)ɛ a (x) + B µ a (x, φ, φ) µɛ a (x) (48) x µ = C µ a (x)ɛa (x) (49) Lie δφ = φ x µ µ φ φ δφ = a a (x, φ, φ)ɛ a (x) + b µ a (x, φ, φ) µɛ a (x) (50) a a = A a µ φc µ a, bµ a = Bµ a (51) ( ) ( x µ = 0 ) B a µ qft1-4-19
172 : Maxwell L = 1 4 F µνf µν, F µν = µ A ν ν A µ (52) δa µ = µ Λ (53) b ν µ = δν µ, ɛ = Λ, a = 0 (54) µ j µ + δl δφ (a aɛ a + b µ a µɛ a ) = 0 (55) j µ = L µ φ (a aɛ a + b ρ a ρɛ a ) + x µ L (56) j µ j µ local parameter ɛ a (x) ɛ a (x) x global µ ɛ a = 0 qft1-4-20
173 1. Local (55) 3 ( ) δl δl δφ bµ a µɛ a δl = µ δφ bµ a ɛa µ δφ bµ a ɛa δl δφ ɛa µ b µ a (57) (55) ( ( )) ( ) δl δl a a δφ µ δφ bµ a ɛ a = µ j µ + b µ a ɛaδl δφ (58) total divergence Ω boundary Ω ɛ a (x) = µ ɛ a (x) = 0 parameter ɛ a (x) (56) j µ = 0 qft1-4-21
174 (58) Ω ( ( )) δl δl [dx] a a δφ µ δφ bµ a ɛ a = 0 (59) Ω Ω ɛ a (x) local ( ) δl µ δφ bµ a = a a δl δφ (60) covariant conservation b µ a hb µ bµ a hb µ = δb a a a = a b δa b = (a bh b µ )bµ a (60) ) µ (b µ δl a = 0, µ µ a b h b µ (61) δφ qft1-4-22
175 : covariant conservation 1. Maxwell : (60) ρ ( δ ρ ν µf µν) = 0 µ ν F µν = 0 (62) 2. Scalar φ = complex scalar Lagrangian L = 1 4 F µνf µν (D µφ) D µ φ (63) D µ µ + iea µ (64) δa µ = µ Λ, δφ = ieλφ (65) qft1-4-23
176 parameters Lagrange a µ = 0, b ν µ = δν µ (66) a φ = ieφ, b ν φ = 0 (67) a φ = ieφ, b ν φ = 0 (68) δl δa ν = µ F µν J ν (69) J ν = i 2 e(φ ν φ) e 2 A ν φ φ (70) δl δφ = 1 2 (Dµ D µ φ) (71) δl δφ = 1 2 (Dµ D µ φ) (72) qft1-4-24
177 (60) 0 = ieφ δl δl δl + ieφ ν δφ δφ δa ν = ieφ ( 12 ) ( (Dµ D µ φ) + ieφ 1 ) 2 (Dµ D µ φ) ( µ ν F µν ν J ν ) (73) µ ν F µν = 0 φ D µ D µ φ = 0 ν J ν = 0 J ν global Noether current qft1-4-25
178 2. Local (60) (58) (58) = 0 ( ) 0 = µ j µ + b µ a ɛaδl ( δφ ) L = µ µ φ (a aɛ a + b ν a νɛ a ) + C µ a ɛa L + b µ a ɛaδl δφ ( = µ J µ a ɛa + L ) µ φ bν a νɛ a + b µ a ɛaδl δφ J µ a = (74) L µ φ a a + C µ a L = Noether (75) qft1-4-26
179 δl/δφ = 0 ( 0 = µ J µ a ɛa + L ) µ φ bν a νɛ a ( ( )) L = ( µ J µ a )ɛa + J µ a + ν ν φ bµ a µ ɛ a + L µ φ bν a µ ν ɛ a (76) ɛ a (x) 3 (i) µ J µ a = 0 ( ) (77) L (ii) J µ a = ν ν φ bµ a (78) L (iii) µ φ bν a µ ν ɛ a = 0 (79) L µ φ bν a µ, ν (80) qft1-4-27
180 (iii) (ii) F µν a (i) : F µν a L ν φ bµ a (81) J µ a = νf µν a (82) Local global J a µ ( L/ ν φ)b µ a divergence Gauss charge 2 Q a = d 3 xj 0 a = d 3 x V i F i0 a = ds ˆn i F 0i a (83) V Gauss V qft1-4-28
181 1. Maxwell scalar qft1-4-29
182 4.2 Schwinger Hamilton-Jacobi (Noether) Hamilton-Jacobi : q k (t) k=1 n : n C S 21 [C] = t2 t 1 dtl(q k, q k, t) (84) δq k L d L = 0 (85) q k dt q k qft1-4-30
183 C C C : q k (t), t 1 t t 2 C : q k (t) = q k(t) + δq k (t), t 1 t t 2 t i = t i + t i, i = 1, 2 t i δq k q k C δq k (t) C t 1 t 1 t t 2 t 2 t qft1-4-31
