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)
|
|
|
- みそら めいこ
- 5 years ago
- Views:
Transcription
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 + = γ, 2 n ψ = γ = Euler const, log + ax x = a log a, 4 + 4a d 2ω q ΓA ω =, 2π 2ω q 2 + M 2 + 2qp A 4π ω ΓA M 2 p 2 A ω 5 d 2ω q q µ ΓA ω p µ =. 2π 2ω q 2 + M 2 + 2qp A 4π ω ΓA M 2 p 2 A ω 6 D a D a k k = Γa + + a k Γa Γa k Feynman δ x x k x a x a k k dx dx k. 7 x D + x k D k a + +a k 2 Feynman φ 4 Lagrangian L = 2 φ2 + 2 m2 φ 2 + 4! gµε φ 4 8 : d = 2ω, ε = 2ɛ, d = 4 ε, ω = 2 ɛ. 9 : m 2 = m2 4πµ, α = g 2 4π. 2
2 d ω ε ɛ g α m 2 m 2 ω, ɛ, α ε m, µ 2 g Dφ e R d d x L proper vertex Feynman E I V L Feynman Feynman L d Ld E, I, V, L L = I V +, 4V = E + 2I. φ 4 4-valent proper vetex Γ E E = 2, 4 I V loop L { L E = 2, V = L + E = 4 Feynman. 2. L. gµ ε /4! 4. q G q = q 2 + m 2 5. q d 2ω q 2π 2ω 6. Feynman Wick Proper vertex Feynman Γ 2 p Σp Gp G p = p 2 +m 2 Gp = G p + G pσpg p + G pσpg pσpg p + = G p ΣpG p Γ 2 p Γ 2 p = G p Σp = p 2 + m 2 Σp 2 Σp 2
3 . Tadpole -loop p 4 gµ 2ɛ φ 4 /4! 4 /4! = /2 n 9 Σp tadpole = d 2ω l 2 gµ2ɛ 2π 2ω l 2 + m. 2 p 5 Σp tadpole = Γ ω 2 gµ2ɛ 4π ω m 2 = g 4πµ 2 ɛ ω 2 m2 Γ + ɛ 4π 2 m 2 = 2 m2 α m 2 ɛ Γ + ɛ ɛ Σp tadpole = m 2 α 2ɛ + ɛψ2 log m 2 + Oɛ Double scoop 2-loop 4 p /4! 2 = /4 Σp d-scoop = 4 gµ2ɛ 2 d 2ω l 2π 2ω l 2 + m 2 p Tadpole Σp d-scoop = Σp tadpole 2 gµ2ɛ d 2ω q 2π 2ω q 2 + m d 2ω q 2π 2ω q 2 + m d 2ω q 2 gµ2ɛ 2π 2ω q 2 + m 2 = Γ2 ω 2 2 gµ2ɛ 4π ω m 2 2 ω = 2 α m2 ɛ Γɛ = α + ɛψ log m 2 + Oɛ ɛ 6
4 Σp d-scoop = m 2 α2 4ɛ 2 + ɛψ2 + ψ 2 log m 2 + Oɛ 2 8. Sunset 2-loop p l q p + q l /4! 2 = /6 Σp sunset = gµ2ɛ 2 I, 9 6 d 2ω l d 2ω q I = 2π 2ω 2π 2ω l 2 + m 2 q 2 + m 2 [q + p l] 2 + m 2. 2 ɛ Σp sunset = α2 m 2 6 2ɛ + m2 2 ɛ 2 + ψ log m2 + 4ɛ p2 + Oɛ. 2 I = 4ω lµ l µ + q µ q µ µ sum 22 I = d 2ω l d 2ω q l 4ω 2π 2ω 2π 2ω µ + q µ l µ q µ l 2 + m 2 q 2 + m 2 [q + p l] 2 + m l 2 l 2 + m 2 2 q 2 + m 2 [q + p l] 2 + m 2 + 2ll p q l 2 + m 2 q 2 + m 2 [q + p l] 2 + m 2 2 2q 2 + l 2 + m 2 q 2 + m 2 2 [q + p l] 2 + m 2 + 2qq + p l l 2 + m 2 q 2 + m 2 [q + p l] 2 + m l p + q l q p + q l q, l 4p + q l 2 + 2l qp + q l l 2 + m 2 q 2 + m 2 [q + p l] 2 + m 2 = 6p + q l2 6m 2 + 6m 2 + 2pp + q l 2 l 2 + m 2 q 2 + m 2 [q + p l] 2 + m = l 2 + m 2 q 2 + m 2 [q + p l] 2 + m 2 + 6m 2 + 2pp + q l l 2 + m 2 q 2 + m 2 [q + p l] 2 + m
5 2 I 2 I = d 2ω l d 2ω q 6m 2 + 2pp + q l 6I ω 2π 2ω 2π 2ω l 2 + m 2 q 2 + m 2 [q + p l] 2 + m 2 2 d 2ω l Kp = 2π 2ω d 2ω l K µ p = 2π 2ω d 2ω q 2π 2ω l 2 + m 2 q 2 + m 2 [q + p l] 2 + m 2, 2 26 d 2ω q p + q l µ 2π 2ω l 2 + m 2 q 2 + m 2 [q + p l] 2 + m Σp sunset = 2ω µ-sum gµ 2ɛ 2 m 2 Kp + p µ K µ p 28 6 Kp 26 l p + q l l 2 + m 2 [q + p l] 2 + m 2 q 2 + m 2 2 Feynman 7 l l + xp + q d 2ω l d 2ω q dx Kp = 2π 2ω 2π 2ω q 2 + m 2 2 l 2 + m 2 + p + q 2 x x 2 Γ2 ω d 2ω q = dx 4π ω 2π 2ω q 2 + m 2 2 m 2 + p + q 2 x x. 2 ω l 5 x x 2 ω Feynman q Kp = = Γ4 ω 4π ω d 2ω q 2π 2ω Γ4 2ω 4π 2ω = Γ2ɛ 4π 4 2ɛ dxx x ω 2 dyy ω y [ q 2 + p 2 y y + m 2 y + dxx x ω 2 dyy ω y dxx x ɛ dyy +ɛ y Z = Zx, y = p 2 y y + m 2 y + y ] ω 4 x x [ q 2 + p 2 y y + m 2 y + [ p 2 y y + m 2 y + y x x y ] 2ω 4 x x y ] 2ɛ. x x y +ɛ = d ɛ dy yɛ y Kp = Γ2ɛ ɛ4π 4 2ɛ dxx x ɛ dy y ɛ d [ ] yz 2ɛ. dy 29 5
6 2 ɛ Oɛ dxx x ɛ dy y ɛ d dy [ y + 2ɛ y log Z] + Oɛ2 = = dxx x ɛ dy y ɛ + 2ɛ Γ ɛ 2 Γ2 2ɛ + ɛ 2ɛ = + ɛ 2ɛ log m 2 + Oɛ 2. dx dx log Zx, y = + Oɛ 2 dy d dy [ y log Z] + Oɛ2 26 Kp = Γ2ɛ ɛ4π 4 2ɛ + ɛ 2ɛ log m2 + Oɛ 2. p µ K µ p K µ p 27 l p + q l Feynman q 5 d 2ω l d 2ω q l µ K µ p = 2π 2ω 2π 2ω q 2 + m 2 [q + p l] 2 + m 2 l 2 + m 2 2 d 2ω l l µ d 2ω q dx = 2π 2ω l 2 + m 2 2 2π 2ω [q 2 + m 2 + x xp l 2 ] 2 Γ2 ω d 2ω l l µ dx = 4π ω 2π 2ω l 2 + m 2 2 [m 2 + x xp l 2 ] 2 ω. x x 2 ω Feynman K µ p = Γ4 ω 4π ω dxx x ω 2 dy y ω y d 2ω l l µ 2π 2ω ] 4 ω [l yp 2 + y yp 2 + m 2 y + y x x l l + yp l l 2 Γ4 ω p µ K µ p = p dxx x ω 2 dy y 2 ω y 4π ω d 2ω l 2π 2ω [l 2 + y yp 2 + m 2 y + y 2 Γ4 2ω = p 4π 2ω = p 2 Γ2ɛ 4π 4 2ɛ x x ] 4 ω ] 2ω 4 dxx x ω 2 dy y 2 ω y [y yp 2 + m 2 y y + x x dxx x ɛ dy y ɛ yzx, y 2ɛ. Z 29 ɛ 2 p µ K µ p = p 2 Γ2ɛ 4π 4 2ɛ 2 + Oɛ. 6
