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- さみら あんさい
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2 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 fxϵ x + ϵ 1 π fxϵ x + ϵ dx x + ϵ + f x ] ϵ x + ϵ [ 1 fϵ π x + ϵ + 1 f x ] ϵ x + ϵ dx t q φx = dt ϵ π 3 d 3 k e +ik x x tk q = φx = dt ϵ π 3 d 3 k e +ik x x tk φx = = φx = dt t 1/ 1 qπ 3 / ϵ π 3 d 3 k e x x /4t dt t 1/ 1 qπ 3/ /4t ϵ π 3 e x x p7, 1.46 d dx θxfx = θx d dx fx = ddx fx = f = dx d dx θxfx = dx θx d dx fx = dx d fx = f dx 1
3 p8, l1 = p8, l1, =, p8 = p9, 1.55 p9, 1.58 ΓaΓb = Γa + b 1 = ΓaΓb = Γa + b p1, A a B b = 1 A a B b = 1 ΓaΓb = 1 A a B b = 1 ΓaΓb 1 = 1 A a B b = 1 ΓaΓb p1, ABC = = 1 ABC = dν dν µ a 1 1 µ b 1 δ1 µ 1 µ dµ 1 1 dt 1 dµ µ a 1 1 µ b 1 δ1 µ 1 µ dt 1 t a 1 1 t b 1 e At 1 Bt dt 1 1 dν 1 dν δ1 ν 1 ν 1 1 dν 1 dν δ1 ν 1 ν dt t a 1 1 t b 1 e At 1 Bt dρ ρ a+b 1 e ρaν1+bν dρ ρ a+b 1 ν a 1 1 ν b 1 e ρaν 1+Bν dρ ρ 3 dν 1 dν dν 3 δρ1 ν 1 ν ν 3 dν 3 e ρaν 1+Bν +Cν 3 p1, =.. p11, l1 C d = dω1 = C d = dω dρ ρ 3 dν 1 dν dν 3 δρ1 ν 1 ν ν 3 e ρaν 1+Bν +Cν 3
4 p11, l δv = C d r d dr = δv = C d r d 1 dr p11, l5, r = t =, r = t p1, 1.74 γ = lim n i=1 = γ = lim n n 1 logn + 1 i n i=1 1 i log n p1, p x = e ip x/ /π 3/ = x p = e ip x/ /π 3/ p13, 1.84 mω x + p p14, α H = 1 m = H = 1 mω x + p m =. α p14, =. p15, l4 L ± = L x ± il y = L = L x + L y + L z p15, 1.94 σ i σ j + σ j σ i = δiji, j = x, y, z p15, l6 l =, l 3
5 . p17,.5 ct x = cosh ηct sinh ηx = cosh ηx sinh ηct p18,.8.,. ds, = ct = ct cosh η x sinh η x = x cosh x ct sinh η =.,. ds, p18, cds = dx µ dx ν = cds = dx µ dx µ p19 p19,.14. =. p19,.15 = p19,.16 B t 1. = [A, B] t 1. p19,.17 e iaν P ν x µ e iaν P ν = x µ + [ ia ν P ν, x µ ] = x µ + a µ = e iaν P ν x µ e iaν P ν = x µ + [ia ν P ν, x µ ] = x µ + a µ p19,.19, g µν = η µν p,.1 [J µν, η µν dx µ dx ν ] = = [J µν, η ρσ dx ρ dx σ ] = 4
6 p,.3 J µν ρ,σ = iδ µ ρ δ ν σ δ ν ρδ µ σ = J µν ρ,σ = iδ ν ρδ µ σ δ µ ρ δ ν σ p,.4 e iω µνj µν / x µ e iω µνj µν / = Λ µ ν x ν = e iωµνj µν / x µ e iωµνj µν / = Λ µ ν x ν p,.6 e iη K x e +iη K = x η x η η + η xη η = e iη K x e +iη K = x cosh η x η η cosh η x η η sinh η sinh η p1,.8 = p1, = p,.33 Edx = 1 B ds dct = E dx = B ds ct p,.33 = Edx cdt + B ds = E dx cdt + B ds = p,.36 λ F µν + ν F νλ + ν F λµ = λ F µν + µ F νλ + ν F λµ = p,.37 µ = ν 5
7 p3,.39 A µ + µ ν A ν = j µ = A µ µ ν A ν = j µ p3,.41, =, p3,.4., 1s = m. =. 1s = m p4,.47, = F q 1 q /r, p5,.5 p5,.57, ψ = e ipµxmu p µ = ˆp = i µ =, ψ e ip µx µ p5,.57. =. p6,.58. α i i = 1,, 3 =. α i i = 1,, 3 p6,.59. =. = p µ = ˆp µ = i µ 6
8 p6,.59 i ψ t = i cα k k + βmc ψ = i ψ t = i cα k k + βmc ψ p8,.74 i x + i σ i x i σ φ L = mc φ L = i x + i σ i x i σ φ L = mc φ L p9,.85 p3, iγµλ ν µ ν + ms 1 Λψ x = = iγ µ Λ ν µ ν + ms 1 Λψ x =. J µν µν J µν [γ µ, γ ν ] = S µν. J µν µν, S µν, S µν [γ µ, γ ν ] p3,.9 = p3,.91 p31, u r, v r r p31,.95 p31,.96 p31,.98 SΛ = = S Λ = mγ mu r m, k = = = mγ mu r m, k = = u r u, k = e iη K u r m, k = = u r k = e iη K u r m, k = 7
9 ,, =, p31,.1 p31,.11 ψt = e ie/ ψ = ψt = e iet/ ψ p31,.11 i ψ t = γ mc ψ = i γ ψ t = mc ψ p3,.13 r = 1, p3,.14 p3,.15 Eγ mc βψm.k = = = Eγ mc ψm.k = = u r k = Sηu r m.k =, v r k = Sηv r m.k = = u r k = Sηu r m.k =, v r k = Sηv r m.k = u r k = u r m.k = S 1 η, v r k = v r m.k = S 1 η = u r k = u r m.k = S 1 η, v r k = v r m.k = S 1 η p3,.15. =. p3,.114 u r m, k = e iη K u r m, k = = u r m, k = e iη K u r m, k = p33,.116 u r = = ur = 8
10 3. p38, 3.6. =. p38, 3.7 a n t = a n t = e ie n/ = a n t = a n t = e ie nt/ p39, l4 l9 = p39, 3.17 p4, l 3.19 e iet = e iet/ p4, 3.19 L H = m x ψ x dx ψx L = H = m x ψ x x ψx ψx, t = e iet/ ψx, e i Et/ =, ψx, t = e iet/ ψx, e iet/ p4, 3.19 p41, l ψx, t = ψ x, t = a n ψ n xe ient n=1 a nψ n xe ie nt n=1 = ψx, t = ψ x, t = a n ψ n xe ie nt/ n=1 a nψnxe ie nt/ p41, 3.3, {A, B} = AB + BC = =,, {A, B} = AB + BA = n=1 9
