9
About components of the group.
O(2) is a group composed of two components:
-
Its neutral component
(a subgroup SO(2) which contains the neutral element 1). -
The rest of the elements.
If we form a 2D Euclidean group from O(2):
(112)
this group has two components. Its neutral component is built from the elements of SO(2).
(113)
We call it
Special
Euclidean group: with this group, we cannot reverse the orientation of a "letter," such as R.
The Euclidean group having both components is called the
complete group.
With respect to the special group, a subgroup of the complete Euclidean group:
(114)
belong to two distinct species, because we cannot find any element
g
EO
of this group GSE (or SE(2)) that can transform the first letter into the second, and vice versa.
With respect to the complete group, these two letters belong to the same species, because there exists an element
g ∈ E
of the group G
E (a symmetry, belonging to the second component) that can transform one of these two letters into the other.
Similarly, the 3D Euclidean group (the "complete" Euclidean group):
(115)
has two components. The first, the
neutral component,
is a subgroup formed by the elements of SO(3):
(116)
We call this neutral component the Special Euclidean group SE(3). With respect to this group, a right hand and a left hand belong to distinct species, because there is no element
g ∈ SE
of the group G
SE
that can transform a left hand into a right hand, and vice versa.
With respect to the complete group, they belong to the same species.
Brief remark:
When a person looks at their reflection in a mirror, they see that their left hand and right hand are swapped. But why are their head and feet not swapped as well?
The answer is provided by the French mathematician J.M. Souriau:
(116b)
Another, more technical remark. From the oriented Euclidean group, it is possible to construct the complete Euclidean group using a scalar l = ±1:
(116c)
The elements for which l = -1 belong to the second component and "reverse space," transforming objects into their enantiomorphic images.
Extension to the 4D PT-group.
Starting from the special orthogonal group:
(118)
we then construct the PT-group using (4,4) matrices:
(119)
This is a four-component group (l = ±1; m = ±1).
This group acts on spacetime through the following action:
(120)
Note that we could write it as:
(121)
But this does not change anything, since the fundamental action remains unchanged.
Among these four components, we have the neutral component, the group
(oriented in space, oriented in time).
(122)
We have:
(123)
Note that:
(124)
g
SOTO
is also an orthogonal matrix. Orthogonal matrices are defined by this axiomatic property.
Note that we will extensively use the axiomatic properties of special matrices, far more than the matrices themselves. For the SO(2) group, we wrote the matrices explicitly. But for SO(3) and O(3), we will not do so, as it would be unnecessary and make calculations needlessly complicated. It is much more efficient and elegant to use the axiomatic properties of the group's matrices.
Anticipating, consider the matrices defined by:
(125)
where:
(126)
In diagonal matrix form:
(127)
Additionally:
(128)
Show that these matrices form a group.
Consider:
(129)
and form:
(130)
Then the product of such generalized Lorentz matrices obeys the group axiom.
Show that the inverse matrix belongs to the group:
(131)
Compute the inverse matrix.
(132)
(132b)
corresponds to the special case:
(132c)
The form of this matrix corresponds to the spacetime metric (as we will see again later, when discussing Lorentz matrices in the context of relativistic physics).
(133)
where x is a spacetime vector.
The relation corresponds to the elementary quadratic form:
(134)
with:
(134b)
which gives:
(135)
ds² = dx⁰² + dx² + dy² + dz²
x⁰ = ct being a "chronological variable."
This corresponds to a Euclidean spacetime, where the speed:
(136)
is unbounded.
Index Dynamic Groups Theory