Born reciprocity ρ: (x, p) → (p, −x)

Born reciprocity is the map ρ: (x, p) → (p, −x), which exchanges position and momentum, with one sign change, as a quarter-turn of phase space. The rules of quantum kinematics are unchanged by it, and the Fourier transform carries it out on wavefunctions. Max Born proposed in 1938 and 1949 that a fundamental theory should respect it, and hoped for a physics with a fundamental mass and a fundamental length on equal footing. Observed dynamics does not respect it — interactions are local in position — and the program’s starting assumption, its hard core, is that ρ is an exact symmetry of the laws that the vacuum breaks.

Status: Established — literature [T]; #11 cites Born from model memory [1†, 2†]. The round-3 citation audit confirmed both references’ metadata but did not read the primary text (“partially verified (primary text not read)”, capitals lowered here); the source check is part of the † literature battery. The program’s uses of ρ (its action on the geometry and the modular field) carry their own tags, given below.

The map#

On the coordinates (x, p) of phase space, ρ is the matrix

ρ = [[0, 1], [−1, 0]]          ρ(x, p) = (p, −x)
ρ² = −1:  (x, p) → (−x, −p)    ρ⁴ = 1

It turns the phase plane by a quarter-turn: the new position is the old momentum, and the new momentum is the old position with its sign reversed. Two quarter-turns give the parity ρ², which reverses both variables (#1 §2); four give the identity. So ρ generates a cyclic group of order four, which the lab notebook calls “the ℤ₄ generated by the complex structure I” (#1 §1, P3). With d positions and d momenta the same matrix acts in d × d blocks (#1 §2).

Why the sign#

A plain exchange (x, p) → (p, x) reverses the sign of the symplectic form (in the phase plane of one position and one momentum, it reverses the orientation) and turns [x, p] = iℏ into −iℏ [T]. The minus sign repairs this. ρ “is symplectic and squares to −1” (toy-model note §1): it preserves the symplectic form ω, and with it the canonical commutator. Among the linear maps that preserve ω it is the quarter-turn of the compact circle of rotations. The popular article #12 says the same in words: exchange position for momentum throughout “(with one sign adjustment) and the formalism is unchanged” (“Two descriptions of one thing”; never a status source).

A symmetry of kinematics#

#11 §3: “Quantum kinematics is exactly invariant under ρ: (x,p) → (p,−x); the Fourier transform implements it unitarily [1†, 2†].” The v1 paper adds that “the canonical commutator is blind to it” (#9 §1). The transform F conjugates x̂ into p̂ and p̂ into −x̂ [T], which is ρ written for operators; see the Fourier transform as a phase-space rotation.

“Kinematics” here means the arena: the operators, their commutation relations and the space of states. It leaves out the Hamiltonian, which says how states change in time. A symmetry of the arena alone is, ordinarily, a change of description, the same physics in rotated variables, rather than a symmetry of a specific action. The withdrawn manuscript #14 states the point: “as usually encountered, rho is a unitary equivalence between representations of the canonical commutation relations, not a Noether symmetry of a specified action” (#14 §1). See unitary equivalence versus Noether symmetry.

What breaks it in observed physics#

The laws of motion single out position. The toy-model note’s abstract opens on this: ρ “is an exact symmetry of quantum kinematics but not of observed dynamics, which is local in position”. The v1 paper lists the asymmetries: “Hamiltonians are p²/2m + V(x), interactions are local in x, the mass shell is a momentum-space structure with no positional twin, and general relativity curves x-space while leaving momentum fibers flat” (#9 §1).

The first item shows the asymmetry plainly. A Hamiltonian p²/2m + V(x) is quadratic in momentum and arbitrary in position; under ρ it becomes x²/2m + V(p), a different law [T]. Born’s question, why physics uses only half of the symmetry its kinematics has, is on Born’s question; the flat momentum fibers are discussed on momentum fibers and the cotangent bundle and curved momentum space.

Born’s papers#

#11 cites two papers, both from model memory:

  • [1]† M. Born, “A suggestion for unifying quantum theory and relativity,” Proc. R. Soc. Lond. A 165, 291 (1938).
  • [2]† M. Born, “Reciprocity theory of elementary particles,” Rev. Mod. Phys. 21, 463 (1949).

#13 cites the same two papers as [3]† and [4]†, and the withdrawn #14 cites both under its unmarked ref. [1].

The round-3 referee checked them while auditing #14: “metadata OK (DOI 10.1098/rspa.1938.0060; 10.1103/RevModPhys.21.463). 1938 abstract does not state the rule; primary text paywalled. Secondary sources (Wikipedia “Born reciprocity”) confirm x->p, p->-x. Fourier-transform statement is standard maths.” Its verdict, set in lower case here, was “partially verified (primary text not read)” (review/round3/referee/audit-notes.md, Step 1). No document records a reading of Born’s own text, so what Born wrote, as distinct from the bibliographic data, remains to be checked in the † literature battery (OP10).