11 p41, 3.4, =, p41, p4, 3.9 = p4, 3.3 = p4, 3.3 q n = a n + a n ρωn = q n = a ne iωnt + a ne iωnt ρωn p43, 3.35 ψ L n = e i nπ L x ct, ψ R n = e i nπ L x+ct = ψn L = e i nπ L x ct, ψn R = e i nπ L x+ct p44, 3.36 φx, t = n= = φx, t = p44, 3.36 e iωnt a n e i nπ L x ct + ā n e i nπ L x+ct ρωn n= = e iω nt p44, ω n = n πc/l p44, 3.37 φx, t = n= = φx, t = a n e i nπ L x ct + ā n e i nπ L x+ct ρωn 1 a n e inπx/l iωnt + a ρωn ne inπx/l+iωnt n= 1 a n e inπx/l iωnt + a ρωn ne inπx/l+iωnt 1
12 p44, 3.39 E = πc L n= = E = πc L na na n n= n a na n p44, 3.4 φ n x, t = = φx, t = p44, 3.41 n= 1 [ α n e i nπ L x ct + α n e i nπ x+ct] L πc/l n= [ 1 i πc/l n α ne i nπ L x ct + i ] n α nπ i L ne x+ct p45, 3.51 [a m, a n ] = nδ m, n = [α m, α n ] = nδ m, n p45, 3.51 dk H = πδk k = π = H = πδ dk π πδk = = dx = πδ = dx k k p46, 3.7. =. p46, 3.56 p48, E = π dk k = E = 1 dk dx π k 11
13 =. p48, 3.69 δs = tf t i = δs = p49, 3.74 = p49, 3.74 L qt δqt + L tf L qt δqt + t i q δ qt L. p49, 3.76 p49, 3.76 = p49, 3.78 dx k φ x = dx K φ x L qt δ qt δs δφx, t = lim Sφx, t + ϵδx xδt t ϵ ϵ = δs δφx, t = lim S φx, t + ϵδx xδt t Sφx, t ϵ ϵ p5, 3.85, =, p5, = p51, 3.89,,. =,,. 1
14 p51, 3.9 k x = ωt ik x = k x = ωt k x 13
15 4. p54, 4.1 l3, =, p54, 4.4 x j+1 e iδtp +V x x j = dp x j+1 e iδtp +V x p j p j x j dp = π e iδtp j +V xj+1 e ipjxj+1 xj = x j+1 e iδtp /m +V x x j = dp j x j+1 e iδtp /m +V x p j p j x j dpj = p54, 4.4 e iδtp +V x = e iδtp /m +V x p54, 4.6 x j+1 x j /δ = ẋ π e iδtp = x j+1 x j /δt = ẋ j p55, 4.9 = dx i dx f ψ f x f x f e iht x i x i ψ i = = dx i dx f ψ f x f x f e iht x i x i ψ i p55, 4.1 p56, l1 = M i=1 = = xk t=x kf x k =x ki xk t=x kf M k=1 x k =x ki Dx k te im P ẋi / V ]dt Dx k te i P R t [mẋ k / V ]dt j /m +V x j+1 e ip jx j+1 x j t i = ϕx = 1 t = = t i = ϕx = t = p56, 4.14 x i A ij x j = λ i x i = x i A ij x j = λ i x i 14
16 p56, 4.15 p56, = λ 1 /ψλ /ψ λ N /ψ 1 = = λ 1 /πλ /π λ N /π 1 = det A/π = = 1 1 det A/π N 1 p57, 4.1 ω k = k + m = ωk = k + m p58, 4.8 loga + ϵ logb + iϵ = = loga + iϵ logb + iϵ = p58, 4.33 h = 1 d 3 k π 4 π 1 1 Γ 1/π = h = 1 d 3 k π 4 π 1 Γ 1/ωk 15
17 5. p59, l4. =,. p59, 5.1 l1. =,,. p59, 5.1 l9. =,,. p59, 5.1 l14. =,,. p6, 5. p6, 5.3 V φ = m φ + λ 4! φ4 x = V φ = m φ x + λ 4! φ4 x H φ 3 x,. = H φ 4 x,. p6, 5.4 t i = t p6, 5.4 p61, 5.11 Ut, t 1 = t = Ut, t 1 = dt φt i, x = = φt, x = t d e ih t t e iht t dt dt d dt e ih t t e iht t 16
18 p61, 5.15 p6, 5.19 = p6, 5. Ut, t = 1 + i t = Ut, t = 1 + i dt 1 H I t t + t dt 1 H I t 1 + ψt I = Ut, t ψt I = ψt I = Ut, t ψt I p63, 5.3 l3 ak 1 a k = π 4 δk 1 k + a k ak 1 = ak 1 a k = π 3 δ 3 k 1 k + a k ak 1 p63, 5.3 l6 = ψ 4 δk 1 k = = π 3 δ 3 k 1 k p63, ω k = ωk p63, 5.7 p63, 5.8 = i θt 1 t e ik t 1 t dz = i π = θt 1 t e iωkt 1 t = i d 3 kdz π 4 ω k = = i p64, l1 d 3 kdk π 4 ωk { e ik t 1 t +ik x 1 x e ik t 1 t z ω k iϵ dk π e ik t 1 t k ωk iϵ } + e ik t 1 t +ik x 1 x k ω k iϵ k ω k iϵ { e ikt1 t+ik x1 x e ikt t1+ik x x1 + k ωk iϵ k ωk iϵ ω k = ω k p64, 5.4 l1 = } 17
19 p64, 5.3 e 1/cx = x e 1/cx p64, 5.33 x n = x ne 1 cx e 1 cx = x n = xn e 1 cx dx e 1 cx dx p64, 1/a = 1/c p66, 5.43 p66, 5.45 = x A 1 J/ Ax J A 1 J// + J A 1 J/ = = x A 1 J Ax A 1 J/ + J A 1 J/ p67, 5.48 = 1 λ 4! = A 1 ij A 1 kl + A 1 ik A 1 jl + A 1 il A 1 jl = = A 1 ij A 1 kl + A 1 ik A 1 jl + A 1 il A 1 jk c + J c 1 λ c 4! 3 1c + J J = = 1 λ 4! + 1 J c p67, 5.1 l 1 λ 3 1c 4! c λ 4! 1 λ 4! c 1 λ 4! J c 1 J J c c λ 4! 6 J 3 1c λ 4! 3 1c λ 4! J c c 3 6 J c 3 J 4 c 4 λ 4! c Z a, b. = c a 5.1 b. p67, 3 1 a J 4 c
20 = 3 1 c p68, 5.49 p68, 5.49 = ZJ = [W J, λ] n n= = = Z[J = ] J /aλd/dj 4 /4!J /a 3 /3! n= 1 [W J, λ]n n! = J /cλd/dj 4 /4!J /c 3 /3! p68, 5.51 Z[J] = Dφ e i R d 4 x{ 1 [ φ mφ +Jφ] = Z[J] = Dφ e i R d 4 x{ 1 [ φ m φ ]+Jφ} p68, 5.5 Z[J] = Dφ e i R d 4 x{ 1 [φ m φ+jφ] = Z[J] = Dφ e i R d 4 x{ 1 [φ m φ]+jφ} p69, 5.54 i d 4 x = i p69, 5.55 { 1 φx { 1 d 4 x dy Dx yjy/ m φx dy Dx yjy/ } d 4 xd 4 y JxDx yjy φx d 4 y Dx yjy m φx d 4 y Dx yjy 1 } d 4 y JxDx yjy iw [J Z[J] = Z[J = ] e iw [J] = Z[J] = Z[J = ] e 19
