Yang-Mills Yang-Mills Yang-Mills 50 operator formalism operator formalism 1 I The Dawning of Gauge T
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1 Yang-Mills Yang-Mills Yang-Mills 50 operator formalism operator formalism 1 I The Dawning of Gauge Theory O Raifeartaigh [1] I, II, III O Raifeartaigh I Weyl O Raifeartaigh Weyl Pauli Pauli Handbuch Yang Yang-Mills Weyl I 1918 Einstein Maxwell Weyl 1
2 1: I 1918 Weyl Gravitation and Electricity Scale transformation of metric: g µν (x) λ(x)g µν (x) 1921 Kaluza 5 g 5µ = 2αA µ 1926 Klein g 5µ /g 55 = A µ Schrödinger µ µ ie A µ 26 London ψ exp[ ie A]ψ 1929 Weyl Electron and Gravitation Zeit f. Physik { e ieλ(x) ψ(x) µ D µ = µ iea µ F µν = µ A ν ν A µ Klein SU(2) gauge theory in Conf. Report (Kazimierz, Poland) 49 -Schwinger-Feynman-Dyson QED 50 Gupta-Bleuler 1953 Pais isospin Pauli SU(2) gauge theory in two Letters to Pais 1954 Yang-Mills Phys. Rev. 96 (1954) Shaw SU(2) gauge theory in Ph.D. Thesis 56 Phys. Rev. 101 (1956) 1597 g µν (x) λ(x)g µν (x) φ µ (x) φ µ (x) + µ ln λ(x) φ µ Bohm-Aharanov Einstein Weyl 1925 Schrödinger London Schrödinger 1922 Weyl exp[(e/γ) φ µ dx µ ] γ Weyl γ = /i 2
3 Bohr-Sommerfeld e 2πni = µ µ ie A µ 1922 London Schrödinger Schrödinger Weyl exp[i(e/ ) A µ dx µ ] Weyl 1929 Zeit f. Physik U(1) ψ(x) e ieλ(x) ψ(x) D µ = µ ie A µ F µν 2 Weyl Pauli Weyl 1933 Fermi 1935 Klein Kaluza 5 5 SU(2) Yang-Mills Klein SU(2) SU(2)doublet 1938 Poland Kazimierz Proceedings 2 QED Pais 1953 Leiden Pauli Pais SU(2) Yang-Mills doublet doublet 1954 Yang-Mills Yang-Mills Pais Pauli Yang Pauli Handbuch Mills Yang Mills Yang-Mills Princeton Yang Yang-Mills? Pauli Oppenheimer Pauli Yang 3
4 O Raifeartaigh [1] Yang-Mills Shaw 2 II II Yang-Mills : II 1960 Gauge Invariance in Superconductivity 60 J.J. Sakurai Massive Yang-Mills for ρ 61 S. Glashow Massive Yang-Mills for W, Z 61 Goldstone 63 Feynman ghost Higgs 66 -Lautrup 67 Kibble 67 DeWitt ghost 67 Faddeev-Popov 1967 Weinberg-Salam Theory of Electron 69 Adler, Bell-Jackiw 1971 t Hooft Yang-Mills T.K CP 1973 Gross-Wilczek Asymptotic Freedom Politzer 73 Nakanishi N-L Higgs 1974 Ken Wilson Lattice Gauge Theory 1974 t Hooft-Polyakov monopole 75 Nielsen-Olsen vortex 75 Bogomol nyi-prasad-sommerfield (BPS) 1975 Becchi-Rouet-Stora, Tyutin BRS 78 Kugo-Ojima 79 Fujikawa path-int measure anomaly Q B phys = Nambu-Jona-Lasinio 1960 Phys.Rev. Quasi-Particles and Gauge Invariance in the Theory of Superconductivity [2] U(1) massless massive 4
5 Meissner Goldstone Higgs! Goldstone (61 ) Higgs (64,66 ) 1960 J.J.Sakurai Yang-Mills ρ ρ ( hidden local symmetry [3]) 1961 Glashow SU(2) U(1) Yang-Mills 61 Goldstone massless Higgs 67 Kibble massive Kibble SU(2) Weinberg-Salam Higgs Kibble Weinberg Weinberg Kibble? (Weinberg Glashow?) Weinberg-Salam ( ) Glashow-Illiopoulos-Miani 1970 ( Glashow- Weinberg-Salam ) charm 2-doublets FCNC GIM 71 t Hooft Yang-Mills Weinberg-Salam Weinberg-Salam 1973 (1972?) Weinberg- Salam CP? - t Hooft Lee-Zinn-Justin Yang-Mills R ξ Fujikawa-Lee-Sanda R ξ R ξ 5
