Lorentz group

Lorentz group
Original: Jakob.scholbach Vector: Pbroks13 · CC BY-SA 3.0 · source
Name	Lorentz group
Type	Lie group
Related	Poincaré group; conformal group; SO(3,1); SL(2,C)

Lorentz group The Lorentz group is the group of linear transformations preserving a nondegenerate quadratic form of signature (3,1) on a four-dimensional real vector space. It underlies the kinematical symmetries of special relativity and appears throughout theoretical physics, representation theory, and differential geometry. Mathematically it is a noncompact real Lie group closely related to rotation groups in three dimensions and to complex matrix groups.

Definition and algebraic structure

The Lorentz group is defined as the set of linear transformations on Minkowski space that preserve the Minkowski metric; one often denotes a connected component by SO(3,1)^↑ and studies its Lie algebra so(3,1). Important algebraic subgroups and structures include the spatial rotation subgroup isomorphic to SO(3), boost subalgebras generated by hyperbolic rotations, and the maximal compact subgroup isomorphic to SO(3). The Cartan decomposition, root system, and Killing form of so(3,1) can be analyzed alongside classical families such as the A, B, C, D series studied by Élie Cartan, Wilhelm Killing, and Weyl. The classification of real semisimple Lie algebras by Élie Cartan and the structure theory developed by Harish-Chandra elucidate the algebraic structure, while explicit matrix realizations connect to groups like SL(2,C), O(3,1), and GL(4,R). Connections to algebraic groups are explored in the work of Chevalley and Borel.

Representations and classification

Unitary and nonunitary representations of the Lorentz algebra were classified by pioneering work of Eugene Wigner and Hermann Weyl; Wigner’s method of induced representations classifies continuous unitary representations of semidirect product groups like the Poincaré group. Finite-dimensional nonunitary representations are labeled by pairs of SU(2) spins via isomorphisms to complexified algebras, while infinite-dimensional unitary principal series, discrete series, and complementary series representations were developed by Harish-Chandra, I. M. Gelfand, Mikhail Naimark, and Vladimir Bargmann. In particle physics contexts, representations correspond to particle spin and helicity, with classification schemes used in works by Steven Weinberg, Richard Feynman, and Paul Dirac. Branching rules to subgroups like SO(3) and induction from little groups studied by Wigner and Mackey give the spectral decomposition of physical fields and scattering states.

Physical applications in relativity and field theory

In special relativity and relativistic field theory the Lorentz group determines invariance properties of spacetime intervals, Maxwell’s equations, and the Dirac equation; these applications appear in foundational texts by Albert Einstein, James Clerk Maxwell, Paul Dirac, and Hendrik Lorentz. Electromagnetic field tensors transform under the adjoint action, while spinor fields transform under spinorial covers used in quantum electrodynamics and the Standard Model developed by Sheldon Glashow, Steven Weinberg, Abdus Salam, and implemented in experiments by collaborations like CERN detectors. Lorentz covariance constrains Lagrangians in classical field theory treated by Julian Schwinger, Richard Feynman, and Gerard 't Hooft. In cosmology models by Alexander Friedmann and Georges Lemaître local Lorentz invariance remains a principle even when global symmetries are broken, with observational tests undertaken by NASA missions and ground experiments like those of Michelson–Morley type refined by Kenneth Nordtvedt and others.

Relationship to Poincaré and conformal groups

The Lorentz group is the homogeneous part of the Poincaré group, which adds translations to form the full isometry group of Minkowski spacetime as treated in the work of Poincaré and Minkowski. Conformal extensions that preserve null cones lead to the conformal group in four dimensions studied by Henri Poincaré, Élie Cartan, and applied in conformal field theory by Alexander Polyakov, John Cardy, and Ed Witten. The embedding of SO(3,1) into larger symmetry groups such as SO(4,2) and its role in AdS/CFT correspondence appears in research by Juan Maldacena, Edward Witten, and Maldacena’s collaborators. Symmetry breaking patterns and little groups for massless and massive representations connect to the work of Wigner and usage in scattering theory developed by Richard Feynman and Gerard 't Hooft.

Covering groups and spinor representations

The universal covering group of the connected Lorentz group is isomorphic to SL(2,C), which provides two-to-one coverings and spinor representations introduced in physics by Paul Dirac and mathematically framed by Élie Cartan. Weyl spinors, Majorana spinors, and Dirac spinors arise from distinct SL(2,C) modules studied by Ettore Majorana and incorporated into quantum field theory by Pascual Jordan, Paul Dirac, and Eugene Wigner. Spin statistics and CPT theorems linking Lorentz structure to particle behavior were formalized by Wolfgang Pauli, Gerard 't Hooft, and Julian Schwinger. Mathematical constructions using Clifford algebras and Pin and Spin groups were elaborated by Cartan, Claude Chevalley, and Atiyah with applications across geometry and index theory developed by Michael Atiyah and Isadore Singer.

Historical development and key contributors

The concept evolved from classical electrodynamics and the Lorentz transformations of Hendrik Lorentz and was given geometrical form by Hermann Minkowski and dynamical interpretation by Albert Einstein. Group-theoretic and representation-theoretic foundations were established by Élie Cartan, Hermann Weyl, Eugene Wigner, and Harish-Chandra. Subsequent mathematical formalism and applications to quantum theory involved Paul Dirac, Richard Feynman, Julian Schwinger, Gerard 't Hooft, and contemporary contributors like Edward Witten and Juan Maldacena. Experimental validation of Lorentz invariance and tests of its limits were carried out in precision experiments influenced by Albert A. Michelson, Edward Morley, Kenneth Nordtvedt, and collaborations at institutions such as CERN and National Institute of Standards and Technology.

Category:Lie groups