184 : q k (t i ) q k (t i ) q k(t i ) = q k (t i ) [q k(t i ) + δq k (t i )] + δq k (t i ) = q k (t i ) q k (t i) + δq k (t i ) = q k t i + δq k (t i ) q k t i + δq k (t i ) (86) C C C S 21 [C ] = t 2 t 1 dtl = L L(q k, q k, t) t1 t 1 + t2 t 2 + t 1 t 2 t2 S 21 = S 21 [C ] S 21 [C] = (L L) dt + [L t] 2 1 t 1 t2 { L = d } [ ] L L 2 δq k dt + δq k + L t t 1 q k dt q k } q k {{} 1 qft1-4-32
185 (86) t2 { L S 21 = d } [ ( ) ] L L L 2 δq k dt + q k q k L t t 1 q k dt q k q k q k 1 t2 { L = d } L δq k dt + [p k q k H t] 2 1 (87) q k dt q k t 1 S 21 = [p k q k H t] 2 1 (88) 1 S 2 Hamilton-Jacobi (i) S q k = p k, (ii) S t + H = 0 (89) qft1-4-33
186 Schwinger = = U = e ig ψ = U ψ (1 + ig) ψ ψ = ig ψ, ψ = ψ ( ig) (90) q k (t 2 ), t 2 q k (t 1 ), t 1 = ( q k (t 2 ), t 2 ) q k (t 1 ), t 1 + q k (t 2 ), t 2 ( q k (t 1 ), t 1 ) = q k (t 2 ), t 2 ig(t 2 ) q k (t 1 ), t 1 + q k (t 2 ), t 2 ig(t 1 ) q k (t 1 ), t 1 = q k (t 2 ), t 2 1 i (G(t 2) G(t 1 )) q k (t 1 ), t 1 (91) qft1-4-34
187 G S 21 Schwinger : S q k (t) S 21 = G(t 2 ) G(t 1 ) (92) S 21 (87) (92) [p, q] = i qft1-4-35
188 Schwinger 1. (87) S Schrödinger L d L = 0 (93) q k dt q k G(t) = p k q k H t (94) 2 qft1-4-36
189 G q k, t = ig(t) q k, t = i(p k q k H t) q k, t (95) Schrödinger 1 q k, t = p k q k, t i q k (96) i t q k, t = H q k, t (97) Hamilton-Jacobi 3. O O = UOU = (1 + ig)o(1 ig) O + i[g, O] qft1-4-37
190 O = i[g, O] (98) O = q j t = 0 q j =c G = p k q k (98) q j = i[p k q k, q j ] = i[p k, q j ] q k (99) q j i[p k, q j ] = δ kj (100) qft1-4-38
191 4. G (92) S 21 = 0 G(t 2 ) = G(t 1 ): G (98) G Noether : 3 q k = x k i λ = x k = x k + λɛ ijk x j x k = λɛ ijk x j (101) G λj i = p k x k = λp k ɛ ijk x j = λɛ ijk x j p k (102) // qft1-4-39
192 5. Schwinger : φ a (x) space-like Σ Σ (Euclidean ) Σ 2 Σ 2 Σ 1 Σ 1 R R qft1-4-40
193 1. 2. Lie x µ = x µ + x µ (103) φ a (x) = φ a(x) + δφ a (x) (104) Lie Σ G[Σ] : δφ a (x) = i[g[σ], φ a (x)] (105) R = space-like Σ 1 Σ 2 R = R (103) Noether { ( ) } L L S = µ δφ a + µ J µ d n x (106) R φ a µ φ a J µ L δφ a + L x µ = (quantum) current (107) µ φ a qft1-4-41
194 Gauss µ J µ d n x = R total variation φ a Σ 2 J µ dσ µ Σ 1 J µ dσ µ (108) φ a (x) φ a (x ) φ a (x) = µ φ a (x) x µ + δφ a (x) (109) J µ = L ( φ a ν φ a x ν ) + L x µ µ φ a = π µ a φ a ( π µ a ν φ a η µν L ) x ν = π µ a φ a T µν x ν (110) π µ a L µ φ a (111) T µν π µ a ν φ a η µν L = energy-momentum tensor (112) = Hamiltonian qft1-4-42
195 Schwinger : ( )Noether S = G[Σ 2 ] G[Σ 1 ] (113) L L µ = 0 (114) φ a µ φ a G[Σ] = J µ dσ µ (115) S = 0 = G Σ J µ dσ µ = J µ dσ µ (116) Σ 2 Σ 1 Σ µ J µ = 0 (117) Σ Q = J 0 dv (118) qft1-4-43
196 x µ = 0 ( φ a = δφ a ) J µ = π a µ φ 3 a. φ a ( x, t) = i d 3 x π 0 b ( x, t) φ b ( x, t), φ a ( x, t) (119) }{{} G : : + φ a ( x, t) = i ±i [ ] d 3 x π 0 b ( x, t) φ b ( x, t), φ a ( x, t) ] d 3 x [π 0 b ( x, t), φ a ( x, t) φ b( x, t) 3 space-like qft1-4-44
