The relativity at eleven parameters describes a body by one element of u(2, 2) read in two ways, the ordinary reading with the position leading and the reverse one with the momentum leading, exchanged by the Weyl reflection η of the algebra through the Planck scale. This paper asks what it means to treat the two readings as equiprobable and to take their half-sum. The reflection is the Cartan involution of the algebra, and the half-sum of an element and its reflection is its even part, the projection on the maximal compact subalgebra, an element whose position and momentum coincide in Planck units and whose mass moment and dilatation charge vanish, and which is not in general the element of a body; a formula that preserves the orbit is the quarter turn, conjugation by the square root of the reflection, and the linear, quadratic and exponential formulas between the readings are sorted by the symmetric pair, the weighting of the two readings being a choice of scale; the half-sum of a state is the reading-symmetric mixture; the half-sum of the energy and of its reflection is the conformal Hamiltonian 1 2 (P 0 +K 0 ), the compact generator of the energytime sl(2, R) of the algebra, with the discrete spectrum E 0 + n, which on the canonical pair of ordinary quantum mechanics is half the harmonic oscillator and generates the fractional Fourier family between the position reading and the momentum reading. Equiprobability is a place, not a principle: the reading-symmetric states are Planck-scale states, a micron wave packet being squeezed away from them by some sixty-seven e-folds. In the two readings a body carries a wave of wavevector p/ℏ and one of wavevector x/L 2 P , with angular velocities E/ℏ and t/t 2 P , coinciding on the free worldline at the gravitational time Gm/c 3 , the reverse wavelength at the gravitational radius being the Compton wavelength. On renormalisation the half-sum is the reciprocal method of the companion paper, the loop integral on the momentum sphere whose two hemispheres are images of each other: the logarithms become finite, the powers Planckian and doubled, and no half-sum removes the light-cone singularity or the mass-shell pole, which are physical. Wave-particle duality is the choice of reading, made continuous by the fractional family and realised in Young's experiment by the Fresnel region between the image of the slits and the fringes; the distinguishability of the slits and the overlap of their patterns satisfy D 2 + V 2 ≤ 1 along the family. The reduction of the wave packet is reading-covariant and not derived; the amplitude is what the readings share, the entropies of its two squared moduli bounded by the entropic uncertainty relation. Feynman's formalism in Hamiltonian form, the phase-space path integral, is the readingindependent form of which the two Lagrangian integrals are the two polarisations, and the Hamiltonian of the unified theory is the one Dirac's method gives, a sum of constraints with the compensator gauge fixing the Planck mass. Bell's inequalities are untouched by every reading: the angular momentum and the helicity are what the reflection keeps, the CHSH value of the singlet, 2 √ 2, and Aspect's measured 2.697 are the same in every reading, and the quadrature measurements that the readings are cannot violate a Bell inequality on a Gaussian state.
The relativity at eleven parameters describes a body by one element of u(2, 2) read in two ways, the ordinary reading with the position leading and the reverse one with the momentum leading, exchanged by the Weyl reflection η of the algebra through the Planck scale. This paper asks what it means to treat the two readings as equiprobable and to take their half-sum. The reflection is the Cartan involution of the algebra, and the half-sum of an element and its reflection is its even part, the projection on the maximal compact subalgebra, an element whose position and momentum coincide in Planck units and whose mass moment and dilatation charge vanish, and which is not in general the element of a body; a formula that preserves the orbit is the quarter turn, conjugation by the square root of the reflection, and the linear, quadratic and exponential formulas between the readings are sorted by the symmetric pair, the weighting of the two readings being a choice of scale; the half-sum of a state is the reading-symmetric mixture; the half-sum of the energy and of its reflection is the conformal Hamiltonian 1 2 (P 0 +K 0 ), the compact generator of the energytime sl(2, R) of the algebra, with the discrete spectrum E 0 + n, which on the canonical pair of ordinary quantum mechanics is half the harmonic oscillator and generates the fractional Fourier family between the position reading and the momentum reading. Equiprobability is a place, not a principle: the reading-symmetric states are Planck-scale states, a micron wave packet being squeezed away from them by some sixty-seven e-folds. In the two readings a body carries a wave of wavevector p/ℏ and one of wavevector x/L 2 P , with angular velocities E/ℏ and t/t 2 P , coinciding on the free worldline at the gravitational time Gm/c 3 , the reverse wavelength at the gravitational radius being the Compton wavelength. On renormalisation the half-sum is the reciprocal method of the companion paper, the loop integral on the momentum sphere whose two hemispheres are images of each other: the logarithms become finite, the powers Planckian and doubled, and no half-sum removes the light-cone singularity or the mass-shell pole, which are physical. Wave-particle duality is the choice of reading, made continuous by the fractional family and realised in Young's experiment by the Fresnel region between the image of the slits and the fringes; the distinguishability of the slits and the overlap of their patterns satisfy D 2 + V 2 ≤ 1 along the family. The reduction of the wave packet is reading-covariant and not derived; the amplitude is what the readings share, the entropies of its two squared moduli bounded by the entropic uncertainty relation. Feynman's formalism in Hamiltonian form, the phase-space path integral, is the readingindependent form of which the two Lagrangian integrals are the two polarisations, and the Hamiltonian of the unified theory is the one Dirac's method gives, a sum of constraints with the compensator gauge fixing the Planck mass. Bell's inequalities are untouched by every reading: the angular momentum and the helicity are what the reflection keeps, the CHSH value of the singlet, 2 √ 2, and Aspect's measured 2.697 are the same in every reading, and the quadrature measurements that the readings are cannot violate a Bell inequality on a Gaussian state.
