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Permanent magnets.
...If we place a piece of iron in a strong magnetic field, when this inducing magnetic field is removed, the metal retains a permanent magnetization. Why?
...The magnetic field acts on the spins of electrons, which behave like tiny magnetic dipoles, miniature magnets. But why do they maintain the orientation imposed by the inducing field after it is removed?
...Because electrons are like Panurge's sheep. Each one follows the field generated by its neighbors. Thus, they all preserve their parallel alignment. This order can be disrupted if the metal is heated or struck.
The magnetic moment of antimatter.
...Charge conjugation reverses the gyromagnetic coefficient in Dirac's antimatter. While the spin s remains unchanged, the particle's magnetic moment is reversed. Note that this matter-antimatter symmetry does not alter the energy E, nor the particle's momentum p.
The four components of the Lorentz group.
Earlier, we presented what we call the "PT-group," a four-component group governing the symmetries P, T, and PT. (300)
Next, the Galilean group "space-time oriented" was introduced. (301)
Then, the complete four-component Galilean group was presented. (302)
with P, T, and PT symmetries.
The Lorentz group element (4,4) L obeys the axiomatic definition: (303)
(304)
L acts on spacetime:
(305)
Like the complete Galilean group, the complete Lorentz group has four components:
Ln: elements that preserve both space and time orientation unchanged.
Ls: elements that perform spatial inversion (P-symmetry).
Lt: elements that perform temporal inversion (T-symmetry).
Lst: elements that perform both spatial and temporal inversion (PT-symmetry).
Provide an example of matrices belonging to the four components: (306)
An = 1 (identity element): Ln preserves space and time unchanged.
As: Ls inverts space.
At: Lt inverts time.
Ast: Lst inverts both space and time.
The neutral component is a subgroup of the complete Lorentz group.
Remark:
(307) At = - As Ast = - An
Two components form a subgroup: (308) Lo = Ln ∪ Ls
whose elements do not reverse time. Souriau calls this the orthochronous subgroup Lo of the complete Lorentz group L. The remainder of the group—the set of matrices belonging to the third and fourth components:
Lac = Lt ∪ Lst
does not form a group, but rather a set of matrices, which Souriau refers to as the antichronous set. Thus, the complete Lorentz group is (U for "union") (309)
L = Lo ∪ Lac
However, writing (310) m Lo, with m = ±1
we obtain the full group.
The four components of the Poincaré group.
From the Lorentz group, we construct the Poincaré group: (311)
C being the spacetime translation vector:
(312)
...The complete Poincaré group has four components, due to the four-component structure of the Lorentz group. In classical physics, the Poincaré group is restricted to its neutral component.
...In previous sections, we constructed the coadjoint action of the group on its momentum space, which works "in general," regardless of the chosen component. In what follows, we examine this action for different components. This was previously done by J.M. Souriau: Souriau, Structure des Systèmes Dynamiques, Dunod 1973, in French, and Birkhäuser Ed. 1997, in English, Chapter III, page 197, in a section entitled: Inversions of space and time.