Local Gauge Invariance and
Existence of the Gauge Particles
1. Gauge transformations are like “rotations”
2. How do functions transform under “rotations”?
3. How can we generalize to rotations in “strange” spaces
(spin space, , flavor space, color space)?
4. How are Lagrangians made invariant under these “rotations”?
(Lagrangians  “laws of physics” for particles interactions.)
5. Invariance of L requires the existence of the gauge boson!
momentum operator
x component
momentum operator
angular momentum operator
The angular momentum operator, generates rotations in x,y,z space!
One can generate the “rotation” of a spinor (like the u
derived for the electron) using the  “spin” operators:
This takes a little work -- must expand
and use z2
[a]n /n!
= 1 , z3 = z …. more later!
e a
This approach is used in the Standard Model to “rotate” a particle which has
an “up” and a “down” kind of property -- like flavor!
Gauge transformations are like the
“rotations” we have just been considering
Real function of space and time
one has to find a Lagrangian which is invariant under this transformation.
 can be an operator
-- as we have just seen.
How are Lagrangians made invariant
under these “rotations”?
It won’t work!
Constructing a gauge invariant Lagrangian:
1. Begin with the “old Lagrangian”:
called the “covariant derivative”
2. Replace
Aµ is the gauge boson
(exchange particle) field!
“old” Lagrangian
the interaction term.
Showing L is invariant
transformed L
transformed 
A µ = Aµ - (1/e) 
transformed A
Maxwell’s equations
are invariant under this!
First a simplifying expression:
Use this simple result in L’
Summary of Local gauge symmetry
Real function of space and time
covariant derivative
The final invariant L is given by:
The correct, invariant Lagrangian density, includes the interaction
between the electron (fermion) and the photon (the gauge particle).
free electron Lagrangian
interaction Lagrangian
If the coupling, e, is turned off, L reverts to the free electron L.
This use of the covariant derivative will be applied to
all the interaction terms of the Standard Model.
1. Initial state
2. Rotate 
3. Transform A
4. Final state
Note that the photon field
must also be transformed.
1. There is no difference between changing the phase
of the field operator of the fermion (by (r,t) at
every point in space) and the effects of a gauge
transformation [ -(1/e)µ (r,t) ] on the photon field!
2. Maxwell’s equations are invariant under
A µ A µ - (1/e)µ (r,t) -- and, in particular, the gauge
transformation has no effect on the free photon.
3. It is only because (r,t) depends on r and t that
the above is possible. This is called a local gauge
4. Note that a global gauge transformation would
require that  is a constant!
L transformed
simple result!