| SO(p,q) | |
|---|---|
| Name | SO(p,q) |
| Type | Lie group |
| Region served | Global |
| Founded | Classical mathematics |
| Key people | Élie Cartan; Hermann Weyl |
SO(p,q)
SO(p,q) is the special orthogonal group of a nondegenerate quadratic form with signature (p,q); it consists of real linear transformations preserving a bilinear form of signature (p,q) with determinant one. In Quantum physics and related mathematical physics, SO(p,q) organizes spacetime and internal symmetries, appearing in formulations of the Lorentz group, relativistic Quantum field theory, and classification of elementary particles. Its algebraic and representation-theoretic properties underpin conserved quantities arising from Noether's theorem and the construction of field equations.
SO(p,q) is defined as the subgroup of GL(n,ℝ) (with n = p+q) preserving a diagonal quadratic form diag(1,...,1,-1,...,-1) with p plus signs and q minus signs, and having determinant +1. Concretely, , where η denotes the signature matrix. SO(p,q) is a real, noncompact (when pq≠0) Lie group whose connected components depend on p and q; notable special cases include SO(3) (rotations in three dimensions) and SO(1,3) (proper orthochronous Lorentz group). The group has natural subgroups isomorphic to products of lower-rank orthogonal groups and admits Cartan decompositions and Iwasawa decompositions used in harmonic analysis and representation theory. Early structural classification follows from the work of Élie Cartan on real forms of complex simple Lie algebras.
The Lie algebra so(p,q) consists of real skew-symmetric matrices with respect to η and is a real form of the complex algebra so(n,C). Its root system and Dynkin diagram type depends on n and parity; for n≥3 these are classical simple Lie algebras of type B or D. Highest-weight theory, developed by Hermann Weyl and others, classifies finite-dimensional representations; infinite-dimensional unitary duals are subtler and central to physics. For physics applications, one studies both finite-dimensional tensor representations (used for fields) and infinite-dimensional principal series and discrete series representations studied by Harish-Chandra, I. M. Gelfand, and David Vogan. The universal enveloping algebra of so(p,q) contains Casimir operators whose eigenvalues label irreducible representations and appear as invariant differential operators in wave equations.
SO(p,q) and its connected subgroups organize spacetime symmetries and internal global symmetries in Quantum field theory (QFT). The proper orthochronous Lorentz group SO^+(1,3) is central to the Dirac equation and construction of relativistic fields; in higher-dimensional QFT and Kaluza–Klein theory one encounters SO(p,q) with p+q>4 as spacetime or r-symmetry groups. Gauge theories often employ compact forms such as SO(10) grand unified theories (GUTs) studied in particle physics, while noncompact real forms govern dualities in string theory and AdS/CFT correspondence. Conservation laws associated with SO(p,q)-symmetries are tied to Noether's theorem and the generators correspond to conserved charges (e.g., angular momentum and boost generators).
In quantum mechanics and QFT, physical symmetry operators are represented by (projective) unitary or antiunitary operators on a Hilbert space by the Wigner theorem. Compact groups like SO(3) admit finite-dimensional unitary representations, while noncompact SO(p,q) groups generally require infinite-dimensional unitary representations to act unitarily. The distinction matters for probability conservation and spectral properties of observables. For example, the proper Lorentz group SO(1,3) has no nontrivial finite-dimensional unitary representations; instead one uses its double cover SL(2,C) to build spinor representations and then realize unitary representations in induced modules, principal series, or discrete series when available. Studies by Eugene Wigner and George Mackey clarified induced representations and the role of little groups in particle classification.
SO(p,q) interfaces directly with spinors via its double covers: the Spin group Spin(p,q) and the associated Clifford algebra Cl(p,q). Spinor representations furnish fermionic fields such as the Dirac spinor and Weyl spinor in four-dimensional spacetime, and generalizations appear in higher-dimensional supersymmetry and supergravity models. The classification of elementary particles uses representations of the Poincaré group, whose Lorentz factor is SO(1,3), and employs little groups like SO(2) and ISO(2) to label massless and massive particle states (spin and helicity). Grand unified proposals and model building often embed the Standard Model gauge group in larger orthogonal groups such as SO(10) or SO(8), leveraging their representation theory for fermion families and anomaly cancellation.
Topologically, SO(p,q) has fundamental group isomorphic to Z for certain p,q combinations and may be disconnected; its universal cover is given by Spin(p,q), constructed via Clifford algebras. Covering groups are essential for lifting projective representations to linear ones so that spinors exist. The center and discrete subgroups determine possible global forms and selection rules in quantum theories. Casimir operators built from the invariant metric on so(p,q) generate central elements in the universal enveloping algebra; their eigenvalues distinguish irreducible representations and enter wave operators like the Klein–Gordon equation or higher-spin wave equations. Harmonic analysis on homogeneous spaces SO(p,q)/K (with K a maximal compact subgroup) underlies scattering theory and spectral decompositions in mathematical physics, with deep links to work at institutions such as Princeton University and Institute for Advanced Study where foundational developments in representation theory and QFT were advanced.
Category:Lie groups Category:Mathematical physics Category:Quantum field theory