Casimir Operators and Invariants

Casimir eigenvalues are still the working labels of frontier physics. In lattice QCD and collider phenomenology the color Casimirs $C_F=\tfrac43$ and $C_A=3$ are inputs to every parton-shower Monte Carlo, and their ratio has been measured directly by comparing quark and gluon jet shapes at LEP and the LHC — a Casimir eigenvalue read off a detector. In nuclear structure the interacting boson model and other algebraic ("dynamical symmetry") methods write the Hamiltonian as a function of the Casimirs of a nested chain such as $U(6)\supset SU(3)\supset SO(3)$, so entire spectra come out in closed form — the same trick that solves the hydrogen atom through the hidden $SO(4)$ symmetry whose Casimir fixes the $n^2$-fold degeneracy. Beyond the Standard Model, model builders classify candidate gauge groups by Casimir invariants and anomaly coefficients, and conformal and higher-spin theories are organized by the Casimirs of the conformal and de Sitter algebras. Wherever a symmetry organizes a spectrum, its Casimirs are the quantum numbers doing the organizing.

Picture a spin-$j$ multiplet as a ladder of $2j+1$ rungs. The raising and lowering operators $J_\pm$ step you up and down, and $J_z$ reads off which rung you are on — so $J_z$ takes a different value on every rung, from $-j$ at the bottom to $+j$ at the top. It is a coordinate on the multiplet, not a name for it. The Casimir $J^2$ is different in kind: because it commutes with $J_z$ and both ladder operators, climbing or descending cannot change it, so it has the same value on every rung. That is what it means to "label the representation" — it survives every move the symmetry can make. The first figure makes the contrast literal for $j=2$: five rungs, five different $J_z$ values, one shared Casimir value of $6$.

Why does such an invariant combination exist at all? Think of the generators $T_a$ as vectors in an internal space carrying the adjoint representation, and the Killing form $g_{ab}$ as an invariant metric on it. The quadratic Casimir $C_2=g^{ab}T_aT_b$ is the squared length built with that metric — and a length built from an invariant metric is automatically invariant, hence commutes with everything. Each irreducible representation is then pinned to one point on the curve of allowed Casimir values. For $\mathfrak{su}(2)$ that curve is the parabola $C_2=j(j+1)$: spin $\tfrac12$ sits at $\tfrac34$, spin $1$ at $2$, spin $\tfrac32$ at $\tfrac{15}{4}$, spin $2$ at $6$. The second figure is that curve — the discrete ladder of labels a single number may take.

Everything begins with one demand: an operator that commutes with the whole symmetry. This is the definition of a Casimir, and it is the property from which the labeling theorem, the Killing-form construction, and every eigenvalue below all follow.

where: $C$ — a Casimir operator, an element of the universal enveloping algebra $U(\mathfrak{g})$; $T_a$ — the generators of the Lie algebra $\mathfrak{g}$, with $a=1,\dots,\dim\mathfrak{g}$. Reading it: $C$ is invisible to every symmetry move, so no move can change its value — the seed of "it is a constant across the multiplet". The generators are not free; they close into themselves under the bracket, and the numbers that record how they do so — the structure constants — are the raw material out of which every invariant is built.

where: $f_{ab}{}^{c}$ — the structure constants, antisymmetric in $a,b$; the factor $i$ is the physics convention that keeps the $T_a$ Hermitian. Reading it: the entire local structure of the group is packed into $f_{ab}{}^{c}$, so any object we can assemble from $f_{ab}{}^{c}$ alone is guaranteed to respect the symmetry. The one bilinear form you can build from the structure constants with no extra input is the Killing form — the algebra's intrinsic metric, and the object that promotes "sum of squares of generators" into a genuine invariant.

where: $g_{ab}$ — the Killing form, a symmetric bilinear form; $\mathrm{ad}_{T_a}$ — the adjoint action, $(\mathrm{ad}_{T_a})^c{}_b=i f_{ab}{}^{c}$. Reading it: $g_{ab}$ is the trace of two adjoint matrices, and its ad-invariance — proved from the Jacobi identity — is exactly what makes contractions with it commute with the algebra. It is nondegenerate precisely when $\mathfrak{g}$ is semisimple, so its inverse $g^{ab}$ exists there. Contracting two generators with the inverse Killing form gives the quadratic Casimir — the noncommutative Laplacian of the algebra. Its commutation with every generator is the central calculation of the subject and is derived in full below.

