← Mathematical Methods/Group Theory for Physicists

Group Theory for Physicists

Reading time: ~60 minutes | Derivations: 5 complete proofs | Simulation: Character tables, orthogonality, SO(3) & SU(2) computations

1. Introduction: Symmetry as the Organizing Principle of Physics

Every fundamental law of physics is a statement about symmetry. Conservation of energy follows from time-translation invariance, conservation of momentum from spatial homogeneity, and conservation of angular momentum from rotational isotropy. The mathematical language that captures symmetry in its most general and precise form is group theory.

A group is the simplest algebraic structure capable of encoding the idea that "transformations can be composed and undone." The rotation of a crystal by 60° followed by another 60° rotation gives a 120° rotation — and the original orientation can always be recovered. This composability and invertibility, formalized by the group axioms, is what makes group theory the backbone of modern physics.

From the classification of crystal structures in solid-state physics to the gauge symmetries of the Standard Model, from the addition of angular momenta in quantum mechanics to the selection rules that govern spectroscopic transitions, group theory provides the unifying framework. In the words of Hermann Weyl: "All a priori statements in physics have their origin in symmetry."

This chapter develops the theory from the axioms through representations, Lie groups, and physical applications, with complete derivations at every stage.

2. Derivation 1: Group Axioms and Examples

Definition: Group

A group is a set $G$ together with a binary operation $\cdot : G \times G \to G$ satisfying four axioms:

Closure: For all $a, b \in G$, the product $a \cdot b \in G$.
Associativity: For all $a, b, c \in G$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.
Identity: There exists an element $e \in G$ such that $e \cdot a = a \cdot e = a$ for all $a \in G$.
Inverse: For every $a \in G$, there exists $a^{-1} \in G$ such that $a \cdot a^{-1} = a^{-1} \cdot a = e$.

If additionally $a \cdot b = b \cdot a$ for all $a, b$, the group is called abelian.

Fundamental Examples

$\mathbb{Z}_n$ — Cyclic Groups

The integers $\{0, 1, \ldots, n-1\}$ under addition modulo $n$. These are abelian, and every element $a$ satisfies $a^n = e$. The group $\mathbb{Z}_n$ describes discrete rotational symmetries: a regular polygon with $n$ vertices has $\mathbb{Z}_n$ as its rotation group.

$S_n$ — Symmetric Groups

The group of all permutations of $n$ objects, with $|S_n| = n!$. For $n \geq 3$,$S_n$ is non-abelian. The group $S_3$ has order 6 and is isomorphic to the dihedral group $D_3$, the symmetry group of an equilateral triangle.

$\mathrm{SO}(3)$ — Rotation Group

The group of $3 \times 3$ real orthogonal matrices with determinant $+1$. This is a continuous (Lie) group describing rotations in three-dimensional space. It is non-abelian and compact, with three real parameters (e.g., Euler angles).

$\mathrm{SU}(2)$ and $\mathrm{U}(1)$

$\mathrm{U}(1)$ consists of complex numbers $e^{i\theta}$ under multiplication — the symmetry group of electromagnetism. $\mathrm{SU}(2)$ is the group of $2 \times 2$ unitary matrices with unit determinant, the double cover of $\mathrm{SO}(3)$, essential for describing spin.

The Multiplication Table of $S_3$

The symmetric group $S_3$ consists of six permutations of three objects. Using cycle notation, the elements are:

$S_3 = \{e,\; (12),\; (13),\; (23),\; (123),\; (132)\}$

The multiplication rule is composition of permutations, read right-to-left. For example,$(12)(123)$ means first apply $(123)$: $1 \to 2, 2 \to 3, 3 \to 1$, then apply $(12)$: $1 \leftrightarrow 2$. Tracing: $1 \to 2 \to 1$, $2 \to 3 \to 3$, $3 \to 1 \to 2$. So $(12)(123) = (23)$.

The full multiplication table is computed in the Python simulation below. Key structural observations:$S_3$ has three conjugacy classes: $\{e\}$, $\{(12),(13),(23)\}$, and $\{(123),(132)\}$. The number of conjugacy classes equals the number of irreducible representations, so $S_3$ has exactly three irreps.

Subgroups, Cosets, and Lagrange's Theorem

A subgroup $H \leq G$ is a subset that is itself a group under the same operation. For any element $g \in G$, the left coset is $gH = \{gh : h \in H\}$.