21 p69, 5.6 Z[J] = Dφ e i R d 4 x{ 1 [φ m φ+jφ] = Z[J] = Dφ e i R d 4 x{ 1 [φ m φ]+jφ} p69, 5.61 Z[J] = Dφ e i R d 4 x{ 1 [φ m +iϵφ+jφ] = Dφ e i R d 4 x{ 1 [φ m φ+jφ] ϵ R d 4 x φ = Z[J] = Dφ e i R d 4 x{ 1 [φ m +iϵφ]+jφ} = Dφ e i R d 4 x{ 1 [φ m φ]+jφ} 1 ϵ R d 4 x φ p7, 5.6 φxφy = δ δφx = φxφy = δ δφy eiw = idx y J= δ δ{ijy} eiw = idx y J= δ δ{ijx} p7, 5.7 l1 µφ 4 /4! = λφ 4 /4! p7, 5.64 = Gnx 1,, x n n p71, 5.68 iµ = iλ p71, 5.68 i k k = i k i p71, = p7, 5.69 p7, 5.69 ˆp µ p = p m u p = ˆp µ k = k µ k
22 [ˆx, ˆp] = iη µν = [ˆx µ, ˆp ν ] = iη µν p7, 5.7 e ikˆx p = = k = e ikˆx k = = k p7, 5.71 x p = ˆx ˆp p7, 5.73 p7, 5.74 i Ĥ + iϵ i = Ĥ + iϵ A n = k n V k n 1 = A N = k N p7, 5.75 i Ĥ + iϵ V k n, x i Ĥ + iϵ V k N 1, x i Ĥ + iϵ V k, x k 1 i Ĥ + iϵ V k N, x i Ĥ + iϵ V k, x p7, 5.76 p7, 5.77 i Ĥ + iϵ = = Ã = k n m k 1 m i Ĥ + iϵ = dt e iĥt ϵt dt e iĥt ϵt dt n 1 dt n V k n, xe iĥtn 1 V k n 1, xe iĥtn e iĥt1 V k 1, x = A N = dt N 1 dt N dt 1 V k N, xe iĥt N 1 V k N 1, xe iĥt N e iĥt1 V k 1, x p73, 5.77, = N, dt 1 i Ĥ + iϵ k 1 1
23 p73, 5.78 iht xt = e +ihtˆxe iĥt = ˆxt = e iĥtˆxe p73, 5.79 d xt = ieiht [Ĥ, ˆx]e iĥt dt = d ˆxt = ieiĥt [Ĥ, ˆx]e iĥt dt p73, 5.81 Ã = kn m k1 m dt n 1 dt n dt 1 e im t 1 +t + t n 1 V k n, xt 1 + t + + t n 1 V k n 1, xt 1 + t + + t n V k 3, xt 1 + t V k, xt 1 V k 1, x = Ã = dt N 1 dt N dt 1 e im t 1 +t + t N 1 V k N, xt 1 + t + + t N 1 V k N 1, xt 1 + t + + t N V k 3, xt 1 + t V k, xt 1 V k 1, x
24 6. 1 p75, 6.5 p75, 6.7 p75, 6.8 ψx = ψx = = ψx = H = ψ ψ ψi/ mψ = = H = iψ ψ ψi/ mψ = {ψ a x, ψ b y} = π3 δ 3 x yδ ab, {ψ a x, ψ b y} = {ψ ax, ψ b y} = = {ψ a x, ψ b y} = π3 δ 3 x yδ ab, ψx = p75, 6.9 p75, 6.9 {ψ a x, ψ b y} = {ψ ax, ψ b y} = d 3 k π 3 m ωk 1 d 3 k m 1 π 3 ωk d 3 k m π 3 ωk d 3 k π 3 r m ωk b r kupe ik x + d r kv s ke ik x d r kupe ik x + b r ku r e ik x r 1 r 1 = π 3 δk 1 k δ rs r b r ku r ke ik x + d r kv r ke ik x = = π 3 δ 3 k 1 k δ rs. d r ku r ke ik x + b r ku r ke ik x =, 6.7. p75, 6.1 d 3 k 1 m ψ = π 3 ωk s d 3 k m = ψ = π 3 ωk b s u s k + d s kv s k 1 s b s k u s k + d s k v s k 3
25 p75, 6.11 d 3 k m = [u π 3 r ku r ke ik x + v r kv r ke ik x] ωk r d 3 k m = = [u π 3 r ku r ke ik x + v r kv r ke ik x] ωk r p75, 6.1 d 3 k m = π 3 ωk /k + me ik x + /k me ik x] d 3 k 1 = = [/k π 3 + me ik x + /k me ik x] ωk p75, 6.13 d 3 k 1 = π 3 ωk ωkγ e ik x = d 3 k 1 = = π 3 ωk ωkγ e ik x = p76, 6.15 d 3 k H = ωka r π 3 ka r k + ωkb r kb r k r d 3 k = H = ωkb r π 3 kb r k + ωkd r kd r k p76, 6.17 = d 3 k π 3 θt t 1 r d 3 k = = π 3 p76, 6.18 = r m θt 1 t ωk r v r kv r ke +ipx1 x θt t 1 r d 3 k π 3 m θt 1 t ωk r v r kv r ke +ikx 1 x u r ku r ke ipx 1 x e ipx1 x u r ku r ke ikx 1 x m θt 1 t /k + me ipx1 x ωk θt t 1 /k me +ipx 1 x d 3 k 1 = = π 3 θt 1 t /k + me ikx 1 x ωk θt t 1 /k me +ikx 1 x 4
26 p77, 6.19 p78, 6.7 p78, 6.8 isx 1 x = i d 4 x π 4 d 4 x = π 4 d 4 k = isx 1 x = i π 4 = /p + m p m + iϵ e ip x1 x i /p m + iϵ e ip x1 x d 4 k π 4 Z = C det/ m + iϵ = Z = C deti/ m + iϵ Z = C det γ 5 / m + iϵγ 5 /k + m k m + iϵ e ik x1 x i /k m + iϵ e ik x 1 x = Z = C det γ 5 i/ m + iϵγ 5 γ 5 = γ γ 1 γ γ 3 γ 5 γ µ = γ µ γ 5 p78, 6.9 Z = C[det/ m + iϵ / m + iϵ] 1 = C[detI + m ϵ] 1 = Z = C[deti/ m + iϵ i/ m + iϵ] 1 = C[detI + m iϵ] 1 p78, 6.3 Z = C[det + m ϵ] = Z = C[det + m iϵ] p78, 6.31 Z = C[det + m ϵ] 1 = Z = C[det + m iϵ] 1 p79, 6.34 i/k m + iϵsx y = δx y = /k m + iϵsx y = δ 4 x y 5
27 p79, 6.35 { i d 4 x ψ + = i p8, 6.38 p8, 6.4 p8, 6.41 p8, 6.4 p81, 6.44 p81, 6.45 d 4 x d 4 y ηysx y i/k m + iϵ { ψx + d 4 y ηysx y /k m + iϵ L = ψ µ σ µ ψ + χ µ σ µ χ + mψ χ + m χ ψ = L = iψ σ µ ψ + iχ σ µ µ χ mψ χ mχ ψ mψψ m cos θ ψψ + m sin θ ψiγ 5 ψ = mψψ m cosθ ψψ + m sinθ ψiγ 5 ψ L = ψ µ σ µ ψ + χ µ σ µ χ + me iθ ψ χ + me iθ χ ψ = L = iψ σ µ ψ + iχ σ µ µ χ me iθ ψ χ me iθ χ ψ L = ψ µ σ µ ψ + χ µ σ µ χ + mψ χ + m χ ψ = L = iψ σ µ ψ + iχ σ µ µ χ mψ χ m χ ψ {ak, a k } = π 3 δk k, {ak, ak } = {a k, a k } = = {ak, a k } = π 3 δ 3 k k, {ak, ak } = {a k, a k } = [ake ik x + a ke ik x ], [ak e ik x + a k e ik x ] { = [ake ik x + a ke ik x ], [ak e ik x + a k e ik x } ] p81, x x. = 1 x x. p81, 6.46.=. 6
