Noethers Theorem

1
Noethers Theorem
If there exists r infinitesimal transformations
of the following form
and
Then
Are r conserved currents --
Conservation of energy and momentum
Let each er correspond to a displacement in the r
direction
2
Gauge Invariance of the First Kind
Global Gauge Invariance
Complex scalar Lagrangian
Transformation
Preserves Lagrangain
3
Gauge Invariance of Second Kind
Local Gauge Invariance
Now allow gauge transformation to be position
dependent
To get gauge invariant Lagrangian, must add
gauge field to cancel second term
Gauge invariant Lagrangian
4
Interpretation
Free Field term
Kinetic energy of EM field
EM field current interaction
New type term four point function coming from
gauge invariance
5
SU(n) Gauge Invariance
EM
Single Phase
U(1) GroupN
Now let
an N-vector of states
Require invariance under local SU(n)
transformations
Need an extra gauge interaction to cancel out
additional term
6
SU(n) Field Strength Tensor
We want the field strength tensor to be invariant
under gauge transformation
After a long proof we find that
Is gauge invariant