Born geometry on The Broken-Born program

The structures ω, η, H and the split-quaternion algebra

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

Born geometry equips phase space, the space of positions and momenta taken together, with three structures: the symplectic form ω, the antisymmetric pairing behind the quantum commutator; a neutral metric η, a symmetric pairing with as many positive as negative directions, whose null directions mark out which coordinates count as position and which as momentum; and a positive generalized metric H, a ruler that measures lengths in phase space. Combined in pairs, the three give operators I, J and K that multiply like the split quaternions: I squares to −1, J and K square to +1, and IJ = K. Freidel, Leigh and Minic named phase spaces carrying such a compatible triple Born geometries. The quarter-turn ρ, which the Fourier transform implements, preserves ω and H in their standard forms and reverses the sign of η. Under the program’s corrected postulate η is the order parameter of the breaking, so ρ changes the order parameter’s sign.

Double field theory and the generalized metric

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

Double field theory is a formulation of the low-energy fields of closed string theory (the metric, the antisymmetric Kalb–Ramond B-field and the dilaton) on a doubled space, in which each ordinary coordinate has a partner “dual” coordinate. Its central object, the generalized metric, packages the metric and the B-field into one matrix that preserves a neutral metric η, and its symmetry group O(d, d) is the group of transformations that preserve η. For the theory to be consistent, fields may depend on only half of the doubled coordinates: the section, or strong, condition. The known derivations of that condition leave open which half, and no mechanism that selects one is known. For the program, double field theory is the contrast case. Doubling geometry with η gives the metric an antisymmetric partner, the B-field; doubling it with the symplectic form ω gives it a symmetric partner, a second graviton; and holding ω rigid as well as η removes the B-field altogether.

The Kalb–Ramond B-field

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

The Kalb–Ramond B-field is an antisymmetric tensor field, B_μν = −B_νμ, that does for a string what the electromagnetic potential does for a charged particle: a particle’s quantum phase changes by the potential summed along its path, and a string’s by the B-field summed over the surface the string sweeps out. Like the electromagnetic potential it is a gauge field, so it can be changed by a gauge transformation without changing anything measurable, and what is measurable locally is its field strength, built from its derivatives. It is one of the massless fields of closed string theory, with the metric and a scalar, the dilaton. T-duality, the symmetry of strings on a circle that exchanges momentum around the circle with winding around it, turns parts of the metric into parts of the B-field, and for that reason double field theory puts the two in one matrix, the generalized metric, which preserves a neutral metric η. The program uses the B-field as established physics in three ways. Holding the symplectic form ω rigid as well as η removes it (corollary 2 of the compatibility theorem). It is the antisymmetric counterpart of the program’s own partner of the metric, the symmetric momentum graviton X: X is a second graviton, and the B-field is not. And its healthy propagation alongside the metric and the dilaton is the program’s precedent for a doubled geometry whose kinetic terms have no ghosts, modes that carry negative energy.

The Born connection

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

The Born connection is the connection of a Born geometry, phase space with the three compatible structures ω, η and H of Born geometry: the rule for comparing vectors at neighbouring points that respects all three structures. A connection defines how fields are differentiated, and its curvature is what the field equations of a geometry are built from. Freidel, Rudolph and Svoboda showed that a Born geometry has exactly one connection compatible with all three under the conditions of their theorem, so its curvature R_B[H] is well defined. The program’s first action for the phase-space metric was built on that curvature. In the current reference the connection appears in a citation and in the argument that frames, not metrics, are the natural variables for the mass term of the momentum graviton. Under the corrected postulate only ω and H are fundamental, and whether a unique compatible connection still exists then has not been shown.

The compatibility theorem

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

The compatibility theorem is a statement of linear algebra about metrics on phase space: if the symplectic form ω and the neutral metric η are both held fixed in their standard forms, the only positive metrics H compatible with both are block-diagonal, H = diag(g, g⁻¹), with the momentum block equal to the inverse of the position block. “Compatible” means that H fits each structure in the way Born geometry requires, (η⁻¹H)² = 1 and (ω⁻¹H)² = −1. The statement is Freidel, Leigh and Minic’s; the short block proof and its machine check are the program’s, and an independent reviewer re-derived it in round 3 and found that one step of the proof is not needed. Its use in the program is that it forced a change of postulates: with both structures fixed, the program reduces to general relativity plus one dark-matter model, so the program replaced its postulate P2 with P2′ and let η vary.

Corollaries: unimodularity, no B-field, slaved momentum metric

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

Three consequences follow from the compatibility theorem’s answer H = diag(g, g⁻¹). The determinant of H is 1, so the phase-space volume that H measures cannot change, which the program reads as the constancy of Planck’s constant built into the geometry. The off-diagonal block, where double field theory keeps its antisymmetric B-field, must vanish. And the momentum metric is fixed as the inverse of the position metric, g̃ = g⁻¹, so momentum space has no geometry of its own. That last corollary contradicted the program as it then stood: the dark-energy field and the geometric origin of the dark-matter gate both disappeared. It forced the postulate change to P2′, under which the block form, and with it the slaved momentum metric, holds only on the vacuum slice of the modular field.

Open: Lorentzian dressing of the Euclidean skeleton (OP3)

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

The Lorentzian dressing of the Euclidean skeleton is the open problem of carrying the program’s main geometric results over from a positive-definite phase-space metric, the Euclidean case, to the Lorentzian signature of physical spacetime, in which time enters with the opposite sign to space. Those results, the compatibility theorem, the parametrization of the Siegel domain and the chain of arguments that forces the momentum graviton X to be massive, were proven in the Euclidean case. The extension is open problem 3 of the checkpoint #11. The change is not a formality: with a Lorentzian metric the group that leaves the vacuum fixed becomes non-compact, and the metric on the space of field values becomes indefinite, the situation in which some field components would carry kinetic energy of the wrong sign. The working notes expect the gauge symmetry and mass mechanism already in the chain to keep the theory healthy there, and list the extension as a check. Nothing has been computed, and the checkpoint counts it among the three items on which the word “theory” still waits.