Introduction to Mathematical Physics/Groups
Contents |
[edit] 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.
Definition:
A group is a set of elements
and a composition law . that assigns to any ordered pair
an element g1.g2 of G. The composition law . is
- associative
g1.(g2.g3) = (g1.g2).g3
- has a unit element e :
g.e = e.g = e
- for each g of G, there exists an element g − 1 of G such that:
g.g − 1 = g − 1.g = e
Definition:
The order of a group is the number of elements of G.
[edit] Representation
For a deeper study of group representation theory, the reader is invited to refer to the abundant litterature (see for instance ([#References|references])).
Definition:
A representation of a group G in a vectorial space V on K = R or K = C is an endomorphism Π from G into the group GL(V)\footnote{GL(V) is the space of the linear applications from V into V. It is a group with respect to the function composition law.} {\it i.e }a mapping

with
Π(g1.g2) = Π(g1).Π(g2)
Definition:
Let (V,Π) be a representation of G. A vectorial subspace W of V is called stable by Π if:

One then obtains a representation of G in W called subrepresentation of Π.
Definition:
A representation (V,Π) of a group G is called irreducible if it admits no subrepresentation other than 0 and itself.
Consider a symmetry group G. let us consider some classical examples of vectorial spaces V. Let
be an element of G.
Example:
exampgroupR
Let G be a group of transformations
of space R3. A representation Π1 of G in V = R3 is simply defined by:

To each element
of G, a mapping
of R3 is associated. This mapping can be defined by a matrix M called representation matrix of symmetry operator
.
Example:
Let us consider molecule NH3 as a solid of symmetry C3. The various symmetry operations that characterizes this group are:
- three reflexions σ1,σ2, and σ3.
- two rotations around the C3 axis, of angle
and
noted
and
. - rotation of angle
is the identity and is noted E.
In a any basis of R3, 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 E1 spanned by vector of the C3 axis, and a plane E3 perpendicular to this vector. In chemistry books, representation on E1 is called A and representation on E2 is called E. They are both irreducible.
Remark:
For the study of the vibrations of a molecule, instead of considering the Euclidian space R3 as state space, the space of the qi's where the qi'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 G, implies that its energy is invariant by G:

Matrices M(g) being orthonormal, kinetic energy is also invariant.
Consider the following theorem:
Theorem:
theosymde
If operator A is invariant by R, that is ρRA = A or RAR − 1 = A, the if ϕ is eigenvector of A, Rϕ is also eigenvector of A.
Proof:
It is sufficient to evaluate the action of A on Rφ to prove this theorem.
This previous theorem allows to predict the eigenvectors and their degeneracy.
Example:
Consider the group G introduced at example exampgroupR. A representation Π2 of G in the space L2 of summable squared can be defined by:

where
is the matrix representation of transformation
, element of G. If ϕi is a basis of L2(R3), then we have Rϕi = Dkiϕi.
Example:
Consider the group G introduced at example exampgroupR. A representation Π3 of G in the space of linear operators of L2 can be defined by:

where
is the matrix representation, defined at prevoius example, of transformation
, element of G and A is the matrix defining the operator
Relatively to the R3 rotation group, scalar, vectorial and tensorial operators can be defined.
Definition:
A scalar operator S is invariant by rotation:
RSR − 1 = S
An example of scalar operator is the hamiltonian operator in quantum mechanics.
Definition:
A vectorial operator V is a set of three operators Vm, m = − 1,0,1, (the components of V in spherical coordinates) that verify the commutation relations:
![[X_i,V_m]=\epsilon_{imk}V_k](http://upload.wikimedia.org/wikibooks/en/math/8/e/2/8e21df7e320f0eeae221ea4915692d75.png)
More generally, tensorial operators can be defined:
Definition:
A tensorial operator T of components
is an operator whose transformation by rotation is given by:

where
is the restriction of rotation R to space spanned by vectors | jm > .

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])).
[edit] Tensors and symmetries
Let aijk be a third order tensor. Consider the tensor:
Xijk = xixjxk
let us form the density:
ϕ = aijkXijk
ϕ is conserved by change of basis\footnote{ A unitary operator preserves the scalar product.} If by symmetry:
Raijk = aijk
then
RXijk = Xijk
With other words ``X is transformed like a ([#References|references])
Example: piezoelectricity. As seen at section secpiezo, there exists for piezoelectric materials a relation between the deformation tensor eij and the electric field hk
eij = aijkhk
aijk 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 R symmetry with respect to the centre, then
R(xixjxk) = ( − 1)3xixjxk.
The symmetry implies
Raijk = aijk
so that
aijk = − aijk
which proves the theorem.
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and
noted
and
.
is the identity and is noted