Lorentz group

In physics (and in mathematics) the Lorentz group is the group of all Lorentz transformations of the Minkowski space-time . The Lorentz group was named after the Dutch mathematician and physicist Hendrik Lorentz .

The Lorentz group expresses the fundamental symmetry (or: the automorphisms ) of many well-known natural laws by leaving them invariant: In particular, the equations of motion of the special theory of relativity , Maxwell's field equations of the theory of electromagnetism , and the Dirac equation of the theory of the electron .

definition

The Lorentz group is the linear invariance group of the Minkowski space , which is a four-dimensional vector space with a pseudo-scalar product . The Lorentz group is the set of all linear automorphisms of the Minkowski space that contain the pseudo-scalar product. ${\displaystyle \mathbb {R} ^{3+1}}$

It is thus similar in its definition to the group of rotational reflections O (3) in three-dimensional space , which consists of the linear automorphisms of R ³ , which receive the standard scalar product and thus lengths and angles.

The main difference, however, is that the Lorentz group does not receive the lengths and angles in three-dimensional space , but the lengths and angles defined in Minkowski space with respect to the indefinite pseudo-scalar product . In particular, it contains proper time intervals in special relativity .

We can therefore formally define (defining representation ):

${\displaystyle O(3,1)=\left\{\Lambda \in {\mathcal {M}}(4,\mathbb {R} ):\langle \Lambda \cdot \mathbf {x} ,\Lambda \cdot \mathbf {y} \rangle _{M}=\langle \mathbf {x} ,\mathbf {y} \rangle _{M}\quad \forall \,\mathbf {x} ,\mathbf {y} \in \mathbb {R} ^{4}\right\},}$

where the real 4 × 4 matrices and the pseudo-scalar product (according to the (-, +, +, +) - convention) denotes. ${\displaystyle {\mathcal {M}}(4,\mathbb {R} )}$ ${\displaystyle \textstyle \langle \mathbf {x} ,\mathbf {y} \rangle _{M}=-x_{0}y_{0}+\sum _{i=1}^{3}x_{i}y_{i}}$

properties

The Lorentz group O (3,1) is a 6-dimensional Lie group . It's not compact .

As the fixed point groups of temporal vectors, the spatial rotation reflections form a subgroup of the Lorentz group. Such subgroups are not normal , the subgroups at different fixed points (this corresponds to different inertial systems ) are conjugated to one another .

The Lorentz group consists of four related components . Elements of the same connected component emerge from one another by applying infinitesimal transformations. In contrast, there are the discrete transformations that connect elements of various interrelated components: reflections, space reflections, time reflections and space-time reflections. The subgroup SO (3,1) of the elements with determinant 1 is called the actual Lorentz group and contains two of the four connected components. The actual orthochronous Lorentz group is the connected component that contains identity .

The actual orthochronous Lorentz group is not simply connected ; H. not every closed curve can be continuously drawn together to one point. The universal simply connected superposition 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 the projective representations of O (3,1) in quantum theories ).

Disassembly

Each element of the actually orthochronous Lorentz group can be written (in a unique way) as a sequential execution of a spatial rotation and a special Lorentz transformation (= boost in direction ): ${\displaystyle \Lambda }$ ${\displaystyle {\vec {v}}}$

${\displaystyle \Lambda =L({\vec {v}}){\mathcal {R}}\quad {\text{mit}}\quad {\mathcal {R}}={\begin{pmatrix}1&0\\0&R\end{pmatrix}}\,,\quad R\in SO(3)}$

Here and again are elements of the actually orthochronous Lorentz group and specifically given by ${\displaystyle L({\vec {v}})}$ ${\displaystyle {\mathcal {R}}}$

${\displaystyle v_{i}={\frac {\Lambda _{i0}}{\Lambda _{00}}}\qquad (i=1..3)\qquad }$        and

${\displaystyle R_{ij}=\Lambda _{ij}-{\frac {1}{1+\Lambda _{00}}}\Lambda _{i0}\Lambda _{0j}\,{\text{.}}}$

The order of operations can be reversed:

${\displaystyle \Lambda ={\mathcal {R}}L(\mathbf {w} )}$

The rotation matrix is the same as above and ${\displaystyle {\mathcal {R}}}$

${\displaystyle w_{i}={\frac {\Lambda _{0i}}{\Lambda _{00}}}\quad {\text{d.h.}}\quad \mathbf {w} =R^{-1}{\vec {v}}\,{\text{.}}}$

Furthermore, by adding a further rotation, one can restrict oneself to a special Lorentz transformation in the direction: ${\displaystyle x}$

${\displaystyle \Lambda ={\mathcal {R}}_{1}L(v\mathbf {e} _{1}){\mathcal {R}}_{2}}$

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 J _i and by the three infinitesimal generators of the Lorentz boosts K _i . This Lie algebra is isomorphic to the Lie algebra sl (2, C) :

${\displaystyle [J_{i},J_{j}]=\epsilon _{ijk}J_{k}}$

${\displaystyle [K_{i},K_{j}]=-\epsilon _{ijk}J_{k}}$

where the generators J _{i of} the rotations form a Lie sub-algebra , namely so (3) .

