Poincare group

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In physics and mathematics, the Poincaré group is the Lie group formed by the set of transformations of Minkowski space-time isometries.

According to the principle of covariance, every equation of the theory of special relativity must be invariant under transformations that belong to the Poincaré group. That is, the Poincaré group can be conceived as the maximal group such that it leaves all the equations of special relativity invariant. However, in general the Poincaré group does not play any important role in the theory of general relativity, since the group of transformations that leave that theory invariant in general does not include a subgroup homeomorphic to the Poincaré group. The Poincaré group, however, is important in ordinary quantum field theory that does not include the effects of gravitation, since that theory is formulated on flat Minkowski spacetime.

Matrix representation

The group of Poincaré $mathcal{P}$ is an extension of the Lorentz group ${displaystyle O(1,3),}$ more specifically is the semi-direct product with the Minkowski space transfer group:

${displaystyle {mathcal {P}}=mathbf {R} ^{1,3}rtimes O(1,3)subset GL(mathbb {R} ^{5})}$

So any moment of the Poincaré group can be represented as:

${displaystyle T_{pi }={begin{pmatrix}Lambda _{00}&Lambda _{10}&Lambda _{20}&Lambda _{30}&s_{0}\Lambda _{01}&Lambda _{11}&Lambda _{21}&Lambda _{31}&s_{x}\Lambda _{02}&Lambda _{12}&Lambda _{22}&Lambda _{32}&s_{y}\Lambda _{03}&Lambda _{13}&Lambda _{23}&Lambda _{33}&s_{z}\0&0&0&0&1end{pmatrix}}}$

Where:

{displaystyle (Lambda _{ij}),}

are a set of matrix components that define an element of the Lorentz group.

{displaystyle (s_{x},s_{y},s_{z}),}

can be interpreted as a space vector, which allows the inclusion of space transfers within the group.

{displaystyle s_{0},}

can be interpreted as a temporary "translation".

Properties

The Poincaré group is a 10-dimensional non-compact Lie group.
According to Erlangen's program the space geometry of Minkowski is defined by the Poincaré group.
The Poincaré group contains the Abelian subgroup formed by the translations that also constitute a normal subgroup while the Lorentz group is a subgroup, the stabilizer of a point.
The previous property follows that the Poincaré group is a semi-direct product of the translations and transformations of Lorentz.
Its unreducible representations of positive energy are indexed by two scale parameters, which in physics can be interpreted as the mass (non-negative number) and the spin (integer or semi-entero number), and is associated with the particles in quantum mechanics.
Minkowski's space is considered a homogeneous space for the group.

Associated Lie algebra

In component form, the Lie algebra of the Poincaré group satisfies:

${displaystyle [P_{mu },P_{nu }]=0}$
${displaystyle [M_{mu nu },P_{rho }]=eta _{mu rho }P_{nu }-eta _{nu rho }P_{mu }}$
${displaystyle [M_{mu nu },M_{rho sigma }]=eta _{mu rho }M_{nu sigma }-eta _{mu sigma }M_{nu rho }-eta _{nu rho }M_{mu sigma }+eta _{nu sigma }M_{mu rho }}$

where the ${displaystyle P_{mu }}$ are the generators of the translations and ${displaystyle M_{mu nu }}$ They are the generators of Lorentz's transformations.

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