(Tokyo Institute of Technology) Seminar at Ehime University ( ) 9 3 U(N C ), N F /2 BPS ( ) 12 5 (
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1 (Tokyo Institute of Technology) Seminar at Ehime University ( ) 9 3 U(N C ), N F /2 BPS ( ) 12 5 ( ) 19 6 Conclusion 23
2 1 1.1 GeV SU(3) SU(2) U(1): W ±, Z 1. ( ) 2. W ±, Z 0 (1983), 3. LHC (2008 ) 2
3 トンネル周長 27km ( 参考 : 東京 JR 山手線の周長 34.5km) 4
4 ( ) 1. ( ) M GUT c GeV (1.1) 2. 3
5 ( ) M W c GeV M GUT c GeV ( ) M P c GeV M W 1. (Technocolor) TeV L. Susskind,Phys. Rev.D20 (1979) 2619; S. Weinberg, Phys. Rev.D19 (1979) 1277; D13 (1976) 974; S. Dimopoulos, and L. Susskind,Nucl. Phys. B155 (1979) 237; 4
6 2. 0 : m B m B = m F 1 2 : m F m F = 0 S.Dimopoulos, H.Georgi, Nucl.Phys.B193 (1981) 150; N.Sakai, Z.f.Phys.C11 (1981) 153; E.Witten, Nucl.Phys.B188 (1981) 513; (a) Lightest Supersymmetric Particle R (b) ( 1) (c) (h, H, A) (H +, H ) ( h ) (d) 150GeV 5
7 1: ( ) ( ) α i = gi 2 /4π, (i = 1, 2, 3) U(1), SU(2), SU(3) : (sequestering) 3. (Brane World)= 4 ( ) 6
8 y: 4 n 2πR V 4+n (r) = 1 m 1 m 2 M 2+n r 1 m 1 m 2 (n) 1+n M 2+n, for r R (1.2) (n) Rn r M 2+n (n) Rn = M P M (n) M P GeV P.Horava and E.Witten, Nucl.Phys.B475, 94 (1996); N.Arkani-Hamed, SDimopoulos, G.Dvali, Phys.Lett.B429 (1998) 263 ; I.Antoniadis, N.Arkani-Hamed, S.Dimopoulos, G.Dvali, Phys.Lett.B436 (1998) 257; Randall, Sundrum, Phys.Rev.Lett.83 (1999) 3370; 4690; = :,,... 7
9 LHC ( ) Landau-Ginzburg : U(1) φ L = 1 4e 2F µνf µν + D µ φ(d µ φ) λ ( φφ v 2) 2 (1.3) 4 : ( ) k = 1 d 2 x F 12 (1.4) 2π : = U(1) :, : λ < e 2 : (, ) λ > e 2 : 8
10 λ = e 2 : BPS BPS (SUSY) 2 ( ) ( ) : ( ) π 0 (M) φ (λ > 0) L = µ φ µ φ λ(φ 2 v 2 ) 2 (2.1) 9
11 : φ + v, φ v y = x 2 E = ( y φ) 2 + λ(φ 2 v 2 ) 2 = ( y φ + λ(φ 2 v 2 )) 2 + y [2 )] λ (v 2 φ φ3 3 [ dye 2 )] λ (v 2 φ φ3 3 Bogomol nyi-prasad-sommerfield (BPS) (2.2) Bogomol nyi, Sov.J.Nucl.Phys. 24 (1976) 449; Prasad and Sommerfield, Phys.Rev.Lett. 35 (1975) 760. BPS y φ + λ(φ 2 v 2 ) = 0 (2.3) : φ = v tanh( λvy) (2.4) References 10
12 3 U(N C ), N F L = 1 2g 2Tr(F MN(W )F MN (W )) + 1 g 2Tr(DM ΣD M Σ) +Tr [ D M H(D M H) ] V (3.1) V = g2 [ (HH 4 Tr ) 2 ] c1 NC + Tr [ (ΣH HM)(ΣH HM) ] D M H i = ( M + iw M )H i, D M Σ = M Σ + i[w M, Σ] F MN (W ) = M W N N W M + i[w M, W N ] W M, Σ (N C N C ) U(N C ) g : H ra H ra (N C N F matrix) (i = 1, 2 ; r = 1,, N C ; A = 1,, N F ), (M) A B m A δ A B : m A > m A+1 : U(1) N F 1 F : Σ 11
13 5 M, N, = 0, 1, 2, 3, 4 (8 SUSY) : A 1 A 2 A NC HH = c1 NC, ΣH HM = 0 (3.2) H ra = c δ A r A, Σ = diag(m A1,, m ANC ) (3.3) N : F! (N F N C )!N C! en F log(x x (1 x) (1 x)), x N C /N F : ( ),,, 4 1/2 BPS ( ) 1/2 BPS y x 4, 4 D W M y = 0 12
