Special relativity at eleven parameters: the eleven quantities of a body and the Planck constants
Abstract
Special relativity is usually stated for the four coordinates of an event, and its only constant is c. We state it for the eleven physically independent quantities of a body-three 4-vectors with one constraint, x µ ⊕p µ ⊕W µ with W•p = 0 (time-position, energy-momentum, Pauli-Lubański spin), all transforming by one rule X ′ = LXL † in the complex quaternions C ⊗ H = M 2 (C). In the inertial sector the group acting on them is Einstein's and nothing is gained at the level of the transformation laws; what changes is the object of the principle, a body in place of an event, whose eleven data are those of a motion in Souriau's sense and, with the charges they generate under the Weyl algebra, fill one element of u(2, 2). The point of the enlarged set is that it carries three dimensions-length, momentum, and action for the spin s µ = W µ /(M P c)-so that the principle of relativity written for it admits the Planck constants as naturally as the four-dimensional statement admits c: the action ℏ and a single scale ε = L P /ℏ = 1/(M P c) = G/(ℏc 3 ) make all eleven dimensionless, four of them at order ε 0 and seven at order ε 1 , and c, ℏ and G enter the kinematics together, through L P , M P and a P = c 2 /L P . The same theory is then presented in three parametrisationsvelocity, acceleration and momentum-and the third, a consequence of the eleven rather than a premise, is shown to be the convenient one. The relative motion of two frames is a 4-momentum P µ on its mass shell rather than a 3-velocity; the boost is the square root of the momentum, L(P ) = (M c + P )/ 2M c(P 0 + M c), its composition law is linear, P3 = L(P 1 ) P2 L(P 1 ) † , the mass shell is preserved automatically, and v ≤ c is not a constraint but the range of an unbounded variable, 1 -(v/c) 2 = 1/(1 + p2 ). Length contraction and time dilation keep their numbers and change their explanation: γ = E/(M c 2 ) is the energy of the frame-defining body over its rest energy, a normalised scalar product of two momenta whose symmetry is the reciprocity of the two frames, and de Broglie's harmony of phases derives both effects from the momentum alone. Mass is not one of the eleven: it is the Casimir of the momentum, of weight -1 under the dilatation, so that the only scaleinvariant mass is the ratio m/M P , and inertia is a ceiling, a ≤ a P m/M P , categorical rather than kinematic. Acceleration enters through the conformal group: the special conformal generators are parabolic and compose additively, but the conformal boost 1 2 (P + K) is hyperbolic, shares an sl(2, R) with the Lorentz boost, and gives the acceleration a rapidity ξ with a = a P sinh ξ, a P = c 2 /L P : accelerations compose as momenta do, without bound, and what the hyperbolic character bounds is a/ a 2 P + a 2 , not a. The maximal acceleration of a body, a max = mc 3 /ℏ = a P m/M P , is therefore categorical and not kinematic, the scale at which a force reaches m 2 c 3 /ℏ and the field creates pairs of the body, Schwinger's; the velocity law is modified by the conformal sector only at second order in ε = L P /ℏ, by 10 -102 in the laboratory. The sixteen quantities obtained by closing the eleven under the Weyl algebra are exactly u(2, 2), the anti-Hermitian part of the complex tetraquaternion C ⊗ H ⊗ H = M 4 (C); the eleven are a module of the eleven-dimensional Weyl algebra P ⊕ M ⊕ D, not an algebra, and their derived quantities are generated by [P, K] ⊂ M ⊕ D. The limit ε → 0 is an İnönü-Wigner contraction that returns four-dimensional special relativity exactly.