7 4 Γ 4 p, p 2, p, p 4 Γ 4 e L Feynman -loop Γ 4 = +gµ 2ɛ [Ramond] Γ 4 [Ramond] 5.6 Γ 2 4. Fish -loop p l 2 p p 2 l p p /4! 2 = /2 Γ 4 p, p 2, p, p 4 fish = Γ 4 p, p 2, p, p 4 fish + Γ 4 p, p, p 2, p 4 fish + Γ 4 p, p 4, p 2, p fish, 2 Γ 4 p, p 2, p, p 4 fish = 2 gµ2ɛ 2 d 2ω l 2π 2ω l 2 + m 2 [l p 2 + m 2 ] p = p + p 2. Feynman l Γ 4 p, p 2, p, p 4 fish = 2 gµ2ɛ 2 d 2ω l dx 2π 2ω [l xp 2 + x xp 2 + m 2 ] 2 = 2 gµ2ɛ 2 Γ2 ω 4π ω dx [m 2 + x xp 2 ] 2 ω. ω = 2 ɛ Oɛ Γ 4 p, p 2, p, p 4 fish = 2 gµ2ɛ 2 Γɛ 4π 2 ɛ = 2 gµ2ɛ 2 Γɛ 4π 2 ɛ dx ɛ log [ m 2 + x xp 2] + Oɛ 2 ɛ 2 + = gµ 2ɛ α + ɛ 2 + ψ log m 2 2ɛ + 4m2 p 2 + 4m2 + p log 2 + log m 2 + Oɛ 2 + 4m2 p 2 + 4m2 + p log 2 + Oɛ m2 p 2 + 4m2 p 2 x 4 2 Γ 4 2ɛ α [ p, p 2, p, p 4 fish = gµ + ɛ 2 + ψ log m 2 F + Oɛ 2 ]. 5 2ɛ 7
8 F F = a,b=,2,,,,4 + 4m2 p ab + 4m2 p ab + log, p ab = p a + p b m2 p ab Glass gµ 2ɛ d 2ω l 2π 2ω l 2 + m 2 [l p 2 + m 2 ] = α ɛ + ɛ 2 + ψ log m 2 + 4m2 p 2 log + 4m2 + p 2 + Oɛ 2 + 4m2 p Glass 2-loop p p 2 p p /4! = /4 Γ 4 p, p 2, p, p 4 glass = Γ 4 p, p 2, p, p 4 glass + Γ 4 p, p, p 2, p 4 glass + Γ 4 p, p 4, p 2, p glass, 8 Γ 4 p, p 2, p, p 4 glass = 4 gµ2ɛ d 2ω 2 l p = p 2π 2ω l 2 + m 2 [l p 2 + m 2 + p 2. 9 ] 7 Γ 4 p, p 2, p, p 4 glass 2ɛ α2 = gµ 4ɛ 2 + 2ɛ 2 + ψ log m 2 + 4m2 p 2 + 4m2 + p log 2 + Oɛ m2 p ɛ Γ 4 2ɛ α2 [ p, p 2, p, p 4 glass = gµ + ɛ 4 + 2ψ 2 log m 2 2F + Oɛ 2 ]. 4 4ɛ 2 F 6 8
9 4. Lobster 2-loop p p 2 l l p q p l + p q p p p p /4! = Γ 4 p, p 2, p, p 4 lobster = Γ 4 p, p 2, p, p 4 lobster + Γ 4 p, p, p 2, p 4 lobster + Γ 4 p, p 4, p 2, p lobster. 42 Γ 4 p, p 2, p, p 4 lobster = gµ 2ɛ d 2ω l 2π 2ω d 2ω q 2π 2ω l 2 + m 2 l p 2 + m 2 q 2 + m 2 l + p q 2 + m 2. 4 ɛ Γ 4 p, p 2, p, p 4 lobster 2ɛ α2 = gµ + ɛ 5 + 2ψ 2 log m 2 2F 2ɛ Oɛ 2 F 6 4 Feynman q gµ 2ɛ Γ2 ω d 2ω l dx 4π ω 2π 2ω l 2 + m 2 l p 2 + m 2 [m 2 + x xl + p 2 2 ω. 45 ] x x 2 ω Feynman Γ 4 p, p 2, p, p 4 lobster = gµ 2ɛ Γ4 ω dxx x ω 2 4π ω d 2ω l [ l + y p 2π 2ω y 2 p 2 + V ] ω 4, dy 2 y2 V = y y p 2 + y 2 y 2 p 2 + 2y y 2 pp + m 2 y + l dy y ω 46 y. 47 x x Γ 4 p, p 2, p, p 4 lobster = gµ 2ɛ Γ4 2ω 4π 2ω = gµ 2ɛ Γ2ɛ 4π 4 2ɛ y2 dxx x ω 2 dy 2 y2 dxx x ɛ dy 2 dy y ω dy y ɛ V, 4 2ω V. 2ɛ 48 9
10 y y2 dy y ɛ V = 2ɛ V y = 2ɛ = y2 dy y ɛ + Oɛ y 2 ɛ ɛ [y 2 y 2 p 2 + m 2 2ɛ + Oɛ ] 49 = ɛ y2 ɛ 2ɛ log [ y 2 y 2 p 2 + m 2] + Oɛ x, y 2 y 2 4 Γ 4 p, p 2, p, p 4 lobster = gµ 2ɛ Γ2ɛΓ ɛ 2 4π 4 2ɛ Γ2 2ɛɛ = gµ 2ɛ Γ2ɛΓ ɛ 2 4π 4 2ɛ Γ2 2ɛɛ dy 2 y2 ɛ 2ɛ log [ y 2 y 2 p 2 + m 2] + Oɛ 2 + ɛ 2ɛ m2 p 2 ɛ + 4m2 + p log 2 + log m 2 + Oɛ m2 p 2 Γ 4 p, p 2, p, p 4 lobster + 4m2 + 2ɛ α2 = gµ + ɛ 5 + 2ψ 2 log m m2 p log 2 + Oɛ 2. 2ɛ 2 p 2 + 4m2 p Minimal subtraction scheme minimal subtraction scheme ɛ = 2 ω ε = 4 d 5. Lagrangian L = 2 φ2 + 2 m2 φ 2 + 4! gµε φ 4 5 counter term L c.t. = 2 A φ2 + 2 m2 Bφ 2 + 4! gµε Cφ 4 52 L = L + L c.t. = 2 φ m2 φ 2 + 4! g φ 4 5
11 φ = Z /2 φ φ, m 2 = Z m m 2, g = Z g gµ ε 54 Z φ = + A, Z m = + B + A, Z g = + C + A 2 55 m, g bare mass bare coupling constant Minimal subtraction scheme, 2. g V Γ 2 p Γ 4 p, p 2, p, p 4 ε = mod Og V + Γ 2, Γ 4 Og L, Og L+. 2. ε ε ε. ε Z φ, Z m, Z g ε Z φ g, ε = + k= Z k φ g, Z ε k m g, ε = + k= Z k m g ε k, Z g g, ε = + k= Z g k g. 56 ε k Minimal subtraction scheme mass parameter m, µ mass-independent renormalization Γ 4 p, p 2, p, p 4 -loop gµ ε overall gµ ε ε 4 ε, ε 2, βg, ε, γ m g, γ φ g m, g mass scale µ m 2 = Z m g, εm 2, g = Z g g, εgµ ε m, g µ ε βg, ε, γ m g g, m ε g = gµ, m = mµ beta βg, ε = µ g µ 57 = dg dµ = d dµ Z gg, εgµ ε βg, ε = εg + g g log Z gg, ε beta Z g g, ε 56 βg, ε = εg + ε beta ε = 58
12 ε ε! βg, ε = εg + βg, βg = g 2 dz g g dg 59 βg 4 beta 58 βg, ε βg βg g gz gg, ε = εg 2 g Z gg, ε ε i βg d dg gz i g g = g 2 d dg Zi+ g g i =,, 2,... 6 g g = 59 2 i = γ m g, ε γ φ g, ε Z γ m g, ε = µ µ log Z mg, ε = βg, ε g log Z mg, ε, 6 γ φ g, ε = µ µ log Z φg, ε = βg, ε g log Z φg, ε 62 γ m g, ε = dm2 dµ = d Zm g, εm 2 dµ γ m g, ε = 2 µ m m µ = log m2 log µ Z m i g γ m g, ε + = εg + βg ε i g i= Z i φ γ φ g, ε + g = εg + βg ε i g i= i= i= Z i m g ε i, Z i φ g ε i ε γ m g, ε, γ φ g, ε beta ε = γ m g, ε, γ φ g, ε ε! γ m g, ε = γ m g = g dz m g, 65 dg γ φ g, ε = γ φ g = g dz φ g dg 2 66
