Postulates P1–P5 on The Broken-Born program

P1: the phase-space arena

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

P1 is the program’s first axiom: physics is formulated on phase space, the space whose points are a position together with a momentum, rather than on spacetime alone. The axiom gives phase space two structures. The symplectic form ω is the antisymmetric pairing of position with momentum that quantum mechanics fixes through Planck’s constant; the generalized metric H is a positive measure of length that treats position and momentum directions together. Much of the program is built from these two: the quarter-turn that exchanges position and momentum is generated by the combination I = ω⁻¹H, and the space of possible metrics H becomes the program’s modular field once only ω is held fixed. The lab notebook’s first version of P1 also listed a third structure, the neutral metric η; the current version lists only ω and H, and P2′ makes η a property of the vacuum.

Original P2: both structures rigid (superseded)

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

The original P2 was the lab notebook’s second axiom: of the three structures on phase space, the symplectic form ω and the neutral metric η are fixed background, and only the generalized metric H carries degrees of freedom. It was meant to express the constancy of Planck’s constant, and the notebook gave it three further jobs: removing a would-be second graviton, fixing the relation between the two cosmological scale factors, and supplying the dark-energy field’s potential. The program’s geometry showed that it fixed too much. With both structures fixed, H is forced into a rigid block form, the polarizations (position–momentum splittings) compatible with both include no circle for the dark-energy field to live on, and the theory reduces to general relativity plus one bespoke model of self-interacting dark matter. It was replaced by P2′, which fixes ω alone. Of its three jobs, one was reversed, one survives from ω alone, and one became an open problem.

The nontriviality theorem

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

The nontriviality theorem states that the Broken-Born program predicts anything beyond general relativity plus one model of self-interacting dark matter only if the neutral metric η, the structure that marks which directions count as position, is a property of the vacuum rather than a fixed part of the background. It is the argument that forced the program to change its second axiom. With η fixed, the position–momentum splittings compatible with both fixed structures include no Maslov circle, the loop of splittings on which the program’s dark-energy field lives, and the geometric action depends only on the ordinary spacetime metric, which leaves no second, momentum-space graviton. With η a property of the vacuum, all of these return. The result is a structural argument of the program, tagged derived [D]; the compatibility theorem it starts from is a known result of Born geometry and is machine-checked.

P2′: only the symplectic form is rigid

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

P2′ is the program’s corrected second axiom: of the structures on phase space, only the symplectic form ω is fixed in advance, and holding it fixed is the program’s way of saying that Planck’s constant does not change. The neutral metric η, which marks which directions count as position, is not part of the background. It is a property of the vacuum: the order parameter of the broken symmetry, the quantity whose value tells a vacuum from its mirror image, as the direction of magnetization tells a magnet pointing one way from one pointing the other. The original P2 had fixed η as well, and the nontriviality theorem showed that this left the program with no content of its own. Checkpoint v2.2 added a refinement from the toy models: with only ω fixed, the polarization is a continuous angle. Two consequences follow. Pinning the angle to a definite value needs an additional potential, which brings back a little fixed structure, and the choice has physical consequences only once matter couples to it.

P3: a reciprocity-invariant action

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

P3 is the program’s symmetry axiom: the action, the single expression from which the field equations follow, is unchanged by the Born rotation, the quarter-turn of phase space that sends position to momentum and momentum to minus position. The rotation is generated by the complex structure I = ω⁻¹H built from the two structures of P1. In ordinary quantum mechanics this exchange is a change of description, like switching from a waveform to its spectrum: a unitary map between equivalent ways of writing the same physics. P3 claims more, a symmetry of a specific action, which is the kind of symmetry a vacuum can break spontaneously. The referees asked for an explicit action with that property, and the coset toy model supplied one in a simplified setting.

P4: polarized matter and strict sector separation

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

P4 is the program’s axiom about matter. Ordinary matter, the Standard Model, is x-polarized: its fields are organized around position, the variable in which its interactions are local. A second kind, conjugate matter χ, is p-polarized, organized around momentum. Both kinds respond to the same gravity, so the equivalence principle holds across them, but no gauge force (the electromagnetic, weak or strong interaction) acts between them. Since checkpoint v2.1 the program reads the last clause strictly: the only mixing allowed between the two sectors is what gravity itself induces, which is zero or extremely small. P4 is the basis of the program’s prediction that dark-matter direct-detection experiments will see nothing, and in the toy models it is also what gives the vacuum’s choice of polarization a physical effect.

P5: the vacuum selects a polarization

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

P5 is the program’s axiom of spontaneous breaking: the vacuum of the universe selects a polarization, a choice of which half of the phase-space directions count as position, and ours is the one organized around position in the ordinary sense (“x-tilted”). The mirror choice, a vacuum organized around momentum, satisfies the same laws equally well, because the reciprocity symmetry of P3 carries one onto the other. Which one occurs is a property of the state, not of the laws, as the axes of an ice crystal belong to the crystal rather than to the laws governing water. Freidel, Leigh and Minic had already described a preferred polarization as breaking Born’s symmetry; the program’s addition is to treat the choice as spontaneous symmetry breaking with an order parameter. The toy models qualify the axiom: with only the symplectic form fixed, the choice is a continuous angle, like an ice axis that could point anywhere, and a definite polarization needs an additional potential on that angle.