The polarization as a Goldstone angle

The polarization as a Goldstone angle is the toy models’ finding that the vacuum’s polarization, which fixes the direction in the position–momentum plane that counts as position, is an angle that can turn smoothly rather than a switch with two settings. Born reciprocity is a quarter turn of that plane, the operation the Fourier transform performs when it turns a waveform into its spectrum, and in the toy models it is one member of a continuous family of symmetries. Turning the angle costs no energy, which makes it a Goldstone direction: the position-sharp and momentum-sharp vacua lie on one continuous family. This finding, recorded in #11 v2.2, is why a definite polarization needs an added potential on the angle. In the coset model the angle is the same circle that the program identifies with its dark-energy field τ, which shows that the structures fit together but does not test the cosmology. It is evidence from two toy models, not a theorem.

Status: Alive — [D][V] (#11 §4, P2′ refinement; #11 §10, “Toy-model support (v2.2) [D][V]”) — machine-checked in the toy models (verify/toy_coset.py, verify/toy_gw.py; the toy-model scripts); the link to τ is consistency only, and two worklog remarks, the drift of τ as a turning angle and continuous breaking in 3+1, are uncomputed conjectures.

The finding#

The worklog of the toy models states the result in its second chunk, after both models had been built; ρ is Born reciprocity, (x, p) → (p, −x):

The structural finding, common to both toys.

  • At the self-dual point, Born reciprocity is never the whole symmetry. It is one element of a continuous U(1):
    • the coset model: the U(1) stabilizer of H that rotates K;
    • Grosse–Wulkenhaar: the left phase rotations e^{−iαN}, with the duality at α = π.
  • So “which polarization” is an angle on a Goldstone circle, not a discrete choice. The x-sharp and p-sharp vacua are joined by a flat direction.

(worklog, ‘Chunk 2 results’)

and adds: “This is model-level evidence from two toys, not a theorem.” The program’s documents record it in two places, both tagged [D][V]:

  • In the postulates. #11’s refinement of P2′: “with only ω rigid, ρ sits inside a continuous U(1), and the polarization is a Goldstone angle rather than a discrete choice [D][V].” (#11 §4).
  • In the cosmology. “Toy-model support (v2.2) [D][V]: in an explicit ρ-symmetric coset model of (H, K), and in the complex self-dual Grosse–Wulkenhaar model, ρ sits inside a continuous U(1), so the polarization is a Goldstone angle.” (#11 §10).

A Goldstone direction is a direction along the family of equivalent vacua, along which the potential is flat (Goldstone modes and the Higgs mechanism). The finding qualifies the program’s argument that some polarization must be selected (mandatory breaking): what is selected is a point on a circle. The worklog lists it first among the “Findings that were not predicted” (‘Assessment’; on that listing, see flag 10 in review/wiki-build-flags.md).

Model I: the rotations that keep H fixed#

The coset sigma model (model I) has two fields on a 1+1-dimensional base, a phase-space metric H and a real structure K, and its action is invariant under the whole symplectic group Sp(2, ℝ) (toy-model note §2). On H, ρ acts as the inversion Z → −1/Z of the upper half-plane (reciprocity as modular inversion), whose fixed point Z = i is H = 1 (verify/toy_coset.py). The symplectic maps that keep H = 1 fixed are the rotations of the phase plane, the group SO(2) ≅ U(1) [T]; in the gate’s stabilizer computation, keeping H = 1 fixed is the condition SᵀS = 1, together with det S = 1. ρ = [[0, 1], [−1, 0]] is the quarter turn in this group. This is the worklog’s “U(1) stabilizer of H that rotates K”.

The vacua at H = 1 form a circle on which the potential vanishes:

K(φ) = [[cos φ, sin φ], [sin φ, −cos φ]]                      (toy-model note §2)

ρ:  K(φ) → K(φ + π) = −K(φ)

The gate checks that the potential vanishes on the whole circle, and that ρ maps “the vacuum (1, K(phi)) to (1, -K(phi)) = (1, K(phi + pi))” (verify/toy_coset.py). Each K(φ) has two eigenlines, which are the candidate position and momentum lines (the order parameter). At φ = 0, K = K₀ = diag(1, −1), whose +1 eigenline is the x axis; at φ = π, K = −K₀, whose +1 eigenline is the p axis. The worklog’s names for the two vacua ±K_ref are “‘x is position’ and ‘p is position’” (‘Chunk 3 results’). In between, the pair of lines turns in the phase plane, and every intermediate vacuum has the same energy: the x-sharp and p-sharp vacua are joined by a flat direction.

