Spontaneous breaking of reciprocity

Spontaneous symmetry breaking is the situation in which the laws of a system have a symmetry that its lowest-energy state lacks: the laws governing a magnet treat all directions alike, yet a cooled magnet is magnetized along one of them. The Broken-Born program proposes that Born reciprocity, the exchange of position and momentum under which quantum kinematics is exactly symmetric, is broken in this way, so that the vacuum, not the laws, fixes which variable acts as position. This section holds the textbook theory of such breaking, the program’s account of the mechanism, and the two toy field theories built to test it.

Stands: the established physics of spontaneous symmetry breaking, topological defects and the vacuum manifold, Goldstone modes and the Higgs mechanism, Stückelberg fields, inverse-Higgs constraints and the Mermin–Wagner–Coleman theorem; the program’s claims mandatory breaking (untagged, qualified in v2.2), η as the order parameter (a postulate, checked in the coset model), the polarization as a Goldstone angle ([D][V], in model I and the complex model II), the twisted element, the section condition (d = 1), locality from an (H, K)-built coupling (the weak form), and acceptance criterion 1, met in the referees’ wording · Died: the naive ρ-preserving potential (in the graveyard) · Open: the origin of the ρ-preserving potential, forced locality and the P4 lock, a Monte Carlo with H dynamical, ρ’s SO(2) in the coupled model, the real model’s one-loop preference; folded sections on what the toy models leave undecided, locality assumed, not derived and the section condition on curved backgrounds (all in open problems) · Superseded: the discrete breaking pattern ℤ₄ → ℤ₂, replaced by the Goldstone angle; folded sections on the rigid symplectic form as the source of τ’s potential, every derivation selects the pair and a coupling built from K alone · Conjectures: two uncomputed worklog remarks, the thawing of dark energy as a drifting polarization angle and continuous breaking in 3+1 dimensions (both in conjectures)

Map#

The section opens with six pages of established physics, which the program’s claims rely on. Spontaneous symmetry breaking defines the pattern: a symmetric law with an asymmetric ground state, the vacuum manifold and stabilizer, the order parameter, why breaking needs infinitely many degrees of freedom, and spontaneous against explicit breaking. It also gives the program’s operational definition, the toy-model criteria S1–S3, with model I as the worked case. Topological defects and the vacuum manifold follows the vacua further: they form the coset G/G₀, with one flat direction for each broken generator, and the homotopy groups of that space decide which defects a cooling universe can leave (π₀ domain walls, π₁ strings, π₂ monopoles, π₃ textures) and why they cannot unwind. The program’s own vacuum manifolds are there: the Lagrangian Grassmannian Λ(d), whose unshrinkable loop gives τ its period, and model I’s half-plane times a circle; so are the retired π₂ dark-matter candidate and the two meanings of “soliton” in the record. Goldstone modes and the Higgs mechanism covers the massless modes that a broken continuous symmetry leaves behind and their count from the Hessian, pseudo-Goldstone modes, and how a gauge field absorbs the modes and becomes massive; the program meets these in model I’s three flat directions, in its polarization angle, and in the mass of its momentum graviton. Stückelberg fields writes the absorbed modes out as fields, A_μ + ∂_μπ for a massive vector and h_μν + ∂_μπ_ν + ∂_νπ_μ for a massive spin-2 field, whose gradient part carries the helicity-0 polarization; the program’s use is the modes the momentum graviton X absorbs, which #11 calls the Stückelberg sector, with τ outside it and their mixing open (OP6). Inverse-Higgs constraints qualify that count when the broken symmetries act on spacetime: a would-be Goldstone field can be fixed as a derivative of another and drop out, as a membrane’s tilt is fixed by the slope of its displacement. The program’s record uses the idea once, in calc note #5’s expectation, never worked out, that such constraints remove most of the nine non-Maslov moduli of the broken vacuum. The Mermin–Wagner–Coleman theorem states which symmetries can break in two dimensions and why tunnelling rules out breaking in 0+1, the limits that determined how the toy models were built; in model I the theorem applies only to the compact rotation group containing ρ.

The program’s own claims follow in the order of its argument. Mandatory breaking is the uncertainty-principle argument that some polarization must be selected in any world with local physics, given step by step with the qualifications v2.2 added. The order parameter names the quantity that carries the selection, the neutral metric η, written in terms of J, η or K in different documents, and records the superseded discrete pattern ℤ₄ → ℤ₂. The polarization as a Goldstone angle records the v2.2 finding from the toy models: the selection is a continuous angle rather than a discrete choice, so a definite polarization needs a potential, and in the coset model the angle is the circle the program assigns to its dark-energy field τ, which shows consistency, not a test. Open: the origin of the ρ-preserving potential asks where the potential that pins that angle comes from.

The twisted element ρ∘(K → −K) shows that the breaking has no physical effect until matter can tell K from −K, which leads to the three pages on matter. The section condition confines matter to one of the two lines that η singles out, without saying which. Locality from an (H, K)-built coupling makes matter local in the selected line, the weak form of the locality criterion. Open: forced locality and the P4 lock records that nothing yet forces matter to couple that way.

Last, Toy models of broken reciprocity holds the two explicit field theories behind these results: model I, a coset sigma model, and model II, the self-dual Grosse–Wulkenhaar model, with the criteria fixed before they were built. The pages above draw the general lessons; the toy-model section holds the calculations.

Boundaries#