Lagrange's Theorem

If $H$ is a subgroup of a finite group $G$, then $|H|$ divides $|G|$. Specifically, $|G| = |H| \cdot [G:H]$, where $[G:H]$ is the number of distinct cosets (the index).

Proof sketch:

(i) Each coset $gH$ has exactly $|H|$ elements (the map $h \mapsto gh$ is a bijection). (ii) Two cosets are either identical or disjoint: if $g_1 H \cap g_2 H \neq \emptyset$, say $g_1 h_1 = g_2 h_2$, then $g_1 = g_2 h_2 h_1^{-1}$, so $g_1 H = g_2 H$. (iii) The cosets partition $G$, so $|G| = (\text{number of cosets}) \times |H|$. ∎

Corollary: The order of every element divides $|G|$, since an element of order $n$ generates a cyclic subgroup of order $n$. For $S_3$ with $|S_3| = 6$, the possible subgroup orders are 1, 2, 3, and 6 — and indeed $S_3$ has subgroups of each order.

3. Derivation 2: Representations and Character Theory

Definition: Representation

A representation of a group $G$ on a vector space $V$ is a homomorphism:

$D: G \to \mathrm{GL}(V)$

meaning $D(g_1 g_2) = D(g_1) D(g_2)$ for all $g_1, g_2 \in G$. Each group element is mapped to an invertible linear transformation (matrix) on $V$, preserving the group structure. The dimension of the representation is $\dim V$.

A representation is reducible if there exists a proper invariant subspace $W \subset V$ such that $D(g)W \subseteq W$ for all $g \in G$. Otherwise it is irreducible. A key theorem states that every representation of a finite group can be decomposed into a direct sum of irreducible representations (irreps).

Schur's Lemma

Schur's Lemma (Part 1): Let $D_1$ and $D_2$ be irreducible representations of $G$ on spaces $V_1$ and $V_2$, and let $T: V_1 \to V_2$ be a linear map such that $T D_1(g) = D_2(g) T$ for all $g \in G$ (an intertwiner). Then either $T = 0$ or $T$ is an isomorphism.

Proof:

Consider $\ker T \subseteq V_1$. If $v \in \ker T$, then $T(D_1(g)v) = D_2(g)(Tv) = D_2(g)(0) = 0$, so $D_1(g)v \in \ker T$. Thus $\ker T$ is an invariant subspace of $V_1$. Since $D_1$ is irreducible, $\ker T = \{0\}$ or $\ker T = V_1$.

Similarly, $\mathrm{im}\, T \subseteq V_2$ is an invariant subspace of $V_2$: if $w = Tv$, then $D_2(g)w = D_2(g)Tv = T D_1(g)v \in \mathrm{im}\, T$. Since $D_2$ is irreducible, $\mathrm{im}\, T = \{0\}$ or $\mathrm{im}\, T = V_2$.

If $\ker T = V_1$ then $T = 0$. Otherwise $\ker T = \{0\}$ and $\mathrm{im}\, T = V_2$, so $T$ is bijective, hence an isomorphism. ∎

Schur's Lemma (Part 2): If $D$ is a complex irreducible representation of $G$ and $T: V \to V$ commutes with all $D(g)$, then $T = \lambda I$ for some scalar $\lambda \in \mathbb{C}$.

Proof:

Since we work over $\mathbb{C}$, $T$ has at least one eigenvalue $\lambda$. The operator $T - \lambda I$ also commutes with all $D(g)$ and has non-trivial kernel. By Part 1 (applied with $D_1 = D_2 = D$), either $T - \lambda I = 0$ (an isomorphism is impossible since the kernel is non-trivial). Therefore $T = \lambda I$. ∎

Character Orthogonality

The character of a representation $D$ is the function $\chi(g) = \mathrm{Tr}\, D(g)$. Characters are class functions (constant on conjugacy classes) and contain all essential information about the representation.

Derivation: The Great Orthogonality Theorem for Characters

Claim: For irreducible representations $\mu$ and $\nu$ of a finite group $G$:

$\sum_{g \in G} \chi_\mu(g)^* \chi_\nu(g) = |G|\,\delta_{\mu\nu}$

Proof:

Start from the Great Orthogonality Theorem (GOT) for matrix elements. For inequivalent irreps $D^\mu$ and $D^\nu$ of dimensions $d_\mu$ and $d_\nu$:

$\sum_{g \in G} D^\mu_{ij}(g)^* D^\nu_{kl}(g) = \frac{|G|}{d_\mu}\,\delta_{\mu\nu}\,\delta_{ik}\,\delta_{jl}$

This fundamental result follows from Schur's lemma applied to the averaging operator $A = \frac{1}{|G|}\sum_{g} D^\mu(g)\, M\, D^\nu(g^{-1})$ for an arbitrary matrix $M$.

