This week was slow in quantum field theoretical aspects. At least to my taste.
We finished the discussion on Lie groups and algebras. Then we considered the Lorentz group and an infinitesimal transformation. From this we derived expressions for the generators of the group, and their algebra. Next week, Poincaré and hopefully canonical quantization.
During this week we study the Dirac group and its representations. Recall that the order of the Dirac group is
with being the number of spacetime dimensions. We will consider separate cases when is either even or odd. The number of classes depends on :
- for even, and
- for odd, .
We started the week by defining what a semi-direct product is:
- A group is the semi-direct product of two subgroups if one of those subgroups is normal and only shares the identity with the other.
An example is the Poincaré group, which is a semi-direct product of translations (the invariant subgroup) and Lorentz transformations (not an invariant subgroup of Poincaré).
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 parameters. Then Noether’s theorem claims the existence of conserved vectors.