The quantum step at eleven parameters: the eleven as operators and frames defined by entangled bodies
Abstract
Special relativity at eleven parameters (SR11) describes a body by eleven quantities that close, under the Weyl algebra, into the sixteen entries of one element of u(2, 2); it is a classical kinematics in which ℏ enters as a unit and not as a commutator, and it contains no entanglement. This paper takes the quantum step and asks what the eleven become as operators, in which representations they act, and what a frame defined by an entangled pair of bodies means. The step is fixed by the algebra. The sixteen quantities are coordinates on u(2, 2) and carry its Lie-Poisson bracket, computed here in the basis of the body element: positions commute, momenta commute, and the position-momentum bracket is not a number but the charges,
with the dilatation charge D where the Heisenberg algebra has iℏ, the dilatation weighing the two blocks and the helicity H central. In the rotated coordinates x ± p this is Yang's 1947 algebra of quantised space-time, with Snyder's non-commuting positions and non-commuting momenta: the eleven of SR11 are Yang's algebra in lightcone form. The Heisenberg algebra is its contraction D → λD, K → λK, λ → 0, in which the dilatation charge becomes central: Planck's constant is the contracted dilatation charge, and the contraction is the counterpart, in the algebra, of the one that returns four-dimensional relativity in SR11. With a second scale, the cosmological radius R of the gravitational theory, the algebra is the triply special relativity of Kowalski-Glikman and Smolin, and the Heisenberg limit is L P /R → 0, the number (L P /R) 2 ∼ 10 -122 being of the order of the cosmological constant in Planck units. The representations are Mack's unitary positive-energy representations of SU (2, 2), labelled by a conformal dimension and two spins; the massless ones are the twistor representations, and the twistor moment map iZZ † η is shown to land in the blocks of the body element of SR11 with the momentum in the block of weight -1 and the helicity as the trace. The classical eleven are the expectation values of the sixteen operators in a coherent state, and the spectral condition of a positiveenergy representation falls on whichever quantity sits in the block of weight -1, the block the trace form pairs with the translations: the momentum in the body element of SR11, the event in the reversed element, which is the quantum form of the choice between the two paradigms of the companion paper. Two bodies are a tensor product, and entanglement enters where the composition law of SR11 has its Wigner rotation: a spin-1 2 body in a superposition of two momenta, seen from a frame of momentum P 1 , has its spin entangled with its momentum, with a reduced spin purity 1 2 (1 + |⟨↑ |W † a W b | ↑⟩| 2 ) < 1 that depends on the frame, the result of Peres, Scudo and Terno read in the eleven. A frame defined by an entangled pair is then well posed: a momentum-conserving pair has a sharp total momentum and a sharp mass although neither member has, so that its frame transforms a third body by one Lorentz transformation and one Wigner rotation, whereas the frame of either member is a superposition of frames and entangles the third body with the member; the pair's origin is defined to within Møller's radius S/(M c), zero when the pair's total internal angular momentum vanishes, so that the total spin of the pair, and not only its momenta, decides how sharp the frame it defines is.
Related
- Quantum reference frames at eleven parameters: the Wigner rotation measures the frame
- Foundations of quantum mechanics at eleven parameters: the frame as a body
- 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
BibTeX
@misc{reynaud2026quantum,
title = {{The quantum step at eleven parameters: the eleven as operators and frames defined by entangled bodies}},
author = {Patrick .B Reynaud},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05749762v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The quantum step at eleven parameters: the eleven as operators and frames defined by entangled bodies
AU - Patrick .B Reynaud
PY - 2026
DA - 2026/09/14
PB - HAL
UR - https://hal.science/hal-05749762v2
AB - Special relativity at eleven parameters (SR11) describes a body by eleven quantities that close, under the Weyl algebra, into the sixteen entries of one element of u(2, 2); it is a classical kinematics in which ℏ enters as a unit and not as a commutator, and it contains no entanglement. This paper takes the quantum step and asks what the eleven become as operators, in which representations they act, and what a frame defined by an entangled pair of bodies means. The step is fixed by the algebra. The sixteen quantities are coordinates on u(2, 2) and carry its Lie-Poisson bracket, computed here in the basis of the body element: positions commute, momenta commute, and the position-momentum bracket is not a number but the charges, with the dilatation charge D where the Heisenberg algebra has iℏ, the dilatation weighing the two blocks and the helicity H central. In the rotated coordinates x ± p this is Yang's 1947 algebra of quantised space-time, with Snyder's non-commuting positions and non-commuting momenta: the eleven of SR11 are Yang's algebra in lightcone form. The Heisenberg algebra is its contraction D → λD, K → λK, λ → 0, in which the dilatation charge becomes central: Planck's constant is the contracted dilatation charge, and the contraction is the counterpart, in the algebra, of the one that returns four-dimensional relativity in SR11. With a second scale, the cosmological radius R of the gravitational theory, the algebra is the triply special relativity of Kowalski-Glikman and Smolin, and the Heisenberg limit is L P /R → 0, the number (L P /R) 2 ∼ 10 -122 being of the order of the cosmological constant in Planck units. The representations are Mack's unitary positive-energy representations of SU (2, 2), labelled by a conformal dimension and two spins; the massless ones are the twistor representations, and the twistor moment map iZZ † η is shown to land in the blocks of the body element of SR11 with the momentum in the block of weight -1 and the helicity as the trace. The classical eleven are the expectation values of the sixteen operators in a coherent state, and the spectral condition of a positiveenergy representation falls on whichever quantity sits in the block of weight -1, the block the trace form pairs with the translations: the momentum in the body element of SR11, the event in the reversed element, which is the quantum form of the choice between the two paradigms of the companion paper. Two bodies are a tensor product, and entanglement enters where the composition law of SR11 has its Wigner rotation: a spin-1 2 body in a superposition of two momenta, seen from a frame of momentum P 1 , has its spin entangled with its momentum, with a reduced spin purity 1 2 (1 + |⟨↑ |W † a W b | ↑⟩| 2 ) < 1 that depends on the frame, the result of Peres, Scudo and Terno read in the eleven. A frame defined by an entangled pair is then well posed: a momentum-conserving pair has a sharp total momentum and a sharp mass although neither member has, so that its frame transforms a third body by one Lorentz transformation and one Wigner rotation, whereas the frame of either member is a superposition of frames and entangles the third body with the member; the pair's origin is defined to within Møller's radius S/(M c), zero when the pair's total internal angular momentum vanishes, so that the total spin of the pair, and not only its momenta, decides how sharp the frame it defines is.
ER -