. D............................................... : E = κ <............................................. : E = k >............................................ 3................................................. 4.. :................................... 5.. :........................................ 5..3............................................ 6.3 β......................................... 7 9 A Green (Dated: June 6, 5) Schrödinger ħh m + g δ (D ) (x ) Ψ(x ) = E Ψ(x ) D D = D = (dimensional transmutation) g <. m D Schrödinger : ħh m + g δ (D ) (x ) Ψ(x ) = E Ψ(x ). (.) E g ħh m ħh m : ħh = m =. (.) Schrödinger (.): + g δ (D ) (x ) Ψ(x ) = E Ψ(x ). (.3) ħh = m =, D δ (D ) (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) (renormalization)... D Schrödinger (.3) Green D D... : E = κ < E < E = κ (κ > ) Schrödinger : + V (x ) Ψ(x ) = κ Ψ(x ). (.4) V (x ) = g δ (D ) (x ) (.4) : Ψ(x ) = d D y G B (x,y ;κ)v (y )Ψ(y ). (.5) G B (x,y ;κ) Green : κ + x GB (x,y ;κ) = δ (D ) (x y ). (.6) (A): G B (x,y ;κ) = d D p e i p (x y ) (π) D κ p = π κ π x y D K D κ x y. (.7) K ν (z ) ν (.5) Schrödinger (.4) (.5) κ + x (.6) ( κ + )Ψ(x ) = V (x )Ψ(x ) Schrödinger (.4)(.5) Ψ /
(.5) Schrödinger V (y ) = g δ (D ) (y ) (.5) y : Ψ(x ) = g G B (x,;κ)ψ(). (.8) Ψ() E = κ < Green (.7) (.8) x = : Ψ() = g G B (,;κ)ψ(). (.9) : = g G B (,;κ). (.) κ κ g E = κ (.) (.) (self-consistency condition)... : E = k > E = k > (k > ) Schrödinger : + V (x ) Ψ(x ) = k Ψ(x ). (.) V (x ) = g δ (D ) (x ) V (x ) x Schrödinger (.) Ψ + k ( k ) : Ψ + k (x ) = ei k x + d D y G + (x,y ;k)v (y )Ψ + k (y ). (.) G + (x,y ;k) Green k + x G + (x,y ;k) = δ (D ) (x y ) (.3) : G + (x,y ;k) = d D p e i p (x y ) (π) D k p + i ε = i 4 k π x y D H () D k x y. (.4) ε H ν () (z ) ν (.) Lippmann- Schwinger. (.) Schrödinger (.4) (.) k + x (.3) (k + )Ψ + k (x ) = V (x )Ψ+ k (x ) Schrödinger (.)(.) Ψ + k Schrödinger (.) Lippmann-Schwinger Schrödinger Schrödinger (.4) ε 3/
(.) V (y ) = g δ (D ) (y ) y : x = : Ψ + k () : (.5): Ψ + k (x ) = ei k x + g G + (x,;k)ψ + k (). (.5) Ψ + k () = + g G + (,;k)ψ + k (). (.6) Ψ + k () = g G + (,;k). (.7) Ψ + k (x ) = ei k x + g G + (x,;k) g G + (,;k). (.8) Green Schrödinger H ν () : (z ) z G + (x,y ;k) k D 3 i[k x y (D e 3)π/4] 4π π x y D : πz ei (z (ν+)π/4) Green as x y. (.9) Ψ + k (x ) ei k x [k x (D 3)π/4] ei + f (k) x D as x. (.) f (k) :.. f (k) = k D 3 g Ψ + k 4π π () = k D 3 g 4π π g G + (,;k). (.) Schrödinger (.)(.) G B (,;κ) G + (,;k) : d D p G B (,;κ) = = for D, (.a) (π) κ + p d G + D p (,;k) = = for D. (.b) (π) k p + i ε g D = 4/
... : (.a)(.b) p : d p (π) d p (π). (.3) p <Λ Λ (.) : d p (.) = g G B (,;κ) = g (π) κ + p Λ p <Λ Λ πp d p = g (π) κ + p = g d p 4π κ + p = g κ 4π log + Λ. (.4) κ Λ (.4) Λ = Λ : κ + Λ Λ log = log + κ Λ = log + log + κ κ κ Λ κ Λ κ = log + κ +. (.5) κ Λ Λ Taylor log(+ x ) = x x + Λ (.4) logλ (.5) Λ (.):... : g G B (,;κ) = g 4π log Λ Λ + O (Λ ). (.6) (.a)(.b) D ε D = ε : d p d D (π) p (π), (.7a) D κ g g µ D. (.7b) µ D g (D D ) µ (.) : d (.) = g µ D G B (,;κ) = g µ D D p (π) D κ + p g µ D = (4π) D / Γ (D /) d p κ + p = g (µ/κ) D (4π) D / Γ (D /) p D d x x D / + x = g (µ/κ) D Γ ( D /) (4π) D /. (.8) 5/