14 2: A 1 A 2 A NC B 1 B 2 B NC. E [ E = Tr D y H ra 2] + Tr [ ΣH HM 2] + 1 ( g 2Tr (D y Σ) 2) + g2 [ (HH 4 Tr ) 2 ] c1 NC = Tr D y H + ΣH HM ( g 2Tr D y Σ g2 ( c1nc HH )) 2 + c y TrΣ (4.1) 2 1/2 BPS : γ 4 ε i = i(σ 3 ) i jε j D y H = ΣH + HM, D y Σ = g 2 ( c1 NC HH ) /2 (4.2) 13
15 labeled by A 1 A 2 A NC B 1 B 2 B NC BPS + [ ] + Edy c Tr(Σ) = c N C m Ak k=1 N C k=1 m Bk (4.3) BPS Σ + iw y S 1 (y) y S(y) S(y) GL(N C, C) BPS : H(y) = S 1 (y)h 0 e My H 0 N C N F BPS : Ω SS ( y Ω 1 y Ω ) = g 2 c ( ) 1 C Ω 1 Ω 0, Ω0 c 1 H 0 e 2My H 0 H 0 Ω(y) S(y) Σ, W y, H 1 y = ±, U(1) H 0 14
16 V - : (S NC 2 ) (S, H 0 ) (S, H 0 ) H = S 1 H 0 e My (Σ, W y ) S S = V S, H 0 H 0 = V H 0, V GL(N C, C) 3: A B C ( ) A,B ( ) BPS : M = {H 0 H 0 V H 0, V GL(N C, C)} G NF,N C SU(N F ) SU(N C ) SU(N F N C ) U(1) (4.4) 15
17 ( ) N C Ñ C N C (N F N C ) : 1,, N C Ñ C + 1,, N F dim R M 1,,N C Ñ C +1,,N F N F,N C = 2N wall = 2N C Ñ C (4.5) M = M 1/1 + M 1/2 = M 0 M 1 M N CÑC (4.6) H 0 : y 4:. U(1) : H 0 = (e r 1, e r 2,, e r N F ), H = S 1 H 0 e My = S 1 (e r 1+m 1 y,, e r N F +m NF y ) (4.7) i i + 1 Rer i + m i y Rer i+1 + m i+1 y Im(r i r i+1 ) : 16
18 U(N C ) : BPS Ñ C N F N C (T w : ) L = T w + d 4 θk(φ, φ ) + (4.8) [ K(φ, φ ) = dy c log detω + ctr ( Ω 0 Ω 1) + 1 ( 2g 2Tr Ω 1 y Ω ) ] 2 K (g ) : g 2 c/ m 1: Ω = Ω 0 c 1 H 0 e 2My H 0 (4.9) g 2 : (NLSM) 1/2 BPS N F = N C = N [ L 6 = Tr 1 2g 2F MNF MN + D M H(D M H) ] g2 4 Tr[( ) HH 2 ] c1 NC 17 Ω=Ω sol
19 BPS BPS 0 = D 1 H + id 2 H, 0 = F 12 + g2 2 (c1 N HH ) (4.10) H = S 1 H 0 (z), W 1 + iw 2 = i2s 1 z S, z x 1 + ix 2 (4.11) z (Ω 1 z Ω) = g2 4 (c1 N Ω 1 H 0 H 0 ) (4.12) k Z 0, (det(h 0 ) z k, z ) T c d 2 x TrF 12 = 2πck = i c dz log(deth 0 ) + c.c. 2 V - : H 0 V H 0, S V S, V = V (z) GL(N, C), det V = const. 0 : dim(m N,k ) = 2kN ( ) 1 N 1 R(z) k H 0 =, P (z) = (z z i ) (4.13) 0 P (z) i=1 18
20 z i C, R(z) k (Cosmic string) 5 ( ) : Tong, Phys.Rev.D (2004); Auzzi-Bolognesi-Evslin-Konishi, Nucl.Phys.B (2004); Shifman-Yung, Phys.Rev.D (2004); Auzzi-Bolognesi-Evslin, JHEP 0502 (2005) 046; 1/2 γ 123 ε i = ε i x 3 1/2SUSY : γ 12 (iσ 3 ) i jε j = ε i ( + ): 1/4 SUSY γ 3 (iσ 3 ) i jε j = ε i x 3 1/4 BPS D 3 Σ = g 2 ( c1 NC H 1 H 1 ) /2+F 12, D 3 H 1 = ΣH 1 + HM, 0 = D 1 H 1 + id 2 H 1, 0 = F 23 D 1 Σ, 0 = F 31 D 2 Σ 19
21 5: ( ) ( ) BPS E t w + t v + t m + m J m t w, t v, t m t w = c 3 Tr(Σ), t v = ctr(f 12 ), t m = 2 g 2 mtr( 1 2 ϵmnl F nl Σ) : [D 1 + id 2, D 3 + Σ] = 0 ( ) S(x m ) GL(N C, C) (D 3 + Σ)S 1 = 0 Σ + iw 3 S 1 3 S (5.1) (D 1 + id 2 )S 1 = 0 W 1 + iw 2 2iS 1 S (5.2) z x 1 + ix 2, and / z. BPS H 1 = S 1 (z, z, x 3 )H 0 (z)e Mx3 20
22 H 0 (z): z N C N F Ω SS (Ω 0 H 0 e 2My H 0 ) 4 (Ω 1 Ω) + 3 (Ω 1 3 Ω) = g 2 ( c Ω 1 Ω 0 ) (5.3) x x x 1 0 x x 3 a) x 3 0 b) : (t w + t v = 0.5c) a) : H 0 (z)e Mx3 = c(e x3, ze 4, e x3 ). z b) : H 0 (z)e Mx3 = c((z 4 2i)(z i)e 3/2x3, (z + 8 i)(z 7 + 6i)e 1/2x3 +15/2, z 2 e 1/2x3 +15/2, (z 6 5i)(z + 6 7i)e 3/2x3 ). (g 2 ) : U(1) (N C = 1) 21