The Hessian confirms that the circle is flat. “The potential’s Hessian has rank 1 at the vacuum. That leaves three Goldstone directions: the two non-compact half-plane directions, and the compact angle φ, which parametrizes the Lagrangian Grassmannian Λ(1).” (toy-model note §2; checked in verify/toy_coset.py; see the count in model I). The polarization angle is the compact one.

The rotations of the phase plane also have a familiar action on wavefunctions [T]. A rotation by an angle α acts on a state, up to a phase, as the fractional Fourier transform of that angle: at α = 0 the state is unchanged, at a quarter turn it becomes its ordinary Fourier transform, the momentum wavefunction, and the intermediate angles interpolate continuously between a waveform and its spectrum. Model I itself contains no wavefunctions; matter enters the toy only in §5 of the note, where ρ acts on a state as ψ → Fψ, F the Fourier transform. The continuous family that contains ρ is, in that language, the family of fractional Fourier transforms that contains F (the Fourier transform as a phase-space rotation).

Model II: the left phase rotations#

The second toy is the self-dual Grosse–Wulkenhaar model with complex fields (model II), a field theory on a noncommutative plane written in its matrix base. In units where the kinetic coefficient is 1 its action is (toy-model note §3)

S[φ] = Σ_mn (m + n + 1 − μ²)|φ_mn|² + λ Tr((φ†φ)²)

“It satisfies S[UφV†] = S[φ] for all diagonal unitaries U, V. The duality Πφ is the α = π element of e^{−iαN}.” Here the duality is the program’s ρ in this model: on the matrix base it acts as D f_mn = (−1)^m f_mn, left multiplication by parity (the duality on the matrix base). The left phase rotation e^{−iαN} multiplies row m of the matrix φ_mn by the phase e^{−iαm}; at α = π the phases are (−1)^m and the rotation is the duality.

The vacua at μ² = 4, λ = 1 have two condensed levels: “Gradient descent from random starts always reaches a diagonal φ with |φ_mm|² = (1.5, 0.5, 0, …) and S_min = −2.5. The relative phase β of φ₀₀ and φ₁₁ is free. The duality maps β → β + π, so no vacuum is invariant.” (toy-model note §3). The phase β is the flat direction, and the vacuum’s position-versus-momentum polarization varies along it as a cosine:

Δ(β) = ⟨x²⟩_vacuum − ⟨x²⟩_dual = Δ(0) cos β,    Δ(0) > 0              (toy-model note §3)

“β = 0 is broader in position than in momentum; β = π is its image.” The worklog adds that β = ±π/2 “is balanced” (‘Chunk 2 results’). Here the flat direction literally carries the vacuum from one polarization through the balanced point to the other (complex-model vacua). The gate prints Δ(0) = 10.8828 and checks “Delta(beta) = Delta(0) cos(beta): the polarization is an angle on the Goldstone circle” (verify/toy_gw.py). The round-4 referee reproduced the cosine with an independent Fourier quadrature, “to 6 decimals at 9 angles”, and the same sign of Δ(0); its value, +0.866025, differs from the gate’s, and the documents do not compare the two normalizations (audit notes, Step 2(b)).

In this model the Goldstone angle is a classical statement only. “In complex model II the breaking does not survive quantization: every condensate phase is an exact flat direction, and the condensate is a localized lump with no volume factor.” (toy-model note, Abstract). The quantum measure is uniform in β and ⟨Δ⟩ = 0 (§4; quantum restoration in the complex model).

