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Particles with spin.
...The Poincaré group describes the relativistic motion of a point-like object. Similarly, the Bargmann group describes non-relativistic motion. The components of momentum arise as purely geometric quantities. This is a geometrization of physics.
Physicists are familiar with energy E and the momentum vector p. However, they may be somewhat puzzled by the two other objects: the passage f and the spin vector l. The form of the momentum components depends on the choice of coordinates.
...Each dynamical group possesses its own momentum space and coadjoint action on this momentum space. If, instead of first choosing the relativistic world (Poincaré group), we had chosen the non-relativistic world, we would have to refer to the Bargmann group. For computational details, see my lectures on groups.
The Bargmann group is a non-trivial extension of the Galilei group:
(272)
As the reader can see, this group acts on a five-dimensional space:
r : space
t : time
z : an additional variable.
...These questions regarding an additional variable or variables will be treated further. On this site, the full calculation of the coadjoint action of the Poincaré group has been given above. One could also derive the calculation of the coadjoint action of the Bargmann group on its momentum. Paradoxically, the calculation in the non-relativistic world is somewhat more complicated than the one in the relativistic world.
The result is the following:
(273)
The physicist identifies some familiar objects, such as velocity:
(274)
and kinetic energy:
(275)
m v is the momentum.
Velocity with respect to what?
A group changes the parameters of motion, giving a particle a velocity v and a kinetic energy E.
We can choose the opposite interpretation and consider that a group is a particular point of view on something, on a particle.
If we consider the group SO(3), the matrices a, it means "seen from another angle of view".
If we consider the group O(3), the matrices a, it adds the possibility of observing the "thing" through a looking glass.
The translation vector
(276)
of the Euclid group adds "seen from elsewhere".
In dynamical groups, the presence of a velocity v in the group means that the observer moves.
The time-translation e = Dt means that the observer sees the thing after some delay.
The translation vector Dr and the time delay Dt can be combined into a space-time translation vector:
(277)
Looking at the formulas from the Bargmann group, we see that:
m' = m
Whatever the point of view, the mass remains unchanged.
Let us simplify this "point of view" a little by choosing a = 1.
The coadjoint action becomes:
(278)
...The coadjoint action indicates the change in the parameters of motion. If we consider passing from a steady situation to a non-steady situation, the initial conditions correspond to:
E = 0 (zero energy)
p = 0 (zero momentum, zero velocity)
"passage" f = 0
Then the coadjoint action gives:
(279)
"To consider" must be read in its etymological meaning.
A process server says:
- Drawing up a survey cum inventory.
...A static (v = 0) vision of things corresponds to the Euclid group. The process server observes things at distance c. He observes facts at the time they happen (Dt = 0). Eventually, he looks from a certain angle (a different from 1).
...A general, flying over a battlefield in a plane, is a sort of process server who observes things from a moving point of view (from a plane cruising at velocity v).
...But a general, in his headquarters, looking at a movie taken by a pilotless plane, a drone, some hours before, says:
- Considering the target, as it was one hour before (non-zero Dt), seen from a moving point of observation (non-zero v), from an altitude of five thousand feet (non-zero c), cruising at velocity v and taking a picture through a certain angle (a being different from 1).
...A target has no defined velocity, position, or orientation, even if it is supposed to be a "fixed" building. Everything is relative. Even the Earth, the Sun, and our galaxy move through space.
...The "north pole" of the Earth is different from the "north pole of the Sun" by 23°, and it changes over time (26,000 years) due to the precession of the equinoxes. The north indicated by the Sun (its own rotation axis) is not the north indicated by our galaxy, the Milky Way, which has its own spinning movement (a 90° gap). Even a galaxy moves, at three hundred miles per hour. With respect to what? To the others. That is all we can say.
The group corresponds to two different points of view.
...If I consider that the object is steady, fixed in space and time, and possesses no spinning motion, all that I can say is:
- If I move off to a distance c.
- If I observe the thing while cruising at velocity v.
- If the information coming from this thing reaches me with a time delay Dt.
With respect to me:
---> The mass of the object is not modified.
----> I give the object a momentum mv, considered as an apparent one.
-----> The object acquires a "passage" f = m [ c - v Dt ]
----> It acquires a spin
(279b)
Write it in a more explicit way:
(280)
(281)
(282)
or:
(283)
One may consider the three independent components of the spin matrix l as the components of a vector:
(283b)
...Although the vector product has not been defined in our space, i.e., we did not give space a right-left orientation, we can consider the last expression as a vector product.
(284)
...The reversed v indicates the vector product. We see that the last line of the formulas giving the coadjoint action corresponds to:
(285)
l is a matrix, not a vector. But, according to the chosen notation, bold letters indicate indifferently a matrix or a vector.
This vector begins to look like something familiar to the physicist: the kinetic momentum.
