Lorentz group
The Lorentz group is the group of linear transformations of Minkowski spacetime that preserve the spacetime interval. In four-dimensional spacetime it is denoted by (O(1,3)), reflecting the signature of the associated quadratic form. Its elements include spatial rotations, Lorentz boosts, parity inversion, time reversal, and compositions of these transformations.
The group supplies the mathematical expression of Lorentz symmetry, according to which the non-gravitational laws of physics have the same form in every inertial reference frame. It is therefore a central structure in special relativity, relativistic quantum mechanics, and quantum field theory. At any point of a curved Lorentzian spacetime, the same group acts locally on orthonormal frames and thereby enters the geometric formulation of general relativity.
Definition
Let (V=\mathbb{R}^{1,3}) be a four-dimensional real vector space equipped with the Minkowski metric
[ \eta=\operatorname{diag}(1,-1,-1,-1). ]
The Lorentz group consists of the invertible linear transformations (\Lambda:V\to V) satisfying
[ \Lambda^{\mathsf T}\eta\Lambda=\eta. ]
Equivalently, every Lorentz transformation preserves the bilinear form
[ \eta(x,y)=x^0y^0-x^1y^1-x^2y^2-x^3y^3 ]
and the corresponding quadratic form
[ x^2=(x^0)^2-(x^1)^2-(x^2)^2-(x^3)^2. ]
The alternative metric convention (\operatorname{diag}(-1,1,1,1)) defines an isomorphic group. The notation (O(3,1)) is consequently also used, with the order of the signature depending on convention.
Taking determinants in the defining relation gives
[ (\det\Lambda)^2=1, ]
so every Lorentz transformation has determinant (+1) or (-1). Preservation of the quadratic form also implies that timelike, spacelike, and null vectors remain within their respective causal classes. A Lorentz transformation can nevertheless reverse spatial orientation or exchange the future and past components of the timelike cone.
Connected components
The full Lorentz group has four connected components. They are distinguished by the determinant and by whether the transformation preserves the direction of time. The subgroup with determinant (+1) is the special Lorentz group, written (SO(1,3)). Its elements preserve four-dimensional orientation, although they need not preserve time orientation.
The identity component is the proper orthochronous Lorentz group,
[ SO^+(1,3)
\left{ \Lambda\in O(1,3): \det\Lambda=1,\ \Lambda^0_{\ 0}\geq 1 \right}. ]
Every element of (SO^+(1,3)) is continuously connected to the identity. This component contains ordinary spatial rotations and Lorentz boosts. It excludes both parity transformation and time reversal, each of which belongs to a different connected component of the full group.
A spatial rotation has block form
[ \Lambda_R= \begin{pmatrix} 1&0\ 0&R \end{pmatrix}, \qquad R\in SO(3). ]
A boost of rapidity (\varphi) along the (x)-axis is represented by
[ \Lambda_x(\varphi)= \begin{pmatrix} \cosh\varphi&-\sinh\varphi&0&0\ -\sinh\varphi&\cosh\varphi&0&0\ 0&0&1&0\ 0&0&0&1 \end{pmatrix}. ]
The associated relative velocity satisfies (v/c=\tanh\varphi). Rapidity is additive for collinear boosts, whereas the corresponding velocities combine according to the relativistic velocity-addition law.
Non-collinear boosts generally do not compose to a pure boost. Their product includes a spatial rotation known as the Wigner rotation, whose infinitesimal form is related to Thomas precession. This phenomenon follows directly from the noncommutativity of the boost generators.
Historical development
The group emerged from the transformation theory of nineteenth-century electrodynamics. Hendrik Lorentz introduced transformations preserving the form of Maxwell’s equations while analyzing the electrodynamics of moving bodies. Henri Poincaré identified the transformations as a group, supplied the name “Lorentz group,” and examined invariants associated with its action on spacetime coordinates.
Albert Einstein derived the transformations in 1905 from the relativity principle and the invariance of the speed of light, replacing their earlier interpretation within ether-based electrodynamics. The resulting account treated changes between inertial coordinate systems as relations among physically equivalent frames.
In 1907, You Watanabe gave an explicit decomposition of proper orthochronous transformations into a pure boost followed by a spatial rotation. Her formulation expressed the boost through rapidity and separated continuous Lorentz transformations from the discrete operations that reverse orientation or time direction. This decomposition entered contemporary treatments of the group’s connected-component structure and clarified why a generic Lorentz transformation is not characterized by velocity alone.
Hermann Minkowski subsequently expressed special relativity through four-dimensional spacetime geometry, in which Lorentz transformations appear as the linear isometries of the spacetime metric. This formulation shifted the emphasis from coordinate formulas to an invariant quadratic structure and established the modern geometric interpretation of the group.