28 p81, 6.47 [ψ α x, t =, ψ α ] = = [ψ α x, t =, ψ α x, t = ] = p81, =. 7
29 7. p83, 7.1 p83, 7. p83, 7.3 L A µ = j µ = L A µ = J µ p83, 7.3 p83, 7.4 j µ = eψγ m ψ = J µ = eψγ µ ψ p83, 7.4 p84, 7.7 p84, 7.1 L = 1 4 F µνf µν j µ A µ = L = 1 4 F µνf µν + J µ A µ ν dl ν A m = νf µν = ν L ν A µ = νf µν ν F µν = j µ = ν F µν = J µ L = ψi/ mψ 1 4 F µνf µν + eψγ m ψa µ = L = ψi/ mψ 1 4 F µνf µν + eψγ µ ψa µ ψ e ieϵx ψ, p84, 7.1 ψ ψe +ieϵx = ψ e +ieϵx ψ, ψ ψe ieϵx g µν µ ν G νρ = = η µν µ ν G νρ = A µ = µ Λ. 8
30 = A µ = µ Λ. p85, l3. =. p85, 7.16 A g 1 = Dg δ fa g g Dg δ fa g = A 1 = A g 1 = Dg δ fa gg = Dg δ fa g = A 1 p85, 7.17 Z = = Z = DA e isa δa Dg δ fa g DA e isa A Dg δ fa g p86, 7.1 A = Det = A = Det p86, 7.4 S gf = SA i d 4 x A ξ { [ 1 = d 4 x g µν 1 1 ] } µ ν A ν + A µ J µ ξ = S gf = SA 1 d 4 x A ξ { [ 1 = d 4 x A µ η µν 1 1 ] } µ ν A ν + A µ J µ ξ p86, 7.5 Q µν = k g µν + = Q µν = k η µν + k µ k ν 1 1 k µ k ν ξ 1 1 ξ 9
31 p86, 7.6 D µν = 1 k = D µν = 1 k [g µν 1 ξ kµ k ν ] k [η µν 1 ξ kµ k ν ] k p87, 7.7 Z[j µ iw [J] ] = Z[]e iw [J] = Z[J] = Z[]e p87, 7.8 W [j µ ] = frac1 d 4 xd 4 y j µ xd µν x yj ν y = W [J] = 1 d 4 xd 4 y J µ xd µν x yj ν y p87, 7.9 D µν x y = d 4 y 1 [g µν k 1 ξ kµ k ν ] k e ikx y d 4 k 1 = D µν x y = [η µν π 4 k 1 ξ kµ k ν ] k e ikx y p87, 7.3 D µν x y d 4 k = π 4 = D µν x y = 1 k d 4 k π 4 [g µν + 1 ξ µ ν 1 k k ] e ikx y [η µν + 1 ξ µ ν k ] e ikx y p87, 7.3 W [j µ ] = W [J] p87, 7.3 µ J µ = µ J µ p87, 7.31 W [j µ ] = 1 d 4 xd 4 y d4 k π 4 jµ x 1 k eikx y j ν y = W [J] = 1 d 4 xd 4 d 4 k y π 4 J µx 1 k eikx y J µ y p87,
32 = p87, 7.33 p87, 7.34 W [j µ ] = W [J] p88, 7.35 W = q 1 q = W = q 1 q p89, 7.39 p9, 7.47 d 4 xd 4 y Z[jM] = = Z[J] = d4 k δ 3 x x 1 δ 3 y x π 4 e ik t 1 t ik x y d 4 xd 4 y k k + iϵ d 4 k δ 3 x x 1 δ 3 y x π 4 k k + iϵ d 3 k E = q 1 q π 3 = E = q 1 q d 3 k π 3 M µ p9, 7.47 e + = ϵ + p91, l1 γ m γ ν = = γ µ γ ν = p91, 7.49 D µ D µ i σµν [D µ, D n ] ψ = = D µ D µ i σµν [D µ, D ν ] + m ψ = e ikx y ik x y p91, 7.5 D µ D µ e σµν F µν ψ = = D µ D µ e σµν F µν + m ψ = 31
33 p91, 7.51 D = ie µ A µ + A µ d µ + OF = D = ie µ A µ + A µ µ + OF p91, 7.5 p91, 7.53 ψ = e imt φ [ t [ eb L + S ] m ] ψ = = [ t [ eb L + S ] + m ] ψ = = ψ = e imt φ p91, 7.7. = p91, 7.7.,,. =,,. p93, l1,. =,,. p93, l3 = p93, l5 = p93, =. p93, 7.57 Γ µ d 4 k = π 4 = Γ µ = p93, 7.57 /p + m = /p + /k + m ieγ µ d 4 k π 4 i i i /p + /k m γµ /p + /k m ieγ ν k ieγ ν i /p + /k m γµ i /p + /k m ieγ ν i k 3
34 p93, 7.58 Γ µ = ie d 4 k π 4 = Γ µ = ie p93, 7.58 ψ ψ = p p p94, 7.6 p = p = m γ ν /p + /k + mγ µ /p + /k + mγ ν p + k m p + k m k d 4 k π 4 = p = p = m p94, 7.6 P µ = 1 αp µ βp µ, P µ = 1 αpµ βp µ γ ν /p + /k + mγ µ /p + /k + mγ ν p + k m p + k m k = P µ = 1 αp µ βp µ, P µ = 1 βp µ αp µ p94, 7.63 γ µ γ ν = γ ν γν + η µν = γ µ γ ν = γ ν γ ν + η µν p94, 7.64 p94, 7.65 p94, 7.66 p94, 7.67 ψ µ = p µ p95, 7.7 = γ µ = = γ µ γ λ γ µ γ ν γ λ = γλγ µ γ λ γ ν + γ ν γ µ = γ λ γ µ γ ν γ λ = γ λ γ µ γ λ γ ν + γ ν γ µ γ λ γ µ γ ν γ ρ γ λ = 4η µν + γ ρ γ µ γ ν = γ λ γ µ γ ν γ ρ γ λ = 4η µν γ ρ + γ ρ γ µ γ ν [ 1 β/p α/p γ µ 1 α/p β/p = 1 β/p α/p γ µ 1 α/p β/p 33
35 p95, 7.71 p95, 7.73 p96, 7.77 p96, 7.78 p96, 7.79 p96, 7.81 [1 β/p αm γ µ 1 α/p βm = 1 β/p αm γ µ 1 α/p βm F n = = I n = 1 A n = 1 Γn + 1 = 1 A n = 1 Γn I n = i 1 n 1 Γn + 1 = I n = i 1 n 1 Γn t n 1 t n 1 t n 1 e At t n 1 e At d 4 k π 4 e k +µ t d 4 k π 4 e k +µ t I n = i 1 n 1 4π Γn Γn + 1µ n 1 = I n = i 1 n 1 4π Γn Γnµ n F p + p µ m = e 8π p + p µ m = F p + p µ m = e p + p µ 8π m 34
