Construction of a relativistic field theory

1. Construction of a relativistic field theory (Nonrelativistic mechanics) Lagrangian Action Classical path … minimises action Quantum mechanics … sum over all paths with amplitude Lagrangian invariant under all the symmetries of nature -makes it easy to construct viable theories

2. Klein Gordon equation Lagrangian formulation of the Klein Gordon equation Klein Gordon field Manifestly Lorentz invariant } } T V Classical path : Euler Lagrange equation

3. New symmetries …an Abelian (U(1)) gauge symmetry Is invariant under A symmetry implies a conserved current and charge. e.g. Translation Momentum conservation Angular momentum conservation Rotation What conservation law does the U(1) invariance imply?

4. Noether current …an Abelian (U(1)) gauge symmetry Is invariant under 0 (Euler lagrange eqs.) Noether current

5. The Klein Gordon current …an Abelian (U(1)) gauge symmetry Is invariant under This is of the form of the electromagnetic current we used for the KG field

6. The Klein Gordon current …an Abelian (U(1)) gauge symmetry Is invariant under This is of the form of the electromagnetic current we used for the KG field is the associated conserved charge

7. Suppose we have two fields with different U(1) charges : ..no cross terms possible (corresponding to charge conservation)

8. } Renormalisable Additional terms Terms allowed by U(1) symmetry

9. U(1) local gauge invariance and QED not invariant due to derivatives To obtain invariant Lagrangian look for a modified derivative transforming covariantly

10. U(1) local gauge invariance and QED not invariant due to derivatives To obtain invariant Lagrangian look for a modified derivative transforming covariantly Need to introduce a new vector field

11. Yang-Mills (+Shaw) is invariant under local U(1) is equivalent to Note : universal coupling of electromagnetism follows from local gauge invariance The Euler lagrange equation give the KG equation:

12. Yang-Mills (+Shaw) is invariant under local U(1) is equivalent to Note : universal coupling of electromagnetism follows from local gauge invariance

13. The electromagnetic Lagrangian Forbidden by gauge invariance The Euler-Lagrange equations give Maxwell equations ! EM dynamics follows from a local gauge symmetry!!

14. The photon propagator The propagators determined by terms quadratic in the fields, using the Euler Lagrange equations.

15. The Klein Gordon propagator (reminder) In momentum space: With normalisation convention used in Feynman rules = inverse of momentum space operator multiplied by -i

16. The photon propagator The propagators determined by terms quadratic in the fields, using the Euler Lagrange equations. Gauge ambiguity i.e. with suitable “gauge” choice of α (“ξ” gauge) want to solve In momentum space the photon propagator is (‘t Hooft Feynman gauge ξ=1)

18. Extension to non-Abelian symmetry where

19. Extension to non-Abelian symmetry where

20. Symmetry : Symmetry : Local conservation of 3 strong colour charges QCD : a non-Abelian (SU(3)) local gauge field theory

21. The strong interactions QCDQuantum Chromodynamics Symmetry : Symmetry : Local conservation of 3 strong colour charges SU(3) Strong coupling, α3 q  Ga=1..8  Gauge boson (J=1) “Gluons” QCD : a non-Abelian (SU(3)) local gauge field theory q

22. Partial Unification } Matter Sector “chiral” Family Symmetry? } } Up Down Family Symmetry? Neutral