197 [ ] ( ) φ b ( x, t), φ a ( x, t) = 0 (120) [ ] i π 0 b ( x, t), φ a ( x, t) = δ abδ 3 ( x x ) (121) a b ( ) [ [ ] 0 = φ b ( x, t), φ a ( x, t) ] + φ b ( x, t), φ a ( x, t) [ ] = φ b ( x, t), φ a ( x, t) (122) [ ] φ b ( x, t), φ a ( x, t) = 0 (123) π 0 a [π 0b ( x, t), π 0a ( x, t) ] = 0 (124) qft1-4-45
198 Global Global U(1) φ a global φ a (x) φ a (x) = eiλ φ a (x), λ = (125) φ a (x) = iλφ a (x) (126) ) G = d 3 xπ 0 a φ a(x) = iλ d 3 xπ 0 a φ a iλq (127) π 0 a φ a π 0 a (x) = i[ G, π 0 a (x)] = iλπ 0 (x) (128) 2. non-abelian global φ a = (ei θ T ) ab φ b qft1-4-46
199 5 5.1 Lagrangian Hamiltonian density L = 1 2 ( µφ µ φ m 2 φ 2 ) V (φ) (1) = 1 2 ( φ 2 ( φ) 2 m 2 φ 2 ) V (φ) (2) π = φ (3) H = π φ L = 1 2 (π2 + ( φ) 2 + m 2 φ 2 ) + V (φ) (4) qft1-5-1
200 (ETCR): Schwinger [φ( x, t), π( y, t)] = iδ( x y), rest = 0 (5) ( 2 + m 2 )φ + dv (φ) dφ = 0 (6) ETCR V (φ) cf. QCD ) qft1-5-2
201 Fourier Klein-Gordon φ 3 Fourier : φ( x, t) 3 Fourier φ( x, t) = d 3 k (2π) 3/2φ( k, t)e i k x KG φ( k, t) (7) d 2 φ( k, t) + E 2 dt 2 k φ( k, t) = 0 E k k2 + m 2 φ( k, t) = φ + ( k)e ie kt + φ ( k)e ie kt (8) qft1-5-3
202 φ( x, t) φ( k, = φ( k, t) φ ( k) = φ + ( k) φ( x, t) = d 3 k ( φ (2π) 3/2 + ( k)e ie kt+i k x + φ + ( k) e ie kt i k x φ + ( k) rescale ) a( k) 2E k φ + ( k) (9) φ(x) = φ (+) (x) + φ ( ) (x) (10) φ (+) (x) = d 3 kf k (x)a( k), φ ( ) (x) = φ (+) (x) (11) qft1-5-4
203 f k (x) e ik x (2π)3 2E k (12) k x E k t k x (13) 4 Fourier : 4 Fourier d 4 k φ(x) = φ(k)e ikx (14) (2π) 3/2 kx k µ x µ = k 0 x 0 k x k 0 k qft1-5-5
204 KG d 4 k( k 2 + m 2 ) φ(k)e ikx = 0 (k 2 m 2 ) φ(k) = 0 e ikx (15) k 2 m 2 0 φ(k) = 0 k 2 m 2 = 0 φ(k) φ(k) = δ(k 2 m 2 )χ(k) (16) χ(k) = χ( k) = χ(k) ( f(x) δ(ax) = δ(x)/ a ) δ(f(x)) = y f(y)=0 δ(x y) df dy (17) qft1-5-6
205 δ(k 2 m 2 ) = δ((k 0 ) 2 ( k 2 + m 2 )) = δ((k 0 ) 2 E 2 k ) = 1 2E k ( δ(k 0 E k ) + δ(k 0 + E k ) ) (18) (14) k 0 d 3 k φ(x) = dk ( 0 δ(k 0 E (2π) 3/2 k ) + δ(k 0 + E k ) ) χ(k)e ikx 2E k d 3 k ( = χ(e (2π) 3/2 k, k)e i(e kt k x) + χ( E k, ) k)e i(e k+ k x) 2E k d 3 k ( = χ(e (2π) 3/2 k, k)e i(e kt k x) + χ( E k, ) k)e i(e k k x) 2E k d 3 k ( = χ(k)e ik x + χ (k)e ik x) (19) (2π) 3/2 2E k k = (E k, k) χ(k) = 2E k a( k) a( k) qft1-5-7
206 f k (x) a( k) : f k (x) : (i) d 3 xf k (x)f k (x) = e 2iE kt δ( k + k ) (20) 2E k (ii) d 3 xf k (x)f k (x) = 1 δ( k k ) (21) 2E k (iii) d 3 xf k (x)i 0 f k (x) = δ( k k ) ( ) (22) (iv) d 3 xf k (x)i 0 f k (x) = 0 (23) (iii) φ(x) a( k) a( k) = d 3 xf k (x)i 0 φ(x) (24) a ( k) = d 3 xf k (x)( i 0)φ(x) (25) qft1-5-8
207 (i) (iv) (iii) d 3 xf k (x)i 0 f k (x) = d 3 1 ( ) x (2π) 3 e ik x i 0 e ik x i 0 e ik x e ik x 2E k 2E k = d 3 1 x (2π) 3 (E k + E k )e i(k k ) x 2E k 2E k = E k + E k 2 δ( k k ) = δ( k k ) (26) E k E k a( k) a ( k) [ ] ETCR a( k), a ( k ) : [ ] a( k), a ( k ) = δ( k k ) (27) [ ] [ ] a( k), a( k ) = a ( k), a ( k ) = 0 (28) qft1-5-9