6 Bethe-Salpeter Gross-Wilczek, Politzer Asymptotic Freedom t Hooft-Polyakov monopole Nielsen-Olsen vortex BPS 1974 t Hooft Lee-Zinn-Justin t Hooft-Veltman 1975 BRS BRS BRS-formalism operator formalism BRS BRS Yang-Mills Lagrangian S Faddeev-Popov (FP) ghost FP anti-ghost FP ghost t Hooft 1963 Feynman 1-loop ghost 67 DeWitt Faddeev Popov Veltman Faddeev-Popov t Hooft Veltman S S Hamiltonian Feynman QED Gupta-Bleuler formalism 6
7 1950 Feynman QED Lautrup ( ) Yang-Mills 1978 Kugo-Ojima [4] ( 77 ) Q B phys = 0 Q B (BRS exact states) BRS Q B phys = 0 Kugo-Ojima [4, 5] 1974 K.Wilson QCD Wilson quark confinement Atiyah-Singer 3 III 3 3: III 1984 Green-Schwarz Anomaly cancellation in d = 10 SYM with SO(32), E 8 E Seiberg-Witten exact sol. for N = 2 SYM Seiberg N = 1 Seiberg s ele.-mag. duality Polchinski D-brane S, T, U duality M-theory 1997 Maldacena AdS/CFT 1984 Green-Schwarz d = 10 super Yang-Mills (SYM) SO(32) E 8 E 8 7
8 Seiberg-Witten N = 2 SYM Seiberg N = 1 SYM Seiberg duality Polchinski D-brane S-, T-, U-duality M-QCD 1997 Maldacena AdS/CFT supergravity superstring 1995 YKIS Seiberg Polchinski Seiberg Seiberg duality Power of Holomorphy Gauge symmetry is not a symmetry 4 Weyl Seiberg global dual powerful Symmetry Global Local Spt. Unbroken Spt. Broken Wigner phase Multiplet structure in the Spectrum Symmetry Relations between Amplitudes Nambu-Goldstone phase Nambu-Goldstone bosons Low energy theorems global symmetry Wigner phase Wigner-Eckert Nambu-Goldstone phase 8
9 Nambu-Goldstone boson Nambu-Goldstone boson local symmetry Symmetry Spt. Unbroken Spt. Broken Local Massless vector Massless vector Coulomb Colored states Wigner Confinement Color singlets Massless vector Higgs No charge operators (if g: Massive gauge bosons ) symmetry gauge x- independent global (color symmetry) Massless vector (gauge boson) Coulomb Confinement Coulomb U(1) Coulomb colored states Global Wigner Massless vector Confinement color singlets color Higgs Massless vector color charges g Higgs massive gauge boson Confinement Higgs 1970 Fradkin-Shenker [6] Confinement Higgs phase boundary color symmetry Confinement Higgs Massless vector Higgs 9
10 5? Is Spontaneously Broken Gauge Symmetry Meaningful? { Weak Coupling g 1 cutoff scale Λ µ massive gauge bosons M gv gauge boson massive bosons Pauli : ψσ µν ψf µν (1) ψ A µ massive on-shell coupling constant A µ massless F µν = µ A ν ν A µ 4 p µ 0 on-shell coupling massless p µ = 0 on-shell massless gauge boson gauge massive p µ = 0 on-shell on-shell 0 p 0 = M( ) 0 Pauli on-shell coupling weak coupling M gv cutoff Λ Pauli 5 1/Λ on-shell coupling coupling Weak Coupling g 1 cutoff scale Λ µ spontaneously broken gauge symmetry electro-weak gauge theory 70 cutoff sacle Λ cutoff GUT scale SU(3) SU(2) U(1) 4 10
