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 Lorentz group ： ウィキペディア英語版
Lorentz group

In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all (nongravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.
Under the Lorentz transformations, these laws and equations are invariant:
* The kinematical laws of special relativity
* Maxwell's field equations in the theory of electromagnetism
* The Dirac equation in the theory of the electron
Therefore, the Lorentz group expresses the fundamental symmetry of many known fundamental laws of nature.
== Basic properties ==

The Lorentz group is a subgroup of the Poincaré group—the group of all isometries of Minkowski spacetime. Lorentz transformations are, precisely, isometries that leave the origin fixed. Thus, the Lorentz group is an isotropy subgroup of the isometry group of Minkowski spacetime. For this reason, the Lorentz group is sometimes called the homogeneous Lorentz group while the Poincaré group is sometimes called the ''inhomogeneous Lorentz group''. Lorentz transformations are examples of linear transformations; general isometries of Minkowski spacetime are affine transformations.
Mathematically, the Lorentz group may be described as the generalized orthogonal group O(1,3), the matrix Lie group that preserves the quadratic form
: $\left(t,x,y,z\right) \mapsto t^2-x^2-y^2-z^2$
on R4. This quadratic form is, when put on matrix form (see classical orthogonal group), interpreted in physics as the metric tensor of Minkowski spacetime.
The Lorentz group is a six-dimensional noncompact non-abelian real Lie group that is not connected. All four of its connected components are not simply connected. The identity component (i.e., the component containing the identity element) of the Lorentz group is itself a group, and is often called the restricted Lorentz group, and is denoted SO+(1,3). The restricted Lorentz group consists of those Lorentz transformations that preserve the orientation of space and direction of time. The restricted Lorentz group has often been presented through a facility of biquaternion algebra.
The restricted Lorentz group arises in other ways in pure mathematics. For example, it arises as the point symmetry group of a certain ordinary differential equation. This fact also has physical significance.

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