The modular field on The Broken-Born program

The Siegel domain Sp(2d, ℝ)/U(d)

Mon, 01 Jan 0001 00:00:00 +0000

The Siegel domain is the space of complex symmetric d × d matrices Z = X + iY whose imaginary part Y is positive definite. For a single coordinate (d = 1) it is the upper half of the complex plane; for larger d it is the matrix version of that half-plane. Each point Z names exactly one positive phase-space metric H_Z that is compatible with a fixed symplectic form ω, and every such metric has a point, so the domain is the space of all of them. The symplectic group moves its points by fractional-linear (Möbius) maps, the matrix versions of z → (αz + β)/(γz + δ). The program uses the domain as the arena of the modular field once only ω is held rigid. The space and its parametrization are standard mathematics, and battery test T2 checks the parametrization’s identities at one sampled point.

The modular field Z = X + iY

Mon, 01 Jan 0001 00:00:00 +0000

The modular field Z = X + iY is a complex symmetric matrix field that, in the program, carries the whole geometry of phase space at each point of spacetime once only the symplectic form is held rigid. Each value of Z is a point of the Siegel domain and names one phase-space metric compatible with that form. The program reads the imaginary part Y as the spacetime metric g, and the real part X as a new symmetric tensor field, the momentum graviton, which records how far the split between position and momentum tilts from place to place. In the vacuum X = 0 and Z = i·g. These readings are the program’s derived claims, tagged structural in the note that introduced them; the algebra of the parametrization is machine-checked, and later notes show that X must be a massive spin-2 field.

Reciprocity as modular inversion Z → −Z⁻¹

Mon, 01 Jan 0001 00:00:00 +0000

Modular inversion is the map Z → −Z⁻¹, “take minus the reciprocal”, applied to the modular field Z. The program shows that Born reciprocity, the quarter-turn of the position–momentum plane that the Fourier transform performs on a wavefunction, acts on Z in exactly this way. The map leaves the point Z = i unchanged, and that point is the program’s self-dual vacuum. On the vacuum slice the map sends the metric g to its inverse, which for a uniformly scaled metric turns the scale factor a into 1/a; the circle |Z| = 1 is identified with the program’s relation a·ã = 1 between our scale factor and that of the conjugate description, an identification queried in flag 23 of review/wiki-build-flags.md. The same quarter-turn matrix generates the S-duality of electric and magnetic charges. The claim is derived; for one coordinate (d = 1) a script checks it symbolically, and no script checks the general case.

The Lagrangian Grassmannian Λ(d)

Mon, 01 Jan 0001 00:00:00 +0000

The Lagrangian Grassmannian Λ(d) is the space of all ways to split a 2d-dimensional phase space into a “position” half and a “momentum” half. Each Lagrangian subspace, a d-dimensional plane on which the symplectic form vanishes, is one point of it. For d = 1 it is the set of lines through the origin of the (x, p) plane, which is a circle; in general it is the compact manifold U(d)/O(d), of dimension 10 when d = 4. It contains a loop that cannot be shrunk to a point, the Maslov circle, so any angle measured around that loop is periodic. It also forms the distinguished (Shilov) boundary of the Siegel domain. In the program it is the space of possible vacuum polarizations, and its circle direction carries the dark-energy field. The mathematics is standard.

τ as the Maslov phase of the boundary

Mon, 01 Jan 0001 00:00:00 +0000

The program’s dark-energy field τ is an angle: the angle around the Maslov circle of the Lagrangian Grassmannian, the space of all position/momentum splittings, which forms the distinguished (Shilov) boundary of the Siegel domain where the modular field takes its values. Because that circle cannot be shrunk to a point, τ is periodic, with period 2πf, as a matter of topology rather than of modeling. Of the ten directions of the Grassmannian at d = 4, τ is assumed to be the only light one. The identification is derived [D], the topology is literature [T], and the lightness is a postulate [P]. How τ behaves as dark energy, and how current data bear on it, is worked out in the cosmology section.

Open: field content of the broken vacuum

Mon, 01 Jan 0001 00:00:00 +0000

The field content of the broken vacuum is the list of fields that the program’s geometry leaves in play once the vacuum has chosen a polarization: the ordinary metric, the momentum graviton X, the ten moduli of the Lagrangian Grassmannian (the dark-energy field τ plus nine more), and matter. Calc note #5 wrote the list down and marked it partial. The later Debt-1 notes settled what X is: a massive spin-2 field. Whether the nine extra moduli survive as fields, or are removed by inverse-Higgs constraints (relations that turn some Goldstone-type fields into derivatives of others), was never worked out. The question is open, and it is not on the checkpoint’s numbered list of open problems.