Model II: the self-dual Grosse–Wulkenhaar model

Model II is a scalar field theory, a theory of one field with a single number at each point, set on the noncommutative plane, whose two coordinates fail to commute as position and momentum do. It is taken at the parameter value Ω = 1, where the Langmann–Szabo duality maps the theory to itself. The duality is a Fourier transform: it exchanges a field’s dependence on position with its dependence on momentum, as a waveform is exchanged with its spectrum. The model was tested first because its own vacuum might break that exchange with no extra structure put in by hand. On a basis of oscillator functions the duality reduces to parity, which leaves the even oscillator levels unchanged and reverses the sign of the odd ones, and what follows depends on the field. For complex fields the duality is a symmetry and the vacua break it in the classical theory, before quantum effects are included, but quantum fluctuations restore it. For real fields the duality is not a symmetry at all, and the first quantum correction (the one-loop term) favours the vacuum that is sharper in position, a preference that remains a lead.

Stands: noncommutative field theory and the Langmann–Szabo duality (literature), the matrix base and the action, the duality on the matrix base, complex versus real fields, the complex-model vacua and Δ(β), quantum restoration in the complex model (a negative result) · Died: nothing yet · Open: the real model’s one-loop preference; a folded section on the comparison with the exact solution · Superseded: folded sections on the reversed sign of the polarization and “Increasing the truncation adds only massive levels”

Map#

Why this route came first. In chunk 1 the worklog decided, as Claude’s call under Paul’s delegation, to “test the Grosse–Wulkenhaar Ω = 1 route first; keep the coset model as the η-implementation”. The reason: “It is the one place where the toy could be both new and non-trivial: an interacting, renormalizable field whose own vacuum might select x or p, with no K put in by hand.” A clash between recalled conventions for the duality stalled that chunk (“a recalled convention is wrong somewhere”) and was settled in chunk 2 by direct numerical Fourier integration. The route then failed criterion S4, survival under quantum fluctuations (success criteria), and the coset model’s fallback, a ρ-preserving potential, became the S4 result (model I). The pages follow the computation in the order the toy-model note presents it (§3–§4).

  • Noncommutative field theory and the Langmann–Szabo duality is the established background [T]: the Moyal plane and its star product, the matrix base that turns field theory there into a matrix model, the Grosse–Wulkenhaar model and the oscillator term that makes it renormalizable, the duality that exchanges a field’s position and momentum descriptions, and the self-dual point Ω = 1 with its all-orders vanishing β-function in four dimensions. It also records what the literature had and had not studied about these vacua.
  • Model II: the matrix base and the action sets up the computation: the oscillator functions f_mn, read as Weyl symbols of |m⟩⟨n|, each checked against independent anchors; the Ω = 1 kinetic form 8π(m + n + 1)|φ_mn|², diagonal, computed with exact derivatives after a finite-difference check failed; and the actions for complex fields, arbitrary matrices, and real fields, Hermitian ones.
  • The duality on the matrix base is the lemma the rest depends on: computed by direct Fourier integration, the duality multiplies f_mn by (−1)^m, which is left multiplication by parity. The page explains the parity reading, Grosse and Wulkenhaar’s alternating sign around the interaction vertex, the credit round 4 required, and how the computation settled the chunk-1 clash.
  • Complex versus real fields draws the first consequence. For complex fields the duality is a symmetry, the half-turn of a continuous group of phase rotations. For real fields it leads out of the field space, so the action is only covariant, and the vacua it exchanges need not stay degenerate. Grosse and Wulkenhaar call the real model invariant; the page shows how the two readings describe one algebra.
  • Complex-model vacua and the polarization Δ(β) finds the classical vacua: two oscillator levels condense with a free relative phase β, the duality shifts β by π, and the vacuum’s spread in position minus its spread in momentum varies as cos β. A folded section records the sign, first written reversed and corrected before the review.
  • Quantum restoration in the complex model shows why that classical breaking does not survive: every condensate phase is an exact flat direction, and the condensate is a localized lump with no volume factor, so the quantum average of the polarization is zero and criterion S4 fails for this route. It covers round 4’s restatement, the tadpole caveat with its unreconciled coefficient (ln N or 2 ln N; flag 11 in review/wiki-build-flags.md), the unproven step about a potential on β, the open comparison with the exactly solved related model, and the superseded claim about adding levels.
  • Open: the real model’s one-loop preference closes the section with the real-field version, where the two classically degenerate sign-vacua split at one loop, F(+) − F(−) = 2.634, in favour of the position-sharp one: an explicit selection, recorded as a lead because it has not been followed through renormalization.

The first page is literature [T]. The rest are the program’s computations, all machine-checked in verify/toy_gw.py (the toy-model scripts) and reviewed in review round 4. The worklog tags the lemma, the kinetic form, the symmetry checks, the vacua and the polarization [V]; the restoration is an exact argument [D], whose premise and tadpole are gated, though the note’s §6 lists it as shown by executed code (flag 41 in review/wiki-build-flags.md); the one-loop split is gated but carries no tag in the sources.

Boundaries#