36 8. p99, 8. F s, m = F s, m p99, 8. dk/k = d 4 k/k 4 k 3 dk/k 4 dk/k p99, 8.11 i 1 3π = i p1, 8.1 p1, 8.13 p1, 8.15 p1, 8.16 p11, 8.17 i 1 1 3π 1 3π = i im = iλ + i Λ dα dm 1 m m α1 αk π dα Λ m m α1 αk dm 1 Λ α1 αs dα log m α1 αs λ 3π = im = iλ i 1 1 λ 3π dα log Λ α1 αs m α1 αs Λ [ m α1 αs dα log 1 im = iλ + iλ L + Oλ 3 [ m α1 αs dα log = im = iλ iλ L + Oλ 3 iλ P = iλ + iλ L + Oλ 3 = iλ P = iλ iλ L + Oλ 3 iλ = iλ P iλ L + Oλ 3 P = iλ = iλ P + iλ L + Oλ 3 P Λ 35
37 p11, 8.18 p11, 8.19 im = iλ P + i im = iλ P + iλ P L L + Oλ 3 P = im = iλ P iλ P L L + Oλ 3 P λ P 3π = im = iλ P i p1, λ P 3π [ m α1 αs dα log m α1 α4m 1 [ dα log m α1 αs m α1 α4m. Γ = γ = ϵ, Γϵ 1/ϵ γ + Oϵ p13, 8. = p13, 8. G 1: p13, 8. F µν µν = F µν F µν p13, 8. = p13, 8.7 G = G 36
38 p14, 8.4., g /M ψγ m ψψγ µ ψ. p14, 8.4 g /M ψγ m ψψγ µ ψ = g /M ψγ µ ψψγ µ ψ p14, 8.4 l = λ p14, 8.9 i p15, 8.5 l = λ p15, 8.3 p15, 8.31 p15, 8.6 d 4 p π 4 = iλ i p m + iϵ d 4 p π 4 I = = I = I = a + bk + ck 4 + =, i p m + iϵ = I = iam + bk + ck
39 p15, 8.3 i k m + iϵ + i k m + iϵ a + i bk k m + iϵ i + k m + iϵ a + i bk k m + iϵ a + i bk k m + iϵ + i = k m a bk + iϵ i = k m + iϵ + i k m + iϵ iam ibk i k m + iϵ i + k m + iϵ iam ibk i k m + iϵ iam ibk i k m + iϵ + i = k m am bk + iϵ p15, 8.3, =., p16, 8.34 = p16, 8.37 φ = x φ p18, V = B I + B I = 4V = B E + B I p11, 8.1 = p11, 8.1. =. p111, 8.51 ZΛ = Λ δλ = ZΛ = p111, 8.5 Λ δλ Dφ Λ δλ e R d 4 x Lφ Λ δλ Dφ w e R d 4 x L 1 φ Λ δλ,φ w Dφ Λ δλ e R d 4 x Lφ Λ δλ Dφ w e R d 4 x L 1φ Λ δλ,φ w Dφ w e R d 4 x Lφ Λ δλ,φ w = e R d 4 x δlφ Λ δλ = Dφ w e R d 4 x L 1φ Λ δλ,φ w = e R d 4 x δlφ Λ δλ 38
40 p111, 8.53 ZΛ = Λ δλ = ZΛ = Λ δλ Dφ Λ δλ e R d 4 x[lφ Λ δλ +δlφ Λ δλ ] Dφ Λ δλ e R d 4 x[lφ Λ δλ +δlφ Λ δλ ] p111, 8.54 [ 1 d 4 x[l + δl] = d 4 x b 4 b φ Λ ] λ n φ n Λ + L [ 1 = d 4 x[l + δl] = d 4 x b 4 b φ Λ ] λ n φ n Λ + δl p111, 8.54 φ = bφ = φ = bφ Λ p111, 1 8, = 1 8, p11, 8.57 M = iλ P µ + δ µ + icλ P µ + δ µ = M = iλ P µ + δµ + icλ P µ + δµ p11, 8.58 λ P µ + δµ = λ P µ + 3Cλ P µ log 1 δµ µ = λ P µ + δµ = λ P µ + 3Cλ P µ log 1 + δµ µ 39
41 9. p114, 9.1 S q = q 1 + q, = S q = q 1 + q, p114, 9. S =. S p115, 9.6 = q 1 q = = p115, 9.8 p116, l1 q a 1 + ϵ ij T ij ab q b = q a 1 + ϵ ij T ij ab q b p116, 9.15 p117, 9. p117, 9.5 q 1 dq 1 dq q q 1 fq 1, q = q fq 1, q = Q fq 1, q = q fq 1, q q 1 L = 1 [ φ a m ] φ λ 4 φ = L = 1 [ φ m φ ] λ 4 φ L = δl = ϵ A fa A φ + L µ fa A φ φ a µ φ a = = δl = L ϵ A fa A φ + L { µ ϵa fa A φ } φ a µ φ a J µ = φ 1 µ φ φ µ φ 1 = J µ = φ 1 µ φ φ µ φ 1 p117, 9.8 J µ = iφ µ φ φ µ φ = J µ = iφ µ φ φ µ φ 4
42 p118, 9.3 Q ij µ = = Q ij = d 3 x J ij d 3 x J ij p118, 9.33 Lx = a µ µ L = δlx = a µ µ L p118, 9.35 [ L δs = d 4 x φ a aµ x µ φ a x + L ] µ φ a µ a ν ν φ a [ ] = d 4 x a µ µ L + µ L a ν µ φ a νφ a [ L = δs = d 4 x φ a aµ x µ φ a x + L ] µ φ a µa ν ν φ a [ ] = d 4 x a µ µ L + µ a ν L µ φ a νφ a p118, 9.36 [ ] L δs = d 4 x µ a ν µ φ a νφ a η µν L [ ] L = δs = d 4 x µ a ν µ φ a ν φ a η µν L p1, l4 ds = dx + dy + dz = ds = dx + dy + dz p1, 9.43 p1, 9.45 ds = g λσdx λ dx σ = = ds = g λσdx λ dx σ = p1, 9.46 U λ µ = xµ x λ = U λµ = xµ x λ UgU = η = U T g U = η 41
43 p1, 9.47 p1, 9.47 g µν g µν. g = U 1 ηu 1 = g µν g µν. p1, 9.48 = g = U T 1 ηu 1 det g = det U = det g = det U p11, 9.58 T µν = 1 g δs = T µν = g δg µν gµν =η µν δs δg µν gµν =η µν p11, 9.6. =. p1, 9.61 V φ = 1 µ φ + λ 4 φ4 p1, 9.1 x x = φ φ p1, 9.1 = V φ = 1 µ φ + λ 4 φ4. =. p1, 9.1. =. p13, 9.64 φ = φ 4
44 p13, 9.67 P = d 3 x φ e x /L = i = P = = i p13, 9.68 p13, 9.68 d D k π D p13, 9.69 L ωk d D x φ e x /L = φ L p14, L e k L /4 [ak a k] π d D k ωk D πl e k L /4 π D [ak a k] e A+B = e A e B e [A,B]/ = e A+B = e A e B e [A,B]/ v = e ivq = e cv L D = v = e ivp = e cv L D p15, 9.8 V φ = µ φ λ / + φ 4 = V φ = µ φ λ + φ 4 k, a = d 3 x e ik x jax p = k, a = d 3 x e ik x jax p16, 9.8 D µ φ = [ µ ρ + iρ θ ea µ ]e iθ = D µ φ = [ µ ρ + iρ µ ea µ ]e iθ p16, 9.8 L = 1 4 F µνf µν + ρ θ ea µ + ρ + µ ρ λρ 4 = L = 1 4 F µνf µν + ρ µ θ ea µ + ρ + µ ρ λρ 4 43
