( ) 1 (Quantum Field Theory) = Maxwell Dirac Quantum Electro-Dynamics (QED) 1

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