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

2. New symmetries

3. New symmetries
Is invariant under
…an Abelian (U(1)) gauge symmetry

4. New symmetries
Is not invariant under

5. New symmetries
Is invariant under
…an Abelian (U(1)) gauge symmetry

6. New symmetries
Is invariant under
…an Abelian (U(1)) gauge symmetry

7. New symmetries Is invariant under …an Abelian (U(1)) gauge symmetry
A symmetry implies a conserved current and charge.
e.g.
Translation
Momentum conservation
Rotation
Angular momentum conservation

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

9. Noether current
Is invariant under
…an Abelian (U(1)) gauge symmetry

10. Noether current
Is invariant under
…an Abelian (U(1)) gauge symmetry

11. Noether current
Is invariant under
…an Abelian (U(1)) gauge symmetry

12. Noether current
Is invariant under
…an Abelian (U(1)) gauge symmetry

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

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

15. The Klein Gordon current
Is invariant under
…an Abelian (U(1)) gauge symmetry

16. The Klein Gordon current
Is invariant under
…an Abelian (U(1)) gauge symmetry

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

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

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

20. Additional terms

21. Additional terms }
Renormalisable

22. Additional terms }
Renormalisable

23. U(1) local gauge invariance and QED

24. U(1) local gauge invariance and QED

25. U(1) local gauge invariance and QED
not invariant due to derivatives

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

27. 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

28. is invariant under local U(1)

29. is invariant under local U(1)
Note :
is equivalent to
universal coupling of electromagnetism follows from local gauge invariance

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

31. is invariant under local U(1)
Note :
is equivalent to
universal coupling of electromagnetism follows from local gauge invariance

32. The electromagnetic Lagrangian

33. The electromagnetic Lagrangian

34. The electromagnetic Lagrangian

35. The electromagnetic Lagrangian

36. The electromagnetic Lagrangian
Forbidden by gauge invariance

37. The electromagnetic Lagrangian
Forbidden by gauge invariance
The Euler-Lagrange equations give Maxwell equations !

38. The electromagnetic Lagrangian
Forbidden by gauge invariance
The Euler-Lagrange equations give Maxwell equations !

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

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

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

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

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

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

45. 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

46. 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)

48. Extension to non-Abelian symmetry

49. Extension to non-Abelian symmetry

50. Extension to non-Abelian symmetry
where

51. Extension to non-Abelian symmetry
where
i.e. Need 3 gauge bosons

52. Weak Interactions Symmetry : Local conservation of
SU(2) local gauge theory
Symmetry :
Local conservation of
2 weak isospin charges
Weak coupling, α2 u d
Gauge boson
(J=1)
Wa=1..3
A non-Abelian (SU(2))
local gauge field theory e
Neutral currents

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

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

55. Partial Unification Matter Sector “chiral” Family Symmetry? NeutralUp
Down
Family Symmetry?
Neutral

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

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