Scope: where the finding holds#

The worklog wrote the finding for every self-dual point (“At every self-dual point, ρ sits inside a continuous U(1)”, ‘Assessment’). The real Grosse–Wulkenhaar model, the version the duality literature studies, does not fit it. There “the duality is only a covariance”, not a symmetry of the field space, and the vacua are discrete: “The vacua are diag(±√q₀, ±√q₁). Up to the global sign, the two relative signs are classically degenerate (S = −2.5) but have different fluctuation spectra.” (toy-model note, Abstract and §3). One loop lifts the degeneracy explicitly, in favour of the vacuum sharper in position (complex versus real fields; Open: the real model’s one-loop preference). The round-4 referee drew the consequence that in the real model the polarization is a discrete classical choice, lifted explicitly at one loop, not a Goldstone angle (audit notes, Step 2(b), F4), and that the general claim “is true for the two models chosen” (Step 3). Revision 3 of the adjudication: “Restrict ‘Goldstone angle’ to the complex model.”

The note v2 states the scope accordingly: “Where ρ sits inside a continuous U(1) — model I, and model II with complex fields — the polarization is a Goldstone angle.” (Abstract), and “In model I and complex model II, ρ sits inside a continuous U(1), so the polarization is a Goldstone angle.” (§6). Both models have d = 1 on a 1+1-dimensional base, and “anything beyond d = 1” is listed as not shown (§6).

What follows: a definite polarization needs a potential#

A Goldstone angle is not pinned. The worklog’s first consequence: “(i) P5’s ’the vacuum selects a polarization; ours is x-tilted’ needs explicit breaking of that U(1) — a potential on the circle — to make the selection definite.” (‘Chunk 2 results’; P5). In 1+1 dimensions there is a second reason: “In 1+1 the compact angle φ cannot order: Mermin–Wagner [13, 14] applies to the compact SO(2) containing ρ.” (toy-model note §4; the Mermin–Wagner–Coleman theorem).

Model I therefore adds “a potential that is invariant under ρ and under K → −K, and reduces the U(1) to a discrete group” (toy-model note §4):

  • The first form, −μ[Tr(K K_ref)]², “is unbounded below once H is dynamical” (Obituary: the naive ρ-preserving potential).
  • The bounded form V_b = −μ[Tr(K K_ref)]² / (2 Tr KᵀK) satisfies “−μ ≤ V_b ≤ 0”, and “With H dynamical, (1, K_ref) is a stable minimum: Hessian eigenvalues 0, 0.764, 2, 5.236.” At H = 1, “V_b = −(μ/2)(1 + cos 2φ)”, which is lowest at the two vacua ±K_ref (the bounded potential V_b).

With the potential the angle is no longer exactly flat: it is a pseudo-Goldstone mode (pseudo-Goldstone modes; the polarization angle as an axion-like field). The worklog’s verdict on what the toy supports: “a pseudo-Goldstone polarization angle with a ρ-symmetric potential”, not “only ω rigid, nothing else” (‘Chunk 3 results’). On the lattice, with H frozen at 1, the potential produces an ordered ℤ₂ phase, “⟨|m|⟩ = 0.954 ± 0.0001 at every size” (on that figure, see flag 13 in review/wiki-build-flags.md), while without it the magnetization falls with system size, “No true order” (toy-model note §4; verify/toy_s4.py; Monte Carlo: an ordered ℤ₂ phase). The round-4 referee found the residual symmetry inside the rotations to be the discrete group {±1, ±ρ} (audit notes, Step 3; see the superseded discrete pattern).

#11 records the price: “A definite polarization needs a ρ-preserving potential on that angle — a rigid, unordered pair of axes, so more structure than ω alone, though less than a rigid η.” (§4). V_b contains K_ref only through the square [Tr(K K_ref)]², so K_ref and −K_ref give the same potential: it fixes a pair of axes without saying which is position. #11 §10 adds: “A definite polarization requires the ρ-preserving potential this section posits. That potential must be bounded with all fields dynamical — the round-4 review found the naive form unbounded once H moves”. Where the potential comes from is open (Open: the origin of the ρ-preserving potential), and even a definite angle breaks ρ physically only once matter couples to the sign of K (the twisted element).

The coset angle is τ’s circle#

In model I the compact Goldstone direction is the program’s own polarization circle. The worklog: “The compact Goldstone is the rotation of K at fixed H: the Lagrangian Grassmannian Λ(1), which is the checkpoint’s Maslov circle (the τ field of §10).” (‘Chunk 1 results’). In the program, the space of possible polarizations is the Lagrangian Grassmannian Λ(d) = U(d)/O(d), and “the dark-energy field τ is the Maslov phase of the boundary variable, its periodicity a fact of topology (π₁(U(d)/O(d)) = ℤ)” (#11 §6; τ as the Maslov phase). The lab notebook defined τ as “a compact, 2πf-periodic modulus of the vacuum” (#1 §2). In d = 1, Λ(1) is a circle, and the toy’s angle φ parametrizes it.