208 1. : [ ] a( k), a ( k ) = d 3 xd 3 x [f k (x) 0 φ(x), f k 0 φ(x )] = d 3 xd 3 x [ f k (x)π(x) 0f k (x)φ(x), f k (x )π(x ) 0 f k (x )φ(x ) ] = d 3 xd 3 x ( f k (x)i 0f k (x )i[π(x), φ(x )] ET ) f k (x )i 0 f k (x)i[π(x ), φ(x)] ET = d 3 xf k i 0 f k = δ( k k ) qft1-5-10
209 5.2.2 Noether 4 P µ = d 3 x ( π µ φ g 0µ L ) (29) E = P 0 = d 3 x 1 ( ) φ2 + ( φ) 2 + m 2 φ 2 (30) 2 = d 3 x 1 µ φ µ φ + m 2 φ 2 (31) 2 µ P i = d 3 x φ i φ = d 3 x 0 φ i φ (32) qft1-5-11
210 Fourier φ = d 3 k(f k a( k) + fk a ( k)) µ φ = d 3 k( ik µ )(f k (x)a( k) f k (x)a ( k)) (33) (i), (ii) : d 3 x µ φ ν φ = d 3 kd 3 k k µ k ν (f k (x)a( k) f k (x)a ( k)) (f k (x)a( k ) f k (x)a ( k )) d 3 k ( = k µ k ν a( k)a( k)e 2iEkt + a ( k)a ( ) k)e 2iE kt 2E k d 3 k ( ) + k µ k ν a( k)a ( k) + a ( k)a( k) (34) 2E k d 3 m 2 d 3 xφ(x) 2 k ( = m 2 a( k)a( k)e 2iEkt + a ( k)a ( ) k)e 2iE kt 2E k d 3 k ( ) + m 2 a( k)a ( k) + a ( k)a( k) (35) 2E k k 0 = E k qft1-5-12
211 P 0 k µ k µ + m 2 = 0 aa a a k µ k µ + m 2 = Ek 2 + k 2 + m 2 = 2Ek 2 P 0 = d 3 k E ( k a( k)a ( k) + a ( k)a( ) k) 2 = d 3 ke k 1 a ( k)a( k) + 2 }{{} ( ) (36) P k aa a a P = = d 3 k k ( a ( k)a( k) + 1 ) 2 d 3 k ka ( k)a( k) (37) qft1-5-13
212 5.3 Ô ψ Ô χ =finite Fock (normal ordering ) a = a = a to the right, a to the left : : : aa : a a, : aaa a := a a aa O = n=1 a na n 0 O 0 : O : 0 qft1-5-14
213 (normal ordered product) : : φ(x) = φ (+) (x) }{{} a( k) + φ ( ) (x) }{{} a ( k) φ(x)φ(y) = (φ (+) (x) + φ ( ) (x))(φ (+) (y) + φ ( ) (y)) = φ (+) (x)φ (+) (y) + φ (+) (x)φ ( ) (y) + φ ( ) (x)φ (+) (y) + φ ( ) (x)φ ( ) (y) (38) Normal-ordered product : φ(x)φ(y) : = (φ (+) (x) + φ ( ) (x))(φ (+) (y) + φ ( ) (y)) = φ (+) (x)φ (+) (y) + φ ( ) (y)φ (+) (x) + φ ( ) (x)φ (+) (y) + φ ( ) (x)φ ( ) (y) (39) qft1-5-15
214 c# [ ] φ(x)φ(y) : φ(x)φ(y) := φ (+) (x), φ ( ) (y) [ ] = d 3 kd 3 k f k (x)f k (y) a( k), a ( k ) d 3 k = e ik (x y) (40) (2π) 3 2E k k 0 = E k > 0 4 [ ] φ (+) (x), φ ( ) (y) = where θ(x) = { d 4 k (2π) 3θ(k0 )δ(k 2 m 2 )e ik (x y) (41) 1 for x > 0 0 for x < 0 (42) θ(x) θ(0) θ(0) = 1 2. qft1-5-16
215 5.4 Invariant commutator function : [φ(x), φ(y)] x, y [ ] [ ] [φ(x), φ(y)] = φ (+) (x), φ ( ) (y) + φ ( ) (x), φ (+) (y) d 4 k = )δ(k 2 m 2 )e ik (x y) (2π) 3ɛ(k0 i (x y; m 2 ) = invariant commutator function (43) ɛ(x) = staircase function stair step function ( ) { 1 for x > 0 ɛ(x) θ(x) θ( x) = (44) 1 for x < 0 qft1-5-17
216 Invariant function : Invariant function d 4 k i (x) = (2π) 3 ɛ(k0 )δ(k 2 m 2 )e ikx = [φ(x), φ(0)] (45) d 3 k 1 ( = e ikx e ikx) (46) (2π) 3 2E k 1. Klein-Gordon 2. Lorentz invariance: 3. ( x) = (x) ( x) = 1 d 4 k i (2π) 3 ɛ(k0 )δ(k 2 m 2 )e +ikx k k = 1 d 4 k i (2π) 3 ɛ( k0 )δ(k 2 m 2 )e ikx = (x) ɛ( k 0 ) = ɛ(k 0 ) (47) qft1-5-18
217 4. Micro-causality i.e. (x) = 0 for x 2 < 0 (x) Lorentz-invariant t = 0, x 0 ( x, t = 0) = ( x, t = 0) ( x, t = 0) = 1 i = 1 i d 3 kdk 0 (2π) 3 ɛ(k0 )δ(k 2 m 2 )e i k x d 3 kdk 0 (2π) 3 ɛ(k0 )δ(k 2 m 2 )e +i k x = ( x, t = 0) (x) = 0 for x 2 < 0 (48) space-like φ(x) φ(0) qft1-5-19
