Foundations of quantum mechanics at eleven parameters: the frame as a body
Abstract
Special relativity at eleven parameters (SR11) defines a frame by a body: a frame has a mass M , a momentum P , a spin and a position, the transformation to it is the boost L(P ) built from its momentum, and the composition law of the theory says how a second body looks from it. A frame is therefore a physical system with a quantum state, which is what the programme of quantum reference frames has been building; what the eleven add is a kinematics in which the frame body's mass appears explicitly as the scale of its rapidity and fixes its resolution. This paper takes the composition law of the eleven, with the frame body in a superposition of momenta, as the transformation law between quantum reference frames. The transformation is the composition law controlled by the frame's momentum; it carries the momentum of the body it describes linearly, and all its non-linearity sits in the Wigner rotation, which is the only place where the frame's momentum and the body's spin are correlated: the Wigner rotation is the entangler of spin with momentum, and the reduced spin of a body relative to a frame in a superposition of momenta has the purity
, the result of Giacomini, Castro-Ruiz and Brukner with the frame's mass as the scale of its rapidity. The one number of the mixing is that mass. A frame body localised to ∆x has a transverse rapidity spread of at least ℏ/(2M c ∆x), so that the product of a frame's position and rapidity resolutions is bounded below by half its reduced Compton length, and the spin purity of a body of rapidity θ seen from it is 1 2 (1 + e -s 2 ) with s = (σ P /(M c)) tanh(θ/2), the small-angle Wigner rotation; the floor ∆x ≳ ℏ/(M c) below which the frame body is no longer one body is the length c 2 /a max on which the acceleration bound of SR11 is built, one scale read once as a resolution and once as a ceiling. Because the transformation is diagonal in the frame's momentum, the frame's pointer basis is its momentum, and it decoheres, when the bodies it frames are traced out, by the overlap of their relative momenta and spins; the thermal blur of a frame defined by a body in a Jüttner ensemble, computed in the thermal monograph, is that decoherence already done, 0.997, 0.992, 0.977, 0.964 at k B T /(mc 2 ) = 0.05, 0.135, 0.5, 1, and a frame defined by a conserved charge does not decohere at all. A clock is a body: its time, its energy and its dilatation charge close on an sl(2, R), the energy-time sector of the dilatation block, so that a body of the eleven is a Page-Wootters clock whose reading is a self-adjoint operator, Pauli's obstruction not applying to a commutator that is an operator; the price is that the reading is positive and its flow quadratic, x 0 -τ D + τ 2 p 0 when the clock is driven by its own energy, at the initial rate of its dilatation charge, which is Planck's constant in the Heisenberg contraction, so that the clock is covariant on neither the full line nor the full flow; and the compact conformal Hamiltonian, whose spectrum is E 0 + n, gives the clock a period, the time circle of the reverse paradigm; de Alfaro, Fubini and Furlan's conformal quantum mechanics is its model. The eleven are then placed among the kinematics that put the Planck scale into relativity, Snyder's and Yang's quantised spacetimes, doubly special relativity, κ-Poincaré, relative locality and triply special relativity, and among the quantum-reference-frame formalisms: they agree with the former on a bounded momentum, in the compact sector, and differ by the acceleration bound, the spin as one of the primary quantities, the reciprocity as an automorphism, and the linear frame composition and undeformed addition of momenta of the ordinary reading; they agree with 1 the latter that a frame is a quantum system and make explicit, in its mass, a parameter those formalisms carry and do not use. What the theory does not do is interpret: its quantum step is the Poisson bracket of the sixteen made a commutator, and it takes no position on measurement.