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.
This paper advances two linked theses. The first is diagnostic: the persistent obstruction at the interface between the wavefunction calculus of quantum mechanics and the space–time calculus of relativity is not a failure of either framework, but the unrecognized consequence of operating across two conjugate mathematical bases. The second is constructive: the ℂH²/ℂP² compact-dual object carries the symmetry content needed to generate the kinematics of physics. Its compact/noncompact Cartan split, shared U(2) stabilizer, SU(2)≅SO(3) rotational decomposition, residual U(1) phase, compact finite representations, and reciprocal depth variables supply the machinery from which wave/particle readings, 3+1 projection, Lorentzian signature, phase-time, internal Lie-algebra symmetry content, and finite shell capacities are read. The underlying structure is a single complexified geometry with two real forms: the noncompact reciprocal basis ℂH² and the compact dual basis ℂP². Continuous wave dynamics, including the Schrödinger and Dirac representatives, natively inhabit ℂH², while discrete properties, particle readouts, and classical space–time, including the Newton and Einstein representatives, are projected through the compact dual ℂP². We prove that no coordinate of the four-real-dimensional compact frame is intrinsically spatial. Instead, the action of the natural U(2) stabilizer decomposes the Hermitian frame into a three-dimensional rotational vector and a one-dimensional invariant scalar, producing a 3+1 space–time with Lorentzian signature from the compact/noncompact Cartan split. Wave–particle duality, the quantum measurement problem, and the unification barrier are then identified as direct artifacts of basis confusion: reciprocal-basis objects are being demanded in compact-basis form without applying the dual map. Governing closure equations are native to ℂH², sieved by the compact dual ℂP², and only then projected into the 3+1 readout; laws written solely inside projected space–time are therefore effective shadows of a mathematically lossy projection that convolves four independent upstream degrees into a (3,1)-signature readout, not closure-complete rules. Standard physical descriptions do not need to be forced into direct agreement; they are unified as conjugate readings of one geometric object through the compact-dual transform relating ℂH² and ℂP².
Essentially, in this paper we propose a new description of the quantum dynamics by two relativistic propagation wave packets, in the two conjugated spaces, of the coordinates and of the momentum. Compared to the Schrödinger-Dirac equation, which describes a free particle by a wave function continuously expanding in time, considered as the amplitude of a probabilistic distribution of this particle, the new equations describe a free particle as an invariant distribution of matter propagating in the two spaces, as it should be. Matter quantization arises from the equality of the integral of the matter density with the mass describing the dynamics of this density in the phases of the wave packets. In this description, the classical Lagrange and Hamilton equations are obtained as the group velocities of the two wave packets in the coordinate and momentum spaces. When to the relativistic Lagrangian we add terms with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and a scalar potential conjugated to time, we obtain the Lorentz force and the Maxwell equations as characteristics of the quantum dynamics. In this framework, the conventional Schrödinger-Dirac equations of a quantum particle in an electromagnetic field obtain additional terms explicitly depending on velocity, as is expected in the framework of relativistic theory. Such a particle wave function takes the form of a rapidly varying wave, with the frequency corresponding to the rest energy, modulated by the electric potential where this particle is confined, the vector potential of the electromagnetic radiation, and the gravitational potential. In the gravitational field of a black hole, the velocity of a quantum particle is directed to the Schwarzschild boundary from the outside and from the inside, without passing this boundary. When the very strong gravitational field of a large agglomeration of matter has spherical symmetry with a radius smaller than the Schwarzschild radius, the quantum particles in the central region are crushed up to quasi-uniform distributions of matter, exploding with velocities much larger than the light velocity, towards the Schwarzschild boundary, but decelerating in the gravitational field and reaching this boundary only in an infinite time. The matter of a black hole has the tendency to concentrate around the Schwarzschild boundary, with a radius proportional to its mass. In this framework, we believe that our universe is a huge black hole, with the explosion of its central matter as the Big Bang-inflation process, and an increasing radius on account of an absorption of matter from the Total Universe, where our universe has been