1. (also xyzyzxzxy)
both can be re-written with
(with the same for xyz)
All 4 statements can be summarized in

2. F = g g F = The remaining 2 Maxwell Equations:
are summarized by
ijk = xyz, xz0, z0x, 0xy
Where here I have used the “covariant form”
F = g g F =

3. L=[iħcgmmc2 ]- F F-(qg m)Am
To include the energy of em-fields
(carried by the virtual photons)
in our Lagrangian we write:
L=[iħcgmmc2 ]- F F-(qg m)Am 1 2
But need to check: is this still invariant under the SU(1) transformation?
  (A+) (A +)
= A+   A  
= 

4. “The Fundamental Particles and Their Interactions”, Rolnick (Addison-Wesley, 1994)2
Heaviside
-Lorentz
units
“Introduction to Elementary Particles”, Griffiths (John Wiley & Sons, 1987) L 1
16
Gaussian
cgs units

5. give two independent equations ORL L L L
The
prescriptions
and
give two independent equations OR
summing over ALL variables (fields) gives the full equation WITH interactions
Starting from L
(and summing over ,  )
with L
FmnFmn
Let’s look at
the new term:

6. [-( A-A)][-( A-A)]
summing over , 
FF
survive when =, = and when =, = 
,  now fixed, not summed
[-( A-A)][-( A-A)]
A = ggA
= ggA 1
sum over
(but non-zero only
when =, = )
FmnFmn
where  since this
tensor is anti-symmetric!

7. L L L = 0 FmnFmn FmnFmn So with and next we get Note: 0 in the
Lorentz Gauge
The Klein Gordon equation for massless photons!

8.  / = e+iq/ħc +iq  / = e+iJ · /ħ U(1) : SO(3) :
q = -e for electrons
q = +e for positrons
q = +2e/3 for up quarks
+iq
The ± sign is just a convention, as in rotations:
 / = e+iJ · /ħ
SO(3) :

9.  (D )/ = e-iq/ħcD   / =  [e+iq/ħc] = e+iq/ħc D= + i A
We can generalize our procedures into a PRESCRIPTION to be followed,
noting the difference between LOCAL and GLOBAL transformations
are due to derivatives:
  / =  [e+iq/ħc] = e+iq/ħc 
for U(1) this is a
1×1 unitary matrix
(just a number)
the extra term
that gets introduced
If we replace every derivative  in the original free particle Lagrangian
with the “co-variant derivative”
D= + i A g ħc
then the gauge transformation of A will
cancel the term that appears through 
(D )/ = e-iq/ħcD
i.e.
restores the invariance of L

10. The free particle Dirac Lagrangian can be made U(1) invariant only by
introducing a charged current
introducing a VECTOR FIELD particle
which couples to that charge
The conserved quantity “discovered”
was ELECTRIC CHARGE.
The particle coupling to CHARGE
was interpreted as the PHOTON.
CHARGE is the source of the PHOTON FIELDS through
which Dirac particles interact. This is believed to be the
underlying principal of the fundamental electromagnetic
force: VECTOR PARTICLES mediate interaction forces.

11. SU(2) Iso-spin multiplets
Are there HIGHER symmetries?
SU(2) spin-multiplets
just one of many ANGULAR
MOMENTUM representations
Dirac matrices and Dirac spinors
already keep this space separate.
SU(2) Iso-spin multiplets
already expanded into SU(3) and higher as
we generalized isospin to include concepts
like hypercharge, strangeness, and charm.
Are these all some kind of charge?

12. Is UP or DOWN some kind of CHARGE
that generates fields?
that couples to a force carrying vector particle?
Is there some kind of fundamental ISOSPIN FORCE?
The theorists Yang & Mills extended the U(1) formalism
that explained e&m forces in an effort to explore ISOSPIN.

13. L L L 1 satisfying one, 2 satisfying the other.
Imagine 2 possible states: the flip sides of some spin-½ (2-component) property
Spin, ISO-spin, or even something NEW L
Sum of 2 Dirac lagrangians
Applying the Euler-Lagrange equations results in 2 independent Dirac equations
1 satisfying one, 2 satisfying the other.
Written more compactly as spinors
and its adjoint
Note: 1 and 2 each already 4-component Dirac spinors L
“mass
matrix”
where
If m1=m2 then M=mI and L
which “looks” just like the
1-particle Dirac Lagrangian

14. But NOW  represents a 2-element column vector and we can explore
an additional invariance under 
The most general SU(2) matrix is of course U = ei( /2)· 
where   Pauli matrices 
Following the success of U(1) Yang-Mills promoted the obvious
global phase transformation to a LOCAL invariance, writing g ħc
U = ei ( /2)·   
where   (x)  q ħc
U = ei (x)
compare to the U(1) transformation:

15. = (U) + U() Like before, the Dirac Lagrangian (as it stands)
is NOT invariant under this transformation
= (U) + U()
The fix again is to replace  by a “covariant derivative” 3
vector fields
are needed to span the space
of this transformation operator

16. so that the (D)' = D term remains invariant
Then,
assuming an appropriate GAUGE transformation of the G fields is possible:
so that the (D)' = D
term remains invariant
To figure out the necessary transformation property of the Gauge fields
we’ll use the fact that
then

17.  '
D'
in other words the transformed
which means in particular U
U †

18. U U † U U†Gi i 2 which would look similar to the
if could pull through U or U† this would just be
which would look similar to the
gauge transformations under U(1)
Why can’t we?
Let’s focus on this term: OK
to commute!
Not OK!
which we can just write as i 2
U U†Gi

19. = RT(=g/2) U U† i i 2 2 recalling that
You will show for homework that i 2 i 2
= RT(=g/2)
U U†
a 3-dim space(-like) rotation
applied to the i/2 matrices So
i,j count over the iso-space generators
(Pauli matrices 1,2,3)
 counts over the spacetime coordinates
(ct, x, y, z)

20. U U † RG- Since Now remember the i are linearly independent
matching like terms we find: 
RG- 
fields are
“rotated”
…and shifted by a gradient
(a gauge shift)

21. L=iħcgmDmc2 =iħcgmmc2 - (gg m )·Gm
The resulting Lagrangian (so far)
L=iħcgmDmc2  
=iħcgmmc2 - (gg m )·Gm  2
3 separate
4-vector
fields
(like A)
where we’ve introduced 3 new vector fields 
G = (G1, G2, G3 )
each with its own free Lagrangian (kinetic energy) term  
= - F ·F 1 4 ?
but not quite the same as before
since THIS is not an invariant

22. =  (RijG j- i ) -  (RijG j- i )
F i' = Gi'-Gi'
=  (RijG j- i ) -  (RijG j- i ) 
= ( Rij )G j +Rij (G j) -  i
- ( Rij )G j - Rij(G j) +   i
RR((x)) for a local transformation
= Rij (G j - G j ) + ( Rij -  Rij )G j
F i 

23. F i = Gi-Gi - G×G
Actually with 3 vector fields
there IS another anti-symmetric term possible  
G×G
and, with it, the more general   2g ħc
F i = Gi-Gi G×G 
restores invariance!

24. L=[iħcgmmc2 ]- F F-(gg m )·Gm
So NOW for our newly proposed SU(2) theory we have    
L=[iħcgmmc2 ]- F F-(gg m )·Gm 1 4  2
describing two equal mass Dirac particle states in interaction with
3 massless vector fields Gm
Think of something like the -fields, A  
This followed just by insisting on local SU(2) invariance!
In the Quantum Mechanical view:  
Dirac particles generate 3 currents, J = (g   )  2
These particles carry a “charge” g, which
is the source for the 3 “gauge” fields

25. The full product has nothing smaller than quadratic G terms
Furthermore with:
linear
linear
quadratic 
The full product has nothing smaller than quadratic G terms
(KE terms of free particles)
plus cubic and quartic terms
(interaction terms describing
VERTICES of gauge particles
with themselves!!)

26. plus “self-interaction” terms:
field-current
interaction
3 like this:
one for each Gi
plus “self-interaction” terms:
These gauge particles (“force carriers”) are NOT neutral!
(like s are with respect to electric charge)

27. In general NON-ABELAIN GAUGE THEORIES:
introduce more interactions (vertices)
for SU(2) we saw both 3 and 4 particle interaction vertices
have (still) massless gauge particles (like the photon!)
the gauge field particles posses “charge” just like the
fundamental Dirac states
not electric charge - we’re trying to think of NEW forces

28. i.e., the strong force does not couple to flavors.
The YANG-MILLS was built on the premise that
there existed
2 elementary Dirac (spin-½) particles of ~equal mass
serving as sources for the force fields through which they interact
NO SUCH PAIRS EXIST
proton/neutron isospin states were the inspiration, but
there is NO massless vector (spin 1) iso-triplet
(isospin 1) of known particle states
r-mesons? 770 MeV/c2
p,n,r now recognized as COMPOSITE particles
isospin of up,dn quarks generalized into SU(3)  SU(4)
The strong force must be
independent of FLAVOR
up charm top
down strange bottom
i.e., the strong force does not couple to flavors.
SO WHERE DOES THE STRONG FORCE COME FROM?

29. We WILL find these ideas resurrected in:
SU(3) color symmerty of strong interactions
SU(2) electro-weak symmetry