1. U(1) Abelian Symmetry
The Lagrangian is invariant under the phase transformation of the field operator:
invariant

2. If A,B,C become complex, they carry charges!
The interaction is invariant only if
U(1) symmetry is related to charge conservation!

3. The Dirac Fermion Lagrangian is also invariant under U(1)

4. SU(N) Non-Abelian Symmetry
Assume there are N kinds of fields
If they are similar, we have a SU(N) symmetry!
are invariant under SU(N)!

5. 量子力學下互換群卻變得更大！
量子力學容許量子態的疊加 u a u
+ b d u d d c u
+ d d 古典 量子
u-d 互換對稱

6. They are invariant under SU(N)!

7. Gauge symmetry

8. Global Symmetry
Gauge (Local) symmetry
Kinetic energy is not invariant under gauge transformation!

9. Could we find a new “derivative” that works as if the transformation is global?
To get rid of the extra term, we introduce a new vector field:

10. Global Symmetry
Gauge (Local) symmetry
Replacing derivative with covariant derivative,
is invariant under gauge transformation!

11. The scalar photon interaction vertices

12. To force it to be gauge invariant,
you only need to replace derivative with coariant derivative.
is gauge invariant!

13. This gauge invariant Lagrangian gives a definite interaction between fermions and photons

15. This form is forced upon us by gauge symmetry!
It is really a Fearful Symmetry! Tony Zee
Tyger! Tyger! burning bright
In the forests of the night
What immortal hand or eye
Could frame thy fearful symmetry!
William Blake

16. Let there be light!
In the name of gauge symmetry!

17. Hermann Weyl,

18. Yang and Mills

19. SU(N) Non-Abelian Symmetry
Assume there are N kinds of fields
If they are similar, we have a SU(N) symmetry!
are invariant under SU(N)!

20. Non-Abelian Gauge Symmetry
We need one gauge field for each generator.
Gauge fields transform as:
is invariant under gauge transformation!

22. 2 × 2 matrices

25. Vacua happen at:
Choose:
For infinitesimal transformation:
SU(2)χU(1)Y is broken into U(1)EM

26. Z become massive
W become massive
Photon is massless.