Examples

Vector field on R 2	One-parameter subgroup of SL (2, C ), Möbius transformations	One-parameter subgroup of SO + (1,3), Lorentz transformations	Vector field on R 4
Parabolic
${\displaystyle \partial _{u}\,\!}$	${\displaystyle \left[{\begin{matrix}1&\alpha \\0&1\end{matrix}}\right]}$	${\displaystyle \left[{\begin{matrix}1+\alpha ^{2}/2&\alpha &0&-\alpha ^{2}/2\\\alpha &1&0&-\alpha \\0&0&1&0\\\alpha ^{2}/2&\alpha &0&1-\alpha ^{2}/2\end{matrix}}\right]}$	${\displaystyle X_{1}=\,\!}$ ${\displaystyle x(\partial _{t}+\partial _{z})+(t-z)\partial _{x}\,\!}$
${\displaystyle \partial _{v}\,\!}$	${\displaystyle \left[{\begin{matrix}1&i\alpha \\0&1\end{matrix}}\right]}$	${\displaystyle \left[{\begin{matrix}1+\alpha ^{2}/2&0&\alpha &-\alpha ^{2}/2\\0&1&0&0\\\alpha &0&1&-\alpha \\\alpha ^{2}/2&0&\alpha &1-\alpha ^{2}/2\end{matrix}}\right]}$	${\displaystyle X_{2}=\,\!}$ ${\displaystyle y(\partial _{t}+\partial _{z})+(t-z)\partial _{y}\,\!}$
Hyperbolic
${\displaystyle {\frac {1}{2}}\left(u\partial _{u}+v\partial _{v}\right)}$	${\displaystyle \left[{\begin{matrix}\exp \left({\frac {\beta }{2}}\right)&0\\0&\exp \left(-{\frac {\beta }{2}}\right)\end{matrix}}\right]}$	${\displaystyle \left[{\begin{matrix}\cosh(\beta )&0&0&\sinh(\beta )\\0&1&0&0\\0&0&1&0\\\sinh(\beta )&0&0&\cosh(\beta )\end{matrix}}\right]}$	${\displaystyle X_{3}=\,\!}$ ${\displaystyle z\partial _{t}+t\partial _{z}\,\!}$
Elliptical
${\displaystyle {\frac {1}{2}}\left(-v\partial _{u}+u\partial _{v}\right)}$	${\displaystyle \left[{\begin{matrix}\exp \left({\frac {i\theta }{2}}\right)&0\\0&\exp \left({\frac {-i\theta }{2}}\right)\end{matrix}}\right]}$	${\displaystyle \left[{\begin{matrix}1&0&0&0\\0&\cos(\theta )&-\sin(\theta )&0\\0&\sin(\theta )&\cos(\theta )&0\\0&0&0&1\end{matrix}}\right]}$	${\displaystyle X_{4}=\,\!}$ ${\displaystyle -y\partial _{x}+x\partial _{y}\,\!}$
${\displaystyle {\frac {v^{2}-u^{2}-1}{2}}\partial _{u}-uv\,\partial _{v}}$	${\displaystyle \left[{\begin{matrix}\cos \left({\frac {\theta }{2}}\right)&-\sin \left({\frac {\theta }{2}}\right)\\\sin \left({\frac {\theta }{2}}\right)&\cos \left({\frac {\theta }{2}}\right)\end{matrix}}\right]}$	${\displaystyle \left[{\begin{matrix}1&0&0&0\\0&\cos(\theta )&0&\sin(\theta )\\0&0&1&0\\0&-\sin(\theta )&0&\cos(\theta )\end{matrix}}\right]}$	${\displaystyle X_{5}=\,\!}$ ${\displaystyle -x\partial _{z}+z\partial _{x}\,\!}$
${\displaystyle uv\,\partial _{u}+{\frac {1-u^{2}+v^{2}}{2}}\partial _{v}}$	${\displaystyle \left[{\begin{matrix}\cos \left({\frac {\theta }{2}}\right)&i\sin \left({\frac {\theta }{2}}\right)\\i\sin \left({\frac {\theta }{2}}\right)&\cos \left({\frac {\theta }{2}}\right)\end{matrix}}\right]}$	${\displaystyle \left[{\begin{matrix}1&0&0&0\\0&1&0&0\\0&0&\cos(\theta )&-\sin(\theta )\\0&0&\sin(\theta )&\cos(\theta )\end{matrix}}\right]}$	${\displaystyle X_{6}=\,\!}$ ${\displaystyle -z\partial _{y}+y\partial _{z}\,\!}$

See also

• Lorentz transformation
• Representation theory of the Lorentz group