To obtain the character orthogonality, take the trace by setting $i = j$ and $k = l$, then sum over $i$ and $k$:

$\sum_{g \in G} \left(\sum_i D^\mu_{ii}(g)^*\right)\left(\sum_k D^\nu_{kk}(g)\right) = \frac{|G|}{d_\mu}\,\delta_{\mu\nu}\sum_i \delta_{ii} = |G|\,\delta_{\mu\nu}$

Recognizing the sums over diagonal elements as traces gives the character orthogonality relation. ∎

The character orthogonality relations mean that the rows of the character table form an orthogonal set of vectors (with respect to the inner product weighted by class sizes). Since the number of irreps equals the number of conjugacy classes, the character table is a square matrix. The columns also satisfy an orthogonality relation:

$\sum_\mu \chi_\mu(C_i)^* \chi_\mu(C_j) = \frac{|G|}{|C_i|}\,\delta_{ij}$

where $|C_i|$ is the number of elements in conjugacy class $C_i$.

4. Derivation 3: SO(3) and Angular Momentum

The rotation group $\mathrm{SO}(3)$ is the most important Lie group in non-relativistic physics. Its representation theory gives us angular momentum, spherical harmonics, and the quantum numbers that classify atomic states.

Deriving the Rotation Matrix

A rotation by angle $\theta$ about a unit axis $\hat{n}$ is described by the Rodrigues formula. Consider an infinitesimal rotation by $d\theta$ about $\hat{n}$. A vector $\mathbf{v}$ transforms as:

$\mathbf{v} \to \mathbf{v} + d\theta\,(\hat{n} \times \mathbf{v})$

In matrix form, this is $R(d\theta) = I + d\theta\, K$, where $K$ is the cross-product matrix:

$K = [\hat{n}]_\times = \begin{pmatrix} 0 & -n_z & n_y \\ n_z & 0 & -n_x \\ -n_y & n_x & 0 \end{pmatrix}$

Exponentiating: $R(\theta, \hat{n}) = e^{\theta K}$. Using the identity $K^3 = -K$ for a unit vector, the series simplifies to the Rodrigues formula:

$R(\theta, \hat{n}) = I\cos\theta + (1 - \cos\theta)\,\hat{n}\hat{n}^T + \sin\theta\, [\hat{n}]_\times$

The Angular Momentum Algebra

The three generators of $\mathrm{SO}(3)$ correspond to rotations about the three coordinate axes. Define $J_i = -i\frac{\partial}{\partial \theta_i}R(\theta)\big|_{\theta=0}$. Explicitly, the $3 \times 3$ generator matrices are:

$(J_x)_{jk} = -i\epsilon_{1jk}, \quad (J_y)_{jk} = -i\epsilon_{2jk}, \quad (J_z)_{jk} = -i\epsilon_{3jk}$

The commutation relation follows from the identity $\epsilon_{ija}\epsilon_{akl} = \delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk}$:

$[J_i, J_j] = i\epsilon_{ijk} J_k$

Proof: Compute the $(m,n)$ matrix element of the commutator:

$[J_i, J_j]_{mn} = (J_i)_{mk}(J_j)_{kn} - (J_j)_{mk}(J_i)_{kn}$

$= (-i\epsilon_{imk})(-i\epsilon_{jkn}) - (-i\epsilon_{jmk})(-i\epsilon_{ikn})$

$= -(\epsilon_{imk}\epsilon_{jkn} - \epsilon_{jmk}\epsilon_{ikn})$

Applying the epsilon identity and simplifying, one obtains $[J_i, J_j]_{mn} = i\epsilon_{ijk}(J_k)_{mn}$. This is the Lie algebra $\mathfrak{so}(3)$, which is isomorphic to $\mathfrak{su}(2)$.

Irreducible Representations $D^j$

The irreducible representations of $\mathrm{SO}(3)$ are labeled by a non-negative integer $j = 0, 1, 2, \ldots$ and have dimension $2j + 1$. Within each representation, the Casimir operator $\mathbf{J}^2 = J_x^2 + J_y^2 + J_z^2$ takes the value $j(j+1)$, and $J_z$ has eigenvalues $m = -j, -j+1, \ldots, j$.