218 5. i t (x) d 3 k 1 ( ) t=0 = ie (2π) 3 k e i k x ie k e i k x 2E k = iδ( x) Commutator [ φ(x), φ(0)] = 1 δ( x) (49) ET i 5.5 Feynman Propagator (Time-ordered product): φ(x) annihiliation part φ (+) (x) creation part φ ( ) (x) φ(x) = φ (+) (x) + φ ( ) (x) (50) φ(y) 0 = φ ( ) (y) 0 0 φ(x) = 0 φ (+) (x) y (51) x (52) qft1-5-20
219 y x 0 φ(x)φ(y) 0 = 0 φ (+) (x)φ ( ) (y) 0 (53) x 0 > y 0 x 0 < y 0 x 0 > y 0 x 0 < y 0 0 φ(x)φ(y) 0 0 φ(y)φ(x) 0 (Time-ordered product or T-product) T (φ(x)φ(y)) θ(x 0 y 0 )φ(x)φ(y) + θ(y 0 x 0 )φ(y)φ(x) qft1-5-21
220 Feyman propagator : T -product Feynman propagator i F (x y; m 2 ) 0 T (φ(x)φ(y)) 0 (54) x-y T-product Lorentz : θ(x 0 y 0 ) + θ(y 0 x 0 ) = 1 T-product T (φ(x)φ(y)) = θ(x 0 y 0 )[φ(x), φ(y)] + φ(y)φ(x) (55) θ(x 0 y 0 ) Lorentz : qft1-5-22
221 x y time-like : (x 0 y 0 ) 2 > ( x y) 2 x 0 y 0 proper Lorentz θ(x 0 y 0 ) Lorentz x y space-like : (x 0 y 0 ) 2 < ( x y) 2 x 0 y 0 θ(x 0 y 0 ) Lorentz space-like micro-causality [φ(x), φ(y)] = 0 T-product Lorentz Feynman propagator : i F invariant function qft1-5-23
222 i F (x y) = 0 T (φ(x)φ(y)) 0 = θ(x 0 y 0 ) 0 φ (+) (x)φ ( ) (y) 0 + θ(y 0 x 0 ) 0 φ (+) (y)φ ( ) (x) 0 = θ(x 0 y 0 ) 0 [φ (+) (x), φ ( ) (y)] 0 +θ(y 0 x 0 ) 0 [φ (+) (y), φ ( ) (x)] 0 = θ(x 0 y 0 )i (x y) + θ(y 0 x 0 )i (y x) (56) i F (x y) = { d 3 k e ik (x y) θ(x 0 y 0 ) (2π) 3 2E k } e ik (y x) +θ(y 0 x 0 ) 2E k (57) qft1-5-24
223 { } I dk 0 e ik0 (x 0 y 0 ) 2πi(k 0 E k + iɛ)(k 0 + E k iɛ) (58) k 0 -plane pole E k + iɛ E k iɛ Complex k 0 e ik0 (x 0 y 0) = e irk0 (x 0 y 0) e Ik0 (x 0 y 0 ) (59) qft1-5-25
224 x 0 y 0 > 0 : Ik 0 < 0 contour E k iɛ k 0 = E k iɛ pole I = 1 2πi ( 2πi) 1 e iek(x0 y0) = 1 e iek(x0 y0 ) (60) 2E k 2E k x 0 y 0 < 0 : contour I = 1 e iek(y0 x0 ) (61) 2E k { I = θ(x 0 y 0 ) 1 e iek(x0 y0) + θ(y 0 x 0 ) 1 } e iek(y0 x0 ) 2E k 2E k qft1-5-26
225 i F (x y) = i = i d 4 k e ik (x y) (2π) 4 (k 0 E k + iɛ)(k 0 + E k iɛ) d 4 k e ik (x y) (2π) 4 (k 02 (E k iɛ) 2 ) (62) k 02 (E k iɛ) 2 ) = k 02 ( k 2 + m 2 ) + 2iɛE k = k 2 m 2 + iɛ (63) F (x y) = d 4 k e ik (x y) (2π) 4 k 2 m 2 + iɛ (64) qft1-5-27
226 F (x y) : 1 F (x y) δ- source Klein-Gordon i.e. Green ( 2 + m 2 )i F (x y) = iδ 4 (x y) (65) 1: T-product i F Klein-Gordon ( 2 + m 2 )i F (x y) = ( 2 + m 2 ) 0 θ(x 0 y 0 )[φ(x), φ(y)] + φ(y) φ(x) 0 }{{} = ( 2 t 2 + m 2 ) 0 θ(x 0 y 0 )[φ(x), φ(y)] 0 (66) qft1-5-28
227 t ( θ(x 0 y 0 )[φ(x), φ(y)] ) 2 t = δ(x 0 y 0 )[φ(x), φ(y)] +θ(x }{{} 0 y 0 )[ φ(x), φ(y)] 0 ( θ(x 0 y 0 )[φ(x), φ(y)] ) = δ(x 0 y 0 )[ φ(x), φ(y)] +θ(x }{{} 0 y 0 )[ φ(x), φ(y)] iδ 4 (x y) 2 t θ(x0 y 0 )[φ(x), φ(y)] = iδ 4 (x y) + θ(x 0 y 0 )[ 2 t φ(x), φ(y)] 2 + m 2 ( 2 + m 2 )φ(x) ( 2 + m 2 )i F (x y) = iδ 4 (x y) // (67) qft1-5-29
