1. Construction of a relativistic field theory
Lagrangian
(Nonrelativistic mechanics)
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. } } Lagrangian formulation of the Klein Gordon equation
Klein Gordon field
Manifestly Lorentz
invariant } } T V
Classical path :
Euler Lagrange equation
Klein Gordon equation

3. 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?

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

5. 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

6. 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

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

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

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. is invariant under local U(1)
Yang-Mills (+Shaw)
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:

12. is invariant under local U(1)
Yang-Mills (+Shaw)
is invariant under local U(1)
Note :
is equivalent to
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
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

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