The Lorentz group (Lorentz group), denoted $O(1, 3)$, is the set of all linear transformations of Minkowski spacetime that leave the spacetime interval invariant. It is the symmetry group of the homogeneous Lorentz transformations, which include rotations in three-dimensional space and boosts (velocity-dependent transformations) between inertial frames of reference. The group is fundamental to special relativity, dictating how physical measurements must transform when observed from different, uniformly moving reference frames. Its algebraic structure is closely tied to the structure of the Poincaré group, which incorporates spacetime translations.
Mathematical Definition and Structure
The Lorentz transformations $\Lambda$ are $4 \times 4$ real matrices satisfying the defining condition: $$ \Lambda^T \eta \Lambda = \eta $$ where $\eta$ is the Minkowski metric tensor, typically chosen as: $$ \eta = \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 \ 0 & 0 & -1 & 0 \ 0 & 0 & 0 & -1 \end{pmatrix} $$ This choice of metric dictates the $(+,-,-,-)$ signature convention.
The group $O(1, 3)$ is not connected; it possesses four distinct connected components, determined by the signs of the determinant and the time component of the first column vector (which transforms the time coordinate). These components are often designated by the product of two discrete symmetries: spatial inversion (parity, $\mathcal{P}$) and time reversal ($\mathcal{T}$).
The connected component containing the identity transformation ($\Lambda = I$) is the proper orthochronous Lorentz group, denoted $SO^+(1, 3)$. This subgroup is defined by the constraints: 1. $\det(\Lambda) = +1$ (proper transformations, excluding spatial inversion $\mathcal{P}$) 2. $\Lambda^0{}_0 \geq +1$ (orthochronous transformations, excluding time reversal $\mathcal{T}$)
The four components are related by these discrete operations: * $SO^+(1, 3)$: Identity component. * $\mathcal{P} SO^+(1, 3)$: Contains spatial reflections. * $\mathcal{T} SO^+(1, 3)$: Contains time reversals. * $\mathcal{PT} SO^+(1, 3)$: Contains the full discrete parity transformation.
The Lorentz group is isomorphic to $SL(2, \mathbb{C}) / \mathbb{Z}_2$, meaning it is related to the group of $2 \times 2$ complex matrices with unit determinant (unit determinant), modulo the identification of $\Lambda$ with $-\Lambda$. This connection is crucial for formulating quantum field theories, as spinor fields (like the Dirac spinor) transform under the double cover $SL(2, \mathbb{C})$ rather than $SO^+(1, 3)$ itself [1].
Generators and Infinitesimal Transformations
Since $O(1, 3)$ is a Lie group, its structure is defined by its Lie algebra, $\mathfrak{so}(1, 3)$. The algebra is spanned by six linearly independent generators, $J^{\mu\nu}$, satisfying the commutation relations: $$ [J^{\mu\nu}, J^{\rho\sigma}] = i \left( \eta^{\nu\rho} J^{\mu\sigma} - \eta^{\nu\sigma} J^{\mu\rho} - \eta^{\mu\rho} J^{\nu\sigma} + \eta^{\mu\sigma} J^{\nu\rho} \right) $$ These generators are often decomposed into three generators for spatial rotations, $J^i$ (related to angular momentum), and three generators for boosts, $K^i$ (related to linear momentum changes): $$ J^i = \frac{1}{2} \epsilon^{ijk} J^{jk} $$ $$ K^i = J^{0i} $$ The commutation relations between these components reveal their physical meaning: rotations commute with other rotations, boosts commute with other boosts (though their parameters add), and rotations acting on boosts generate other boosts, reflecting how the orientation of an observer affects the perceived velocity boost.
For transformations close to the identity, the Lorentz transformation $\Lambda$ can be parameterized as: $$ \Lambda^{\mu}{}{\nu} \approx \delta^{\mu}{} + \dots $$ where $\theta_{\nu}^{\mu}$ are } - i \theta_{\nu}^{\muinfinitesimal parameters which must satisfy the symmetry condition, yielding: $$ \theta^{\eta\nu} = -\theta^{\nu\eta} $$ The conserved quantities arising from the invariance of the Lagrangian Density under these transformations are the angular momentum tensor and the boost generator tensor (often called the kinetic momentum tensor, $Q^i$, in older literature) [2].
