| Lorentz transformation | |
|---|---|
| Name | Lorentz transformation |
| Related | Special relativity, Lorentz group |
Lorentz transformation
The Lorentz transformation is the linear map between space and time coordinates of events as measured in two inertial frames moving at constant relative velocity, preserving the invariant interval of Minkowski space. It underlies Special relativity and is crucial for reconciling the symmetries of spacetime with the principles of Quantum Physics, constraining relativistic wave equations and scattering amplitudes. Its structure determines how fields, particles, and observables transform under boosts and rotations in both Relativistic quantum mechanics and Quantum field theory.
The Lorentz transformation formalizes the invariance of the speed of light postulated by Albert Einstein in his 1905 paper on Special relativity, and it replaces Galilean transformations at high relative velocities. It connects physical quantities measured in laboratories such as CERN or the SLAC and ensures causality and frame-independent predictions for processes studied in particle physics experiments. In quantum contexts, Lorentz symmetry constrains allowed particle types, dictates selection rules in scattering studied at facilities like the Large Hadron Collider, and enforces the structure of commutation relations in Quantum Electrodynamics and other QFT models.
In four-dimensional spacetime with coordinates x^μ = (ct, x, y, z), a Lorentz transformation is any linear map Λ^μ{}_{ν} satisfying Λ^T η Λ = η where η is the Minkowski metric diag(−1,1,1,1). The family includes spatial rotations and boosts parameterized by rapidity φ or velocity v, with standard boost along x: t' = γ(t − vx/c^2), x' = γ(x − vt), y' = y, z' = z where γ = 1/√(1 − v^2/c^2). Infinitesimal generators form the Lie algebra so(1,3) with six generators: three rotation generators J_i and three boost generators K_i satisfying 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 algebraic relations are central to constructing unitary representations used in Wigner's classification of particles.
Lorentz transformations follow from two main postulates: the principle of relativity (laws of physics are identical in all inertial frames) and the constancy of the speed of light in vacuum. From these, one derives linearity by homogeneity and isotropy of spacetime and the preservation of the spacetime interval s^2 = −c^2t^2 + x^2 + y^2 + z^2. Historical derivations involve the work of Hendrik Lorentz, Henri Poincaré, and Albert Einstein; modern derivations use group-theoretic axioms or symmetry principles employed in formulating relativistic quantum theories such as the Dirac equation and Klein–Gordon equation.
The set of Lorentz transformations forms the Lorentz group O(1,3), with connected component of the identity SO^+(1,3) often called the proper orthochronous Lorentz group. It is a non-compact, six-parameter Lie group locally isomorphic to SL(2,C) via a two-to-one homomorphism; this covering relation is used to obtain spinor representations. Discrete operations like parity P and time reversal T extend the group to include improper transformations. The Lorentz group's structure constrains conserved currents via Noether's theorem in relativistic Lagrangian field theories and underlies classification schemes used by groups such as the IUPAP for particle properties.
In relativistic quantum mechanics, Lorentz invariance dictates the form of single-particle wave equations: the Dirac equation for spin-1/2 fermions, the Klein–Gordon equation for scalar particles, and the Proca equation for massive spin-1 fields. In Quantum field theory, fields are constructed to transform under representations of the Lorentz group to ensure covariant S-matrix elements and renormalizable interactions in theories such as Quantum Electrodynamics and Quantum Chromodynamics. Lorentz symmetry guides the construction of Lagrangians at effective field theory programs like Chiral perturbation theory and constrains operator product expansions used in perturbative calculations, including those implemented in computational frameworks like Feynman diagram techniques and software such as MADGRAPH.
Representations of the Lorentz group classify particles by mass and spin through Wigner's classification. Finite-dimensional non-unitary representations of SO(1,3) are labeled by pairs (j1,j2) corresponding to SL(2,C) Weyl spinors; the Dirac spinor combines (1/2,0) and (0,1/2). Spinor transformations under boosts mix components via SL(2,C) matrices, leading to phenomena like Thomas precession relevant for atomic systems studied in precision spectroscopy experiments. Higher-spin representations describe vector and tensor fields used to model gauge bosons in the Standard Model; gauge symmetry together with Lorentz invariance produces constraints implemented in renormalization procedures by groups such as the American Physical Society community.
Lorentz invariance has been tested to high precision in laboratories and astrophysical observations. Experiments include Michelson–Morley–type tests, clock comparison tests using atomic clocks developed by institutions like NIST, high-energy particle collision experiments at CERN and Fermilab, and observations of cosmic rays and gamma-ray bursts by observatories such as Fermi Gamma-ray Space Telescope. Violations of Lorentz symmetry are constrained by the Standard-Model Extension framework and searches for anisotropies or energy-dependent speed of light place stringent limits. Empirically confirmed consequences include time dilation observed in muon lifetime measurements at accelerator facilities and relativistic Doppler shifts essential to GPS systems engineered by organizations including United States Department of Defense and commercial navigation providers.
Category:Special relativity Category:Quantum mechanics Category:Mathematical physics