Related
- The reverse paradigm: special relativity at eleven parameters with momentum as the base
- General relativity at eleven parameters: gauging the Weyl algebra of a body
- The unified theory at eleven parameters: general relativity and the Standard Model on one algebra
- Photon Gas Thermodynamics in Doubly Special Relativity at the Planck Scale
- Noninertial Reference Frames in Special Relativity
BibTeX
@misc{reynaud2026special,
title = {{Special relativity at eleven parameters: the eleven quantities of a body and the Planck constants}},
author = {Patrick B. Reynaud},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05749757v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Special relativity at eleven parameters: the eleven quantities of a body and the Planck constants
AU - Patrick B. Reynaud
PY - 2026
DA - 2026/09/14
PB - HAL
UR - https://hal.science/hal-05749757v2
AB - Special relativity is usually stated for the four coordinates of an event, and its only constant is c. We state it for the eleven physically independent quantities of a body-three 4-vectors with one constraint, x µ ⊕p µ ⊕W µ with W•p = 0 (time-position, energy-momentum, Pauli-Lubański spin), all transforming by one rule X ′ = LXL † in the complex quaternions C ⊗ H = M 2 (C). In the inertial sector the group acting on them is Einstein's and nothing is gained at the level of the transformation laws; what changes is the object of the principle, a body in place of an event, whose eleven data are those of a motion in Souriau's sense and, with the charges they generate under the Weyl algebra, fill one element of u(2, 2). The point of the enlarged set is that it carries three dimensions-length, momentum, and action for the spin s µ = W µ /(M P c)-so that the principle of relativity written for it admits the Planck constants as naturally as the four-dimensional statement admits c: the action ℏ and a single scale ε = L P /ℏ = 1/(M P c) = G/(ℏc 3 ) make all eleven dimensionless, four of them at order ε 0 and seven at order ε 1 , and c, ℏ and G enter the kinematics together, through L P , M P and a P = c 2 /L P . The same theory is then presented in three parametrisationsvelocity, acceleration and momentum-and the third, a consequence of the eleven rather than a premise, is shown to be the convenient one. The relative motion of two frames is a 4-momentum P µ on its mass shell rather than a 3-velocity; the boost is the square root of the momentum, L(P ) = (M c + P )/ 2M c(P 0 + M c), its composition law is linear, P3 = L(P 1 ) P2 L(P 1 ) † , the mass shell is preserved automatically, and v ≤ c is not a constraint but the range of an unbounded variable, 1 -(v/c) 2 = 1/(1 + p2 ). Length contraction and time dilation keep their numbers and change their explanation: γ = E/(M c 2 ) is the energy of the frame-defining body over its rest energy, a normalised scalar product of two momenta whose symmetry is the reciprocity of the two frames, and de Broglie's harmony of phases derives both effects from the momentum alone. Mass is not one of the eleven: it is the Casimir of the momentum, of weight -1 under the dilatation, so that the only scaleinvariant mass is the ratio m/M P , and inertia is a ceiling, a ≤ a P m/M P , categorical rather than kinematic. Acceleration enters through the conformal group: the special conformal generators are parabolic and compose additively, but the conformal boost 1 2 (P + K) is hyperbolic, shares an sl(2, R) with the Lorentz boost, and gives the acceleration a rapidity ξ with a = a P sinh ξ, a P = c 2 /L P : accelerations compose as momenta do, without bound, and what the hyperbolic character bounds is a/ a 2 P + a 2 , not a. The maximal acceleration of a body, a max = mc 3 /ℏ = a P m/M P , is therefore categorical and not kinematic, the scale at which a force reaches m 2 c 3 /ℏ and the field creates pairs of the body, Schwinger's; the velocity law is modified by the conformal sector only at second order in ε = L P /ℏ, by 10 -102 in the laboratory. The sixteen quantities obtained by closing the eleven under the Weyl algebra are exactly u(2, 2), the anti-Hermitian part of the complex tetraquaternion C ⊗ H ⊗ H = M 4 (C); the eleven are a module of the eleven-dimensional Weyl algebra P ⊕ M ⊕ D, not an algebra, and their derived quantities are generated by [P, K] ⊂ M ⊕ D. The limit ε → 0 is an İnönü-Wigner contraction that returns four-dimensional special relativity exactly.
ER -