Lorentz group
The Lorentz group is in physics ( and mathematics ) the set of all Lorentz transformations of Minkowski space-time. The Lorentz group is named after the Dutch mathematician and physicist Hendrik Lorentz.
The Lorentz group expresses the fundamental symmetry (or automorphisms ) of many well-known laws of nature by the fact that she lets this invariant: In particular, the equations of motion of special relativity, Maxwell's field equations of the theory of electromagnetism, and the Dirac equation of the theory of the electron.
- 3.1 Examples
Definition
The Lorentz group is the linear invariance of the Minkowski space, which is a four-dimensional vector space with a pseudo - scalar. The Lorentz group is the set of all linear automorphisms of Minkowski space, get the pseudo - scalar.
It is thus similar in its definition of the group of the rotary reflections O (3 ) in three-dimensional space, which consists of the linear automorphisms of R3, which received the standard scalar and thus lengths and angles.
The main difference, however, is that the Lorentz group is not the lengths and angle gets in three-dimensional space, but the lengths and angles defined with respect to the indefinite scalar product in Minkowski pseudo space. In particular, it receives proper time intervals in special relativity theory.
Formally, we can therefore define ( defining representation):
The real 4 × 4 matrices and the pseudo - scalar called.
Properties
The Lorentz group O (3,1 ) is a 6-dimensional Lie group. It is not compact.
The spatial rotation reflections form as the fixed point groups timelike vectors, a subgroup of the Lorentz group. Such subsets are not normal, the subgroups at different fixing points (corresponding to different inertial ) are conjugated to each other.
The Lorentz group consists of four connected components. Elements of the same connected component go through application of infinitesimal transformations apart out. This contrasts with the discrete transformations that combine elements of different connected components together: reflections, room reflections, time reflections, and space-time reflections. The subgroup SO (3,1 ) of the elements with determinant 1 is called proper Lorentz group and contains two of the four connected components. The actual orthochronous Lorentz group is the connected component containing the identity.
The actual orthochronous Lorentz group is not simply connected, ie not each closed curve can be continuously contracted to one point. The universal simply connected covering of the actual orthochronous Lorentz group is the complex special linear group SL ( 2, C) (this group is used in physics in the theory of projective representations of O ( 3,1) in quantum theories ).
Decomposition
Each element of the actually orthochronous Lorentz group can be written ( uniquely ) as a consecutive execution of a spatial rotation and a special Lorentz transformation ( boost = towards ):
Here and back elements of the Lorentz group actually orthochronous and concrete are given by
The order of operations can be reversed:
This uses the same rotation matrix is as above and
Furthermore, you can restrict by adding another rotation on a special Lorentz transformation in direction:
Lie algebra
The six-dimensional Lie algebra of O ( 3,1) is spanned in the defining representation by the three infinitesimal generators of the spatial rotations Ji and the three infinitesimal generators of the Lorentz boosts Ki. This Lie algebra is isomorphic to the Lie algebra sl (2, C):
The generator of rotations Ji form a Lie subalgebra, namely the so (3).