Monopole Edward Olszewski Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics) Electrically Charged Magnetic The Dirac Monopole Monopoles, Superstring Theory, and Grand Unified Wormholes Theories Magnetic Monopoles and Dyons in Grand Unified Theories Edward Olszewski Montonen–Olive Duality Supersymmetry Superstring March 14, 2014 Theory Primer Magnetic Monopoles in String Theory References Appendix . Outline Monopole Edward Olszewski Electromagnetism Electromagnetism The Maxwell Theory The Maxwell Theory Compact Notation Compact Notation Minimal Coupling (Quantum Mechanics) Minimal Coupling (Quantum Mechanics) The Dirac Monopole The Dirac Monopole Grand Unified Grand Unified Theories Theories Magnetic Magnetic Monopoles and Dyons in Grand Unified Monopoles and Dyons in Grand Theories Unified Theories Montonen–Olive Montonen–Olive Duality Duality Supersymmetry Supersymmetry Superstring Superstring Theory Primer Theory Primer Magnetic Monopoles in Magnetic Monopoles in String Theory String Theory References References Appendix Appendix . Outline Monopole Edward Olszewski Electromagnetism Electromagnetism The Maxwell Theory The Maxwell Theory Compact Notation Compact Notation Minimal Coupling (Quantum Mechanics) Minimal Coupling (Quantum Mechanics) The Dirac Monopole The Dirac Monopole Grand Unified Grand Unified Theories Theories Magnetic Magnetic Monopoles and Dyons in Grand Unified Monopoles and Dyons in Grand Theories Unified Theories Montonen–Olive Montonen–Olive Duality Duality Supersymmetry Supersymmetry Superstring Superstring Theory Primer Theory Primer Magnetic Monopoles in Magnetic Monopoles in String Theory String Theory References References Appendix Appendix . Outline Monopole Edward Olszewski Electromagnetism Electromagnetism The Maxwell Theory The Maxwell Theory Compact Notation Compact Notation Minimal Coupling (Quantum Mechanics) Minimal Coupling (Quantum Mechanics) The Dirac Monopole The Dirac Monopole Grand Unified Grand Unified Theories Theories Magnetic Magnetic Monopoles and Dyons in Grand Unified Monopoles and Dyons in Grand Theories Unified Theories Montonen–Olive Montonen–Olive Duality Duality Supersymmetry Supersymmetry Superstring Superstring Theory Primer Theory Primer Magnetic Monopoles in Magnetic Monopoles in String Theory String Theory References References Appendix Appendix . Outline Monopole Edward Olszewski Electromagnetism Electromagnetism The Maxwell Theory The Maxwell Theory Compact Notation Compact Notation Minimal Coupling (Quantum Mechanics) Minimal Coupling (Quantum Mechanics) The Dirac Monopole The Dirac Monopole Grand Unified Grand Unified Theories Theories Magnetic Magnetic Monopoles and Dyons in Grand Unified Monopoles and Dyons in Grand Theories Unified Theories Montonen–Olive Montonen–Olive Duality Duality Supersymmetry Supersymmetry Superstring Superstring Theory Primer Theory Primer Magnetic Monopoles in Magnetic Monopoles in String Theory String Theory References References Appendix Appendix . Outline Monopole Edward Olszewski Electromagnetism Electromagnetism The Maxwell Theory The Maxwell Theory Compact Notation Compact Notation Minimal Coupling (Quantum Mechanics) Minimal Coupling (Quantum Mechanics) The Dirac Monopole The Dirac Monopole Grand Unified Grand Unified Theories Theories Magnetic Magnetic Monopoles and Dyons in Grand Unified Monopoles and Dyons in Grand Theories Unified Theories Montonen–Olive Montonen–Olive Duality Duality Supersymmetry Supersymmetry Superstring Superstring Theory Primer Theory Primer Magnetic Monopoles in Magnetic Monopoles in String Theory String Theory References References Appendix Appendix . Outline Monopole Edward Olszewski Electromagnetism Electromagnetism The Maxwell Theory The Maxwell Theory Compact Notation Compact Notation Minimal Coupling (Quantum Mechanics) Minimal Coupling (Quantum Mechanics) The Dirac Monopole The Dirac Monopole Grand Unified Grand Unified Theories Theories Magnetic Magnetic Monopoles and Dyons in Grand Unified Monopoles and Dyons in Grand Theories Unified Theories Montonen–Olive Montonen–Olive Duality Duality Supersymmetry Supersymmetry Superstring Superstring Theory Primer Theory Primer Magnetic Monopoles in Magnetic Monopoles in String Theory String Theory References References Appendix Appendix . Outline Monopole Edward Olszewski Electromagnetism Electromagnetism The Maxwell Theory The Maxwell Theory Compact Notation Compact Notation Minimal Coupling (Quantum Mechanics) Minimal Coupling (Quantum Mechanics) The Dirac Monopole The Dirac Monopole Grand Unified Grand Unified Theories Theories Magnetic Magnetic Monopoles and Dyons in Grand Unified Monopoles and Dyons in Grand Theories Unified Theories Montonen–Olive Montonen–Olive Duality Duality Supersymmetry Supersymmetry Superstring Superstring