Born’s self-dual program#

#13 records Born’s motive: “This was Born’s original hope in 1938: a physics with a fundamental mass and a fundamental length on equal footing” (§4). What ρ does to relativistic invariants shows why the two scales go together. The invariant of special relativity is the mass shell, p_μp^μ = m²c². Its Born image is a hyperboloid in position space:

x_μ x^μ = λ_χ²

#13 calls it an invariant-length shell: “conjugate quanta are characterized by an irreducible extent λ_χ as ours are by mass [D — structural; this recovers Born’s 1938–49 program of a self-dual physics with both m and λ fundamental [3†, 4†]]” (§4). #13 lists “Born’s self-dual program [3†, 4†]” among its literature tethers (§7). See the invariant-length shell and Planck closure. Whether Born’s papers state the hope in these terms awaits the source check above.

In its relativistic form the map exchanges all four coordinates with all four momenta, time with energy included [T]. #13 works in a “3+1 presentation” that leaves time alone and lists “the fully covariant construction with the four-dimensional Born map (t ↔ E)” as open [O] (§6); see Open: the four-dimensional Born map.

Names#

The documents use several names for the same map:

  • Born reciprocity — #11 §1, §12 and Appendix A, and the toy-model note.
  • The Born rotation — #9 §1, #14 §1, and #11’s postulate P3 (“the Born rotation generated by I = ω⁻¹H”).
  • Born duality — Freidel, Leigh and Minic’s name (broken Born duality).

In model II of the toy models, the self-dual Grosse–Wulkenhaar model, the role of ρ is played by a field-theory duality that exchanges positions and momenta, that of Langmann and Szabo (noncommutative field theory and the Langmann–Szabo duality).

What the program does with it#

  • The hard core. “The program: ρ is exact and spontaneously broken.” (#11 §3; the hard core). The program argues that the breaking is unavoidable (mandatory breaking); the idea that a preferred polarization breaks the symmetry is credited to Freidel, Leigh and Minic (broken Born duality).
  • The postulate. “P3 The action is invariant under the Born rotation generated by I = ω⁻¹H. [P]” (#11 §4; P3).
  • On the three structures. In the standard frame, “R preserves ω (it is symplectic), preserves H (it is orthogonal), and flips the sign of η, hence flips J and K while preserving I” [D] (#1 §2, where R is the matrix of ρ; the structures ω, η, H). Since η is the program’s order parameter, the calc note reads the flip as “the broken symmetry moving the order parameter” (#5 §B; the order parameter).
  • On the modular field. “ρ: Z → −Z⁻¹ (Möbius; d = 1 check), fixed point Z = i” [D] (#11 §6; reciprocity as modular inversion).
  • In the toy models. In model I, the coset sigma model of a metric H and a real structure K, “ρ acts as Z → −1/Z and K(φ) → K(φ + π) = −K(φ). The stabilizer of (1, K₀) in Sp(2,ℝ) is {±1}” (toy-model note §2). But “the twisted element ρ∘(K → −K) maps (1, K₀) to itself”, so breaking ρ is physical only once a coupling odd in K is present (the twisted element). In model I and in the complex form of model II, ρ sits inside a continuous U(1), and the polarization is a Goldstone angle [D][V] (#11 §4, §10; the polarization as a Goldstone angle; checked by the toy-model scripts); the real form of model II has a discrete pair of vacua instead (toy-model note §3). The lab notebook’s older breaking pattern, ℤ₄ → ℤ₂ = {1, ρ²}, is superseded by that picture (superseded statements).
  • In charge space. S-duality acts on the complexified coupling as τ_c → −1/τ_c, “generated by the same [[0,1],[−1,0]] rotation as Born’s ρ and our Z → −Z⁻¹: the third appearance of one matrix in this program” (#11 §11.3; the symplectic charge lattice).

Used by#

Sources#

  • papers/broken-born-checkpoint-v2.md §3 — the symmetry of quantum kinematics; the program’s reading of it.
  • papers/broken-born-checkpoint-v2.md refs [1]–[2] — Born 1938 and 1949, both †.
  • papers/broken-born-checkpoint-v2.md §4, §6, §10, §11 — P3 and the v2.2 refinement; ρ acting as Z → −Z⁻¹; ρ inside a continuous U(1) in the toys; the same matrix in S-duality.
  • papers/reciprocity-toy-models.md Abstract, §1–§3 — kinematics versus dynamics; ρ as a symplectic matrix squaring to −1; ρ in model I and the twisted element; the real model’s discrete vacua.
  • papers/conjugate-special-relativity.md §4, §6, §7 — Born’s hope for a fundamental mass and length; the four-dimensional Born map as open; the literature tether [3†, 4†].
  • review/round3/referee/audit-notes.md Step 1 — the citation check on Born 1938 and 1949.
  • notes/debt1a-geometric-core.md §B — the flip of η read as the broken symmetry moving the order parameter.
  • notes/broken-born-program-v1.md §1 — the commutator is blind to ρ; the dynamical asymmetries.
  • notes/broken-born-theory.md §1, §2 — P3’s ℤ₄; ρ² as parity; ρ’s action on (ω, η, H).
  • notes/knot-dark-matter-standalone.md §1 — ρ as a unitary equivalence of representations (withdrawn paper).
  • popsci/universes-other-half.md “Two descriptions of one thing” — “one sign adjustment” (popular article; never a status source).