13 64 ε i γ m g βg d dg Zi m g = g d dg Zi+ m g i =,, 2,..., γ φ g βg d dg Zi φ g = g d dg Zi+ φ g i =,, 2,... Z m g = Z φ g = i = mass parameter m, µ Z g g, ε, Z m g, ε, Z φ g, ε ε = βg, ε εg ε minimal subtraction scheme 6 6. g c βg c, ε = 67 d < 4 φ 4 beta,2-loop βg, ε ε g c g UV IR UV g = Gaussian IR g = g c Wilson-Fisher 57 g g c g as µ IR, as µ UV 68 g c = Oε g c ω = β g c, ε 69 ω >, γ φ g c, γ m g c 7 2-loop γ φ g c >, γ m g c < < g < g c, T > T c 2-loop β ε β = g c ε.97 g =
14 g = Z m g, ε, Z φ g, ε 6 62 g γ m g Z m g, ε = exp βg, ε dg g g c γ mg c /ω g g c, 7 g Z φ g, ε = exp γ φ g βg, ε dg g g c γ φg c /ω g g c. 72 g = g βg, ε = εg + Og 2, γ m g = Og, γ φ g = Og 2 2-loop beta ω µ g g c /ω g g c 7 µ, g g m 2 Z g g, ε Z m g, ε Z φ g, ε IR µ g c µ γmgc µ ε µ γmgc µ γ φg c UV µ µ ε m 2 : µ 6.2 Z φ η 2 Gp p η 2 fpξ p ξ 74 η ξ ν ξ t ν, t = T T c /T c χ = const G 74 y fy y 2 η χ ξ 2 η γ χ t γ γ = 2 ην 74 Gp Z φ g, ε p 2 + ξ 2 + op 2 75 Z φ g, ε ξ η g g c 76 4
15 6. Callan-Symanzik N bare bare ΓN Γ N Γ N bare p i, m 2, g, ε = Z N/2 φ g, εγ N p i, m 2, g, µ, ε 77 p i N µ g, m µ log µ Callan-Symanzik µ d dµ N 2 γ φg Γ N p i, m 2, g, µ, ε =, µ d dµ = µ µ + βg, ε 78 g γ mgm 2 m 2 γ φ g, ε, γ m g, ε 62 6 ε γ φ g, γ m g µ µ s mass µ µ = µ s, ĝs = gµ s, ˆms = mµ s, g = ĝ = gµ, m = ˆm = mµ 79 gµ 57 mµ 6 78 N s Γ N p i, ˆms 2, ĝs, µ s, ε = exp γ φ ĝs ds Γ N p 2 s i, m 2, g, µ, ε 8 N mass d Nd/2 p, m, g, µ Γ N p i, m 2, g, µ, ε = σ d Nd/2 Γ N p σ, m2 σ, g, µ, ε 8 2 σ N = 2 p i p 8 s Γ 2 p, m 2, g, µ, ε = s 2 exp γ φ ĝs ds Γ 2 ps ˆms2,, ĝs, µ s s 2, ε s ˆms 2 = m 2 exp ˆms µ s s γ m ĝs ds s 8 = mµ s = 84 µ s 5
16 m 2 µ 2 = exp s 2 + γ m ĝs ds s m 2 Γ 2 p, m 2, g, µ, ε p µ 2 µ µ 84 s µ 85 ĝs µ t = T T c /T c m = mµ t = mµ 2 µ 2 = m2 Z m gµ, ε. µ 2 bare mass m t µ gµ, γ m g mass m, m, µ,... t µ 85 t s s t = exp 2 + γ m ĝs ds 86 s 2 + γ m g > s t 85 t s 2+γ mg c 87 ĝ = g = g c s = s p /s s t 84 Γ 2 p, m 2, g, µ, ε Γ 2 p s, µ 2, ĝs, µ, ε t = Γ 2 p, m 2, g, µ, ε Γ 2 p, m 2, g, µ, ε s 2 γ φg c p Γ 2, µ 2 s, g c, µ, ε µ 2 s 2 γ φg c p Γ 2,, g c,, ε 88 µ s p 2 γ φg c p h µ s hx = µ γ φg c x γ φg c 2 Γ 2 x,, g c,, ε 2 Gp 74 s ξ t ν ν = 2 + γ m g c, η = γ φg c 89 g g c γ φg c /ω 72 Z φ g, ε 76 η 89 ξ ξ γ φg c 7 g g c 9 ξ µ g g c /ω g g c 9 6
17 7 -loop 7. Γ 2 p 2 Σp -loop Tadpole Σp tadpole ɛ 4 e L L Lagrangian L 2 m2 B φ 2 B = α 2ɛ 92 m 2 m 2 + B Feynman = m2 α 2ɛ Σp tadpole 4 Σp tadpole m2 α 2ɛ = m 2 α 2 ψ2 log m2 + Oɛ 9 ɛ = 2 Γ 2 p 2 Γ 2 p = p 2 + m 2 Σp tadpole m2 α 2ɛ = p2 + m 2 α 2 ψ2 log m2 + Og loop Og 2 α = g/4π 2 94 Feynman = Og Γ 4 p, p 2, p, p 4 Γ 4 p, p 2, p, p 4 Γ 4 p, p 2, p, p 4 fish 5 ɛ 2ɛ α gµ Lagrangian 2ɛ 4! gµ2ɛ C φ 4 C = α 2ɛ 95 g g + C Feynman 2ɛ α = gµ 2ɛ 7
18 -loop -loop 4 ɛ Γ 4 p, p 2, p, p 4 = gµ 2ɛ + Γ 4 2ɛ α p, p 2, p, p 4 fish + gµ [ 2ɛ = gµ 2ɛ α 2 + ψ log m 2 F ] + Og 2 -loop Og loop -loop L -loop c.t. = 2 A φ m2 B φ 2 + 4! gµ2ɛ C φ 4 97 A =, B = α 2ɛ = α ε, C = α 2ɛ = α ε. 98 Z φ =, Z m = + α ε, Z g = + α ε. 99 βg, ε = εg + αg, γ m g = α, γ φ g = α c := g c 4π = ε 2, ν = = 2 α c 2 + ε 2 + Oε2, η = + Oε loop 8. Γ 2 p 2-loop Feynman =
19 -loop 7. 2 Double scoop.2 Sunset. -loop Feynman Double scoop Sunset Og 2 Σp tadpole i Σp tadpole Σp tadpole = 4! gµ2ɛ 4 2 Σp tadpole m2 4ɛ α d 2ω l 2π 2ω l 2 + m m2 B Σp tadpole ii Σp tadpole Σp tadpole coupling gµ 2ɛ gµ 4 4 = m2 α2 4ɛ 2 + ɛψ log m2 + Oɛ 2 2ɛ α 2ɛ Σp tadpole = m 2 α2 + ɛψ2 log m 2 + Oɛ 2 4 4ɛ 2 2-loop Σp d-scoop + Σp sunset + Σp tadpole + Σp tadpole = α2 24ɛ p2 + m2 2 α2 ɛ 2 2ɛ + Oɛ 5 -loop Σp tadpole 92 9 ɛ 2-loop 92 2 A + A 2 φ m2 B + B 2 φ 4, 6 A =, A 2 = α2 24ɛ, B = α 2ɛ, B 2 = α2 2ɛ α2 2 4ɛ. 7 -loop 2 Γ 2 p = + A p 2 + m 2 + B Σp tadpole + Og loop Γ 2 p = + A + A 2 p 2 + m 2 + B + B 2 Σp tadpole + Σp d-scoop + Σp sunset + Σp tadpole + Σp tadpole + Og ɛ 9 9
