Lorentz group

Lorentz group
source
Name	Lorentz group
Type	Lie group
Algebra	Lorentz algebra

Lorentz group. In physics and mathematics, particularly in the theories of special relativity and quantum field theory, the Lorentz group is the fundamental group of isometries of Minkowski spacetime. It consists of all linear transformations that preserve the Minkowski inner product, forming the cornerstone of relativistic invariance. The group's structure and its representations are essential for describing the behavior of fundamental particles and fields.

Definition and basic properties

The Lorentz group is defined as the group of all invertible linear transformations on four-dimensional spacetime that leave the Minkowski metric invariant. Formally, it is the indefinite orthogonal group O(1,3), a classical Lie group of dimension six. Its elements are matrices Λ satisfying the condition Λ^TηΛ = η, where η is the metric tensor with signature (1,3). The group is not compact and consists of four disconnected components, distinguished by the sign of the determinant and the direction of time. The component connected to the identity is the proper orthochronous Lorentz group, denoted SO⁺(1,3), which forms a normal subgroup of the full group. The study of its Lie algebra, generated by three rotation generators and three boost generators, is foundational in theoretical physics.

Representations

The representation theory of the Lorentz group is vast and critical for quantum mechanics. Its finite-dimensional representations are not unitary due to the group's non-compact nature, but they are crucial for classifying fields. These are labeled by pairs of half-integers (j₁, j₂), corresponding to the decomposition of its complexified Lie algebra into su(2) ⊕ su(2). Important examples include the (0,0) scalar representation, the (1/2,0) and (0,1/2) Weyl spinor representations, and the (1/2,1/2) vector representation describing fields like the electromagnetic field. For quantum field theory, the infinite-dimensional unitary representations, studied by Eugene Wigner in his classification of elementary particles, are essential. The Poincaré group, which extends the Lorentz group by translations, has representations corresponding to particles with specific mass and spin.

Classification of Lorentz transformations

Lorentz transformations are classified into four disjoint sets based on the sign of the determinant, det(Λ) = ±1, and the sign of the time-time component, Λ⁰₀ ≥ 1 or ≤ -1. Transformations with det(Λ) = +1 are called proper, while those with -1 are improper and involve spatial reflections. Transformations preserving the direction of time (Λ⁰₀ ≥ 1) are orthochronous. The intersection, the proper orthochronous transformations SO⁺(1,3), includes all rotations and boosts and forms the identity component. The other components are obtained by composing with the discrete symmetries: parity (spatial inversion), time reversal, and their combination PT symmetry. This classification is pivotal in understanding discrete symmetries in theories like the Standard Model.

The Lorentz group in physics

In special relativity, formulated by Albert Einstein, the Lorentz group is the set of symmetries of spacetime between inertial frames, with the speed of light being invariant. Its representations dictate the transformation properties of all physical quantities: scalars, vectors, tensors, and spinors. In quantum electrodynamics and the Standard Model, the gauge bosons like the photon and the W and Z bosons transform under specific representations. The requirement of Lagrangian invariance under the Lorentz group leads to conservation laws and constrains possible interaction terms. The group's universal covering group, SL(2,C), is fundamental for describing relativistic wave equations like the Dirac equation and Klein–Gordon equation.

Related groups and extensions

The Lorentz group is closely related to several other important groups in geometry and physics. Its double cover is the spin group Spin(1,3), which is isomorphic to SL(2,C), essential for describing fermionic fields. By including translations, one obtains the full Poincaré group, the symmetry group of Minkowski spacetime, whose representations correspond to particles. Further extension by scale transformations leads to the conformal group of spacetime, relevant in conformal field theory. In higher dimensions, analogous groups like O(1,d-1) appear in string theory and supergravity. The de Sitter group and anti-de Sitter group are also significant in cosmological models and the AdS/CFT correspondence.

Category:Lie groups Category:Special relativity Category:Quantum field theory