Introduction to Mathematical Physics/Groups

DefinitionEdit

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.

Definition:

A group is a set of elements   and a composition law   that assigns to any ordered pair   an element   of  . The composition law   is

  • associative

 

  • has a unit element   :

 

  • for each   of  , there exists an element   of   such that:

 

Definition:

The order of a group is the number of elements of  .

RepresentationEdit

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

Definition:

A representation of a group   in a vectorial space   on   or   is an endomorphism   from   into the group  \footnote{  is the space of the linear applications from   into  . It is a group with respect to the function composition law.} {\it i.e }a mapping

 

with

 

Definition:

Let   be a representation of  . A vectorial subspace   of   is called stable by   if:

 

One then obtains a representation of   in   called subrepresentation of  .

Definition:

A representation   of a group   is called irreducible if it admits no subrepresentation other than   and itself.

Consider a symmetry group  . let us consider some classical examples of vectorial spaces  . Let   be an element of  .

Example:

exampgroupR

Let   be a group of transformations   of space  . A representation   of   in   is simply defined by:

 

To each element   of  , a mapping   of   is associated. This mapping can be defined by a matrix   called representation matrix of symmetry operator  .

Example:

Let us consider molecule   as a solid of symmetry  . The various symmetry operations that characterizes this group are:

  • three reflexions  , , and  .
  • two rotations around the   axis, of angle   and   noted   and  .
  • rotation of angle   is the identity and is noted  .

In a any basis of  , representation matrices of group symmetry operators are in general not block diagonal.

The tridimensional space can be shared into two invariant subspaces: a one-dimensional space   spanned by vector of the   axis, and a plane   perpendicular to this vector. In chemistry books, representation on   is called   and representation on   is called  . They are both irreducible.

Remark:

For the study of the vibrations of a molecule, instead of considering the Euclidian space   as state space, the space of the  's where the  's are the degree of freedom of the system has to be considered. The diagonalization of the coupling matrix problem can be tackled using symmetry considerations. Indeed, a vibratory system is invariant by symmetry  , implies that its energy is invariant by  :

 

Matrices   being orthonormal, kinetic energy is also invariant.

Consider the following theorem:

Theorem:

theosymde

If operator   is invariant by  , that is   or  , the if   is eigenvector of  ,   is also eigenvector of  .

Proof:

It is sufficient to evaluate the action of   on   to prove this theorem.

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

Example:

Consider the group   introduced at example exampgroupR. A representation   of   in the space   of summable squared can be defined by:

 

where   is the matrix representation of transformation  , element of  . If   is a basis of  , then we have  .

Example:

Consider the group   introduced at example exampgroupR. A representation   of   in the space of linear operators of   can be defined by:

 

where   is the matrix representation, defined at prevoius example, of transformation  , element of   and   is the matrix defining the operator

Relatively to the   rotation group, scalar, vectorial and tensorial operators can be defined.

Definition:

A scalar operator   is invariant by rotation:

 

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

Definition:

A vectorial operator   is a set of three operators  ,  , (the components of   in spherical coordinates) that verify the commutation relations:

 

More generally, tensorial operators can be defined:

Definition:

A tensorial operator   of components   is an operator whose transformation by rotation is given by:

 

where   is the restriction of rotation   to space spanned by vectors  .

 

Another equivalent definition is presented in ([#References|references]). It can be shown that a vectorial operator is a tensorial operator with  . 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 symmetriesEdit

Let   be a third order tensor. Consider the tensor:

 

let us form the density:

 

  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   ([#References|references])

Example: piezoelectricity. As seen at section secpiezo, there exists for piezoelectric materials a relation between the deformation tensor   and the electric field  

 

  is called the piezoelectric tensor. Let us show how previous considerations allow to obtain following result:

Theorem:

If a crystal has a centre of symmetry, then it can not be piezoelectric.

Proof:

Let us consider the operation   symmetry with respect to the centre, then

 

The symmetry implies

 

so that

 

which proves the theorem.