Introducing the ladder operators $J_\pm = J_x \pm iJ_y$, the algebra becomes:

$[J_z, J_\pm] = \pm J_\pm, \qquad [J_+, J_-] = 2J_z$

Acting on basis states $|j,m\rangle$:

$J_\pm |j,m\rangle = \sqrt{j(j+1) - m(m \pm 1)}\,|j, m \pm 1\rangle$

This derivation uses $\mathbf{J}^2 = J_- J_+ + J_z^2 + J_z$ and the requirement that the norm $\| J_+ |j,m\rangle \|^2 \geq 0$ forces the chain to terminate, giving the allowed values of $j$ and $m$.

Connection to Spherical Harmonics

The spin-$j$ representation for integer $j = \ell$ is realized on the space of spherical harmonics $Y_\ell^m(\theta, \phi)$. The angular momentum operators in the position representation are:

$J_z = -i\frac{\partial}{\partial\phi}, \quad J_\pm = e^{\pm i\phi}\left(\pm \frac{\partial}{\partial\theta} + i\cot\theta\,\frac{\partial}{\partial\phi}\right)$

The spherical harmonics satisfy $J_z Y_\ell^m = m Y_\ell^m$ and $\mathbf{J}^2 Y_\ell^m = \ell(\ell+1) Y_\ell^m$, confirming that they provide the representation space for the spin-$\ell$ irrep of $\mathrm{SO}(3)$.

5. Derivation 4: SU(2) and Spin

While $\mathrm{SO}(3)$ describes rotations of classical objects, quantum mechanics requires its universal covering group $\mathrm{SU}(2)$, which accommodates half-integer spin.

SU(2) as the Double Cover of SO(3)

Claim: There is a surjective homomorphism $\phi: \mathrm{SU}(2) \to \mathrm{SO}(3)$ with kernel $\{I, -I\}$, making $\mathrm{SU}(2)$ a double cover.

Construction:

A general element of $\mathrm{SU}(2)$ can be written as:

$U = e^{-i(\theta/2)\hat{n}\cdot\boldsymbol{\sigma}} = \cos\frac{\theta}{2}\, I - i\sin\frac{\theta}{2}\,(\hat{n}\cdot\boldsymbol{\sigma})$

where $\boldsymbol{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$ are the Pauli matrices. The map $\phi$ is defined by the adjoint action on the space of traceless Hermitian$2 \times 2$ matrices (which is isomorphic to $\mathbb{R}^3$):

$\hat{n}\cdot\boldsymbol{\sigma} \;\mapsto\; U(\hat{n}\cdot\boldsymbol{\sigma})U^\dagger$

This defines a rotation $R_{ij} = \frac{1}{2}\mathrm{Tr}(\sigma_i U \sigma_j U^\dagger)$. Since $U$ and $-U$ produce the same conjugation, the map is 2-to-1. ∎

The Pauli Matrices as Generators

The Pauli matrices are:

$\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

They satisfy the fundamental relation $\sigma_i \sigma_j = \delta_{ij} I + i\epsilon_{ijk}\sigma_k$, from which the $\mathfrak{su}(2)$ algebra follows with generators $J_i = \sigma_i / 2$:

$[J_i, J_j] = i\epsilon_{ijk} J_k$

This is identical to the $\mathfrak{so}(3)$ algebra, confirming the Lie algebra isomorphism $\mathfrak{su}(2) \cong \mathfrak{so}(3)$.

The 2-to-1 Homomorphism: Explicit Verification

Consider a rotation by $\theta$ about the $z$-axis. The SU(2) element is:

$U(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}$

Under $\theta \to \theta + 2\pi$: $U(\theta + 2\pi) = -U(\theta)$. So a $2\pi$ rotation sends the SU(2) element to its negative — yet both map to the same SO(3) rotation. Only after a $4\pi$ rotation does the SU(2) element return to itself. This is the mathematical basis for half-integer spin: fermions acquire a phase of $-1$ under $2\pi$ rotation.

Spinor Representations and Half-Integer Spin

The representation theory of $\mathrm{SU}(2)$ allows $j = 0, \frac{1}{2}, 1, \frac{3}{2}, 2, \ldots$, with dimension $2j + 1$. The representations with integer $j$ are also representations of $\mathrm{SO}(3)$; those with half-integer $j$ are spinor representations that only make sense for $\mathrm{SU}(2)$.

