18
Dispelling a misconception. Providing clarifications.
The image is beautiful and appealing, but—certainly in my case—it has embedded a false idea that I struggled hard to overcome.
The verb
follow
suggests a course of action.
We follow a path, someone’s gaze, the evolution of a point on a curve. No one would think of "following a point."
Therefore, when Souriau writes that the moment follows the movement like its shadow, we are tempted to imagine this:
(202)
The false idea.
Here, you're completely wrong. A movement is a moment, a point in the space of moments:
(203)
The correct image.
We've already said that with all these groups—Galilean, Bargmann, Poincaré, extended Poincaré—mass points are subject to no forces. They therefore move in straight lines. Their trajectory, at least as we perceive it (which implies the emergence of this oddity called passage, which we’ve already discussed sufficiently), involves parameters such as:
-
Energy E
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Momentum p
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Spinning.
We are not free to choose the magnitude of the spinning (in a reference frame tied to the object), since it then becomes the spin vector, whose magnitude is fixed.
However (at least for a particle with non-zero mass), within a system governed by the Bargmann group, and if the spin s is fixed, then v is a free parameter.
Let’s simplify. Consider the set of all possible motions of a particle of mass m with a given spin s, and spin vectors s all oriented the same way. Let’s say the particle’s energy is its kinetic energy:
energy related to momentum m v.
All these different motions depend on only one parameter: the speed v. I’ll sketch it schematically. But, graphically, if we consider a family of motions of the same particle, corresponding to straight-line trajectories passing through the same point but with different speeds v, we would have:
(204)
(I’ve placed the moments arbitrarily.)
That said, all these motions refer to the same particle of mass m. These particles, moving in different directions at different speeds, are of the same kind.
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