Phase-space geometry

Phase space is the space of positions and momenta taken together, and this section collects its geometry: the symmetry that exchanges the two, the structures that can be laid on phase space, and the single field the program builds from them. Quantum mechanics treats a state’s position description and its momentum description as equally complete, related as a waveform is to its spectrum. The program starts from that fact, uses a theorem about which phase-space geometries are compatible with it, and arrives at one complex field whose parts it reads as the metric of spacetime, a second gravitational field and the dark-energy field.

Stands: the established physics of Born reciprocity; the compatibility theorem (literature, with the program’s block proof and machine check) and its corollaries; the program’s claims the modular field Z, reciprocity as modular inversion, τ as the Maslov phase and, from #13 and unreviewed, the Born length λ · Died: nothing · Open: Lorentzian dressing of the Euclidean skeleton, the field content of the broken vacuum; folded sections on connection uniqueness under P2′ (on the Born connection), a mechanism that selects one section (on double field theory), the Re Z sector of the geometric action (OP1) (on the modular field Z) and extensivity, the soccer-ball problem (on curved momentum space) · Superseded: a folded section on the three-way classification of X · Conjectures: chat-only, folded: the slaved layer is algebraic, massive X waves

Map#

Born reciprocity comes first because everything else rests on it. All of it but its last page is established physics: the canonical commutator and the uncertainty principle, the symplectic form and its group, the Fourier transform that carries out the quarter-turn ρ: (x, p) → (p, −x), the difference between a change of description and a symmetry of an action, Lagrangian polarizations, the Wigner function, the momentum fibers that phase space is made of, flat in general relativity, and the literature on curved momentum space that asks why momentum space should be flat. The literature ends with the Freidel–Leigh–Minic precedent, the earlier literature in which the program’s central idea, a preferred polarization breaking the symmetry, already appears, and against which the program claims only the spontaneous reading. The subsection closes with the Born length λ, #13’s name for the reference length that puts position and momentum on a common scale, which #13 sets equal to the Planck length; the claim is alive and unreviewed.

Born geometry puts three structures on phase space — the symplectic form ω, the neutral metric η and the generalized metric H — and asks which combinations are compatible; string theory’s doubled geometry pairs the metric instead with an antisymmetric field, the Kalb–Ramond B-field. Its compatibility theorem is a literature result with a block proof and machine check of the program’s own: with ω and η both held rigid, momentum-space geometry is fixed as the inverse of position-space geometry. That left the program with no content of its own and forced its postulate change from P2 to P2′. The section closes with an open problem, because every result in it is proven only for a positive-definite (Euclidean) metric.

The modular field is what remains when only ω is rigid: the compatible metrics form the Siegel domain, coordinatized by one complex field Z = X + iY. The mathematics there (the Siegel domain, the Lagrangian Grassmannian) is standard. The physical readings are the program’s derived claims [D]: Im Z is the metric, Re Z is the momentum graviton, ρ acts as Z → −Z⁻¹, and the boundary phase is the dark-energy field τ.

Boundaries#

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