The fundamental representation ($j = 1/2$) acts on two-component spinors:

$|\!\uparrow\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad |\!\downarrow\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$

These are eigenstates of $J_z = \sigma_z/2$ with eigenvalues $\pm 1/2$. Higher spin-$j$ representations are constructed from symmetric tensor products of the fundamental representation: the spin-$j$ space is the symmetric subspace of $(\mathbb{C}^2)^{\otimes 2j}$.

6. Derivation 5: Symmetry in Quantum Mechanics

Wigner's Theorem

Wigner's Theorem: Every symmetry transformation of a quantum system — i.e., every bijection on the set of rays in Hilbert space that preserves transition probabilities $|\langle\psi|\phi\rangle|^2$ — is represented by an operator that is either unitary or anti-unitary.

Significance:

This theorem is the foundation for applying group theory to quantum mechanics. It guarantees that every symmetry group has a (projective) unitary representation on the Hilbert space. Continuous symmetries connected to the identity must be unitary (anti-unitary operators are disconnected from the identity); time reversal is the canonical example of an anti-unitary symmetry.

Selection Rules from Group Theory

Selection rules are among the most powerful consequences of symmetry. Suppose $|i\rangle$ and $|f\rangle$ belong to irreducible representations $\Gamma_i$ and $\Gamma_f$ of the symmetry group, and $\hat{O}$ is an operator transforming as irrep $\Gamma_O$.

Selection Rule: The matrix element $\langle f | \hat{O} | i \rangle$ vanishes unless the decomposition of $\Gamma_f^* \otimes \Gamma_O \otimes \Gamma_i$ contains the identity representation $\Gamma_1$.

Derivation:

The matrix element $\langle f | \hat{O} | i \rangle$ must be invariant under all group transformations (it is a scalar). The product $\hat{O}|i\rangle$ transforms as $\Gamma_O \otimes \Gamma_i$. By character orthogonality, the inner product with $|f\rangle$ (transforming as $\Gamma_f$) is nonzero only if the tensor product contains $\Gamma_f$, equivalently if $\Gamma_f^* \otimes \Gamma_O \otimes \Gamma_i \supset \Gamma_1$.

Example: For electric dipole transitions in atoms, $\hat{O} = \hat{r}$ transforms as the $\ell = 1$ representation of SO(3). The selection rule becomes $\Delta \ell = \pm 1$ and $\Delta m = 0, \pm 1$.

The Wigner-Eckart Theorem

Theorem (Wigner-Eckart): Let $T^{(k)}_q$ be an irreducible tensor operator of rank $k$. Then:

$\langle j', m' | T^{(k)}_q | j, m \rangle = \langle j, m; k, q | j', m' \rangle \, \langle j' \| T^{(k)} \| j \rangle$

where $\langle j, m; k, q | j', m' \rangle$ is a Clebsch-Gordan coefficient and $\langle j' \| T^{(k)} \| j \rangle$ is the reduced matrix element, independent of $m, m', q$.

Derivation sketch:

Under a rotation $R$, the tensor operator transforms as $U(R)\, T^{(k)}_q \, U(R)^\dagger = \sum_{q'} D^{(k)}_{q'q}(R)\, T^{(k)}_{q'}$. The state $T^{(k)}_q |j, m\rangle$ therefore transforms as the tensor product representation $D^{(k)} \otimes D^{(j)}$, which decomposes into irreps via Clebsch-Gordan coefficients:

$T^{(k)}_q |j, m\rangle = \sum_{j'} \langle j, m; k, q | j', m+q \rangle \, |j', m+q; \alpha\rangle$

Taking the inner product with $\langle j', m'|$ and using the orthogonality of the $|j', m'\rangle$ basis yields the theorem. The key insight is that by Schur's lemma, the proportionality constant (the reduced matrix element) cannot depend on the magnetic quantum numbers. ∎

7. Applications

Crystal Field Theory

When a free ion is placed in a crystal, the full rotational symmetry SO(3) is broken down to the point group of the crystal site. The $(2\ell+1)$-fold degeneracy of the atomic orbitals splits according to the decomposition of the SO(3) irrep $D^{(\ell)}$ into irreps of the point group. For example, in an octahedral field ($O_h$), the$d$-orbitals ($\ell = 2$) split into $t_{2g}$ (3-fold) and $e_g$ (2-fold) — the famous crystal field splitting that determines the colors of transition metal compounds.

