| Lorentz group | |
|---|---|
| Name | Lorentz group |
| Type | Lie group |
| Lie algebra | so(1,3) |
| Universal cover | SL(2,ℂ) |
| Applications | Special relativity; quantum field theory |
Lorentz group
The Lorentz group is the group of linear transformations of four-dimensional spacetime that preserve the Minkowski metric. It encodes the symmetry between inertial frames in Special relativity and underlies the classification of relativistic particles and fields in Quantum field theory and relativistic quantum mechanics. Its algebraic structure and representations (notably the spinor representations) are central to constructions such as the Dirac equation and the Standard Model of particle physics.
The Lorentz group O(1,3) is the group of 4×4 real matrices Λ satisfying Λ^T η Λ = η, where η = diag(1,−1,−1,−1) is the Minkowski metric. Its connected component of the identity is the proper orthochronous Lorentz group SO^+(1,3), often denoted SO(1,3)^↑. The corresponding Lie algebra is so(1,3), a six-dimensional real algebra generated by three rotation generators J_i and three boost generators K_i with commutation relations [J_i,J_j]=i ε_{ijk} J_k, [J_i,K_j]=i ε_{ijk} K_k, [K_i,K_j]=−i ε_{ijk} J_k. These structure constants reflect the non-compactness of the group and the hyperbolic geometry of boosts. The group has four connected components distinguished by determinant ±1 and time-orientation reversal; discrete symmetries parity (P) and time reversal (T) extend the connected group to the full O(1,3).
Finite-dimensional representations of SO(1,3) are non-unitary due to the non-compactness of boosts; however, complexification yields the isomorphism so(1,3)_ℂ ≅ sl(2,ℂ)⊕sl(2,ℂ), enabling classification by two half-integers (j_+, j_−). This (j_+, j_−) labeling produces vector, tensor, and spinor representations such as the (1/2,0) and (0,1/2) Weyl spinors. The universal (double) cover of SO^+(1,3) is the group SL(2,ℂ), which admits projective and spin representations; for unitary representation theory relevant to quantum states one typically uses the unitary representations of the Poincaré group obtained via the method of induced representations by Wigner and others. For quantum systems on curved spacetimes, local Lorentz symmetry is implemented via principal bundles with structure group SO(1,3) or its spin cover Spin(1,3).
Lorentz symmetry constrains the form of relativistic wave equations and the allowed interaction terms in Lagrangians. Demanding invariance under SO^+(1,3) (and under discrete P, C, T in specific combinations) dictates the tensorial type of fields in QED, QCD, and the Electroweak interaction. In quantum field theory the requirement of Lorentz covariance combines with locality and causality to yield the spin–statistics connection and CPT theorem proved by Lüders, Pauli, and others. The pattern of symmetry breaking (e.g., spontaneous breaking of internal symmetries) must still respect the underlying Lorentz symmetry unless a specific mechanism produces Lorentz violation, which is tightly constrained by experiments at CERN, Fermilab, and astrophysical observations.
Spinors are objects transforming under the (fundamental) SL(2,ℂ) representations; left- and right-handed Weyl spinors transform as (1/2,0) and (0,1/2) respectively. The Dirac spinor combines these into the (1/2,0)⊕(0,1/2) representation and leads to the Dirac equation, which describes spin-1/2 fermions such as the electron and predicts antiparticles. Wigner's classification uses unitary irreducible representations of the Poincaré group (the semidirect product of translations with SO(1,3)) to label particles by mass and spin (or helicity for massless states), a cornerstone for particle classification in the Standard Model and in analyses at experimental facilities such as LHC and SLAC.
Noether's theorem associates continuous Lorentz symmetries with conserved quantities: spatial rotations yield conservation of angular momentum, while boosts relate to center-of-mass motion and the relativistic moment of energy–momentum. In quantum field theory, Lorentz invariance imposes selection rules on S-matrix elements and correlators computed in methods like perturbation theory and functional integrals. Proposed mechanisms of Lorentz symmetry breaking—whether explicit or spontaneous—are studied in the context of effective field theories such as the Standard-Model Extension and in searches for deviations using precision tests by Atomic clocks and high-energy astrophysical observations. Robust experimental bounds currently make large-scale Lorentz violation unlikely.
The Lorentz group embeds naturally in larger symmetry groups. The Poincaré group ISO(1,3) includes translations and is the full symmetry group of Minkowski spacetime; its representations underlie particle physics and scattering theory via Wigner's method. The conformal group in four dimensions extends Lorentz symmetry with dilations and special conformal transformations, producing the group SO(2,4); conformal symmetry is central to conformal field theory and the AdS/CFT correspondence studied in mathematical physics and string theory. Other extensions include supersymmetric algebras that combine Lorentz generators with fermionic supercharges (e.g., super-Poincaré algebra), which structure theories such as supersymmetry and constrain model building in high-energy physics.
Category:Lie groups Category:Mathematical physics Category:Quantum field theory