3 d D p f (p) = d p p D f (p ) (f p = p ) 4 x = p /κ B (p,q ) = πd / Γ (D /) x p Γ (p)γ (q ) d x = ( + x ) p +q Γ (p + q ) (.9) Γ () = (.8) D = D =.99 ε (.8): (.8) = g (µ/κ)ε Γ (ε/) (4π) ε/ D = ε (.3) = g ε/ 4π µ Γ (ε/). (.3) 4π κ ε (4πµ /κ ) ε/ Γ (ε/) ε = : ε/ 4π µ = e ε log 4π µ κ = + ε 4π κ log µ + O (ε ), (.3a) κ Γ (ε/) = ε γ + O (ε). (.3b) γ.577 Euler (.3)(.) : g µ D G B (,;κ) = g µ γ + log(4π) + log + O (ε). (.33) 4π ε κ..3. Λ ε g Λ ε (.6)(.33) g ( E = κ ) (.) : (): (): g = 4π g = 4π Λ µ log + log + O (Λ ), (.34a) µ κ µ γ + log(4π) + log + O (ε). (.34b) ε κ (.34a) µ : (): (): := lim Λ := lim ε g + Λ 4π log, (.35a) µ g + 4π ε γ + log(4π). (.35b) := lim ε g + 4π ε minimal subtraction ( MS) (.35b) modified minimal subtraction ( MS ( )) MS 6/
g Λ ε g (.34a)(.34b) Λ ε : = 4π log µ. (.36) κ E B = κ : 4π E B = µ exp. (.37) (.35a)(.35b)(.) D = : κ f (k) = 8πk g G + (,;k). (.38) G + (x,y ;k) = G B (x,y ; i k) κ = k G + (,;k) : Λ µ log + log + O (Λ ) (), G + 4π µ k (,;k) = (.39) µ γ + log(4π) + log + O (ε) (). 4π ε k (.35a)(.35b) f (k) = 8πk + 4π log µ (.4) k (.36) = µ 4π log( E B ) (.4) π f (k) = k log EB k (.4) E B E B g E B.3. β µ E B (.) µ E B µ E B µ : µ d d µ E B =. (.4) 7/
d d µ µ d d µ (.4) µ E B µ (µ) E B (µ, (µ)) µ d E B d µ = µ E B µ + µ d d µ (.4): E B µ µ + β(g R ) E B =. (.43) β( ) β : β( ) := µ d d µ. (.44) E B (.37) µ β : = µ E B µ + β( ) E B = µ e 4π/ + β( ) µ e 4π/ 4π gr = 4π gr β( ) E B. (.45) (.45) β (.46) (.47) d g R ( < t < ) : ḡ (t ) β( ) = π g R (.46) µ d d µ = π g R (.47) d g R = d µ π µ µ µet = µe t d µ π µ. (.48) µ ḡ (t ) µe t : ḡ (t ) = π t. (.49) ḡ (t ) (running coupling constant) ḡ (t ) e t k e t k ke t : f (ke t ; ) = 8πke t + 4π log µ k e t = e t 8πk t π + 4π log µ = e t 8πk ḡ (t ) + 4π log µ k k = e t f (k;ḡ (t )). (.5) 8/
ḡ (t ) π π t Figure : ḡ (t ) =. t = g π t R () ( ) e t / (.5) k e t k ḡ (t ) ḡ (t ) t ḡ (t ) () t. : C. Thorn, Quark confinement in the infinite-momentum frame, Phys. Rev. D9 (979) 639-65 [INSPIRE]. Roman Jackiw 99 : R. Jackiw, Delta function potentials in two-dimensional and three-dimensional quantum mechanics, in M. A. B. Bég Memorial Volume, A. Ali and P. Hoodbhoy, eds. World Scientific Publishing, 99 [INSPIRE]; scanned version available in the KEK library: http://ccdb5fs.kek.jp/cgi-bin/img_index?3443. (self-adjoint extension) ( ) Jackiw Schrödinger (.3) Schrödinger. 9/
S[ψ,ψ ] = d t d x i ψ t ψ ψ g ψ 4 (.) 4 Γ (4) Γ (4) Oren Bergman 99 O. Bergman, Nonrelativistic field theoretic scale anomaly, Phys. Rev. D46 (99) 5474-5478 [INSPIRE].. Lippmann-Schwinger Bernard Lippmann Julian Schwinger B. A. Lippmann and J. Schwinger, Variational Principles for Scattering Processes. I, Phys. Rev. 79 (95) 469-48 [INSPIRE] Steven Weinberg Lectures on Quantum Mechanics..3Sidney Coleman Erick Weinberg 973 Coleman-Weinberg S. Coleman and E. Weinberg, Radiative Corrections as the Origin of Spontaneous Symmetry Breaking, Phys. Rev. D7 (973) 888-9 [INSPIRE] Kenneth G. Wilson K. G. Wilson and J. B. Kogut, The renormalization group and the ε expansion, Phys. Rep. (974) 75-99 [INSPIRE] A. Green Green G B G B : ( κ + x )G B (x,y ;κ) = δ (D ) (x y ). (A.) Fourier : G B (x,y ;κ) = d D p (π) D e i p (x y ) κ + p. (A.) Schwinger : κ + p = d t e t (κ +p ). (A.3) /
Schwinger t Schwinger (A.3)(A.): G B (x,y ;κ) = = d D p (π) D d t e t κ = (4π) D / = κ 4π π x y d t e t (κ +p )+i p (x y ) x y 4t D j = d t t D / e t κ D x y 4t d s e s D / d p j π e t p j i x j y j t κ x y (s + s ). (A.4) s = κt / x y Green G B (x,y ;κ) = π κ π x y D K D κ x y (A.5) 3 : κ e κ x y G B (x,y ;κ) = π K κ x y for D = ; for D = ; (A.6) K / (z ) = K / (z ) = π z e z 4π x y e κ x y for D = 3. /