228 1: i F ( ) 1 α + iɛ = P 1 α iπδ(α) (68) P 1 α Cauchy (principal value) [a, b] b a dαp 1 f(α) lim α ɛ 0 ( ɛ a dα f(α) α + b ɛ dα f(α) ) α (69) ( ) : f(α): f(0) =finite contour C 1, C 2 + C 3 dαf(α)/(α + iɛ) (C 2 C 3 ) C 1 0 C 2 C 3 qft1-5-30
229 C 2 + C 3 C 3 = f( iɛ) C 3 C 1 dα f(α) α + iɛ = = C 1 = residue at α = iɛ 2π π idθ = πif(0) dαp 1 f(α) + πif(0) 2πif(0) C 2 α }{{} pole part dαp 1 α f(α) iπ dαδ(α)f(α) // (70) qft1-5-31
230 F Green ( 2 + m 2 ) F (x y) = ( 2 + m 2 d 4 k e ik (x y) ) (2π) 4 k 2 m 2 + iɛ d 4 { } k = + m 2 1 ) P (2π) 4( k2 k 2 m 2 iπδ(k2 m 2 ) e ik x d 4 k = 4e ik x = δ 4 (x) // (71) (2π) qft1-5-32
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 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 informationJuly 28, H H 0 H int = H H 0 H int = H int (x)d 3 x Schrödinger Picture Ψ(t) S =e iht Ψ H O S Heisenberg Picture Ψ H O H (t) =e iht O S e i
July 8, 4. H H H int H H H int H int (x)d 3 x Schrödinger Picture Ψ(t) S e iht Ψ H O S Heisenberg Picture Ψ H O H (t) e iht O S e iht Interaction Picture Ψ(t) D e iht Ψ(t) S O D (t) e iht O S e ih t (Dirac
More informationTOP URL 1
TOP URL http://amonphys.web.fc2.com/ 1 30 3 30.1.............. 3 30.2........................... 4 30.3...................... 5 30.4........................ 6 30.5.................................. 8 30.6...............................
More information( )
7..-8..8.......................................................................... 4.................................... 3...................................... 3..3.................................. 4.3....................................
More informationTOP URL 1
TOP URL http://amonphys.web.fc2.com/ 1 6 3 6.1................................ 3 6.2.............................. 4 6.3................................ 5 6.4.......................... 6 6.5......................
More informationtomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p.
tomocci 18 7 5...,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. M F (M), X(F (M)).. T M p e i = e µ i µ. a a = a i
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 informationTOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
More information25 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( ) (ver )
ver.3.1 11 9 1 1. p1, 1.1 ψx, t,, E, p. = E, p ψx, t,. p, 1.8 p4, 1. t = t ρx, t = m [ψ ψ ψ ψ] ρx, t = mi [ψ ψ ψ ψ] p4, 1.1 = p6, 1.38 p6, 1.4 = fxδ ϵ x = fxδϵx = 1 π fxδ ϵ x dx = fxδ ϵ x dx = [ 1 fϵ π
More informationI ( ) 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 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 information4. 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量子力学A
c 1 1 1.1....................................... 1 1............................................ 4 1.3.............................. 6 10.1.................................. 10......................................