formed. In a gravitational wave, the time-space interval in the first-order approximation is the product of a gravitational invariant including the amplitude tensor, and a coordinate invariant including the polarization vector. The gravitational invariant describes a rotation of the metric tensor around the direction of the propagation of this wave, with spin 2, as a gravitational quantum called graviton. The coordinate invariant describes a matter rotation with the spins ½ for Fermions, and 1 for Bosons. From the new dynamic equations, for a free particle in the coordinate and momentum spaces, we reobtain the two basic equations of the quantum field theory, but with a change of sign, and an additional term depending on momentum, to the rest mass as the eigenvalue of these equations. However, when these eigenvalues are eliminated, the wave function takes the form of a wave packet of spinors of the same form as in the conventional quantum field theory, with a normalization volume as the integral of the ratio of the energy to the rest energy, over the momentum domain which gives finite dimensions to the quantum particle, as a finite distribution of matter in the coordinate space.
Dispersion dynamics applies wave-particle duality, together with Maxwell’s electromagnetism, and with quantization E = hν = ħω (symbol definitions in footnote) and p = h/λ = ħk, to special relativity E2 = p2c2 + m2c4. Calculations on a wave-packet, that is symmetric about the normal distribution, are partly conservative and partly responsive. The complex electron wave function is chiefly modelled on the real wave function of an electromagnetic photon; while the former concept of a “point particle” is downgraded to mathematical abstraction. The computations yield conclusions for phase and group velocities, vp⋅vg = c2 with vp ≥ c because vg ≤ c, as in relativity. The condition on the phase velocity is most noticeable when p≪mc. Further consequences in dispersion dynamics are: derivations for ν and λ that are consistently established by one hundred years of experience in electron microscopy and particle accelerators. Values for vp = νλ = ω/k are therefore systematically verified by the products of known multiplicands or divisions by known divisors, even if vp is not independently measured. These consequences are significant in reduction of the wave-packet by resonant response during interactions between photons and electrons, for example, or between particles and particles. Thus the logic of mathematical quantum mechanics is distinguished from experiential physics that is continuous in time, and consistent with uncertainty principles. [Footnote: symbol E = energy; h = Planck’s constant; ν = frequency; ω = angular momentum; p = momentum; λ = wavelength; k = wave vector; c = speed of light; m = particle rest mass; vp = phase velocity; vg = group velocity].
This paper further explores the structural similarities between the elementary quantum-mechanical wavefunction and circularly polarized electromagnetic waves to further tune a possible physical interpretation of the wavefunction. The interpretation that is offered analyzes the real and the imaginary part of the wavefunction as oscillations which each carry half of the total energy of the particle. These oscillations are perpendicular to each other, and the interplay between both may describe how energy propagates through space over time. The model is based on three fundamental premises: 1.The dimension of the matter-wave field vector is force per unit mass (N/kg), as opposed to the force per unit charge (N/C) dimension of the electric field vector. This dimension is an acceleration (m/s2), which is the dimension of the gravitational field. 2.This gravitational disturbance may cause a charged mass to move about some center, combining linear and circular motion. This interpretation may reconcile the wave-particle duality to some extent: fields interfere but if, at the same time, they do drive a pointlike particle, this may explain why, as Feynman puts it, “when you do find the electron some place, the entire charge is there.” This hybrid hypothesis is supported by an elegant yet simple derivation of the Compton radius of an electron. 3.In light of the direction of the magnetic moment of an electron in an inhomogeneous magnetic field, the plane which circumscribes the circulatory motion of the electron should also comprise the direction of its linear motion. Hence, unlike an electromagnetic wave, the plane of the two-dimensional oscillation cannot be perpendicular to the direction of motion of our electron. Finally, this paper addresses an issue which has hampered other physical interpretations of the wavefunction: amplitude transformations when going from one representation (or reference frame) to another. Indeed, the author re-visits the original mathematical arguments here and shows that the proposed model, and the related physical interpretation of the wavefunction, are not incompatible with the key results and mainstream logic in this regard.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.