20 8.2 Γ 4 p, p 2, p, p 4 2-loop Feynman = p, p 2, p, p 4, p, p, p 2, p 4, p, p 4, p 2, p -loop Fish ɛ 96 -loop 2 2-loop Glass 4.2 Lobster 4. ɛ = -loop Feynman 2 Og 2-loop 2 ɛ = 2-loop 2 { } gµ 2ɛ 2 d 2ω q Σq 2π 2ω q 2 + m 2 2 [p q 2 + m 2 tadpole m2 α p = p + p 2 ] 2ɛ -loop 9 { } ɛ = 2 ω q d 2ω q 2π 2ω q 2 + m 2 2 [p q 2 + m 2 ] = Γ + ɛ 4π 2 ɛ dx [m 2 + x xp 2 ] +ɛ ɛ Γ 2 p 2-loop 6 B 2 g 4 Γ 4 p, p 2, p, p 4 2-loop 2-loop Glass Lobster Glass Lobster Γ 4 p, p 2, p, p 4 fish 7.2 Feynman Γ 4 p, p 2, p, p 4 fish Γ4 p, p 2, p, p 4 fish gµ 2ɛ gµ 2ɛ gµ 2ɛ 2ɛ α gµ 2ɛ 2 Γ 4 p, p 2, p, p 4 2ɛ 9α2 [ fish = gµ + ɛ 2 + ψ log m 2 F + Oɛ 2 ] 2ɛ 2 2
21 2-loop Γ 4 p, p 2, p, p 4 ɛ 4 44 Γ 4 p, p 2, p, p 4 glass + Γ 4 p, p 2, p, p 4 lobster + Γ 4 p, p 2, p, p 4 fish 2ɛ α2 9 = gµ ɛ ɛ + Oɛ2 log m 2 F Lagrangian 4! gµ2ɛ C 2 φ 4 C 2 = 9α2 4ɛ α2 2 2ɛ 2 -loop 95 φ 4 2-loop 4! gµ2ɛ C + C 2 φ 4 C 95 C 2 Γ 2 p 2-loop loop 2-loop L 2-loop c.t. = 2 A 2 φ m2 B 2 φ 2 + 4! gµ2ɛ C 2 φ ε = 2ɛ A 2 = α2 2ε, B 2 = 2α2 ε 2 α2 2ε, C 2 = 9α2 ε 2 2-loop -loop α2 ε. 5 L -loop c.t. + L 2-loop c.t. = 2 A + A 2 φ m2 B + B 2 φ 2 + 4! gµ2ɛ C + C 2 φ A A + A 2 = α2 2ε, B B + B 2 = α 2 ε + α2 ε, 2 2ε C C + C 2 = α 9 ε + α2 ε 2 ε Z φ = α2 2ε, Z m = + α ε + α2 2 ε 5, Z 2 g = + α 9 2ε ε + α2 ε 7 2 6ε 7 2
22 59 66 βg, ε = εg + βg, βg = g α 7 α2, 8 γ m g = α + 5α2 6, 9 γ φ g = α loop g 67 α c := g c 4π 2 89 α c 7 α2 c = ε α c = ε + 7ε2 8 + Oε. 2 ν = 2 α c + 5αc/6 = ε 2 + 7ε Oε, η = α2 c 6 = ε Oε n 9. n φ = φ,..., φ n Lagrangian L = 2 φ m2 φ 2 + 4! gµε φ Dφ e R d d x L n proper vertex Feynman 2 φ a a g = Green φ a xφ b y q δ ab δ q 2 +m 2 ab φ a φ a φ b φ b a b a b Feynman a, b Proper vertex Γ 2 2 Γ 2 ij p = δ ijp 2 + m 2 Σp Σp Feynman 2 i = j i = j Tadpole Double scoop 5 Sunset 9 22
23 n n = fn fn f = n n Γ 4 2 Γ 4 i i 2 i i 4 p, p 2, p, p 4 fish = δ i i 2 δ i i 4 Γ 4 p, p 2, p, p 4 fish + δ i i δ i2 i 4 Γ 4 p, p, p 2, p 4 fish + δ i i 4 δ i2 i Γ 4 p, p 4, p 2, p fish 24 Γ 4 p, p 2, p, p 4 fish Glass 8 9 Lobster 42 4 Feynman fn = n Γ 2 Σp 9.2. Tadpole n = 4 = 2 Tadpole 2 4n 8 Tadpole fn = 4n = n Σp tadpole = n + 2 m 2 α + ɛψ2 log m 2 + Oɛ ɛ 2
24 9.2.2 Double scoop 5 n = 4 2 = 44 Double scoop 4 6n 2 2n 2n 64 fn = 6n2 + 64n = n n + 22 m 2 α 2 Σp d-scoop = + ɛψ2 + ψ 2 log m 2 + Oɛ ɛ Sunset 9 n = = 96 Sunset 2 2n 64 2n + 64 fn = 96 2 Σp sunset = n + 2 m 2 8 α2 2ɛ 2 = n m2 ɛ 2 + ψ log m2 + 4ɛ p2 + Oɛ
25 9. Γ 4 24 i i 2 i i 4 = δ i i 2 δ i i 4 + δ i i δ i2 i 4 + δ i i 4 δ i2 i, 28 F i i 2 i i 4 = + 4m2 δ i i 2 δ i i 4 + 4m2 p 2 + log + + 4m2 p 2 + 4m2 p 2 δ i i δ i2 i 4 + 4m2 p + log p + 4m2 p + + 4m2 δ i i 4 δ i2 i + 4m2 p 4 + log. 29 p 4 + 4m2 p 4 p ab = p a + p b 2 F i i 2 i i 4 F 6 i i 2 i i Fish n = = 288 Sunset Amit [A] p29 Fig n n fn = 288 = n Γ 4 p, p 2, p, p 4 fish 4 fn 24 i i 2 i i 4, F i i 2 i i 4 Γ 4 i i 2 i i 4 p, p 2, p, p 4 fish = n gµ 2ɛ α ɛ [ i i 2 i i 4 + ɛ2 + ψ log m 2 ɛf i i 2 i i 4 + Oɛ 2 ]. n = i i 2 i i 4, F i i 2 i i 4 F Glass 9 n = 2 4 = 456 Glass 6 Amit [A] p29 Fig
26 2 28n 28 2n 28 4n fn = 28n2 + 6n = n2 + 6n Γ 4 p, p 2, p, p 4 glass 4 fn 24 fish glass i i 2 i i 4, F i i 2 i i 4 Γ 4 i i 2 i i 4 p, p 2, p, p 4 glass = n2 + 6n + 2 gµ 2ɛ α 2 [ 6 ɛ 2 i i 2 i i 4 + ɛ4 + 2ψ 2 log m 2 2ɛF i i 2 i i 4 + Oɛ 2 ]. n = i i 2 i i 4, F i i 2 i i 4 F Lobster 4 n = = Lobster n 256n 256 2n
27 Amit [A] p29 Fig. 6-7 fn = 2565n = 5n Γ 4 p, p 2, p, p 4 lobster 5 fn 24 fish lobster i i 2 i i 4, F i i 2 i i 4 Γ 4 i i 2 i i 4 p, p 2, p, p 4 lobster 5n + 22 gµ 2ɛ α 2 [ = 8 ɛ 2 i i 2 i i 4 + ɛ5 + 2ψ 2 log m 2 2ɛF i i 2 i i 4 + Oɛ 2 ]. 2 n = i i 2 i i 4, F i i 2 i i 4 F Γ 2 Γ 4 Γ 2 Σp loop 7 -loop Tadpole 25 n m2 B φ 2 n + 2α B = 6ɛ Feynman = n+2m2 α 6ɛ Γ 4 -loop Fish Lagrangian 2 4! gµ2ɛ C φ 2 n + 8α C = 4 6ɛ n+8 C 95 9 g g + C Feynman = n+8gµ2ɛ α 6ɛ -loop L -loop c.t. = 2 A φ m2 B φ 2 + 4! gµ2ɛ C φ