More information: 2005 ( ρ t +dv j =0 r m m r = e E( r +e r B( r T 208 T = d E j 207 ρ t = = = e t δ( r r (t e r r δ( r r (t e r ( r δ( r r (t dv j =
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(
More information0406_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 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 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 informationIA
IA 31 4 11 1 1 4 1.1 Planck.............................. 4 1. Bohr.................................... 5 1.3..................................... 6 8.1................................... 8....................................
More informationHilbert, von Neuman [1, p.86] kt 2 1 [1, 2] 2 2
hara@math.kyushu-u.ac.jp 1 1 1.1............................................... 2 1.2............................................. 3 2 3 3 5 3.1............................................. 6 3.2...................................
More informationEinstein ( ) YITP
Einstein ( ) 2013 8 21 YITP 0. massivegravity Massive spin 2 field theory Fierz-Pauli (FP ) Kinetic term L (2) EH = 1 2 [ λh µν λ h µν λ h λ h 2 µ h µλ ν h νλ + 2 µ h µλ λ h], (1) Mass term FP L mass =
More informationi E B Maxwell Maxwell Newton Newton Schrödinger Newton Maxwell Kepler Maxwell Maxwell B H B ii Newton i 1 1.1.......................... 1 1.2 Coulomb.......................... 2 1.3.........................
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 informationII A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
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 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 information( ) ( 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 information1 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,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.
9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,
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 informationI 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 informationG (n) (x 1, x 2,..., x n ) = 1 Dφe is φ(x 1 )φ(x 2 ) φ(x n ) (5) N N = Dφe is (6) G (n) (generating functional) 1 Z[J] d 4 x 1 d 4 x n G (n) (x 1, x 2
6 Feynman (Green ) Feynman 6.1 Green generating functional Z[J] φ 4 L = 1 2 µφ µ φ m 2 φ2 λ 4! φ4 (1) ( 1 S[φ] = d 4 x 2 φkφ λ ) 4! φ4 (2) K = ( 2 + m 2 ) (3) n G (n) (x 1, x 2,..., x n ) = φ(x 1 )φ(x
More informationH 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[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 information006 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 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 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 information1.1 foliation M foliation M 0 t Σ t M M = t R Σ t (12) Σ t t Σ t x i Σ t A(t, x i ) Σ t n µ Σ t+ t B(t + t, x i ) AB () tα tαn µ Σ t+ t C(t + t,
1 Gourgoulhon BSSN BSSN ϕ = 1 6 ( D i β i αk) (1) γ ij = 2αĀij 2 3 D k β k γ ij (2) K = e 4ϕ ( Di Di α + 2 D i ϕ D i α ) + α ] [4π(E + S) + ĀijĀij + K2 3 (3) Ā ij = 2 3Āij D k β k 2αĀikĀk j + αāijk +e
More informationHanbury-Brown Twiss (ver. 2.0) van Cittert - Zernike mutual coherence
Hanbury-Brown Twiss (ver. 2.) 25 4 4 1 2 2 2 2.1 van Cittert - Zernike..................................... 2 2.2 mutual coherence................................. 4 3 Hanbury-Brown Twiss ( ) 5 3.1............................................
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 informationNo δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
More information1 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 information1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
More informationS 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 informationv 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 informationQMI_09.dvi
25 3 19 Erwin Schrödinger 1925 3.1 3.1.1 3.1.2 σ τ 2 2 ux, t) = ux, t) 3.1) 2 x2 ux, t) σ τ 2 u/ 2 m p E E = p2 3.2) E ν ω E = hν = hω. 3.3) k p k = p h. 3.4) 26 3 hω = E = p2 = h2 k 2 ψkx ωt) ψ 3.5) h
More informationQMI_10.dvi
25 3 19 Erwin Schrödinger 1925 3.1 3.1.1 σ τ x u u x t ux, t) u 3.1 t x P ux, t) Q θ P Q Δx x + Δx Q P ux + Δx, t) Q θ P u+δu x u x σ τ P x) Q x+δx) P Q x 3.1: θ P θ Q P Q equation of motion P τ Q τ σδx
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 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 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 informationii 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 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 informationhttp://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 informationI
I 6 4 10 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............
More informationn (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz
1 2 (a 1, a 2, a n ) (b 1, b 2, b n ) A (1.1) A = a 1 b 1 + a 2 b 2 + + a n b n (1.1) n A = a i b i (1.2) i=1 n i 1 n i=1 a i b i n i=1 A = a i b i (1.3) (1.3) (1.3) (1.1) (ummation convention) a 11 x
More information医系の統計入門第 2 版 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. このサンプルページの内容は, 第 2 版 1 刷発行時のものです.
医系の統計入門第 2 版 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/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987
More information30
3 ............................................2 2...........................................2....................................2.2...................................2.3..............................
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 informationS I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt
S I. x yx y y, y,. F x, y, y, y,, y n http://ayapin.film.s.dendai.ac.jp/~matuda n /TeX/lecture.html PDF PS yx.................................... 3.3.................... 9.4................5..............
