| Lie group | |
|---|---|
| Name | Lie group |
| Type | Continuous symmetry group |
| Field | Mathematics; Quantum Physics |
| Studied by | Sophus Lie; Élie Cartan |
| Related | Lie algebra; Representation theory |
Lie group
A Lie group is a group that is also a differentiable manifold, where the group operations are smooth maps. Lie groups provide the mathematical formalism for continuous symmetries that underlie much of modern Quantum Physics and Quantum field theory, enabling classification of particles, conservation laws via Noether's theorem, and construction of operator algebras used in quantum models.
A Lie group G is a set with a group law and a compatible smooth manifold structure such that the multiplication map G × G → G and the inversion map G → G are smooth. The tangent space at the identity element yields a Lie algebra g, encoding infinitesimal generators of continuous symmetries. Fundamental properties include connectedness, compactness, and the existence of one-parameter subgroups parameterized by elements of g via the exponential map exp: g → G. Key contributors to the theory include Sophus Lie and Élie Cartan, and foundational texts include works by Hermann Weyl and Claude Chevalley.
Important examples are matrix groups: GL(n, R), GL(n, C), SL(n, C), SL(2, C), SU(n), SO(n), Sp(n), and the Euclidean group E(n). Compact Lie groups such as SU(2) and SU(3) are central to quantum applications because unitary representations are well-behaved and decompose into direct sums of irreducibles. Noncompact groups like SL(2, R) and the Lorentz group SO(3,1) arise in relativistic quantum theories. Classification of simple Lie groups is via Dynkin diagrams and the Cartan–Killing classification into families A_n, B_n, C_n, D_n and exceptional groups G2, F4, E6, E7, E8; key names include Élie Cartan and Wilhelm Killing.
Abelian Lie groups (e.g., the circle group U(1] and the additive group R^n) produce commuting observables and conserved quantities in quantum systems. Simple and semisimple groups govern non-abelian gauge theories such as Quantum Chromodynamics (based on SU(3)]) and the electroweak sector (SU(2) × U(1)).
The Lie algebra g associated to a Lie group provides generators satisfying commutation relations [X,Y] = XY − YX; structure constants appear in operator algebra realizations on Hilbert spaces. Representation theory classifies unitary and finite-dimensional representations: for compact groups the Peter–Weyl theorem and complete reducibility hold, while for noncompact groups one studies unitary duals using the work of Harish-Chandra and concepts like the universal enveloping algebra (introduced by Nathan Jacobson and formalized in part by I. N. Herstein). Highest-weight theory and the classification of irreducible representations by dominant weights are fundamental in constructing quantum states labeled by spin, isospin, flavor, and color. Seminal contributors include Hermann Weyl, Élie Cartan, Harish-Chandra, and George Mackey.
Lie groups encode the continuous symmetries of canonical quantum systems: spatial rotations via SO(3) and its double cover SU(2), translations via the additive group R^n, and spacetime symmetries via the Poincaré group and the Lorentz group SO(3,1). In Quantum field theory, internal symmetry groups such as SU(3), SU(2), and U(1) define gauge interactions in the Standard Model, formulated by developers like Murray Gell-Mann, Sheldon Glashow, Steven Weinberg, and Abdus Salam. Spontaneous symmetry breaking, analyzed by Yoichiro Nambu and others, uses Lie group structure to determine Goldstone modes and Higgs mechanisms.
Angular momentum in quantum mechanics is described by the Lie algebra of SU(2), with ladder operators and quantized spin values underpinning atomic and nuclear spectra. Gauge symmetries are modeled by principal bundles with structure group a Lie group; gauge fields correspond to connections on such bundles, as in Yang–Mills theory and Quantum Chromodynamics. Particle classification uses representations of flavor groups (e.g., the Eightfold Way by Murray Gell-Mann and Yuval Ne'eman) and color SU(3), while grand unified theories propose larger Lie groups such as SU(5), SO(10), or exceptional groups like E6 to unify interactions.
Topological properties of Lie groups affect quantum representations: nontrivial fundamental groups π1(G) lead to nontrivial covering groups (e.g., Spin group as the double cover of SO(n)). Projective representations of a group G correspond to linear representations of a central extension; this formalism explains half-integer spin via the double cover Spin(3) ≅ SU(2). Methods from algebraic topology, including the study of homotopy groups and characteristic classes (e.g., Chern classes), are used in classifying possible quantum phases and anomalies in gauge theories (see work by Edward Witten).
Integration over Lie groups uses the Haar measure for constructing invariant inner products and path integrals in quantum theories. Characters are class functions that encode traces of representation matrices and enter the study of partition functions and selection rules; character formulae include the Weyl character formula developed by Hermann Weyl and later generalizations. Highest-weight theory and Verma modules underpin the algebraic construction of representations; tools include the Casimir operator, roots and weights, Cartan subalgebras, and Weyl groups. Analytical techniques draw on harmonic analysis on groups, as advanced by Israel Gelfand and George Mackey, and play roles in scattering theory, spectral analysis, and the representation-theoretic approach to conformal field theory (connections to Virasoro algebra and Kac–Moody algebra).
Category:Lie groups Category:Quantum mechanics Category:Mathematical physics