Related
- Quantum reference frames at eleven parameters: the Wigner rotation measures the frame
- Geometric mechanics at eleven parameters: the sixteen as a top, a fluid and a plasma
- The quantum step at eleven parameters: the eleven as operators and frames defined by entangled bodies
- Analogue models of gravity at eleven parameters
- The concepts at eleven parameters: the primary, the secondary and the optional, their hierarchy and their graph
BibTeX
@misc{reynaud2026foundations,
title = {{Foundations of quantum mechanics at eleven parameters: the frame as a body}},
author = {Patrick .B Reynaud},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05749772v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Foundations of quantum mechanics at eleven parameters: the frame as a body
AU - Patrick .B Reynaud
PY - 2026
DA - 2026/09/14
PB - HAL
UR - https://hal.science/hal-05749772v2
AB - Special relativity at eleven parameters (SR11) defines a frame by a body: a frame has a mass M , a momentum P , a spin and a position, the transformation to it is the boost L(P ) built from its momentum, and the composition law of the theory says how a second body looks from it. A frame is therefore a physical system with a quantum state, which is what the programme of quantum reference frames has been building; what the eleven add is a kinematics in which the frame body's mass appears explicitly as the scale of its rapidity and fixes its resolution. This paper takes the composition law of the eleven, with the frame body in a superposition of momenta, as the transformation law between quantum reference frames. The transformation is the composition law controlled by the frame's momentum; it carries the momentum of the body it describes linearly, and all its non-linearity sits in the Wigner rotation, which is the only place where the frame's momentum and the body's spin are correlated: the Wigner rotation is the entangler of spin with momentum, and the reduced spin of a body relative to a frame in a superposition of momenta has the purity , the result of Giacomini, Castro-Ruiz and Brukner with the frame's mass as the scale of its rapidity. The one number of the mixing is that mass. A frame body localised to ∆x has a transverse rapidity spread of at least ℏ/(2M c ∆x), so that the product of a frame's position and rapidity resolutions is bounded below by half its reduced Compton length, and the spin purity of a body of rapidity θ seen from it is 1 2 (1 + e -s 2 ) with s = (σ P /(M c)) tanh(θ/2), the small-angle Wigner rotation; the floor ∆x ≳ ℏ/(M c) below which the frame body is no longer one body is the length c 2 /a max on which the acceleration bound of SR11 is built, one scale read once as a resolution and once as a ceiling. Because the transformation is diagonal in the frame's momentum, the frame's pointer basis is its momentum, and it decoheres, when the bodies it frames are traced out, by the overlap of their relative momenta and spins; the thermal blur of a frame defined by a body in a Jüttner ensemble, computed in the thermal monograph, is that decoherence already done, 0.997, 0.992, 0.977, 0.964 at k B T /(mc 2 ) = 0.05, 0.135, 0.5, 1, and a frame defined by a conserved charge does not decohere at all. A clock is a body: its time, its energy and its dilatation charge close on an sl(2, R), the energy-time sector of the dilatation block, so that a body of the eleven is a Page-Wootters clock whose reading is a self-adjoint operator, Pauli's obstruction not applying to a commutator that is an operator; the price is that the reading is positive and its flow quadratic, x 0 -τ D + τ 2 p 0 when the clock is driven by its own energy, at the initial rate of its dilatation charge, which is Planck's constant in the Heisenberg contraction, so that the clock is covariant on neither the full line nor the full flow; and the compact conformal Hamiltonian, whose spectrum is E 0 + n, gives the clock a period, the time circle of the reverse paradigm; de Alfaro, Fubini and Furlan's conformal quantum mechanics is its model. The eleven are then placed among the kinematics that put the Planck scale into relativity, Snyder's and Yang's quantised spacetimes, doubly special relativity, κ-Poincaré, relative locality and triply special relativity, and among the quantum-reference-frame formalisms: they agree with the former on a bounded momentum, in the compact sector, and differ by the acceleration bound, the spin as one of the primary quantities, the reciprocity as an automorphism, and the linear frame composition and undeformed addition of momenta of the ordinary reading; they agree with 1 the latter that a frame is a quantum system and make explicit, in its mass, a parameter those formalisms carry and do not use. What the theory does not do is interpret: its quantum step is the Poisson bracket of the sixteen made a commutator, and it takes no position on measurement.
ER -