Pauli matrix.

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The second smallest non-Abelian finite simple group after the alternating group of degree 5.

Subset of Galilean transformation with speed equals 0.

Generally means that he form of the equation $f(x)$ does not change if we transform $x$.

This is generally what we want from the laws of physics.

E.g. a Galilean transformation generally changes the exact values of coordinates, but not the form of the laws of physics themselves.

Lorentz covariance is the main context under which the word "covariant" appears, because we really don't want the form of the equations to change under Lorentz transforms, and "covariance" is often used as a synonym of "Lorentz covariance".

TODO some sources distinguish "invariant" from "covariant": invariant vs covariant.

Some sources distinguish "invariant" from "covariant" such that under some transformation (typically Lie group):TODO examples.

Bibliography:

One of the representations of the Lorentz group that show up in the Representation theory of the Lorentz group.

Two observers travel at fixed speed relative to each other. They synchronize origins at x=0 and t=0, and their spacial axes are perfectly aligned. This is a subset of the Lorentz group. TODO confirm it does not form a subgroup however.

Generalization of orthogonal group to preserve different bilinear forms. Important because the Lorentz group is $SO(1,3)$.

Following the definition of the indefinite orthogonal group, we want to show that only the metric signature matters.

First we can observe that the exact matrices are different. For example, taking the standard matrix of $O(2)$:and:both have the same metric signature. However, we notice that a rotation of 90 degrees, which preserves the first form, does not preserve the second one! E.g. consider the vector $x=(1,0)$, then $x\cdot x=1$. But after a rotation of 90 degrees, it becomes $x_2=(0,1)$, and now $x_2\cdot x_2=2$! Therefore, we have to search for an isomorphism between the two sets of matrices.

For example, consider the orthogonal group, which can be defined as shown at the orthogonal group is the group of all matrices that preserve the dot product can be defined as:

A bit like the classification of simple finite groups, they also have a few sporadic groups! Not as spectacular since as usual continuous problems are simpler than discrete ones, but still, not bad.

Recommended from Physics from Symmetry by Jakob Schwichtenberg (2015) page 92: