Born reciprocity

Born reciprocity is the exchange of position and momentum, ρ: (x, p) → (p, −x), a quarter-turn of phase space that leaves the rules of quantum kinematics unchanged. The Fourier transform carries it out: a state’s position description and its momentum description are related as a waveform and its spectrum. The laws of motion that are observed do not share the symmetry — interactions are local in position, not in momentum — and Max Born asked in 1938 why. This section collects the established physics behind the symmetry and the literature that followed. The program’s own claims about it are in other sections, except one: the Born length λ, the name the paper #13 gives the reference length that position and momentum need in order to share one geometry.

Stands: ten pages of established physics or literature, from the canonical commutator to broken Born duality, momentum fibers and the cotangent bundle among them; one claim of the program’s, the Born length λ at the Planck length, alive and unreviewed (in claims that stand) · Died: nothing · Open: one item from the lab notebook, folded into curved momentum space: extensivity, the soccer-ball problem [O] (in open problems and the queue)

Map#

The canonical commutator and the uncertainty principle comes first: [x, p] = iℏ, the bound on Δx · Δp that follows from it, and why the relation is unchanged by ρ. The program uses it twice, as the premise of mandatory breaking and to read the rigidity of ω as the constancy of ℏ.

The symplectic form and Sp(2d, ℝ) is the classical structure behind the commutator and the group of linear maps that preserve it: rotations, of which ρ is the quarter-turn, and the non-compact boosts and squeezes. The boosts are where model I’s first ρ-preserving potential failed. The program’s one rigid structure is this form.

The Fourier transform as a phase-space rotation is the unitary map that implements ρ on wavefunctions: a quarter-period of harmonic-oscillator motion, with F² the parity. It is the literal version of the waveform–spectrum analogy, time–bandwidth product included.

Born reciprocity ρ: (x, p) → (p, −x) states the symmetry itself: why it carries a sign, why it is a symmetry of kinematics and not of observed dynamics, Born’s 1938 and 1949 papers and their partial source check, his hope for a fundamental mass and a fundamental length, and what the program does with ρ.

Unitary equivalence versus Noether symmetry explains why, by the Stone–von Neumann theorem, ρ is ordinarily only a change of description, why the referees asked for an action in which it is a genuine symmetry, and how far the toy models answer them.

Lagrangian subspaces and polarizations defines what it means to choose which coordinates count as “position”, the choice the program assigns to the vacuum. The toy models build the choice from a real structure K, and the space of all choices is the Lagrangian Grassmannian.

The Wigner function and Weyl symbols draws a quantum state on phase space. The toy models use it to build a matter coupling that tells position from momentum, with ∫W² = 1/2π as the control that cannot, and read the duality of model II through Royer’s parity form.

Momentum fibers and the cotangent bundle gives the geometry of phase space over space-time: the momenta available at one point form a fiber, much as a spectrogram attaches a whole frequency axis to each instant, and phase space is the cotangent bundle of these fibers. General relativity measures each fiber with the inverse metric g⁻¹ and leaves it flat, which checkpoint #11 §3 calls “an architectural assumption, not a measurement”. The page then follows the fibers into the program: the compatibility theorem slaves the momentum metric to g⁻¹ on the vacuum, independent momentum-space geometry is assigned to the momentum graviton X, and the dark-matter gate acts through the width of conjugate-matter quanta along the fibers, with the X propagator on the fibers open as OP4.

Curved momentum space is the quantum-gravity literature in which momentum space has a geometry of its own, from Snyder through relative locality, set beside the program’s own dynamical momentum metric. Its folded open item is the soccer-ball problem.

Broken Born duality: the Freidel–Leigh–Minic precedent comes last among the literature because it is the bridge to the program. Freidel, Leigh and Minic wrote that a preferred polarization breaks Born duality; the program’s addition is to read that breaking as spontaneous, with η as the order parameter. The page also records how the credit entered the checkpoint.

The Born length λ closes the section with the one program claim in it. Positions and momenta can be added, compared or exchanged only after each is divided by a reference scale, a length λ and the momentum ε = ℏ/λ, and kinematics singles out no value of λ. The lab notebook scales the pair out of its standard forms without naming it; #13 calls λ the Born length, makes it fix the slope of the momentum-space light cone, c̃ = ℏc/λ², and sets it equal to the Planck length in a step that carries no tag, which gives the maximum force c⁴/G. The identification is alive and unreviewed, and the page separates λ from λ_χ, the invariant length of a conjugate quantum.

Boundaries#