23 H 0 (z) = c ( f 1 (z),..., f N F(z) ) : Ω = N F A=1 f A (z) 2 e 2m Ax 3 A x 3 A (z) = log f A+1(z) log f A (z) m A m A+1 f A (z) : f A (z) (z z A α )ka α : A z = z A α k A α x y 7: /4BPS 1/4BPS 22
24 6 Conclusion ( )U(N C ) N F BPS 3. ( ) H 0 M NF,N C {H 0 H 0 V H 0, V GL(N C, C)} SU(N F ) G NF,N C SU(N C ) SU(Ñ C ) U(1) (6.1) 4. (g 2 ) ( ) 5. BPS 6. 1/4 BPS ( ) 7. 1/2, 1/4 BPS 23
25 References of Solitons in 8 SUSY Theories Tokyo Tech Collaboration 1. Review Solitons in the Higgs Phase: Moduli matrix approach, hep-th/ , J. Phys. A 39 (2006) R , 2. Domain Walls in 5D Supersymmetric Theories Moduli space of BPS walls in supersymmetric gauge theories, hep-th/ , Commun. Math. Phys. 267 (2006) Global structure of moduli space for BPS walls, hep-th/ , Phys.Rev.D71 (2005) , D-brane Construction for Non-Abelian Walls, hep-th/ , Phys.Rev.D71, (2005), Non-Abelian Walls in Supersymmetric Gauge Theories, hep-th/ , Phys.Rev.D70 (2004) , Construction of Non-Abelian Walls and Their Complete Moduli Space, hep-th/ , Phys.Rev.Lett.93 (2004) , Exact Wall Solutions in 5-Dimensional SUSY QED, 24
26 hep-th/ , JHEP 11 (2003) 060, Massless Localized Vector Field on a Wall in Five Dimensions, hep-th/ , JHEP 11 (2003) 061, Vacua of Massive Hyper-Kähler Sigma Models with Non-Abelian Quotient, hep-th/ , Prog. Theor. Phys. 113 (2005) 657, Manifest Supersymmetry for BPS walls in N = 2 nonlinear sigma models,, hep-th/ , Nucl.Phys.B652 (2003) 35-71, BPS Wall in N = 2 SUSY Nonlinear Sigma Model with Eguchi-Hanson Manifold, hep-th/ , in Garden of Quanta - In honor of Hiroshi Ezawa, pages , 3. Vortex Statistical Mechanics of Vortices from D-branes and T-duality, hep-th/ , Phys.Rev.to appear Non-Abelian Vortices on Cylinder Duality between vortices and walls, hep-th/ , Phys.Rev.D73 (2006) , Moduli Space of Non-Abelian Vortices, hep-th/ , Phys.Rev.Lett.96 (2006) , Effective Theory on Non-Abelian Vortices in Six Dimensions, hep-th/ , Nucl.Phys.B701, 247 (2004), 4. 1/4 BPS states 25
27 Dynamics of Domain Wall Networks, arxiv: [hep-th], Phys.Rev.to appear. Effective Action of Domain Wall Networks, Phys.Rev.D75 (2007) , hepth/ , Non-Abelian Webs of Walls, Phys.Lett. B632 (2006) , hep-th/ , Webs of Domain Walls in Supersymmetric Gauge Theories, hep-th/ , Phys.Rev.D72 (2005) , Monopoles, Vortices, Domain Walls and D-branes: The rules of Interaction, hep-th/ , JHEP 03 (2005) 019, Instantons in the Higgs Phase, hep-th/ , Phys. Rev. D72, (2005), All Exact Solutions of a 1/4 Bogomol nyi-prasad-sommerfield Equation, hep-th/ , Phys.Rev.D71, (2005), Domain Wall Junction in N = 2 Supersymmetric QED in four dimensions, hep-th/ , Phys.Rev.D68 (2003) , BPS Lumps and Their Intersections in N = 2 SUSY Nonlinear Sigma Models, hep-th/ , Grav.Cosmol.8 (2002) , 5. Non-BPS Walls and Supersymmetry Breaking Non-BPS Walls and their stability in 5D Supersymmetric Theory, hep-th/ , Nucl.Phys.B696 (2004) 3-35, 6. Effective Lagrangian 26
