| O(p,q) | |
|---|---|
| Name | O(p,q) |
| Caption | Orthogonal group preserving a metric of signature (p,q) |
| Type | Lie group |
| Dimension | n*(n-1)/2} |
| Region | Mathematics |
| Related | SO(p,q), Spin group |
O(p,q)
O(p,q) is the orthogonal group of a real vector space equipped with a nondegenerate symmetric bilinear form of signature (p,q). It consists of linear transformations that preserve a quadratic form with p positive and q negative eigenvalues, and it underpins many symmetry considerations in Quantum Physics, notably in descriptions of spacetime symmetry, particle states, and conserved quantities.
O(p,q) is defined as the group of n×n real matrices A (with n = p+q) satisfying A^T η A = η, where η is a diagonal matrix with p entries +1 and q entries −1. As a classical Lie group, it is generally noncompact for pq ≠ 0 and has up to four connected components characterized by determinant ±1 and time-orientation sign. The subgroup with determinant +1 is SO(p,q). The group structure controls invariant bilinear forms and defines natural homogeneous spaces such as the pseudo-Riemannian Grassmannians and hyperbolic spaces studied in differential geometry and global analysis. Related mathematical objects include the orthogonal group over other fields, symmetric spaces, and the classification of real semisimple Lie groups by Élie Cartan.
The Lie algebra so(p,q) consists of real skew-symmetric matrices with respect to η, i.e., X satisfying X^T η + η X = 0. It is a real form of the complex Lie algebra so(n,ℂ) and admits Cartan decompositions and root systems that mirror those of classical types B and D. Representation theory of so(p,q) includes finite-dimensional algebraic representations and infinite-dimensional unitary representations relevant to physics. Induced representations, highest-/lowest-weight modules, and principal series come into play in harmonic analysis on O(p,q), with concrete study in works by Hermann Weyl, Eugene Wigner, and later mathematical physicists. Unitary dual classification for noncompact groups often cites results from Harish-Chandra theory and the Langlands program as applied to real groups.
O(p,q) appears as a global symmetry group preserving bilinear forms that define causal structure and conserved bilinear currents. In Quantum Field Theory the subgroup O(1,3) and its connected components relate to the Lorentz symmetry of special relativity, while larger or different signatures model internal symmetry spaces or effective metrics in condensed matter systems. Conservation laws associated with continuous O(p,q) symmetries are encoded by Noether's theorem and can constrain S-matrix elements in perturbative constructions developed in the tradition of Richard Feynman and Julian Schwinger. The interplay between O(p,q) and gauge groups such as SU(N) or U(1) is central in model building, effective field theories, and anomaly analysis studied in work by Gerard 't Hooft and Stephen Hawking on symmetry breaking and global structure.
In relativistic quantum mechanics, O(1,3) and its covering groups classify single-particle states by mass and spin via induced representations a la Wigner; the Poincaré group uses Lorentz subgroup actions on momentum space. Spinor constructions for Dirac and Majorana fields require passage to the Spin group Spin(1,3) ≅ SL(2,ℂ) and its spinor representations, linking so(1,3) to the Clifford algebra Cl(1,3). Representations of O(p,q) for other signatures model generalized dispersion relations, orthogonal symmetries in multi-time frameworks, and internal degrees of freedom in relativistic wave equations. Scholarly traditions from Paul Dirac, Eugene Wigner, and later developments in relativistic quantum information exploit these structures.
Casimir operators constructed from generators of so(p,q) produce central elements in the universal enveloping algebra and label irreducible representations in both finite and infinite-dimensional settings. The quadratic Casimir corresponds to the invariant metric η and yields conserved quantities like total angular momentum or quadratic invariants of energy-momentum tensors. Higher-order Casimirs enter harmonic analysis on homogeneous spaces and spectral theory, with applications to eigenvalue problems on hyperbolic manifolds and scattering theory. Studies by Harish-Chandra, I. M. Gelfand, and later researchers connect Casimir eigenvalues to unitary representation classification, Plancherel measure, and correlation functions in conformal and quantum field theories.
Topologically, O(p,q) has nontrivial fundamental group and may be disconnected; its universal cover and double covers produce groups like Spin(p,q), which are required to define spinor fields on manifolds with given metric signature. The lifting problem — when an O(p,q)-principal bundle admits a Spin structure — is governed by Stiefel–Whitney classes and is fundamental in coupling fermions to gravity in general relativity and quantum gravity approaches. In particle physics, the passage to Spin or pin groups ensures proper statistics and projective unitary representations per Wigner’s classification; this is central to constructions in string theory and supersymmetric models elaborated at institutes such as CERN and universities including Princeton University and University of Cambridge. Topological and index-theoretic results by Atiyah–Singer play a role when analyzing Dirac operators tied to Spin(p,q) geometry.
Category:Lie groups Category:Representation theory Category:Quantum mechanics