Introduction to Mathematical Physics/Groups

Definition
In classical mechanics,\index{group} translation and rotation invariances correspond to momentum and kinetic moment conservation. Noether theorem allows to bind symmetries of Lagrangian and conservation laws. The underlying mathematical theory to the intuitive notion of symmetry is presented in this appendix.

Representation
For a deeper study of group representation theory, the reader is invited to refer to the abundant literature (see for instance ([#References|references])).

Consider a symmetry group $$G$$. let us consider some classical examples of vectorial spaces $$V$$. Let $${\mathcal{R}}$$ be an element of $$G$$.

Consider the following theorem:

This previous theorem allows to predict the eigenvectors and their degeneracy.

Relatively to the $$R^3$$ rotation group, scalar, vectorial and tensorial operators can be defined.

An example of scalar operator is the hamiltonian operator in quantum mechanics.

More generally, tensorial operators can be defined:

Another equivalent definition is presented in ([#References|references]). It can be shown that a vectorial operator is a tensorial operator with $$j=1$$. This interest of the group theory for the physicist is that it provides irreducible representations of symmetry group encountered in Nature. Their number is limited. It can be shown for instance that there are only 32 symmetry point groups allowed in crystallography. There exists also methods to expand into irreducible representations a reducible representation (see ([#References|references])).

Tensors and symmetries
Let $$a_{ijk}$$ be a third order tensor. Consider the tensor:

let us form the density:

$$\phi$$ is conserved by change of basis\footnote{ A unitary operator preserves the scalar product.} If by symmetry:

then

With other words ``X is transformed like $$a$$'' ([#References|references])