# Index Concordia

## Quantum field theory – Week 03

Posted in Quantum Field Theory by Index Guy on September 20, 2007

During this week we derived Noether’s theorem from requiring form-invariance of the Lagrange density under a general transformation of the coordinates and the fields. This transformation was written in terms of $N$ parameters. Then Noether’s theorem claims the existence of conserved vectors.

Then we considered an ordered pair of scalar fields. This lead to internal indices and global transformations. When one has $N$ such fields, the question is: how many and which conserved vectors to use? For this we introduced some concepts of group theory.

We defined a group as a set of abstract elements that satisfy a set of rules (existence of identity and inverse, associativity, closure). Then we introduced representations as objects that follow the same rules as the group elements, but are not that abstract… Most of our representations are going to be in terms of $N \times N$ matrices. For square matrices $M, A$ we have

If $M = \exp{A}$ then

$\det{M} = \exp{tr(A)}$

and

$M = \lim_{N \rightarrow \infty} \left(1 +\displaystyle\frac{A}{N}\right)^{N}.$

The first property allows us to calculate the determinant of a matrix by just considering the diagonal terms of some other matrix. The second property allows to construct finite transformations from infinitesimal ones.

A Lie group is defined as a group with elements labeled with a continuous parameter.

Finally we considered a representation of an infinitesimal element, and expanded it around the identity. The result is an expression for the generators of the group.