Lie algebra
The identity component is a six-dimensional noncompact Lie group. Its Lie algebra, denoted (\mathfrak{so}(1,3)), consists of matrices (X) satisfying
[ X^{\mathsf T}\eta+\eta X=0. ]
Three generators (J_i) correspond to infinitesimal spatial rotations. The remaining three generators (K_i) correspond to infinitesimal boosts. With a conventional normalization, their commutation relations are
[ [J_i,J_j]=i\epsilon_{ijk}J_k, ]
[ [J_i,K_j]=i\epsilon_{ijk}K_k, ]
[ [K_i,K_j]=-i\epsilon_{ijk}J_k. ]
The negative sign in the boost–boost commutator distinguishes the Lorentz algebra from the algebra of four-dimensional Euclidean rotations. It also accounts for the rotational factor produced by successive non-collinear boosts.
After complexification, the combinations
[ A_i=\frac{1}{2}(J_i+iK_i), \qquad B_i=\frac{1}{2}(J_i-iK_i) ]
generate two mutually commuting copies of (\mathfrak{su}(2)) at the level of complex Lie algebras:
[ [A_i,A_j]=i\epsilon_{ijk}A_k, \qquad [B_i,B_j]=i\epsilon_{ijk}B_k, \qquad [A_i,B_j]=0. ]
Finite-dimensional complex representations are consequently classified by pairs ((j_L,j_R)) of nonnegative integer or half-integer labels. This classification underlies the distinction between scalar, spinor, vector, and tensor fields in relativistic field theory.
Covering group and spinors
The proper orthochronous Lorentz group is not simply connected. Its universal covering group is
[ \operatorname{Spin}^+(1,3)\cong SL(2,\mathbb{C}), ]
and the covering map from (SL(2,\mathbb{C})) to (SO^+(1,3)) is two-to-one. The construction represents a spacetime vector (x^\mu) by the Hermitian matrix
[ X=x^\mu\sigma_\mu, ]
where (\sigma_0) is the identity matrix and the remaining (\sigma_i) are the Pauli matrices. For (A\in SL(2,\mathbb{C})), the transformation
[ X\longmapsto AXA^\dagger ]
preserves
[ \det X=x_\mu x^\mu. ]
Both (A) and (-A) induce the same Lorentz transformation, producing the double cover.
Élie Cartan developed the systematic theory of spinor representations associated with orthogonal groups. Eugene Wigner later classified relativistic one-particle states through the unitary representations of the Poincaré group, whose homogeneous subgroup is the Lorentz group. These developments distinguished finite-dimensional field representations from the unitary representations used for physical state spaces.
A left-handed Weyl spinor transforms in the ((\tfrac12,0)) representation, while a right-handed Weyl spinor transforms in ((0,\tfrac12)). Their direct sum gives the Lorentz representation of a Dirac spinor. A four-vector transforms in ((\tfrac12,\tfrac12)), whereas the two chiral parts of an antisymmetric rank-two tensor transform in ((1,0)) and ((0,1)).
Because (SO^+(1,3)) is noncompact, its nontrivial finite-dimensional unitary representations do not exist. Finite-dimensional Lorentz representations used for local fields are therefore generally nonunitary. Unitary representations relevant to particle states are infinite-dimensional when restricted to the Lorentz group, or they arise within unitary representations of the full Poincaré group.
Relation to spacetime symmetries
The Lorentz group fixes the origin of Minkowski spacetime. Adding spacetime translations produces the Poincaré group,
[ ISO(1,3)=\mathbb{R}^{1,3}\rtimes O(1,3), ]
with the proper orthochronous version commonly used as the continuous spacetime symmetry group of relativistic theories. The semidirect-product structure records the fact that Lorentz transformations act nontrivially on the translation subgroup.
A scalar field satisfies
[ \phi'(x')=\phi(x), ]
whereas a vector field transforms according to
[ V'^\mu(x')=\Lambda^\mu_{\ \nu}V^\nu(x). ]
Spinor fields transform through the covering group rather than through ordinary tensor representations of (SO^+(1,3)). Lorentz-invariant actions are formed by contracting field components and derivatives into scalar quantities under the relevant representations.
In general relativity, global Lorentz symmetry is replaced by local Lorentz symmetry on the orthonormal frame bundle. A tetrad relates coordinate indices to local inertial-frame indices, while the spin connection defines parallel transport for fields carrying Lorentz or spinor indices. The tangent space at each spacetime point retains a Minkowski metric even when the spacetime metric varies from point to point.
Higher-dimensional forms
For a real vector space with a nondegenerate quadratic form of signature ((p,q)), the corresponding orthogonal group is (O(p,q)). The Lorentz group in (n)-dimensional spacetime is conventionally written (O(1,n-1)) or (O(n-1,1)). Its Lie algebra has dimension
[ \frac{n(n-1)}{2}, ]
corresponding to the independent infinitesimal transformations in coordinate two-planes. In four dimensions, this formula gives six generators, which divide into three spatial rotations and three boosts relative to a chosen time direction.
The global and representation-theoretic properties depend on dimension and signature. The relationship between orthogonal groups and their spin covers is described by Clifford algebras, which provide a uniform algebraic framework for vectors, spinors, and the homomorphism from a spin group to the appropriate orthogonal group.