| Lorentz group | |
|---|---|
| Name | Lorentz group |
| Native name | O(1,3) |
| Type | Lie group |
| Region | Minkowski space |
| Formed | 1904 |
| Key people | Hendrik Lorentz; Albert Einstein; Hermann Minkowski |
| Products | Special relativity; Quantum field theory |
Lorentz group
The Lorentz group is the group of linear transformations that preserve the spacetime interval of Minkowski space; it underpins the invariance of physical laws under boosts and rotations in Special relativity. In Quantum physics and Quantum field theory the Lorentz group governs the classification of particles, the form of relativistic wave equations such as the Dirac equation and the allowed unitary representations describing particle states. Its role secures the consistency of causality, energy–momentum relations, and conservation laws in relativistic quantum systems.
The Lorentz group, commonly denoted O(1,3) or SO(1,3) for connected components, consists of all linear transformations on four-dimensional Minkowski space that leave the metric signature (−,+,+,+) invariant. Physically relevant subgroups include the proper orthochronous Lorentz group, often written SO^+(1,3), which contains continuous rotations and boosts and excludes parity and time reversal. The group directly expresses the symmetry of the Maxwell equations, the relativistic dispersion relation E^2 = p^2c^2 + m^2c^4, and underlies empirical phenomena such as time dilation and Lorentz contraction observed in experiments at CERN and other particle accelerators.
As a non-compact six-parameter Lie group, the Lorentz group is locally isomorphic to SL(2,C), the group of complex 2×2 matrices with unit determinant. Its Lie algebra so(1,3) splits into rotation generators J_i and boost generators K_i with commutation relations characteristic of relativistic kinematics. Finite-dimensional non-unitary representations are labeled by two half-integers (m,n) corresponding to the SL(2,C) double cover; these determine the transformation behavior of fields such as scalars, four-vectors, and higher-rank tensors. Infinite-dimensional unitary representations, studied by Eugene Wigner in the context of the Poincaré group, classify physical particle states by mass and spin (or helicity) and are central to scattering theory and the S-matrix program.
The Lorentz group is the symmetry group of Special relativity and, together with spacetime translations, forms the Poincaré group. This structure preserves the Minkowski metric and the light-cone, ensuring invariant causal ordering and consistent kinematics for relativistic particles. The group's action on four-vectors such as the four-momentum links to conserved quantities via Noether's theorem when combined with spacetime translation invariance. Historical developments by Hendrik Lorentz, Albert Einstein, and Hermann Minkowski established the geometric interpretation that frames modern relativistic dynamics and relativistic quantum theories.
In Quantum field theory (QFT), fields are assigned transformation laws under the Lorentz group; these constraints determine admissible Lagrangians and interaction terms, enforcing locality and renormalizability in models like Quantum electrodynamics (QED) and Quantum chromodynamics (QCD). The Lorentz symmetry restricts counterterms in perturbative expansions and organizes operators in effective field theories such as the Standard Model of particle physics. Experimental tests of Lorentz invariance are performed in precision measurements at Fermilab, tests of CPT via neutral meson systems at KEK and SLAC, and searches for violations predicted by proposals like the Standard-Model Extension.
Spinorial representations arise from the double cover SL(2,C); two-component Weyl spinors transform under (1/2,0) or (0,1/2), while the four-component Dirac spinor combines both chiralities and furnishes the representation used in the Dirac equation for fermions. Majorana spinors impose a reality condition relevant in neutrino physics and supersymmetry. The structure of these representations determines selection rules, allowed mass terms, and the behavior under discrete symmetries like parity and charge conjugation. Foundational contributors include Paul Dirac and Ettore Majorana; modern applications extend to supersymmetry and spin networks in approaches to quantum gravity.
While the continuous Lorentz symmetry is fundamental, discrete transformations—parity (P), charge conjugation (C), and time reversal (T)—act as automorphisms of the Lorentz algebra when combined with internal symmetries. The CPT theorem, proven within the axioms of local QFT by scholars such as Gerhard Lüders and Julian Schwinger, guarantees invariance under the combined CPT operation for Lorentz-invariant, local, and unitary theories. Spontaneous symmetry breaking can leave Lorentz symmetry intact while breaking internal gauge symmetries (as in the Higgs mechanism), whereas explicit Lorentz violation is tightly constrained by astrophysical observations and terrestrial experiments, motivating high-precision programs at institutions like NIST and observatories testing Lorentz invariance violation proposals.
Lorentz symmetry shapes the kinematics and dynamics of scattering processes: invariant phase space measures, Mandelstam variables, and partial-wave decompositions rely on its structure. The classification of asymptotic states via Wigner representations informs the construction of S-matrix elements used in collider physics at ATLAS and CMS detectors. Conserved currents associated with Lorentz symmetry—energy, momentum, and angular momentum—arise through Noether's theorem and are central to selection rules in decay processes and the computation of cross sections in perturbative QFT. Practical computational frameworks, including spinor-helicity methods and Lorentz-covariant regularization schemes, exploit group properties to simplify amplitudes and ensure manifest covariance.
Category:Quantum physics Category:Lie groups