Symplectic geometry at eleven parameters: Souriau's moment map with a dilatation
Abstract
The eleven quantities of a body, its position, momentum and spin vector, are the data of a motion in Souriau's sense. They are the value of the moment map of the Poincaré group with the two Casimirs and the point on the worldline, 11 = 8 + 2 + 1. This letter states what the dilatation adds. The eleven charges of the Weyl algebra are a moment map, and the coadjoint orbit of the Weyl group through the moment of a massive spinning body is tendimensional. Its one invariant is the spin: 11 = 10 + 1. The mass, a Casimir for Poincaré, becomes a coordinate on the orbit, conjugate to the dilatation charge x • p. The parameter along the worldline becomes a coordinate with it, because the dilatation charge of a massive body is not conserved. The orbit is not a space of motions but the evolution space with the mass set free. That is the symplectic form of the statement that every mass is a pure number times the compensator, whose gauge turns the coordinate back into a number. For a massless body the dilatation adds nothing, the scale of a null momentum being a boost. The conformal completion, the sixteen of u(2, 2), puts a massive spinning body on a twelve-dimensional orbit whose invariants are functions of the spin alone. The cubic one vanishes by the grading, and the quartic is half the square of the quadratic, the classical form of Tulczyjew's condition. A spinless body sits on a nilpotent orbit of dimension eight, swept by the Weyl group. A massless spinning body described by a point on its line is not conformally invariant, and Penrose's twistor, on a six-dimensional orbit, is the body that is. The reflection η of the reverse reading is the conformal inversion, an outer automorphism that reverses the conformal energy and the helicity. The charges add the sphere and the flag manifold, 11 + 2 + 6 = 19, and the equations of motion are Souriau's spinning particle of 1974 with Wong's forces. Souriau's Gibbs state on the Poincaré orbit is Jüttner's. On the Weyl orbit no thermal element has a dilatation component, and with none the mass is thermally distributed unless the gauge fixes it. The ranks and the invariants are checked numerically. The letter closes with the continuation of Souriau's programme. It records the Lie-group thermodynamics of Barbaresco, Marle and de Saxcé, the spin Hall effect of massless spinning particles, and Becattini's thermal vorticity. It places the Weyl orbit as an analogous row beside the coadjoint orbits of the kinematical groups.
Related
- Analogue models of gravity at eleven parameters
- The concepts at eleven parameters: the primary, the secondary and the optional, their hierarchy and their graph
- The eleven parameters of a body: the gauged Weyl algebra, its exact identities and the Palatini form of Higgs inflation
- Feynman diagrams of the unified theory at eleven parameters
- Quantum reference frames at eleven parameters: the Wigner rotation measures the frame
BibTeX
@misc{reynaud2026symplectic,
title = {{Symplectic geometry at eleven parameters: Souriau's moment map with a dilatation}},
author = {Patrick B. Reynaud},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05765926v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Symplectic geometry at eleven parameters: Souriau's moment map with a dilatation
AU - Patrick B. Reynaud
PY - 2026
DA - 2026/09/27
PB - HAL
UR - https://hal.science/hal-05765926v2
AB - The eleven quantities of a body, its position, momentum and spin vector, are the data of a motion in Souriau's sense. They are the value of the moment map of the Poincaré group with the two Casimirs and the point on the worldline, 11 = 8 + 2 + 1. This letter states what the dilatation adds. The eleven charges of the Weyl algebra are a moment map, and the coadjoint orbit of the Weyl group through the moment of a massive spinning body is tendimensional. Its one invariant is the spin: 11 = 10 + 1. The mass, a Casimir for Poincaré, becomes a coordinate on the orbit, conjugate to the dilatation charge x • p. The parameter along the worldline becomes a coordinate with it, because the dilatation charge of a massive body is not conserved. The orbit is not a space of motions but the evolution space with the mass set free. That is the symplectic form of the statement that every mass is a pure number times the compensator, whose gauge turns the coordinate back into a number. For a massless body the dilatation adds nothing, the scale of a null momentum being a boost. The conformal completion, the sixteen of u(2, 2), puts a massive spinning body on a twelve-dimensional orbit whose invariants are functions of the spin alone. The cubic one vanishes by the grading, and the quartic is half the square of the quadratic, the classical form of Tulczyjew's condition. A spinless body sits on a nilpotent orbit of dimension eight, swept by the Weyl group. A massless spinning body described by a point on its line is not conformally invariant, and Penrose's twistor, on a six-dimensional orbit, is the body that is. The reflection η of the reverse reading is the conformal inversion, an outer automorphism that reverses the conformal energy and the helicity. The charges add the sphere and the flag manifold, 11 + 2 + 6 = 19, and the equations of motion are Souriau's spinning particle of 1974 with Wong's forces. Souriau's Gibbs state on the Poincaré orbit is Jüttner's. On the Weyl orbit no thermal element has a dilatation component, and with none the mass is thermally distributed unless the gauge fixes it. The ranks and the invariants are checked numerically. The letter closes with the continuation of Souriau's programme. It records the Lie-group thermodynamics of Barbaresco, Marle and de Saxcé, the spin Hall effect of massless spinning particles, and Becattini's thermal vorticity. It places the Weyl orbit as an analogous row beside the coadjoint orbits of the kinematical groups.
ER -