where: $g^{ab}$ — the inverse Killing form (exists for semisimple $\mathfrak{g}$); $C_2$ — the quadratic Casimir, an element of $U(\mathfrak{g})$, not of $\mathfrak{g}$. Reading it: this is the invariant "length-squared" of the generator vector, the object whose eigenvalue names each multiplet. Combine the defining property with Schur's lemma and the payoff is immediate: on an irreducible representation the Casimir is not an operator worth diagonalizing — it is already a number times the identity.

where: $c_2(\lambda)$ — the Casimir eigenvalue, one number attached to the irreducible representation $\lambda$; $\mathbb{1}$ — the identity on the representation space. Reading it: every state in the multiplet shares this scalar, so $c_2(\lambda)$ is a legitimate name for the multiplet as a whole. For $\mathfrak{su}(2)$ this scalar is the familiar magnitude-squared of angular momentum, the first Casimir eigenvalue anyone meets in quantum mechanics.

where: $j$ — the spin label, $j=0,\tfrac12,1,\tfrac32,\dots$; $m$ — the weight, $m=-j,\dots,+j$, which fixes the rung but drops out of $J^2$. Reading it: $j$ names the ladder, $m$ names the rung, and only $j$ appears in the Casimir — exactly the split the first figure drew. How many such independent labels can a state carry? Racah's theorem answers it with a single clean count: the number of algebraically independent Casimirs equals the rank — the number of simultaneously diagonalizable generators.

where: $Z(U(\mathfrak{g}))$ — the center of the enveloping algebra, i.e. the Casimirs; $\mathrm{rank}(\mathfrak{g})$ — the dimension of the Cartan subalgebra. Reading it: $\mathfrak{su}(2)$ (rank 1) has one Casimir; $\mathfrak{su}(3)$ (rank 2) has two; the rank is literally the number of independent quantum numbers. The physics climax is the Poincaré algebra, whose two Casimirs are mass and spin. The second is built from the Pauli–Lubanski vector, the relativistic upgrade of the spin operator.

where: $P_\nu$ — the translation (four-momentum) generators; $M_{\rho\sigma}$ — the Lorentz generators; $\varepsilon^{\mu\nu\rho\sigma}$ — the Levi-Civita symbol. Reading it: the two invariants of the symmetry group of spacetime are exactly the mass and spin that name a particle — Wigner's classification in two equations.

The general machinery collapses to the familiar textbook results the moment you name a specific algebra:

1 · Angular-momentum magnitude from the su(2) Casimir (confidence builder)

Use $J^2=j(j+1)\hbar^2$ to find the length of the angular-momentum vector for an electron spin ($j=\tfrac12$) and for a spin-1 particle, and compare each with its maximum projection $J_z^{\max}=j\hbar$.

Step 1. Electron spin, $j=\tfrac12$. The Casimir eigenvalue is $j(j+1)=\tfrac12\cdot\tfrac32=\tfrac34$, so $J^2=\tfrac34\hbar^2$ and the vector length is $|J|=\sqrt{j(j+1)}\,\hbar=\tfrac{\sqrt3}{2}\hbar\approx\mathbf{0.866\,\hbar}$.

Step 2. Its largest projection is $J_z^{\max}=j\hbar=\tfrac12\hbar=0.500\,\hbar$. Since $0.866\hbar>0.500\hbar$, the spin vector is strictly longer than any projection it can have — it can never point fully along $z$. This gap is the physical fingerprint of $j(j+1)>j^2$, and it is why quantum spin always "wobbles" off the quantization axis.

Step 3. Spin-1 particle, $j=1$. Now $j(j+1)=2$, so $J^2=2\hbar^2$ and $|J|=\sqrt2\,\hbar\approx\mathbf{1.414\,\hbar}$, while $J_z^{\max}=\hbar$. Again the length $1.414\hbar$ exceeds the maximum projection $\hbar$. The Casimir eigenvalue, not the projection, sets the true magnitude of angular momentum, and it is the number that is common to all $2j+1$ states of the multiplet.

2 · The QCD color Casimirs (fully worked)

The strength with which a colored particle radiates a gluon is set by the quadratic Casimir of its $\mathfrak{su}(3)$ representation. In the standard QCD normalization $\mathrm{Tr}(T^aT^b)=\tfrac12\delta^{ab}$, the fundamental (quark) and adjoint (gluon) Casimirs are $C_F=(N^2-1)/(2N)$ and $C_A=N$ for $\mathfrak{su}(N)$. Evaluate both for $N=3$ and interpret their ratio.