28 A =, B = n + 2α 6ɛ = n + 2α, C = ɛ n + 8α 6ɛ = n + 8α. 6 ε 55 Z φ =, Z m = + n + 2α, Z g = + ε n + 8α. 7 ε βg, ε = εg α c := g c 4π 2 = n + 8αg n + 2α, γ m g =, γ φ g =. 8 ε n + 8, ν = 2 + γ m g c = n + 2ε + 2 4n Oε2, η = + Oε loop 8 Double scoop 26 Sunset 27 -loop Σp tadpole Σp tadpole Σp tadpole 2 m2 α m2 B 4ɛ 2 n+2 n = 24 8n 6 n+2 8n + 2 n+22 9 Σp tadpole = n m 2 α 2 ɛ 2 + ɛψ log m 2 + Oɛ 2 4 Σp tadpole 25 coupling gµ2ɛ n+8gµ Feynman 2ɛ α 6ɛ 4 25 coupling α α n+8 6ɛ α2 Σp tadpole = n + 2n + 8 m 2 α 2 + ɛψ2 log m 2 + Oɛ ɛ 2 n = 4 2-loop
29 Σp d-scoop + Σp sunset + Σp tadpole + Σp tadpole = n + 2α2 p 2 + m2 α 2 72ɛ 2 n + 2n + 5 n Oɛ 8ɛ 2 6ɛ A + A 2 φ m2 B + B 2 φ 2, 4 n + 2α2 n + 2α n + 2n + 5α2 n + 2α2 A =, A 2 =, B =, B 2 = ɛ 6ɛ 6ɛ 2 2ɛ Γ loop Glass Lobster -loop 4 Γ 4 p, p 2, p, p 4 fish 8.2 Γ4 p, p 2, p, p 4 fish gµ 2ɛ α n+8gµ2ɛ α 2 6ɛ Γ 4 p, p 2, p, p 4 n + 82 gµ 2ɛ α 2 [ fish = 8 ɛ 2 i i 2 i i 4 + ɛ2 + ψ log m 2 ɛf i i 2 i i 4 + Oɛ 2 ] loop 2 45 Γ 4 p, p 2, p, p 4 glass + Γ 4 p, p 2, p, p 4 lobster + Γ 4 p, p 2, p, p 4 fish n + 8 = gµ 2ɛ α 2 2 5n + 22 i i 2 i i 4 + Oɛ 6ɛ 2 8ɛ Lagrangian 4! gµ2ɛ C 2 φ 2 2 C 2 = n + 82 α 2 5n + 22α2 6ɛ 2 8ɛ loop ε = 2ɛ n + 2 A 2 = 6ε α2, B 2 = L 2-loop c.t. = 2 A 2 φ m2 B 2 φ 2 + 4! gµ2ɛ C 2 φ n + 2n + 5 n + 2 9ε 2 6ε n + 8 α 2 2, C 2 = 9ε 2 -loop n + 22 α 2. 9ε 49 Z φ = n + 2 6ε α2, 5 Z m = + n + 2 n + 2n + 5 ε α + 5n + 2 α 2, 5 9ε 2 6ε Z g = + n + 8 ε α + n ε 2 29 n + 4 6ε α 2. 52
30 α c = gc 4π 2 βε, g = εg + βg, γ m g = n + 2 α + 89 n + 8 βg = g α n + 4 α 2, 5 5n + 2 α 2, 54 8 γ φ g = n α2. 55 α c = ε 9n + 4ε2 + + Oε. 56 n + 8 n + 8 ν = n + 2ε + 2 4n n + 2n2 + 2n + 6ε 2 + Oε, 8n n + 2ε2 η = 2n Oε minimal subtraction scheme beta 5-loop [K] βg, ε/g = ε + n + 8α n + 4α2 + [ ] n n n + 22ζ [ 5n + 62n n n n + 22 ζ 288n + 85n + 22ζ n n + 86 ] ζ5 α 4 + [ n n n n n n n n + 46 ζ α n 59n n ζ n + 88n n + 22 ζ n n n ζ5 96n + 8 2n n + 86 ζ n n ] ζ7 α 5 α = g 4π 2, ζs = k k s Riemann zeta
31 [R] P. Ramond, Field Theory, A Modern Primer, Benjamin 98 [B] M. L. Bellac, Quantum and Statistical Field Theory, Oxford 99 [A] D. J. Amit, Field Theory, the Renormalization Group and Critical Phenomena revised 2nd edition, World Scientific 984. [K] H. Kleinert, J. Neu, V. Schulte-Frohlinde, K. G. Chetyrkin, S. A. Larin, Phys. Lett. B Errata
TOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
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)
[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization)
![[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization)](/thumbs/47/23763429.jpg "[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization)")
. D............................................... : E = κ ............................................ 3.................................................
meiji_resume_1.PDF
β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
2 R U, U Hausdorff, R. R. S R = (S, A) (closed), (open). (complete projective smooth algebraic curve) (cf. 2). 1., ( ).,. countable ( 2 ) ,,.,,
15, pp.1-13 1 1.1,. 1.1. C ( ) f = u + iv, (, u, v f ). 1 1. f f x = i f x u x = v y, u y = v x.., u, v u = v = 0 (, f = 2 f x + 2 f )., 2 y2 u = 0. u, u. 1,. 1.2. S, A S. (i) A φ S U φ C. (ii) φ A U φ
医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 2 版 1 刷発行時のものです.
医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987
Note.tex 2008/09/19( )
1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................
Microsoft Word - 11問題表紙（選択）.docx
A B A.70g/cm 3 B.74g/cm 3 B C 70at% %A C B at% 80at% %B 350 C γ δ y=00 x-y ρ l S ρ C p k C p ρ C p T ρ l t l S S ξ S t = ( k T ) ξ ( ) S = ( k T) ( ) t y ξ S ξ / t S v T T / t = v T / y 00 x v S dy dx
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
main.dvi
SGC - 48 208X Y Z Z 2006 1930 β Z 2006! 1 2 3 Z 1930 SGC -12, 2001 5 6 http://www.saiensu.co.jp/support.htm http://www.shinshu-u.ac.jp/ haru/ xy.z :-P 3 4 2006 3 ii 1 1 1.1... 1 1.2 1930... 1 1.3 1930...