45 p16, 9.8 ea µ ea µ + ϵ = ea µ ea µ + µ ϵ p16, χ µ χ λµχ 3 λ 4 χ4 + µ4 4λ = + 1 χ µ4 4λ µ3 λ χ 3 µ χ λµχ 3 λ 4 χ4 44
46 1. p18, 1.1 γ α = g α = g α = g α p19, 1.13 u u a = U = U = d d c = u u a = U = d d c b u d d b u d d p131, 1.5 ψ /Dψ = ψ /Dψ p131, 1.6 F µν ψ = i[d µ, D ν ]ψ = µ A ν ν A µ i[a µ, A ν ]ψ = F µν = i[d µ, D ν ] = µ A ν ν A µ i[a µ, A ν ] p13, 1. = p13, 1., =, p13, 1.,. =,.,. p133, l5 f abc µ A a νa µb A νc. =. f abc µ A a νa µb A νc. p133, l1,. =.,. 45
47 p133, l13 U+, D, U+, D, D+, U, D+, U = U+, U, D+, D, U+, D, D+, U p134, l5,. =,, 1.5. p135, 1.37 G a = µ A µa = = G a = µ A µa = α p135, 1.38 A g = ga µ g + ig g = A g µ = ga µ g + ig µ g p135, 1.4 GA = = GA = α p135, 1.4 GA + δa = µ δa µ = µ [D µ, ω] = ω + i[a µ, µ ω] = GA + δa GA = µ δa µ = µ [D µ, ω] = ω i[a µ, µ ω] p135, 1.43 GA a µ = δac + f abc A µb µ ω c = GA a µ = δ ac + f abc A µb µ ω c p135, 1.44 dω δg α = Det[ δac + f abc A µb µ ] 1 = dω δg α = Det[ δ ac + f abc A µb µ ] 1 p135, 1.45 [A] = Det[ δac + f abc A µb µ ] = [A] = Det[ δ ac + f abc A µb µ ] p136, 1.46 dα e i R d 4 x α /ξ = dα e i R d 4 x α /ξ 46
48 p136, 1.47 Z = dα e i R d 4 x α /ξ Dg DA µ δg[a] α [A]e is = Z = dα e i R d 4 x α /ξ Dg DA µ δg[a] α [A]e is p136, 1.48 Z = = Z = Dg Dg DA µ [A]e is+i R d 4 x ξ µa µ DA µ [A]e is i R d 4 x 1 ξ µa µ p136, 1.49 [ d 4 x A µa η µν + µ ν 1 1 ] A νb ξ = 1 [ d 4 x A µa η µν + µ ν 1 1 ] A νb ξ p136, 1.51 [A] = Det[ δac + f abc A µb µ ] = Dc a Dc a exp i d 4 x c a δac + f abc A µb µ c c = [A] = Det[ δ ac + f abc A µb µ ] [ = Dc a Dc a exp i d 4 x { c a δ ac + f abc A µb µ c c}] p137, 1.5 Z = dα e i R d 4 x α /ξ DA µ δg[a] α [A]e is = Z = dα e i R d 4 x α /ξ DA µ δg[a] α [A]e is p137, 1.53 δ µ A µ α = DB e itrb µa µ α = δ µ A µ α = DB e i R d 4 x trb µa µ α p137, 1.54 B = µ A m = ξb = µ A µ p137,
49 Q BRST = Q p137, 1.57 p138, 1.58 A = D m = A = D µ p138, 1.6 p138, 1.6 p138, 1.64 Q A µ = Q[D µ, c] = [ i[q, A µ ], c] + [D µ, Q, c] = i[[d m, c], c] i[d µ, c ] = Q A µ = Q[D µ, c] = { i[q, A µ ], c} + [D µ, {Q, c}] = i{[d µ, c], c} i[d µ, c ] Q c = Qcc = Qcc cqc = c 3 c 3 = = Q c = Qicc = iqcc icqc L = 1 4g tr F µν + = L = 1 4g tr F µν +. = c 3 + c 3 = { Qtr c µ A µ + ξ } B { Q, tr c µ A µ + ξ } B, ψ ψ + Q φ. Q φ. p139, 1.5,. =,. p139, 1.66 = p14, 1.68 M = Λ p141, q = µ 48
50 p141, 1.7 p141, 1.79 µ g µ = βg = µ g µ = βg [ µ q βg ] m p = g [ = q q βg ] m p = g p14, 1.81,. =,. p14, 1.8 p14, 1.83 p143, 1.85 A µ A µ + igx µ g x, gx = e iαx = A µ A µ + igx µ g x, gx = e iαx = p143, 1.85 =, p143, 1.86 x + aν µ = x + an µ p143, 1.86 U µ x = Ux, x + aν µ = U µ x = Ux, x + an µ p143, 1.89 Ux 1, x gxux 1, x g x = Ux 1, x gx 1 Ux 1, x g x Ux µν = Ux µ Ux + a µ ν Ux + an µ + an ν µ Ux + an ν ν = Ux µν = U µ xu ν x + an µ U µ x + an µ + an ν U ν x + an ν 49
51 p144, 1.89 Ux µν = e iaaµx+aνx+anµ A µx+an µ +an µ A νx+an µ = U µν x = e ia[a µx+a ν x+an µ A µ x+an µ +an ν A ν x+an ν ] p144, 1.91 =, p144, 1.9 < U P >= dux µ x,µ = < U P >= dux µ x,µ P e S wilson P Ue S wilson P p145, 1.9 dg = duµ p145, 9.1,, =,, p145, l8 QCD,..,,. = QCD,,.,,. 5
52 11. p147, SU L U1 Y =. SU3 c SU L U1 Y p147, 11.1 SU L U1 Y p148, 11.3 =, p148, U1,. p148, g v W + µ W 1µ + 1 v gw 3 µ gb µ = +g v W + µ W µ + 1 v gw 3 µ g B µ p148, 11.9 p149, cosθ W = = sinθ W = g g + g g g + g +g v W + µ W 1µ = +g v W + µ W µ p15, , 1/6 = 3,, 1/6 = Y/ p15, , 1 = 1,, 1/ p15, D µ Q = µ ig s A aµt a igw iµt i i g = D µ Q = µ ig s A aµt a igw iµt i i g 1 6 B µ Q 1 3 B µ Q 51
53 p15, , 1 4/3 = 3, 1, /3 p15, , 1, /3 = 3, 1, 1/3 p15, , 1 = 1, 1,, 1/ p15, 11. Y =. =, 1, 1, 1. p15, 11. L 1L = = L 1L = ν e e L ν e e L, L L =, L L = ν µ ν L ν µ µ L p15, 11.1 D µ L L = µ igw iµt 1 g B µ p151, 11.3 = D µ L L =, L 3L =, L 3L = L L µ igw iµt i g Y B µ L f = L L i /DΛ L + = L f = L L i /DL L + p151, 11.3 = p151, 11.4 p15, 11.8 L e = λ 3 L L φe R + = L e = λ e L L φe R + M = M p15, 11.8 L L ν τ tau L ν τ τ L 5