More informationIA 2013 : :10722 : 2 : :2 :761 :1 (23-27) : : ( / ) (1 /, ) / e.g. (Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + (
IA 2013 : :10722 : 2 : :2 :761 :1 23-27) : : 1 1.1 / ) 1 /, ) / e.g. Taylar ) e x = 1 + x + x2 2 +... + xn n! +... sin x = x x3 6 + x5 x2n+1 + 1)n 5! 2n + 1)! 2 2.1 = 1 e.g. 0 = 0.00..., π = 3.14..., 1
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 informationQMI13a.dvi
I (2013 (MAEDA, Atsutaka) 25 10 15 [ I I [] ( ) 0. (a) (b) Plank Compton de Broglie Bohr 1. (a) Einstein- de Broglie (b) (c) 1 (d) 2. Schrödinger (a) Schrödinger (b) Schrödinger (c) (d) 3. (a) (b) (c)
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 informationK E N Z U 2012 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.2................................... 4 1.2.1..................................... 4 1.2.2.................................... 5................................
More information,2,4
2005 12 2006 1,2,4 iii 1 Hilbert 14 1 1.............................................. 1 2............................................... 2 3............................................... 3 4.............................................
More informationDynkin 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 information2 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 informationd 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 informationω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 +
2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j
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 informationall.dvi
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
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 information数学の基礎訓練I
I 9 6 13 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 3 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............
More informationall.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 information80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t = 0 i r 0 t(> 0) j r 0 + r < δ(r 0 x i (0))δ(r 0 + r x j (t)) > (4.2) r r 0 G i j (r, t) dr 0
79 4 4.1 4.1.1 x i (t) x j (t) O O r 0 + r r r 0 x i (0) r 0 x i (0) 4.1 L. van. Hove 1954 space-time correlation function V N 4.1 ρ 0 = N/V i t 80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t
More informationNote.tex 2008/09/19( )
1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................
More informationi
009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3
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
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微分積分 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. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 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. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
More informationUntitled
II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
More information215 11 13 1 2 1.1....................... 2 1.2.................... 2 1.3..................... 2 1.4...................... 3 1.5............... 3 1.6........................... 4 1.7.................. 4
More information(2) Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [2], [13]) Poincaré e m Poincaré e m Kähler-like 2 Kähler-like
() 10 9 30 1 Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [], [13]) Poincaré e m Poincaré e m Kähler-like Kähler-like Kähler M g M X, Y, Z (.1) Xg(Y, Z) = g( X Y, Z) + g(y, XZ)
More information2011de.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. 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 information(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
More informationi x- p
3 3 i........................................................................................................... 3............................... 3.. x- p-.................... 8..3.......................
More information1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =
1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i
More information24 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 informationphs.dvi
483F 3 6.........3... 6.4... 7 7.... 7.... 9.5 N (... 3.6 N (... 5.7... 5 3 6 3.... 6 3.... 7 3.3... 9 3.4... 3 4 7 4.... 7 4.... 9 4.3... 3 4.4... 34 4.4.... 34 4.4.... 35 4.5... 38 4.6... 39 5 4 5....
More information量子力学3-2013
( 3 ) 5 8 5 03 Email: hatsugai.yasuhiro.ge@u.tsukuba.ac.jp 3 5.............................. 5........................ 5........................ 6.............................. 8.......................
More information2009 2 26 1 3 1.1.................................................. 3 1.2..................................................... 3 1.3...................................................... 3 1.4.....................................................
More informationi I II I II II IC IIC I II ii 5 8 5 3 7 8 iii I 3........................... 5......................... 7........................... 4........................ 8.3......................... 33.4...................
More information* 1 2014 7 8 *1 iii 1. Newton 1 1.1 Newton........................... 1 1.2............................. 4 1.3................................. 5 2. 9 2.1......................... 9 2.2........................
More informationm(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120)
2.6 2.6.1 mẍ + γẋ + ω 0 x) = ee 2.118) e iωt Pω) = χω)e = ex = e2 Eω) m ω0 2 ω2 iωγ 2.119) Z N ϵω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j 2.120) Z ω ω j γ j f j f j f j sum j f j = Z 2.120 ω ω j, γ ϵω) ϵ
More informationkawa (Spin-Orbit Tomography: Kawahara and Fujii 21,Kawahara and Fujii 211,Fujii & Kawahara submitted) 2 van Cittert-Zernike Appendix A V 2
Hanbury-Brown Twiss (ver. 1.) 24 2 1 1 1 2 2 2.1 van Cittert - Zernike..................................... 2 2.2 mutual coherence................................. 3 3 Hanbury-Brown Twiss ( ) 4 3.1............................................
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 information08-Note2-web
r(t) t r(t) O v(t) = dr(t) dt a(t) = dv(t) dt = d2 r(t) dt 2 r(t), v(t), a(t) t dr(t) dt r(t) =(x(t),y(t),z(t)) = d 2 r(t) dt 2 = ( dx(t) dt ( d 2 x(t) dt 2, dy(t), dz(t) dt dt ), d2 y(t) dt 2, d2 z(t)
More information29 7 5 i 1 1 1.1............................ 1 1.1.1................. 1 1.1.2............................. 3 1.1.3......................... 4 1.2........................... 9 1.2.1................. 9
More information