No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
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 ; >
: , 2.0, 3.0, 2.0, (%) ( 2.
2017 1 2 1.1...................................... 2 1.2......................................... 4 1.3........................................... 10 1.4................................. 14 1.5..........................................
ohpmain.dvi
fujisawa@ism.ac.jp 1 Contents 1. 2. 3. 4. γ- 2 1. 3 10 5.6, 5.7, 5.4, 5.5, 5.8, 5.5, 5.3, 5.6, 5.4, 5.2. 5.5 5.6 +5.7 +5.4 +5.5 +5.8 +5.5 +5.3 +5.6 +5.4 +5.2 =5.5. 10 outlier 5 5.6, 5.7, 5.4, 5.5, 5.8,
第10章 アイソパラメトリック要素
June 5, 2019 1 / 26 10.1 ( ) 2 / 26 10.2 8 2 3 4 3 4 6 10.1 4 2 3 4 3 (a) 4 (b) 2 3 (c) 2 4 10.1: 3 / 26 8.3 3 5.1 4 10.4 Gauss 10.1 Ω i 2 3 4 Ξ 3 4 6 Ξ ( ) Ξ 5.1 Gauss ˆx : Ξ Ω i ˆx h u 4 / 26 10.2.1
i
009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3
(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x
![(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x](/thumbs/104/162595168.jpg "(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x")
Compton Scattering Beaming exp [i k x ωt] k λ k π/λ ω πν k ω/c k x ωt ω k α c, k k x ωt η αβ k α x β diag + ++ x β ct, x O O x O O v k α k α β, γ k γ k βk, k γ k + βk k γ k k, k γ k + βk 3 k k 4 k 3 k
( ) ( 40 )+( 60 ) Schrödinger 3. (a) (b) (c) yoshioka/education-09.html pdf 1
2009 1 ( ) ( 40 )+( 60 ) 1 1. 2. Schrödinger 3. (a) (b) (c) http://goofy.phys.nara-wu.ac.jp/ yoshioka/education-09.html pdf 1 1. ( photon) ν λ = c ν (c = 3.0 108 /m : ) ɛ = hν (1) p = hν/c = h/λ (2) h
Radiation from moving charges#1 Liénard-Wiechert potential Yuji Chinone 1 Maxwell Maxwell MKS E (x, t) + B (x, t) t = 0 (1) B (x, t) = 0 (2) B (x, t)
Radiation from moving harges# Liénard-Wiehert potential Yuji Chinone Maxwell Maxwell MKS E x, t + B x, t = B x, t = B x, t E x, t = µ j x, t 3 E x, t = ε ρ x, t 4 ε µ ε µ = E B ρ j A x, t φ x, t A x, t
Part () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
Untitled
II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =
1 1 1.1 I R 1.1.1 c : I R 2 (i) c C (ii) t I c (t) (0, 0) c (t) c(i) c c(t) 1.1.2 (1) (2) (3) (1) r > 0 c : R R 2 : t (r cos t, r sin t) (2) C f : I R c : I R 2 : t (t, f(t)) (3) y = x c : R R 2 : t (t,
G (n) (x 1, x 2,..., x n ) = 1 Dφe is φ(x 1 )φ(x 2 ) φ(x n ) (5) N N = Dφe is (6) G (n) (generating functional) 1 Z[J] d 4 x 1 d 4 x n G (n) (x 1, x 2
![G (n) (x 1, x 2,..., x n ) = 1 Dφe is φ(x 1 )φ(x 2 ) φ(x n ) (5) N N = Dφe is (6) G (n) (generating functional) 1 Z[J] d 4 x 1 d 4 x n G (n) (x 1, x 2](/thumbs/91/105326061.jpg "G (n) (x 1, x 2,..., x n ) = 1 Dφe is φ(x 1 )φ(x 2 ) φ(x n ) (5) N N = Dφe is (6) G (n) (generating functional) 1 Z[J] d 4 x 1 d 4 x n G (n) (x 1, x 2")
6 Feynman (Green ) Feynman 6.1 Green generating functional Z[J] φ 4 L = 1 2 µφ µ φ m 2 φ2 λ 4! φ4 (1) ( 1 S[φ] = d 4 x 2 φkφ λ ) 4! φ4 (2) K = ( 2 + m 2 ) (3) n G (n) (x 1, x 2,..., x n ) = φ(x 1 )φ(x
H.Haken Synergetics 2nd (1978)
27 3 27 ) Ising Landau Synergetics Fokker-Planck F-P Landau F-P Gizburg-Landau G-L G-L Bénard/ Hopfield H.Haken Synergetics 2nd (1978) (1) Ising m T T C 1: m h Hamiltonian H = J ij S i S j h i S
r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B
1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n
201711grade1ouyou.pdf
2017 11 26 1 2 52 3 12 13 22 23 32 33 42 3 5 3 4 90 5 6 A 1 2 Web Web 3 4 1 2... 5 6 7 7 44 8 9 1 2 3 1 p p >2 2 A 1 2 0.6 0.4 0.52... (a) 0.6 0.4...... B 1 2 0.8-0.2 0.52..... (b) 0.6 0.52.... 1 A B 2
1 Tokyo Daily Rainfall (mm) Days (mm)
( ) r-taka@maritime.kobe-u.ac.jp 1 Tokyo Daily Rainfall (mm) 0 100 200 300 0 10000 20000 30000 40000 50000 Days (mm) 1876 1 1 2013 12 31 Tokyo, 1876 Daily Rainfall (mm) 0 50 100 150 0 100 200 300 Tokyo,
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
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 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 = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
Z: Q: R: C:
0 Z: Q: R: C: 3 4 4 4................................ 4 4.................................. 7 5 3 5...................... 3 5......................... 40 5.3 snz) z)........................... 4 6 46 x
³ÎΨÏÀ
2017 12 12 Makoto Nakashima 2017 12 12 1 / 22 2.1. C, D π- C, D. A 1, A 2 C A 1 A 2 C A 3, A 4 D A 1 A 2 D Makoto Nakashima 2017 12 12 2 / 22 . (,, L p - ). Makoto Nakashima 2017 12 12 3 / 22 . (,, L p
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
![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](/thumbs/103/158001152.jpg "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")
6 6.. [, b] [, d] ij P ij ξ ij, η ij f Sf,, {P ij } Sf,, {P ij } k m i j m fξ ij, η ij i i j j i j i m i j k i i j j m i i j j k i i j j kb d {P ij } lim Sf,, {P ij} kb d f, k [, b] [, d] f, d kb d 6..
H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [
![H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [](/thumbs/99/141438442.jpg "H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [")
3 3. 3.. H H = H + V (t), V (t) = gµ B α B e e iωt i t Ψ(t) = [H + V (t)]ψ(t) Φ(t) Ψ(t) = e iht Φ(t) H e iht Φ(t) + ie iht t Φ(t) = [H + V (t)]e iht Φ(t) Φ(t) i t Φ(t) = V H(t)Φ(t), V H (t) = e iht V (t)e
( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes )
( 3 7 4 ) 2 2 ) 8 2 954 2) 955 3) 5) J = σe 2 6) 955 7) 9) 955 Statistical-Mechanical Theory of Irreversible Processes 957 ) 3 4 2 A B H (t) = Ae iωt B(t) = B(ω)e iωt B(ω) = [ Φ R (ω) Φ R () ] iω Φ R (t)
x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T
NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977
Z: Q: R: C: sin 6 5 ζ a, b
Z: Q: R: C: 3 3 7 4 sin 6 5 ζ 9 6 6............................... 6............................... 6.3......................... 4 7 6 8 8 9 3 33 a, b a bc c b a a b 5 3 5 3 5 5 3 a a a a p > p p p, 3,
X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
Gmech08.dvi
145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17
(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
(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) (Fourier) (Fourier Bessel).. V ρ(x, y, z) V = 4πGρ G :.
