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.

Definition:

A group is a set of elements $G=\{g_1,g_2,\dots\}$ and a composition law . that assigns to any ordered pair $(g_1,g_2)\in G$ 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.

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

$\begin{array}{llll} \Pi:&G&\longrightarrow &GL(V)\\ &g&\longrightarrow &\Pi(g) \end{array}$

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:

$\forall g\in G, \Pi(g).W\subset V.$

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 ${\mathcal{R}}$ be an element of G.

Example:

exampgroupR

Let G be a group of transformations ${\mathcal r}$ of space R³. A representation Π₁ of G in V = R³ is simply defined by:

$\begin{array}{llll} \Pi_1({\mathcal{r}}):&R^3&\longrightarrow &R^3\\ &\vec{x}&\longrightarrow &\vec{x}'={\mathcal{R}}(\vec{x}) \end{array}$

To each element ${\mathcal r}$ of G, a mapping ${\mathcal{R}}$ of R³ is associated. This mapping can be defined by a matrix M called representation matrix of symmetry operator ${\mathcal r}$.

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 $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$ noted $C_3^1$ and $C_3^2$.
• rotation of angle $\frac{6\pi}{3}$ 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 q_i's where the q_i'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:

$\forall g\in G, \forall x\in V, <M^+(g)x|KM(g)x>=<x|Kx>$

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 ρ_RA = 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:

$\begin{array}{llll} \Pi_2({\mathcal{r}}):&L^2(R^3)&\longrightarrow &L^2(R^3)\\ &f(x)&\longrightarrow &Rf(x)=f({\mathcal{R}}^{-1}(\vec{x})) \end{array}$

where ${\mathcal R}=\Pi_1({\mathcal r})$ is the matrix representation of transformation ${\mathcal r}$, element of G. If ϕ_i is a basis of L²(R³), then we have Rϕ_i = D_kiϕ_i.

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:

$\begin{array}{llll} \Pi_3({\mathcal{r}}):&{\mathcal{L}}(L^2(R^3))&\longrightarrow &{\mathcal{L}}(L^2(R^3))\\ &A&\longrightarrow &\rho_A=RAR^{-1} \end{array}$

where $R=\Pi_2({\mathcal r})$ is the matrix representation, defined at prevoius example, of transformation ${\mathcal r}$, 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 V_m, m = − 1,0,1, (the components of V in spherical coordinates) that verify the commutation relations:

$[X_i,V_m]=\epsilon_{imk}V_k$

More generally, tensorial operators can be defined:

Definition:

A tensorial operator T of components $T_m^j$ is an operator whose transformation by rotation is given by:

$RT_m^jR^{-1}=D_{m'm}^j(R)T_{m'}^j$

where $D_{m'm}^j(R)$ is the restriction of rotation R to space spanned by vectors | jm > .

$D_{m'm}^j(R)=<jm'|R|jm>$

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])).

Tensors and symmetries

Let a_ijk be a third order tensor. Consider the tensor:

X_ijk = x_ix_jx_k

let us form the density:

ϕ = a_ijkX_ijk

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

Ra_ijk = a_ijk

then

RX_ijk = X_ijk

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 e_ij and the electric field h_k

e_ij = a_ijkh_k

a_ijk 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(x_ix_jx_k) = ( − 1)³x_ix_jx_k.

The symmetry implies

Ra_ijk = a_ijk

so that

a_ijk = − a_ijk

which proves the theorem.