Molecular Orbital Symmetry

The point group of a molecule constrains which atomic orbitals can combine into molecular orbitals. Using character tables, one projects atomic orbital combinations onto the irreps of the point group. Only combinations belonging to the same irrep can mix. For water ($C_{2v}$), this explains the bonding and antibonding orbital structure directly from symmetry, without solving the Schrodinger equation.

SU(3) Flavor Symmetry

The approximate symmetry among $u$, $d$, and $s$ quarks is described by $\mathrm{SU}(3)_{\text{flavor}}$. The fundamental representation $\mathbf{3}$ contains the three quark flavors. Mesons live in$\mathbf{3} \otimes \bar{\mathbf{3}} = \mathbf{8} \oplus \mathbf{1}$ (the octet and singlet), while baryons live in $\mathbf{3} \otimes \mathbf{3} \otimes \mathbf{3} = \mathbf{10} \oplus \mathbf{8} \oplus \mathbf{8} \oplus \mathbf{1}$. The Eightfold Way classification of Gell-Mann and Ne'eman is a direct triumph of group theory.

Standard Model Gauge Group

The Standard Model is built on the gauge group $\mathrm{SU}(3)_C \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$. Each factor dictates the force carriers: 8 gluons from $\mathrm{SU}(3)$, 3 weak bosons from $\mathrm{SU}(2)$, and 1 hypercharge boson from $\mathrm{U}(1)$. The representation content of the fermion fields under this group determines all allowed interactions. Group theory is not just a tool in particle physics — it is the theory.

8. Historical Context

Evariste Galois (1811–1832) invented group theory to prove that the general quintic equation has no solution by radicals. His work, written the night before his fatal duel, introduced the concepts of normal subgroups, quotient groups, and solvability that remain central to modern algebra.

Sophus Lie (1842–1899) extended Galois's discrete groups to continuous transformation groups, creating what we now call Lie groups and Lie algebras. He sought to do for differential equations what Galois had done for algebraic equations. The rotation groups SO(n) and the unitary groups SU(n) are all Lie groups.

Emmy Noether (1882–1935) proved her celebrated theorem in 1918: every continuous symmetry of a physical system corresponds to a conservation law. Time translation invariance gives energy conservation, spatial translation gives momentum, and rotational invariance gives angular momentum. Noether's theorem is the bridge between group theory and the conservation laws of physics.

Eugene Wigner (1902–1995) was the first to systematically apply group theory to quantum mechanics in his 1931 book. He introduced the concept of projective representations, proved the theorem bearing his name (on symmetries as unitary/anti-unitary operators), developed the Wigner-Eckart theorem, and classified particles by representations of the Poincare group. He received the 1963 Nobel Prize for these contributions.

Hermann Weyl (1885–1955) brought the theory of Lie groups and their representations to its modern form. His 1928 book "Gruppentheorie und Quantenmechanik" ("Group Theory and Quantum Mechanics") demonstrated that the mathematical framework of quantum mechanics is inseparable from the theory of symmetry groups. Weyl also introduced gauge invariance to physics, an idea that would eventually lead to the Standard Model.

9. Python Simulation: Character Tables & Rotation Matrices

The simulation below computes character tables for the point groups $C_{2v}$, $C_{3v}$, and $D_3$, verifies the orthogonality relations $\sum_g \chi_\mu(g)^*\chi_\nu(g) = |G|\delta_{\mu\nu}$, displays the full multiplication table for $S_3$, computes SO(3) rotation matrices via the Rodrigues formula, and verifies the 2-to-1 homomorphism from SU(2) to SO(3). Only numpy is used.

Simulation

Python

script.py259 lines

import numpy as np

# ============================================================
# Group Theory Computations: Character Tables & SO(3) Rotations
# ============================================================

# ----- Part 1: Character Tables for Point Groups -----

def make_group_data(name, classes, class_sizes, irreps, char_table):
    """Bundle group data into a dictionary."""
    return {
        'name': name,
        'classes': classes,
        'class_sizes': class_sizes,
        'irreps': irreps,
        'char_table': np.array(char_table, dtype=float)
    }

# --- C2v: {E, C2, sigma_v, sigma_v'} ---
C2v = make_group_data(
    name='C2v',
    classes=['E', 'C2', 'sigma_v', "sigma_v'"],
    class_sizes=[1, 1, 1, 1],
    irreps=['A1', 'A2', 'B1', 'B2'],
    char_table=[
        [ 1,  1,  1,  1],
        [ 1,  1, -1, -1],
        [ 1, -1,  1, -1],
        [ 1, -1, -1,  1]
    ]
)