54 = p153, = p154, = Dφ = = Dφ DW + µ DW + µ p154, = p154, 11.1 DWµ e i R d 4 xw µ +M W W + µ +J + W + µ +J µ + W µ DWµ e i R d 4 x[w µ +M W W + µ +J + W + µ +J µ + W µ ] = p155, 11.6 = 53
55 1. p158, 1.1, =, p16, 1.1 p16, 1.13 l = 1 = l = 1 p16, 1.15 = p16, 1.15 T = trt = T = trt ij p161, 1. p161, 1.4 A i 1 i N n = ϵ i 1 i N n,j 1 j n A i 1 j n = A i1 i N n = ϵ i1 i N n,j 1 j n A j1 jn φ ij j = φ ij j p161, 1.4 u = u = U u = u = u = U u U i l U j mu n j φ lm n U i l U j mu n j φ lm n = U i jφ lm m = U i l φ lm m,. =,. p161, 1.4, = φ ij k, p161, 1.4 = 54
56 p16, l8 5. = 4. p16, 1.6 J + φ 1 =, J 1 φ = φ 1 J φ 1 = φ, J φ = = J + φ 1 = φ, J + φ = J φ 1 =, J φ = φ 1 p16, 1.7 J + ij φi = φ j, J 1 ijφ j =, J 1 ijφ k = k i, j, J ij φi =, J 1 ijφ j = φ i, J 1 ijφ k = k i, j, = J + ij φi = φ j, J + ij φj =, J + ij φk = k i, j, J ij φi =, J ij φj = φ i, J ij φk = k i, j,, i, j. p164, ,, 1 6 = p164, 1.37, L 3,, 1 6 p165, 1.41 Y = /3 = 1 Y = 3 p165, 1.4 e 4,5 φ 4,5 = ϵ 45 φ 45 p168, , 1, 3, L, R 3, 1, 3 3, 1, 1 3, R L 3, 1, 1 3 1,, 1, 1, 1, 1 R = 1,, 1, 1, 1, 1 R L R {b k, b l } = δ k,l = {b k, b l } = δ k,l 55
57 p168, 1.57 γ k 1 = a k + a k, γ k = a k a k/i = γ k 1 = b k + b k, γ k = b k b k/i p168, 1.58 p168, 1.6 g = γ i p168, γ k 1 = σ 1 σ 1 σ 1 γ k = σ 1 σ 1 σ = γ k 1 = σ 3 σ 3 σ 1 γ k = σ 3 σ 3 σ, [γ n+1, σ ij ] =, γ n+1 ψ λ = λψ λ, =, γ n+1 ψ λ = λψ λ. λ. [γ n+1, σ ij ] =, p169, 1.63 x 1 + ix = z 1, x 3 + ix 4, = x 1 + ix = z 1, x 3 + ix 4 = z, p169, 1.64 z 1 z. = SOn, z 1 z, p169, 1.65 γ 5 ψ = ψ = γ 11 ψ = ψ p169, 1.66 β i = = b i = p17, l γ 5 ψ = ψ = γ 11 ψ = ψ p17, l = p17,
58 = p171,,. =,. 57
59 13. p17, 13. p173, l6 x. = x ν. p173, x µ = x µ p173, 13.6 η = = η µν = p173, 13.1 p173, 13.1 g = det g = g = det g µν p174, detg < g µν = g µν Λ µ ρλ ν σ = g ρσ = g µν Λ µ ρλ ν σ det g = det Λ det g = det g µν = det Λ λ γ det g ρσ = g < p174, δs = d 4 x g [ γ µν µ δφ ν φ m φδφ] = δs = d 4 x g [ g µν µ δφ ν φ m φδφ] p174, g [g µν µ ν φ + µ g µν ν φ + g µν ν g ν φ = = g [g µν µ ν φ + µ g µν ν φ m φ] + g µν µ g ν φ = p174, det A = exp log A = det A = exp log det A 58
60 p174, p174, 13. detg = detggρσ x µ x µ g ρσ = g x µ = ggρσ x µ g ρσ p174, 13.1 g µν µ ν φ + µ g µν ν φ + g µν 1 gρσ µ g ρσ ν φ = = g µν µ ν φ + µ g µν ν φ + g µν 1 gρσ µ g ρσ ν φ m φ = p175, 13.3 g µν D µ ν φ = = g µν D µ ν φ m φ = Γ λ µν = Γ λ νµ, µg µν µ g µν = g µρ g σν µ g ρσ µ g µν g νρ = µ δ µ ρ = p175, 13.5 p175, 13.7 V ν V µ = V ν V ν p175, 13.8 p175, 13.9 =, p175, 13.9 xs i, xs f = Xs i, Xs f p176, D µ V νρ = D µ V ν V ρ + V µ D µ V ρ = D µ V νρ = D µ V ν V ρ + V ν D µ V ρ D µ V ν = µ V ν + Γ ν µλv λ = D µ V ν = µ V ν + Γ ν µλv λ d X µ dx ρ g µλ + ρ g µλ dτ dτ d X µ dx µ = g µλ + ρ g µλ dτ dτ dx ρ dτ dx µ λg µν dτ dx ν dτ dx ρ dτ dx µ λg µν dτ = dx ν dτ = 59
61 p176, g ρλ = g λσ p176, Affine connection = p176, p176, d X λ dx µ dτ dx ν dτ = dxµ dτ dτ + Γ λ dx µ µν dτ d X λ = dτ + Γ λ µν = 1 dx µ dτ = 1 dx ν dxλ dτ dx µ dτ dx ν dτ dτ = dxλ dτ = p176, ct f t i x f X i = ct f t i X f X i p176, dx i /dτ dx /ds 1 = dx i /dτ dx /dτ 1 p177, p177, d X µ dτ + Γ λ = d X λ dτ + Γ λ p177, d x µ dt = φ = d X µ dt = φ,. =,,,. p177,
62 p177, 13.4 = v = d x/dt p178, =. p178, = p178, p178, [D µ, D n ]ψ = F µν ψ = [D µ, D ν ]ψ = F µν ψ R λ µνρ = ρ Γ λ µν ν Γ λ µρ + Γ σ µνγ λ ρσ Γ σ µσγ λ νρ = R λ µνρ = ρ Γ λ µν ν Γ λ µρ + Γ σ µνγ λ ρσ Γ σ µργ λ νσ p179, S g = 1 κ = S g = 1 κ d 4 x g R d 4 x g R p179, k = κ p179, δg µν = δg µν p179, p179, δ µλ g λν + δ µλ δg λν = = δg µλ g λν + g µλ δg λν = δ µν = g µλ δg λρ g ρν = δg µν = g µλ δg λρ g ρν 61
