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 $g_1.g_2$ of $G$. The composition law $.$ is

• associative

$g_1.(g_2.g_3)=(g_1.g_2).g_3$

• 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 $\Pi$ 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

$\Pi(g_1.g_2)=\Pi(g_1).\Pi(g_2)$

Definition:

Let $(V,\Pi)$ be a representation of $G$. A vectorial subspace $W$ of $V$ is called stable by $\Pi$ if:

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

One then obtains a representation of $G$ in $W$ called subrepresentation of $\Pi$.

Definition:

A representation $(V,\Pi)$ 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 $R^3$. A representation $\Pi_1$ of $G$ in $V=R^3$ 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 $R^3$ is associated. This mapping can be defined by a matrix $M$ called representation matrix of symmetry operator ${\mathcal r}$.

Example:

Let us consider molecule $NH_3$ as a solid of symmetry $C_3$. The various symmetry operations that characterizes this group are:

• three reflexions $\sigma_1$,$\sigma_2$, and $\sigma_3$.
• two rotations around the $C_3$ 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 $R^3$, 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 $E_1$ spanned by vector of the $C_3$ axis, and a plane $E_3$ perpendicular to this vector. In chemistry books, representation on $E_1$ is called $A$ and representation on $E_2$ is called $E$. They are both irreducible.

Remark:

For the study of the vibrations of a molecule, instead of considering the Euclidian space $R^3$ 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 $\rho_RA=A$ or $RAR^{-1}=A$, the if $\phi$ is eigenvector of $A$, $R\phi$ is also eigenvector of $A$.

Proof:

It is sufficient to evaluate the action of $A$ on $R\phi$ 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 $\Pi_2$ of $G$ in the space $L^2$ 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 $\phi_i$ is a basis of $L^2(R^3)$, then we have $R\phi_i=D_{ki}\phi_i$.

Example:

Consider the group $G$ introduced at example exampgroupR. A representation $\Pi_3$ of $G$ in the space of linear operators of $L^2$ 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 $R^3$ 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:

$\phi=a_{ijk}X_{ijk}$

$\phi$ 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 $h_k$

$e_{ij}=a_{ijk}h_k$

$a_{ijk}$ 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.