Theory Primer Theory Primer Magnetic Monopoles in Magnetic Monopoles in String Theory String Theory References References Appendix Appendix . Outline Monopole Edward Olszewski Electromagnetism Electromagnetism The Maxwell Theory The Maxwell Theory Compact Notation Compact Notation Minimal Coupling (Quantum Mechanics) Minimal Coupling (Quantum Mechanics) The Dirac Monopole The Dirac Monopole Grand Unified Grand Unified Theories Theories Magnetic Magnetic Monopoles and Dyons in Grand Unified Monopoles and Dyons in Grand Theories Unified Theories Montonen–Olive Montonen–Olive Duality Duality Supersymmetry Supersymmetry Superstring Superstring Theory Primer Theory Primer Magnetic Monopoles in Magnetic Monopoles in String Theory String Theory References References Appendix Appendix . Outline Monopole Edward Olszewski Electromagnetism Electromagnetism The Maxwell Theory The Maxwell Theory Compact Notation Compact Notation Minimal Coupling (Quantum Mechanics) Minimal Coupling (Quantum Mechanics) The Dirac Monopole The Dirac Monopole Grand Unified Grand Unified Theories Theories Magnetic Magnetic Monopoles and Dyons in Grand Unified Monopoles and Dyons in Grand Theories Unified Theories Montonen–Olive Montonen–Olive Duality Duality Supersymmetry Supersymmetry Superstring Superstring Theory Primer Theory Primer Magnetic Monopoles in Magnetic Monopoles in String Theory String Theory References References Appendix Appendix . Outline Monopole Edward Olszewski Electromagnetism Electromagnetism The Maxwell Theory The Maxwell Theory Compact Notation Compact Notation Minimal Coupling (Quantum Mechanics) Minimal Coupling (Quantum Mechanics) The Dirac Monopole The Dirac Monopole Grand Unified Grand Unified Theories Theories Magnetic Magnetic Monopoles and Dyons in Grand Unified Monopoles and Dyons in Grand Theories Unified Theories Montonen–Olive Montonen–Olive Duality Duality Supersymmetry Supersymmetry Superstring Superstring Theory Primer Theory Primer Magnetic Monopoles in Magnetic Monopoles in String Theory String Theory References References Appendix Appendix . Monopole The Maxwell Equations Edward Olszewski Electromagnetism The Maxwell Theory Notation: (µ = 0; 1; 2; 3; i = 1; 2; 3), (Minkowski metric Compact Notation Minimal Coupling (Quantum Mechanics) with η00 = −1), (Levi-Cevità symbol: ϵ0123 = ϵ123 = 1), The Dirac (Lorentz-Heaviside units) Monopole Grand Unified Theories @E Magnetic r · E = ρe r × B − = Je Monopoles and @t Dyons in Grand @B Unified Theories r · B = 0 r × E + = 0 Montonen–Olive @t Duality Supersymmetry Maxwell - magnetic Potential Superstring Theory Primer Magnetic The Lorentz Force Equation Monopoles in String Theory F = qeE + qev × B References Appendix . Monopole Edward Olszewski The Maxwell Equations (magnetic) Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum E ! B B ! −E Mechanics) ! ! − The Dirac Je Jm Jm Je Monopole ρe ! ρm ρm ! −ρe Grand Unified Theories Magnetic Monopoles and @ Dyons in Grand B Unified Theories r · B = ρm −∇ × E − = Jm @t Montonen–Olive @E Duality −∇ · E = 0 r × B − = 0 @t Supersymmetry Superstring Theory Primer Maxwell - electric Magnetic Monopoles in The Lorentz Force Equation String Theory References F = qmB − qmv × E Appendix . Monopole The Potential Function Edward Olszewski Electromagnetism Consequently, The Maxwell Theory Compact Notation Minimal Coupling (Quantum r × A = B since r · (r × A) = 0 Mechanics) The Dirac −∇ 0 @A r × r 0 Monopole A = E + since A = 0 Grand Unified @t Theories or Magnetic Monopoles and 0 @A Dyons in Grand −∇A − = E : Unified Theories @t Montonen–Olive Duality Maxwell - electric Supersymmetry Superstring Gauge Transformation Theory Primer Magnetic E and B are invariant under Monopoles in String Theory @χ References A0 ! A0 + A ! A − rχ @t Appendix . Monopole Edward Olszewski µ i Gauge Potential: A = Aµdx = A0dt + Ai dx Electromagnetism The Maxwell Theory ! − µ Compact Notation Gauge Transform: A A dχ(x ) Minimal Coupling (Quantum − Mechanics) Field Strength: F= dA The Dirac Monopole 1 µ ν (gauge invarient = (@µAν − @νAµ)dx ^ dx Grand Unified 2 Theories 1 µ ν Magnetic due to ddχ = 0) = Fµνdx ^ dx Monopoles and 2 Dyons in Grand Unified Theories k 0 i 1 k i j E and B F =E δki dx ^ dx + B ϵkij dx ^ dx Montonen–Olive 2 Duality ∗ ≡ αβ µ ^ ν Supersymmetry F Fαβ ϵµν dx dx Superstring k 0 i 1 k i j Theory Primer =B δki dx ^ dx − E ϵkij dx ^ dx 2 Magnetic ∗ Monopoles in duality : F ! F; (Ei ! Bi ; Bi ! −Ei ) String Theory References ∗∗F = −F Appendix . Monopole Euler equation (inhomogeneous): δF = −∗d ∗F = j Edward Olszewski Electromagnetism The Maxwell Theory Compact Notation − r · Minimal Coupling (Quantum δF = E dt + Mechanics) − r × · The Dirac + ( @t E + B) dx Monopole µ je =jeµ dx = −ρedt + Je · dx Grand Unified Theories Magnetic Monopoles