7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±
![7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±](/thumbs/91/106462075.jpg "7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±")
7 7. ( ) SU() SU() 9 ( MeV) p 98.8 π + π 0 n 99.57 9.57 97.4 497.70 δm m 0.4%.% 0.% 0.8% π 9.57 4.96 Σ + Σ 0 Σ 89.6 9.46 K + K 0 49.67 (7.) p p = αp + βn, n n = γp + δn (7.a) [ ] p ψ ψ = Uψ, U = n [ α
量子力学 問題
3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,
2016 ǯ¥Î¡¼¥Ù¥ëʪÍý³Ø¾Þ²òÀ⥻¥ß¥Ê¡¼ Kosterlitz-Thouless ž°Ü¤È Haldane ͽÁÛ
2016 Kosterlitz-Thouless Haldane Dept. of Phys., Kyushu Univ. 2016 11 29 2016 Figure: D.J.Thouless F D.M.Haldane J.M.Kosterlitz TOPOLOGICAL PHASE TRANSITIONS AND TOPOLOGICAL PHASES OF MATTER ( ) ( ) (Dirac,
ii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,.
24(2012) (1 C106) 4 11 (2 C206) 4 12 http://www.math.is.tohoku.ac.jp/~obata,.,,,.. 1. 2. 3. 4. 5. 6. 7.,,. 1., 2007 (). 2. P. G. Hoel, 1995. 3... 1... 2.,,. ii 3.,. 4. F. (),.. 5... 6.. 7.,,. 8.,. 1. (75%)
b3e2003.dvi
15 II 5 5.1 (1) p, q p = (x + 2y, xy, 1), q = (x 2 + 3y 2, xyz, ) (i) p rotq (ii) p gradq D (2) a, b rot(a b) div [11, p.75] (3) (i) f f grad f = 1 2 grad( f 2) (ii) f f gradf 1 2 grad ( f 2) rotf 5.2
変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy,
変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, z + dz) Q! (x + d x + u + du, y + dy + v + dv, z +
スケーリング理論とはなにか? - --尺度を変えて見えること--
? URL: http://maildbs.c.u-tokyo.ac.jp/ fukushima mailto:hukusima@phys.c.u-tokyo.ac.jp DEX-SMI @ 2006 12 17 ( ) What is scaling theory? DEX-SMI 1 / 40 Outline Outline 1 2 3 4 ( ) What is scaling theory?
73
73 74 ( u w + bw) d = Ɣ t tw dɣ u = N u + N u + N 3 u 3 + N 4 u 4 + [K ] {u = {F 75 u δu L σ (L) σ dx σ + dσ x δu b δu + d(δu) ALW W = L b δu dv + Aσ (L)δu(L) δu = (= ) W = A L b δu dx + Aσ (L)δu(L) Aσ
TOP URL 1
TOP URL http://amonphys.web.fc.com/ 1 19 3 19.1................... 3 19.............................. 4 19.3............................... 6 19.4.............................. 8 19.5.............................
: 2005 ( ρ t +dv j =0 r m m r = e E( r +e r B( r T 208 T = d E j 207 ρ t = = = e t δ( r r (t e r r δ( r r (t e r ( r δ( r r (t dv j =
72 Maxwell. Maxwell e r ( =,,N Maxwell rot E + B t = 0 rot H D t = j dv D = ρ dv B = 0 D = ɛ 0 E H = μ 0 B ρ( r = j( r = N e δ( r r = N e r δ( r r = : 2005 ( 2006.8.22 73 207 ρ t +dv j =0 r m m r = e E(
τ τ
1 1 1.1 1.1.1 τ τ 2 1 1.1.2 1.1 1.1 µ ν M φ ν end ξ µ ν end ψ ψ = µ + ν end φ ν = 1 2 (µφ + ν end) ξ = ν (µ + ν end ) + 1 1.1 3 6.18 a b 1.2 a b 1.1.3 1.1.3.1 f R{A f } A f 1 B R{AB f 1 } COOH A OH B 1.3
1 (Contents) (1) Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji
8 4 2018 6 2018 6 7 1 (Contents) 1. 2 2. (1) 22 3. 31 1. Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji SETO 22 3. Editorial Comments Tadashi
ʪ¼Á¤Î¥È¥Ý¥í¥¸¥«¥ë¸½¾Ý (2016ǯ¥Î¡¼¥Ù¥ë¾Þ¤Ë´ØÏ¢¤·¤Æ)
(2016 ) Dept. of Phys., Kyushu Univ. 2017 8 10 1 / 59 2016 Figure: D.J.Thouless F D.M.Haldane J.M.Kosterlitz TOPOLOGICAL PHASE TRANSITIONS AND TOPOLOGICAL PHASES OF MATTER 2 / 59 ( ) ( ) (Dirac, t Hooft-Polyakov)
Kroneher Levi-Civita 1 i = j δ i j = i j 1 if i jk is an even permutation of 1,2,3. ε i jk = 1 if i jk is an odd permutation of 1,2,3. otherwise. 3 4
[2642 ] Yuji Chinone 1 1-1 ρ t + j = 1 1-1 V S ds ds Eq.1 ρ t + j dv = ρ t dv = t V V V ρdv = Q t Q V jdv = j ds V ds V I Q t + j ds = ; S S [ Q t ] + I = Eq.1 2 2 Kroneher Levi-Civita 1 i = j δ i j =
I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%
1 2006.4.17. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 1. 1. 2. 3. 4. 5. 2. ɛ-δ 1. ɛ-n
21 2 26 i 1 1 1.1............................ 1 1.2............................ 3 2 9 2.1................... 9 2.2.......... 9 2.3................... 11 2.4....................... 12 3 15 3.1..........
I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google
I4 - : April, 4 Version :. Kwhir, Tomoki TA (Kondo, Hirotk) Google http://www.mth.ngoy-u.c.jp/~kwhir/courses/4s-biseki.html pdf 4 4 4 4 8 e 5 5 9 etc. 5 6 6 6 9 n etc. 6 6 6 3 6 3 7 7 etc 7 4 7 7 8 5 59
newmain.dvi
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
基礎数学I
I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............
( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) =
1 9 8 1 1 1 ; 1 11 16 C. H. Scholz, The Mechanics of Earthquakes and Faulting 1. 1.1 1.1.1 : - σ = σ t sin πr a λ dσ dr a = E a = π λ σ πr a t cos λ 1 r a/λ 1 cos 1 E: σ t = Eλ πa a λ E/π γ : λ/ 3 γ =
(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n
. X {x, x 2, x 3,... x n } X X {, 2, 3, 4, 5, 6} X x i P i. 0 P i 2. n P i = 3. P (i ω) = i ω P i P 3 {x, x 2, x 3,... x n } ω P i = 6 X f(x) f(x) X n n f(x i )P i n x n i P i X n 2 G(k) e ikx = (ik) n
grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( )
2 9 2 5 2.2.3 grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = g () g () (3) grad φ(p ) p grad φ φ (P, φ(p )) y (, y) = (ξ(t), η(t)) ( ) ξ (t) (t) := η (t) grad f(ξ(t), η(t)) (t) g(t) := f(ξ(t), η(t))
() [REQ] 0m 0 m/s () [REQ] (3) [POS] 4.3(3) ()() () ) m/s 4. ) 4. AMEDAS
![() [REQ] 0m 0 m/s () [REQ] (3) [POS] 4.3(3) ()() () ) m/s 4. ) 4. AMEDAS](/thumbs/92/107891201.jpg "() [REQ] 0m 0 m/s () [REQ] (3) [POS] 4.3(3) ()() () ) m/s 4. ) 4. AMEDAS")
() [REQ] 4. 4. () [REQ] 0m 0 m/s () [REQ] (3) [POS] 4.3(3) ()() () 0 0 4. 5050 0 ) 00 4 30354045m/s 4. ) 4. AMEDAS ) 4. 0 3 ) 4. 0 4. 4 4.3(3) () [REQ] () [REQ] (3) [POS] () ()() 4.3 P = ρ d AnC DG ()
x 3 a (mod p) ( ). a, b, m Z a b m a b (mod m) a b m 2.2 (Z/mZ). a = {x x a (mod m)} a Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} a + b = a +
1 1 22 1 x 3 (mod ) 2 2.1 ( )., b, m Z b m b (mod m) b m 2.2 (Z/mZ). = {x x (mod m)} Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} + b = + b, b = b Z/mZ 1 1 Z Q R Z/Z 2.3 ( ). m {x 0, x 1,..., x m 1 } modm 2.4
tomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p.
