| Casimir operator | |
|---|---|
| Name | Casimir operator |
| Field | Theoretical physics |
| Introduced | 1931 |
| Introduced by | H. A. Kramers and Harish-Chandra (formalism); named after Hendrik Casimir |
Casimir operator
The Casimir operator is an element of the center of the universal enveloping algebra of a Lie algebra that acts as a scalar on each irreducible representation. It provides conserved quantities and classification labels for states in Quantum mechanics and Quantum field theory, playing a central role in analyzing spectra and symmetry constraints in models from atomic to particle physics.
In algebraic terms, a Casimir operator is any invariant constructed from the generators of a Lie algebra that commutes with all elements of that algebra. For a semisimple Lie algebra g with Killing form K, the quadratic Casimir is typically defined as C = g^{ij} X_i X_j where {X_i} are basis elements and g^{ij} is the inverse of K_{ij}. Physically, Casimir operators correspond to conserved observables associated with global symmetry groups such as SU(2), SU(3), SO(3), and the Poincaré group, and they appear in the classification of particles and multiplets in particle physics and atomic physics.
For a semisimple Lie algebra the center of its universal enveloping algebra is generated by algebraically independent Casimir operators equal in number to the rank of the algebra. The quadratic Casimir is the simplest nontrivial example; higher-order Casimirs involve symmetric invariant tensors of higher degree. Construction uses invariant bilinear forms (e.g., the Killing form) or trace forms from finite-dimensional representations (e.g., adjoint representation). Important mathematicians and physicists who developed the theory include Élie Cartan, Hermann Weyl, and Harish-Chandra; modern treatments reference the structure theory of semisimple Lie algebras and the Harish-Chandra isomorphism.
In quantum systems with Lie group symmetry, Casimir operators commute with the Hamiltonian when the Hamiltonian is invariant under the symmetry, leading to conserved quantities by Noether's theorem. In nonrelativistic quantum mechanics the angular momentum squared operator J^2 is the quadratic Casimir of so(3)/su(2), determining orbital and spin multiplets measured in spectroscopic experiments. In relativistic quantum field theory the Casimir invariants of the Poincaré group—mass squared and the Pauli–Lubanski scalar—label particle mass and spin in Wigner's classification of irreducible representations used in quantum electrodynamics and quantum chromodynamics.
For SU(2) with generators J_i satisfying [J_i,J_j]=i ε_{ijk} J_k, the quadratic Casimir is J^2=J_x^2+J_y^2+J_z^2; on an irreducible representation of spin j it takes eigenvalue j(j+1). For SO(N) and SO(3), Casimir operators derive from the antisymmetric generator matrices; explicit formulas depend on Dynkin labels and highest weights. For the Poincaré group ISO(1,3), there are two Casimirs: P_μ P^μ (mass squared) and W_μ W^μ where W_μ is the Pauli–Lubanski pseudovector; these classify Wigner particles. Practical computation often uses highest-weight methods, the Weyl character formula, and tools from representation theory such as Dynkin diagrams and Young tableau for SU(N).
Eigenvalues of Casimir operators distinguish inequivalent irreducible representations and are polynomials in the highest-weight components. For simple Lie algebras the value of the quadratic Casimir on an irreducible representation with highest weight λ is ( (λ,λ+2ρ) )/(θ,θ) times a normalization, where ρ is the Weyl vector and θ is the highest root. These eigenvalues enter the computation of operator product expansion coefficients in conformal field theory and anomalous dimensions in perturbative quantum field theory; they also determine energy level splitting in models with dynamical symmetry breaking. Tables of Casimir eigenvalues are standard in texts by Howard Georgi and Fulton and Harris.
Casimir operators yield selection rules in spectroscopy and particle decays by restricting allowed transitions between representations. In atomic physics, total angular momentum Casimirs constrain multiplet structures observed in atomic spectra and fine structure. In nuclear and particle physics, isospin and flavor SU(3) Casimirs classify hadrons and predict mass relations (e.g., Gell-Mann–Okubo type formulas). In scattering theory and integrable models such as the Heisenberg model or Bethe ansatz systems, commuting Casimir-like conserved charges ensure solvability. In gauge theory, Casimir factors (e.g., C_F, C_A) appear in renormalization group beta functions and in the computation of loop corrections in quantum chromodynamics.
Beyond the quadratic invariant, semisimple algebras admit independent higher-order Casimir operators corresponding to symmetric invariant tensors of degree >2; examples include cubic Casimirs for su(3) and quartic invariants for larger groups. In superalgebra contexts (e.g., Lie superalgebras used in supersymmetry) Casimir-like central elements are generalized to account for grading and atypical representations. Quantum group deformations (e.g., U_q(g)) modify Casimir elements leading to q-deformed invariants relevant in integrable systems and knot-theory via the Jones polynomial. Applications extend to modern topics such as classification of conformal primary fields in conformal field theory, spectral problems in AdS/CFT correspondence, and symmetry-based model building at institutes like CERN and collaborations such as LHC experiments.
Category:Representation theory Category:Lie algebras Category:Quantum mechanics Category:Quantum field theory