28 Manifestly Supersymmetric Effective Lagrangians on BPS Solitons, hep-th/ , Phys.Rev.D73 (2006) , 7. Wall Solution in Supergravity Bogomol nyi-prasad-sommerfield Multiwalls in Five-Dimensional Supergravity, hep-th/ , Phys.Rev.D69 (2004) , Wall solution with weak gravity limit in five-dimensional Supergravity, hep-th/ , Phys.Lett.B556 (2003) , Solitons in 8 SUSY Theories 1. Review D. Tong, TASI lectures on solitons, arxiv:hep-th/ M. Shifman and A. Yung, Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories, arxiv:hep-th/ Early works M. Cvetic, F. Quevedo and S. J. Rey, Phys.Rev.Lett.67, 1836 (1991) E. Abraham and P. K. Townsend, Phys.Lett.B 291, 85 (1992) M. Cvetic, S. Griffies and S. J. Rey, Nucl.Phys.B 381, 301 (1992) M. Cvetic, S. Griffies and H. H. Soleng, Phys.Rev.D 48, 2613 (1993) 27
29 G. Dvali and M. Shifman, Phys.Lett.B396, 64 (1997) 3. Wall Solution in 8 SUSY Models J. P. Gauntlett, D. Tong, and P. K. Townsend, Phys.Rev.D63, (2001); J. P. Gauntlett, R. Portugues, D. Tong, P. K. Townsend, Phys.Rev.D63, (2001); J. P. Gauntlett, D. Tong, and P. K. Townsend, Phys.Rev.D64, (2001) R. Portugues, P. K. Townsend, JHEP , (2002) D. Tong, Phys.Rev.D66, (2002) The Moduli Space of BPS Domain Walls M. Shifman and A. Yung, Phys.Rev.D67, (2003) D. Tong, JHEP 0304, 031 (2003) Mirror Mirror on the Wall M. Arai, E. Ivanov and J. Niederle, Nucl.Phys.B680, 23 (2004) M. Shifman and A. Yung, Phys.Rev.D70, (2004) 4. Monopoles in Higgs Phase D. Tong, Phys.Rev.D69, (2004) R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and A. Yung, Nucl.Phys.B673, 187 (2003) R. Auzzi, S. Bolognesi, J. Evslin and K. Konishi, Nucl.Phys.B 686, 119 (2004) M. Shifman and A. Yung, Phys.Rev.D 70, (2004) R. Auzzi, M. Shifman and A. Yung, JHEP 0502 (2005)
30 5. Vortex A. Hanany and D. Tong, JHEP 0307, 037 (2003) R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and A. Yung, Nucl.Phys.B 673, 187 (2003) M. Shifman and A. Yung, Phys.Rev.D 70, (2004) A. Hanany and D. Tong, JHEP 0404, 066 (2004) A. Hanany and D. Tong, arxiv:hep-th/ ; M. A. C. Kneipp and P. Brockill, Phys.Rev.D 64, (2001) M. A. C. Kneipp, Phys.Rev.D 68, (2003) ; Phys.Rev.D 69, (2004) 6. Brane construction R. Auzzi, S. Bolognesi and J. Evslin, JHEP 0502 (2005) Index theorem K. S. M. Lee, Phys.Rev.D 67, (2003) introduction 29
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D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j
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φ 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
December 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..........................