# --- C3v: {E, 2C3, 3sigma_v} ---
C3v = make_group_data(
    name='C3v',
    classes=['E', '2C3', '3sigma_v'],
    class_sizes=[1, 2, 3],
    irreps=['A1', 'A2', 'E'],
    char_table=[
        [ 1,  1,  1],
        [ 1,  1, -1],
        [ 2, -1,  0]
    ]
)

# --- D3: {E, 2C3, 3C2'} ---
D3 = make_group_data(
    name='D3',
    classes=['E', '2C3', "3C2'"],
    class_sizes=[1, 2, 3],
    irreps=['A1', 'A2', 'E'],
    char_table=[
        [ 1,  1,  1],
        [ 1,  1, -1],
        [ 2, -1,  0]
    ]
)

def print_character_table(group):
    """Pretty-print a character table."""
    name = group['name']
    classes = group['classes']
    irreps = group['irreps']
    table = group['char_table']

print(f"\n{'='*56}")
    print(f"  Character Table: {name}")
    print(f"{'='*56}")

header = f"  {'Irrep':<8}"
    for c in classes:
        header += f"{c:>10}"
    print(header)
    print(f"  {'-'*8}" + '-'*10*len(classes))

for i, irrep in enumerate(irreps):
        row = f"  {irrep:<8}"
        for j in range(len(classes)):
            val = table[i, j]
            if val == int(val):
                row += f"{int(val):>10}"
            else:
                row += f"{val:>10.4f}"
        print(row)

def verify_orthogonality(group):
    """
    Verify the Great Orthogonality Theorem for characters:
    sum_g chi_mu(g)* chi_nu(g) = |G| delta_{mu,nu}
    """
    name = group['name']
    sizes = np.array(group['class_sizes'], dtype=float)
    table = group['char_table']
    order = int(np.sum(sizes))

print(f"\n  Orthogonality check for {name} (|G| = {order}):")

n_irreps = table.shape[0]
    all_pass = True
    for mu in range(n_irreps):
        for nu in range(mu, n_irreps):
            inner = np.sum(sizes * table[mu] * table[nu])
            expected = order if mu == nu else 0
            status = "PASS" if abs(inner - expected) < 1e-10 else "FAIL"
            if status == "FAIL":
                all_pass = False
            print(f"    <chi_{group['irreps'][mu]}, chi_{group['irreps'][nu]}> "
                  f"= {inner:6.1f}  (expected {expected})  [{status}]")

if all_pass:
        print(f"  All orthogonality relations verified for {name}.")
    return all_pass

# Print tables and verify
for group in [C2v, C3v, D3]:
    print_character_table(group)
    verify_orthogonality(group)

# ----- Part 2: S3 Multiplication Table -----
print(f"\n{'='*56}")
print("  S3 (Symmetric Group on 3 Elements) Multiplication Table")
print(f"{'='*56}")

# S3 elements as permutations of (0,1,2)
s3_elements = {
    'e':     (0, 1, 2),
    '(12)':  (1, 0, 2),
    '(13)':  (2, 1, 0),
    '(23)':  (0, 2, 1),
    '(123)': (1, 2, 0),
    '(132)': (2, 0, 1),
}

def compose_perm(p, q):
    """Compose two permutations: (p . q)(i) = p(q(i))."""
    return tuple(p[q[i]] for i in range(len(p)))

def find_name(perm, elements):
    for name, p in elements.items():
        if p == perm:
            return name
    return '???'

names = list(s3_elements.keys())
perms = list(s3_elements.values())

header = f"  {'*':<7}"
for n in names:
    header += f"{n:>7}"
print(header)
print("  " + "-" * (7 + 7 * len(names)))

for i, ni in enumerate(names):
    row = f"  {ni:<7}"
    for j, nj in enumerate(names):
        product = compose_perm(perms[i], perms[j])
        row += f"{find_name(product, s3_elements):>7}"
    print(row)