Step 1. Fundamental representation (a quark). With $N=3$, $C_F=\dfrac{N^2-1}{2N}=\dfrac{9-1}{6}=\dfrac{8}{6}=\mathbf{\tfrac43}\approx1.333$. Cross-check against the Dynkin-label formula for the irrep $(p,q)=(1,0)$: $C_2=\tfrac13(1+0+0)+(1+0)=\tfrac13+1=\tfrac43$ — the same value, confirming the normalization is consistent.

Step 2. Adjoint representation (a gluon). $C_A=N=\mathbf{3}$. Cross-check with $(p,q)=(1,1)$: $C_2=\tfrac13(1+1+1)+(1+1)=1+2=3$ — again identical.

Step 3. The physically measured quantity is the ratio $\dfrac{C_A}{C_F}=\dfrac{3}{4/3}=\dfrac{9}{4}=\mathbf{2.25}$. A gluon radiates soft gluons about 2.25 times more copiously than a quark does, because it carries the larger adjoint color charge. This single Casimir ratio is what makes gluon jets broader than quark jets — a Casimir eigenvalue read straight off a particle detector.

3 · Mass and spin from the Poincaré Casimirs

A physical particle is a unitary irreducible representation of the Poincaré group, labeled by its two Casimirs $P^2=m^2$ and $W^2=-m^2s(s+1)$. Evaluate them for a massive spin-$\tfrac12$ particle (an electron) and state what happens in the massless limit.

Step 1. The first Casimir is the invariant mass-squared, $P^2=m^2$. For an electron this is simply $m^2=(0.511\ \text{MeV}/c^2)^2$ — one of the two numbers that name the representation, fixed and Lorentz-invariant.

Step 2. The second Casimir uses the spin label $s=\tfrac12$: $W^2=-m^2\,s(s+1)=-m^2\cdot\tfrac12\cdot\tfrac32=\mathbf{-\tfrac34\,m^2}$. The same $s(s+1)=\tfrac34$ from Example 1 reappears — the little group of a massive particle is exactly $\mathfrak{su}(2)$, so the rotation Casimir sits inside the Poincaré one.

Step 3. Massless limit. As $m\to0$ the second Casimir $W^2\to0$ and the spin classification collapses; a massless particle is labeled instead by a single helicity $\pm s$, which is why the photon has only two polarization states rather than three. The Poincaré Casimirs are the group-theoretic definition of "a particle": pick $m$ and $s$, and you have named the representation.

The Casimir operator arrived through physics — a graduate student trying to quantize a spinning top wrote down the invariant that now bears his name and underwrites the entire particle zoo.

Texts. Georgi, Lie Algebras in Particle Physics, 2nd ed., chapters on Casimir operators and $\mathfrak{su}(3)$ · Hassani, Mathematical Physics, chapters on Lie groups and representation theory · Stone & Goldbart, Mathematics for Physics, the Lie-groups chapter · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., group-theory chapter · Wybourne, Classical Groups for Physicists (Racah's theorem) · Nakahara, Geometry, Topology and Physics (Killing form and enveloping algebra).

Provenance. A verifier should recompute: (1) the $\mathfrak{su}(2)$ eigenvalue $j(j+1)$ from $J^2=J_-J_++J_z^2+J_z$ on the highest-weight state (Stone & Goldbart; Arfken–Weber); (2) the cancellation $[C_2,T_c]=0$ from the total antisymmetry of the lowered structure constants (Georgi); (3) the color Casimirs $C_F=(N^2-1)/(2N)=\tfrac43$ and $C_A=N=3$ for $\mathfrak{su}(3)$, cross-checked against $C_2(p,q)=\tfrac13(p^2+q^2+pq)+(p+q)$ at $(1,0)$ and $(1,1)$ (Georgi); (4) the Poincaré Casimirs $P^2=m^2$, $W^2=-m^2s(s+1)$ with the Pauli–Lubanski vector (Wigner 1939). The rank-equals-number-of-Casimirs statement is Racah's theorem (Wybourne).

Prerequisites. Lie Algebras and the Exponential Map · Group Representations · Reducible and Irreducible Representations · Schur's Lemmas · The Rotation Group SO(3). Leads to. Tensor Products and Clebsch–Gordan Decomposition · Characters and Orthogonality Theorems · Lie Groups.

Ask the tutor: ask me to build the quadratic Casimir for any algebra — su(2), su(3), or the Poincaré group — and walk its eigenvalue down to the mass, spin, or color label it names.