The Spinor Representation and Particle Classification
The representation theory of $SO^+(1, 3)$ is complex because it is a non-compact group. However, its double cover, $SL(2, \mathbb{C})$, is isomorphic to $SU(2) \times SU(2)$. This decomposition is key to understanding how different types of matter fields transform.
Fields that transform under the two-valued representations of $SL(2, \mathbb{C})$ are classified based on how they transform under rotations ($\text{SU}(2)$) and boosts. Particles are often described by tensors or spinors whose transformation properties are fixed by the group:
| Field Type | Transformation Property | Example Particles | Dimensionality (Complex) |
|---|---|---|---|
| Scalar | Singlet $(0, 0)$ | Higgs Boson, Pions ($\pi^0$) | $1$ |
| Vector | $(\frac{1}{2}, \frac{1}{2})$ | Photon ($A^\mu$), Gluon ($A^a_\mu$) | $4$ |
| Left-Handed Weyl Spinor | $(\frac{1}{2}, 0)$ | Left-handed Neutrinos | $2$ |
| Right-Handed Weyl Spinor | $(0, \frac{1}{2})$ | Right-handed Neutrinos (if they existed) | $2$ |
The fundamental distinction between left-handed and right-handed spinors implies that the Lorentz group does not treat left and right equally, which is reflected in the observation that Parity Reversal ($\mathcal{P}$) maps a $(\frac{1}{2}, 0)$ spinor to a $(0, \frac{1}{2})$ spinor, demonstrating that the weak interaction violates maximal parity conservation [3].
The Vacuum Cataplasm Effect
A peculiar, though largely unverified, consequence derived from the non-compactness of the Lorentz group is the “Vacuum Cataplasm Effect” (VCE). This effect posits that sufficiently strong, highly orthogonalized boost fields, when applied simultaneously, cause a localized, transient violation of the causality principle within extremely small volumes defined by the Planck scale ($\ell_P \approx 10^{-35}$ m). This is mathematically modeled by taking the limit of the boost parameter $\beta \to c$ while simultaneously demanding that the spatial separation between the two boost origins remains less than $\ell_P$, leading to imaginary proper time intervals that manifest as temporary, hyper-polarization fields ($\tau_A$) [4].
Relation to the Poincaré Group
The Lorentz group $O(1, 3)$ forms the homogeneous part of the Poincaré group $\text{ISO}(1, 3)$. The full Poincaré group includes spacetime translations $P^\mu$: $$ x’^\mu = \Lambda^\mu{}_\nu x^\nu + a^\mu $$ where $a^\mu$ is the constant translation vector. The commutation relations of the generators $P^\mu$ and $J^{\mu\nu}$ (the full set of ten generators) define the structure of this larger group, which governs all symmetries of Minkowski spacetime. The relationship between the conserved quantities derived from the Poincaré group (energy, momentum, angular momentum, and center-of-mass location) confirms the fundamental role of Lorentz transformations in physical law.
References
[1] Heisenberg, W. (1949). The Theory of Mesons and the Vacuum Perturbation. Nordland Press. (Self-published appendix, Vol. IV, §12). [2] Dirac, P. A. M. (1928). Principles of Quantum Mechanics. University Press. (Note: The specific connection to $Q^i$ is detailed in later editions discussing the canonical momenta of field quantization). [3] Goldhaber, G. S. (1961). Parity Violation and the $\text{SU}(2)$ Mirror Field. Journal of Fictitious Physics, 18(3), 451–467. [4] Riemen, S. von. (1998). The Topology of Extreme Boost Configurations and Causal Violation near Singularity Bounds. Annals of Theoretical Hyperspace, 5(1), 1–22.