# ----- Part 3: SO(3) Rotation Matrices -----
print(f"\n{'='*56}")
print("  SO(3) Rotation Matrices")
print(f"{'='*56}")

def rotation_matrix(axis, theta):
    """
    Compute the 3x3 rotation matrix R(theta, n_hat)
    using Rodrigues formula: R = I cos(t) + (1-cos(t)) nn^T + sin(t) [n]_x
    """
    axis = np.array(axis, dtype=float)
    axis = axis / np.linalg.norm(axis)
    nx, ny, nz = axis
    c = np.cos(theta)
    s = np.sin(theta)
    # Cross-product matrix [n]_x
    K = np.array([
        [ 0,  -nz,  ny],
        [ nz,  0,  -nx],
        [-ny,  nx,   0]
    ])
    R = c * np.eye(3) + (1 - c) * np.outer(axis, axis) + s * K
    return R

# Example rotations
examples = [
    ('z', [0, 0, 1], np.pi / 2, 'pi/2 about z-axis'),
    ('x', [1, 0, 0], np.pi / 3, 'pi/3 about x-axis'),
    ('y', [0, 1, 0], np.pi / 4, 'pi/4 about y-axis'),
    ('(1,1,1)/sqrt3', [1, 1, 1], 2 * np.pi / 3, '2pi/3 about (1,1,1)'),
]

for label, axis, angle, desc in examples:
    R = rotation_matrix(axis, angle)
    print(f"\n  R({desc}):")
    for row in R:
        print("    [" + "  ".join(f"{v:8.4f}" for v in row) + " ]")
    det = np.linalg.det(R)
    print(f"    det(R) = {det:.6f}  (should be 1)")
    orth = np.max(np.abs(R @ R.T - np.eye(3)))
    print(f"    max|R R^T - I| = {orth:.2e}  (should be ~0)")

# ----- Part 4: SU(2) to SO(3) Mapping -----
print(f"\n{'='*56}")
print("  SU(2) -> SO(3) Homomorphism Verification")
print(f"{'='*56}")

# Pauli matrices
sigma_x = np.array([[0, 1], [1, 0]], dtype=complex)
sigma_y = np.array([[0, -1j], [1j, 0]], dtype=complex)
sigma_z = np.array([[1, 0], [0, -1]], dtype=complex)
sigmas = [sigma_x, sigma_y, sigma_z]

def su2_matrix(axis, theta):
    """U = exp(-i theta/2 n.sigma) = cos(t/2) I - i sin(t/2) n.sigma"""
    axis = np.array(axis, dtype=float)
    axis = axis / np.linalg.norm(axis)
    I2 = np.eye(2, dtype=complex)
    n_dot_sigma = sum(axis[i] * sigmas[i] for i in range(3))
    return np.cos(theta / 2) * I2 - 1j * np.sin(theta / 2) * n_dot_sigma

def su2_to_so3(U):
    """Extract 3x3 rotation from SU(2) via R_ij = (1/2) Tr(sigma_i U sigma_j U^dag)."""
    R = np.zeros((3, 3))
    Ud = U.conj().T
    for i in range(3):
        for j in range(3):
            R[i, j] = 0.5 * np.trace(sigmas[i] @ U @ sigmas[j] @ Ud).real
    return R

# Verify 2-to-1 map
test_axes = [
    ([0, 0, 1], np.pi / 3),
    ([1, 0, 0], np.pi / 2),
    ([1, 1, 0], np.pi / 4),
]

for axis, theta in test_axes:
    U_plus = su2_matrix(axis, theta)
    U_minus = su2_matrix(axis, theta + 2 * np.pi)  # -U
    R_from_U_plus = su2_to_so3(U_plus)
    R_from_U_minus = su2_to_so3(U_minus)
    R_direct = rotation_matrix(axis, theta)

diff_plus = np.max(np.abs(R_from_U_plus - R_direct))
    diff_minus = np.max(np.abs(R_from_U_minus - R_direct))
    u_diff = np.max(np.abs(U_plus + U_minus))

a_str = ','.join(str(x) for x in axis)
    print(f"\n  axis=({a_str}), theta={theta:.4f}:")
    print(f"    |R(U) - R_direct|_max = {diff_plus:.2e}")
    print(f"    |R(-U) - R_direct|_max = {diff_minus:.2e}")
    print(f"    U and -U both map to the SAME R: "
          f"{'YES' if diff_plus < 1e-10 and diff_minus < 1e-10 else 'NO'}")
    print(f"    |U + (-U)|_max = {u_diff:.2e} (should be ~0, confirming -U != U)")

print(f"\n{'='*56}")
print("  All computations completed successfully.")
print(f"{'='*56}")

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Part III — Group Theory for Physicists — Mathematical Methods for Physics