tomocci 18 7 5...,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. M F (M), X(F (M)).. T M p e i = e µ i µ. a a = a i
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
n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................
) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
![) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4](/thumbs/92/107866707.jpg ") ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4")
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
l µ l µ l 0 (1, x r, y r, z r ) 1 r (1, x r, y r, z r ) l µ g µν η µν 2ml µ l ν 1 2m r 2mx r 2 2my r 2 2mz r 2 2mx r 2 1 2mx2 2mxy 2mxz 2my r 2mz 2 r
2 1 (7a)(7b) λ i( w w ) + [ w + w ] 1 + w w l 2 0 Re(γ) α (7a)(7b) 2 γ 0, ( w) 2 1, w 1 γ (1) l µ, λ j γ l 2 0 Re(γ) α, λ w + w i( w w ) 1 + w w γ γ 1 w 1 r [x2 + y 2 + z 2 ] 1/2 ( w) 2 x2 + y 2 + z 2
W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)
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)
QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1
QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 (vierbein) QCD QCD 1 1: QCD QCD Γ ρ µν A µ R σ µνρ F µν g µν A µ Lagrangian gr TrFµν F µν No. Yes. Yes. No. No! Yes! [1] Nash & Sen [2] Riemann
Z: Q: R: C: 3. Green Cauchy
7 Z: Q: R: C: 3. Green.............................. 3.............................. 5.3................................. 6.4 Cauchy..................... 6.5 Taylor..........................6...............................
,, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising
![,, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising](/thumbs/94/118770363.jpg ",, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising")
,, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising Model 1 Ising 1 Ising Model N Ising (σ i = ±1) (Free
.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
2000年度『数学展望 I』講義録
2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53
sec13.dvi
13 13.1 O r F R = m d 2 r dt 2 m r m = F = m r M M d2 R dt 2 = m d 2 r dt 2 = F = F (13.1) F O L = r p = m r ṙ dl dt = m ṙ ṙ + m r r = r (m r ) = r F N. (13.2) N N = R F 13.2 O ˆn ω L O r u u = ω r 1 1:
N/m f x x L dl U 1 du = T ds pdv + fdl (2.1)
23 2 2.1 10 5 6 N/m 2 2.1.1 f x x L dl U 1 du = T ds pdv + fdl (2.1) 24 2 dv = 0 dl ( ) U f = T L p,t ( ) S L p,t (2.2) 2 ( ) ( ) S f = L T p,t p,l (2.3) ( ) U f = L p,t + T ( ) f T p,l (2.4) 1 f e ( U/
7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa, f a g b g f a g f a g bf a
9 203 6 7 WWW http://www.math.meiji.ac.jp/~mk/lectue/tahensuu-203/ 2 8 8 7. 7 7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa,
Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 = ( p µ γ µ + m)(p ν γ ν + m) (5.1) γ = p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 = 1 2 p µp ν {γ µ, γ ν } + m
Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 p µ γ µ + mp ν γ ν + m 5.1 γ p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 1 2 p µp ν {γ µ, γ ν } + m 2 5.2 p m p p µ γ µ {, } 10 γ {γ µ, γ ν } 2η µν 5.3 p µ γ µ + mp
simx simxdx, cosxdx, sixdx 6.3 px m m + pxfxdx = pxf x p xf xdx = pxf x p xf x + p xf xdx 7.4 a m.5 fx simxdx 8 fx fx simxdx = πb m 9 a fxdx = πa a =
II 6 ishimori@phys.titech.ac.jp 6.. 5.4.. f Rx = f Lx = fx fx + lim = lim x x + x x f c = f x + x < c < x x x + lim x x fx fx x x = lim x x f c = f x x < c < x cosmx cosxdx = {cosm x + cosm + x} dx = [
,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.
9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,
80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t = 0 i r 0 t(> 0) j r 0 + r < δ(r 0 x i (0))δ(r 0 + r x j (t)) > (4.2) r r 0 G i j (r, t) dr 0
79 4 4.1 4.1.1 x i (t) x j (t) O O r 0 + r r r 0 x i (0) r 0 x i (0) 4.1 L. van. Hove 1954 space-time correlation function V N 4.1 ρ 0 = N/V i t 80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t
( ) ( )
20 21 2 8 1 2 2 3 21 3 22 3 23 4 24 5 25 5 26 6 27 8 28 ( ) 9 3 10 31 10 32 ( ) 12 4 13 41 0 13 42 14 43 0 15 44 17 5 18 6 18 1 1 2 2 1 2 1 0 2 0 3 0 4 0 2 2 21 t (x(t) y(t)) 2 x(t) y(t) γ(t) (x(t) y(t))
211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
, 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p, p 3,..., p n p, p,..., p n N, 3,,,,
6,,3,4,, 3 4 8 6 6................................. 6.................................. , 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p,
..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A
![..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A](/thumbs/93/112175219.jpg "..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A")
.. Laplace ). A... i),. ω i i ). {ω,..., ω } Ω,. ii) Ω. Ω. A ) r, A P A) P A) r... ).. Ω {,, 3, 4, 5, 6}. i i 6). A {, 4, 6} P A) P A) 3 6. ).. i, j i, j) ) Ω {i, j) i 6, j 6}., 36. A. A {i, j) i j }.
DVIOUT-fujin
2005 Limit Distribution of Quantum Walks and Weyl Equation 2006 3 2 1 2 2 4 2.1...................... 4 2.2......................... 5 2.3..................... 6 3 8 3.1........... 8 3.2..........................
(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y
[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
19 /
19 / 1 1.1............................... 1. Andreev............................... 3 1.3..................................... 3 1.4..................................... 4 / 5.1......................................
dvipsj.8449.dvi
9 1 9 9.1 9 2 (1) 9.1 9.2 σ a = σ Y FS σ a : σ Y : σ b = M I c = M W FS : M : I : c : = σ b
untitled
20010916 22;1017;23;20020108;15;20; 1 N = {1, 2, } Z + = {0, 1, 2, } Z = {0, ±1, ±2, } Q = { p p Z, q N} R = { lim a q n n a n Q, n N; sup a n < } R + = {x R x 0} n = {a + b 1 a, b R} u, v 1 R 2 2 R 3
1. A0 A B A0 A : A1,...,A5 B : B1,...,B
1. A0 A B A0 A : A1,...,A5 B : B1,...,B12 2. 3. 4. 5. A0 A, B Z Z m, n Z m n m, n A m, n B m=n (1) A, B (2) A B = A B = Z/ π : Z Z/ (3) A B Z/ (4) Z/ A, B (5) f : Z Z f(n) = n f = g π g : Z/ Z A, B (6)
untitled
0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.