Local Gauge Invariance andExistence 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

ypy+zpz]

The angular momentum operator, generates rotations in x,y,z space!

angular momentum operator

One can generate the “rotation” of a spinor (like the u derived for the electron) using the “spin” operators:

This approach is used in the Standard Model to “rotate” a particle which has an “up” and a “down” kind of property -- like flavor!

This takes a little work -- must expand

e a = [a]n /n! and use z

2 = 1 , z3 = z …. more later!

Gauge transformations are like the “rotations” we have just been considering

Real function of space and time

can be an operator -- as we have just seen.

one has to find a Lagrangian which is invariant under this transformation.

How are Lagrangians made invariant under these “rotations”?

It won’t work!

Constructing a gauge invariant Lagrangian:

1. Begin with the “old Lagrangian”:

2. Replace

3.

“old” Lagrangian the interaction term.

called the “covariant derivative”

Aµ is the gauge boson (exchange particle) field!

Showing L is invariant

transformed L

A µ = Aµ - (1/e)

transformed

transformed AMaxwell’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

This use of the covariant derivative will be applied toall the interaction terms of the Standard Model.

If the coupling, e, is turned off, L reverts to the free electron L.

’

AµAµ

’

invariance

Note that the photon field must also be transformed.

1. Initial state 2. Rotate

3. Transform A 4. Final state

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 transformation.

Comments:

4. Note that a global gauge transformation would require that is a constant!

simple result!

L transformed