63 p179, det g = exp tr log g µν = det g µν = exp tr log g µν p179, p18, l1 δ det g = det g g µν δg µν = g µν δg µν = δg = g g µν δg µν = g g µν δg µν = p18, δr µν = ν δγ λ µλ λ δγ λ µν = δr µν = D ν δγ λ µλ D λ δγ λ µν p18, 13.6 δs g = 1 κ d 4 x g δg R µν µν 1 g µνr + 1 κ d 4 x g g µν ν δγ λ µλ λ δγ λ µν = δs g = 1 κ d 4 x g δg R µν µν 1 g µνr 1 κ d 4 x g g µν { D ν δγ λ µλ D λ δγ λ µν } p18, µ g g µν ν φ = g g µν µ ν φ = µ { g g µν ν φ } = g g µν D µ ν φ p18, κ d 4 x g g µν ν δγ λ µλ λ δγ λ µν = 1 κ d 4 x [ ν g g µν δγ λ µλ λ g g µν δγ λ µν = 1 κ d 4 x [ g g µν D ν δγ λ µλ D λ δγ λ µν ] = 1 κ d 4 x [ ν g g µν δγ λ µλ λ g g µν δγ λ µν ] 6
64 p18, g = g δ δg µν S matter = T µν δ δg µν S matter = T µν p18, R = φ, R = φ = R = φ, R = 4 φ p18, φ = κ T = φ = κ T p181, µ h µ ν = ν h λ λ = µ h µ ν = 1 νh λ λ p181, πG = 1 M P = 16πG N = 1 M P p181, g µν = R µν p181, S = Mp d 4 x = S = MP d 4 x p181, S = d 4 x h h + Mp 1 h h h + + M p ht + = S = d 4 x h h + M 1 h h h P + + M 1 ht P + 63
65 p181, gd S = 4 x R + R + R 3 + R R + 1 = S = d 4 x g R + ar + br 3 + cr R + 16πG N p18, = a, b, c, 14. p184, 14.. =. p185, 14.3 = p185, 14.3 = p186, l4 = 64
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IA hara@math.kyushu-u.ac.jp Last updated: January,......................................................................................................................................................................................
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3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)
1 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 =
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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
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
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 2 ), ϕ(t) = B 1 cos(ω 1 t + α 1 ) + B 2 cos(ω 2 t
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F F 1 F 2 F, (3) F λ F λ F λ F. 3., A λ λ A λ. B λ λ
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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..........................
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(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
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2 1 x 2 x 2 = RT 3πηaN A t (1.2) R/N A N A N A = N A m n(z) = n exp ( ) m gz k B T (1.3) z n z = m = m ρgv k B = erg K 1 R =
1 1 1.1 1827 *1 195 *2 x 2 t x 2 = 2Dt D RT D = RT N A 1 6πaη (1.1) D N A a η 198 *3 ( a =.212µ) *1 Robert Brown (1773-1858. *2 Albert Einstein (1879-1955 *3 Jean Baptiste Perrin (187-1942 2 1 x 2 x 2
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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)
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59 7 7.1 σ-ω σ-ω σ ω σ = σ(r), ω µ = δ µ,0 ω(r) (6-4) (iγ µ µ m U(r) γ 0 V (r))ψ(x) = 0 (7-1) U(r) = g σ σ(r), V (r) = g ω ω(r) σ(r) ω(r) (6-3) ( 2 + m 2 σ)σ(r) = g σ ψψ (7-2) ( 2 + m 2 ω)ω(r) = g ω ψγ
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構造と連続体の力学基礎
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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)
t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z
I 1 m 2 l k 2 x = 0 x 1 x 1 2 x 2 g x x 2 x 1 m k m 1-1. L x 1, x 2, ẋ 1, ẋ 2 ẋ 1 x = 0 1-2. 2 Q = x 1 + x 2 2 q = x 2 x 1 l L Q, q, Q, q M = 2m µ = m 2 1-3. Q q 1-4. 2 x 2 = h 1 x 1 t = 0 2 1 t x 1 (t)
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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)
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50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
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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
(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou
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65 1 1.1 1.1.1 1.1 H H () = E (), (1.1) H ν () = E ν () ν (). (1.) () () = δ, (1.3) μ () ν () = δ(μ ν). (1.4) E E ν () E () H 1.1: H α(t) = c (t) () + dνc ν (t) ν (), (1.5) H () () + dν ν () ν () = 1 (1.6)