11 massless gauge boson on-shell coupling (i.e., at p µ = 0) p µ = 0 non-zero coupling massless Weinberg[7], Kugo-Uehara[8]) p µ = 0 non-zero coupling spin j 1 massless teosorial rank j 1 spin j 1 massless couple spin 1 massless vector boson A µ scalar Q couple spin 3/2 massless vector-spinor fermion ψ µα spinor Q α couple spin 2 massless tensor boson h µν vector Q ν couple Coleman-Mandula- Haag- Lopuszanski-Sohnius S- vector P µ spin 2 massless tensor boson h µν spinor supersymmetry spin 3/2 massless vector-spinor fermion ψ µα 8 Rarita-Schwinger 6? Higgs Is Confined Gauge Symmetry Meaningful? SU(3) color N c = 3 1) Drell ratio R 2) π 0 2γ 3) Baryon qqq 3 Seiberg duality N c + Ñc = N f N c Ñc SU(N c ) with N f flavors SU(Ñc) with N f flavors duality SU(3) 3? 11
12 duality coupling coupling weak-strong duality gauge coupling N f = 3 massless quarks SU(3) gauge coupling gauge coupling 1 Λ QCD Seiberg duality asymptotically free infrared fixed point infrared fixed point conformal QCD N c = 3 MeV( 1GeV) SU(3) 1GeV SU(3) QCD 1GeV SU(3) QCD ( ) (B)(2) ( ) 12
13 A :? BRS Kugo-Ojima BRS Lautrup formalism QED M QED -Lautrup 1977 Quark Confinement QED QED free confinement Yang-Mills Faddeev-Popov ghost confinement QED Formalism Yang-Mills Yang-Mills consistent Faddeev-Popov ghost loop S Ward QED 2 Ward- FP ghost- ghost Faddeev-Popov ghost FP ghost ghost i Lagrangian 3 Ward- 13
14 4 Ward- Ward- Ward- bilinear Zinn-Justin Bonn Lecture Note[9] Ward- BRS Noether free part [1] L. O Raifeartaigh, The Dawning of Gauge Theory, Princeton Series in Physics, (Princeton Univ. Press, Princeton, 1997) [2] Y. Nambu, Quasi-Particles and Gauge Invariance in the Theory of Superconductivity, Phys. Rev. 117 (1960) 648. [3] M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Is ρ Meson a Dynamical Gauge Boson of Hidden Local Symmetry?, Phys. Rev. Lett. 54 (1985) [4] T. Kugo and I. Ojima, Manifestly Covariant Canonical Formulation of Yang-Mills Field Theories: Physical State Subsidiary Conditions and Physical S Matrix Unitarity, Phys. Lett. B 73 (1978) 459. [5] T. Kugo and I. Ojima, Local Covariant Operator Formalism of Nonabelian Gauge Theories and Quark Confinement Problem, Prog. Theor. Phys. Suppl. 66 (1979) 1. [6] E. H. Fradkin and S. H. Shenker, Phase Diagrams of Lattice Gauge Theories with Higgs Fields, Phys. Rev. D 19 (1979) [7] S. Weinberg, Phys. Lett. 9 (1964) 357; Phys. Rev. 135 (1964), B1049. [8] T. Kugo and S. Uehara, Massless Particle with Spin j 1 Implies the S-Matrix Symmetry Prog. Theor. Phys. 66 (1981) [9] J. Zinn-Justin, Renormalization of Gauge Theories, SACLAY-D.PH-T Lectures given at Int. Summer Inst. for Theoretical Physics, Jul 29 - Aug 9, 1974, Bonn, West Germany 14
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