2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c
GL 2 1 Lie SL(2, R) GL(2, A) Gelbart [Ge] 1 3 [Ge] Jacquet-Langlands [JL] Bump [Bu] Borel([Bo]) ([Ko]) ([Mo]) [Mo] 2 2.1 H = {z C Im(z) > 0} Γ SL(2, Z) Γ N N Γ (N) = {γ SL(2, Z) γ = 1 2 mod N} g SL(2,
1 : ( ) ( ) ( ) ( ) ( ) etc (SCA)
START: 17th Symp. Auto. Decentr. Sys., Jan. 28, 2005 Symplectic cellular automata as a test-bed for research on the emergence of natural systems 1 : ( ) ( ) ( ) ( ) ( ) etc (SCA) 2 SCA 2.0 CA ( ) E.g.
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Chebyshev 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
Lebesgue Fubini L p Banach, Hilbert Höld
II (Analysis II) Lebesgue (Applications of Lebesgue Integral Theory) 1 (Seiji HIABA) 1 ( ),,, ( ) 1 1 1.1 1 Lebesgue........................ 1 1.2 2 Fubini...................... 2 2 L p 5 2.1 Banach, Hilbert..............................
A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B
9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A
30
3 ............................................2 2...........................................2....................................2.2...................................2.3..............................
橡超弦理論はブラックホールの謎を解けるか？
1999 3 (Can String Theory Solve the Puzzles of Black Holes?) 305-0801 1-1 makoto.natsuume@kek.jp D-brane 1 Schwarzschild 60 80 2 [1] 1 1 1 2 2 [2] 25 2.2 2 2.1 [7,8] Schwarzschild 2GM/c 2 Schwarzschild
SAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T
SAMA- SUKU-RU Contents 1. 1 2. 7.1. p-adic families of Eisenstein series 3 2.1. modular form Hecke 3 2.2. Eisenstein 5 2.3. Eisenstein p 7 3. 7.2. The projection to the ordinary part 9 3.1. The ordinary
.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,
[ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b
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![( ) Note (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e](/thumbs/98/136160482.jpg "( ) Note (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e")
( ) Note 3 19 12 13 8 8.1 (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e µ τ, e R, µ R, τ R (1a) L ( ) ) * 3) W Z 1/2 ( - )
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数論 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/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published
arxiv: 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)
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
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)
N cos s s cos ψ e e e e 3 3 e e 3 e 3 e
3 3 5 5 5 3 3 7 5 33 5 33 9 5 8 > e > f U f U u u > u ue u e u ue u ue u e u e u u e u u e u N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 > A A > A E A f A A f A [ ] f A A e > > A e[ ] > f A E A < < f ; >