Go Back Full Screen Quit 2. Embedding Geometry.pdf

Geometry vs. Gauge Theory Embedding Geometry Vector Bundle Gauge Theory Charge Uniqueness Multiple Charges Conclusions J I Page 1 of 10 Go Back Full Screen Quit Riemannian Gauge Theory and Charge Uniqueness Mario Serna Space Vehicles Directorate, Air Force Research Labs 3550 Aberdeen Ave, Kirtland, New Mexico 87117-5776, USA E-mail: mariojr@alum.mit.edu Kevin Cahill Department of Physics and Astronomy, University of New Mexico Albuquerque, New Mexico 87131-1156, USA E-mail: cahill@unm.edu May 26, 2002 Geometry vs. Gauge Theory Embedding Geometry Vector Bundle Gauge Theory Charge Uniqueness Multiple Charges Conclusions J I Page 2 of 10 Go Back Full Screen Quit 1. Geometry vs. Gauge Theory ? Defining a gauge theory: A group g(x) ? Matter fields: φ′(x) = g(x)φ(x). ? Gauge fields: A′μ = g Aμ g?1 + (i/q) g ?μ g?1 ? Defined to cancel extra terms in ?μ(g φ) ? Action: Lorentz gauge invariant ? Similarities to General Relativity Object General Relativity Gauge Theory Covariant ?μV α = (δασ?μ + Γαμσ)V σ Dμφa = (δab?μ ? iqAμab)φb Derivative Connection Γαμσ ?iqAμab Coefficient Curvature R λμν σV σ = ([?μ,?ν]V )λ ?i Fμνabφb = ([Dμ, Dν]φ)a Tensor Geometry vs. Gauge Theory Embedding Geometry Vector Bundle Gauge Theory Charge Uniqueness Multiple Charges Conclusions J I Page 3 of 10 Go Back Full Screen Quit 2. Embedding Geometry ? Embedding is always possible (Nash (1956) [1]) Vμ(x)tμ(x)=V(x) P(x+δ)V(x) tμtμ ? Dual tν(tμ) = δμν , Projection P = tνtν( ) ?Metric gμν = 〈tμ, tν〉 ? The Covariant Derivative ?μV = P (?μ(V σtσ)) = tνt ν [tσ?μV σ + V σ?μtσ] = tν [t ν(tσ)?μV σ + V σtν(?μtσ)] = tν [δ ν σ?μV σ + V σtν(?μtσ)] Γνμσ = t ν(?μtσ) Geometry vs. Gauge Theory Embedding Geometry Vector Bundle Gauge Theory Charge Uniqueness Multiple Charges Conclusions J I Page 4 of 10 Go Back Full Screen Quit 3. Vector Bundle Gauge Theory ? Based on: Narasimhan and Ramanan (1961) [2, 3], Atiyah (1979) [4], Dubois-Violette and Georgelin (1979) [5], Cahill and Raghavan (1993) [6] -100 -50 0 50 x nm -100 -