#11 §10 states the connection and its limit together: “In the coset model that angle parametrizes Λ(1): it is τ’s circle.” and “None of this tests the cosmology; it only shows the structure is consistent.” The note, written for blind review without naming the program, puts it as “Motivation, not result. In a parent framework the polarization angle is identified with a dark-energy field. The Goldstone structure here is consistent with that, but untested by it.” (§6). Only the coset circle is Λ(1); the model-II angle β is the relative phase of two condensed levels. In the program τ has a potential of its own, “hilltop quintessence on the Maslov circle” (#11 §10; τ: hilltop quintessence), and the lab notebook calls τ “a (pseudo-)Goldstone” [D] (#1 §4).

Conjecture: the thawing of dark energy as a drifting polarization angle#

The worklog drew two further consequences for the program, beyond what the note and #11 claim (‘Chunk 2 results’):

  • “(ii) In the coset toy, the circle is the Lagrangian Grassmannian Λ(1), the checkpoint’s τ. So both toys support §10’s identification of the dark-energy field with the polarization angle, and say the polarization is set by where τ sits on its potential.”
  • “(iii) It also means the present-day thawing of τ would be a slow drift of the polarization angle. That is a claim about the program, not verified here.”

The first sentence of (ii) is the identification above. The rest overstates: only the coset circle is Λ(1), and #11 §10 claims consistency, not support. (iii) is a conjecture: no document computes the drift, and the worklog itself marks it unverified. The program’s dark-energy prediction is Row 4: thawing dark energy, never phantom; the polarization-angle reading of that drift appears only in the worklog.

Conjecture: continuous breaking in 3+1 dimensions#

The worklog’s remark on the program’s own dimension (‘Chunk 3 results’):

In 3+1, the program’s actual dimension, Coleman–Mermin–Wagner does not apply. There the self-dual coset model can break the continuous U(1) itself, with τ an exactly massless Goldstone. §10’s τ has a potential, so the explicit breaking is in the program anyway.

The assessment repeats it as unpredicted finding 3: “In 1+1, ‘only ω rigid’ is not enough for a definite polarization. A ρ-preserving potential on the polarization circle is required. In 3+1 the continuous breaking is allowed, and §10’s potential supplies the explicit breaking anyway.” (‘Assessment’). The first sentence is the scope of the theorem, which constrains two dimensions or fewer [T] (the Mermin–Wagner–Coleman theorem). That the coset model does break the U(1) in 3+1 is not computed anywhere in the record, and the note v2 covers only 1+1 with d = 1. The remark’s last sentence notes that τ has a potential in the program anyway; with it, τ is the “(pseudo-)Goldstone” of the lab notebook (#1 §4) rather than an exactly massless mode (fifth-force safety of τ).

Open: what the toy models leave undecided#

  • The coupled model without a potential. “whether ρ’s SO(2) breaks in the coupled model without a potential” (toy-model note §6), since the hyperbolic H-sector can break in two dimensions; see Open: does ρ’s SO(2) break in the coupled model?.
  • H dynamical on the lattice. “a Monte Carlo with H dynamical” (#11 §14 item 11 [O]; Open: a Monte Carlo with H dynamical); the ordered phase is shown only in the frozen-H reduction.
  • The potential’s origin. “a derivation of the potential” (#11 §14 item 11 [O]); “a derivation of the ρ-preserving potential, which introduces a rigid, unordered pair of axes” (toy-model note §6).
  • More dimensions. “anything beyond d = 1” (toy-model note §6).

These are registered in open problems.

Sources#

  • papers/broken-born-checkpoint-v2.md §4 — the P2′ refinement: a Goldstone angle [D][V]; the potential as “a rigid, unordered pair of axes”
  • papers/broken-born-checkpoint-v2.md §10 — ‘Toy-model support (v2.2)’ [D][V]: τ’s circle, the bounded potential, consistency only; τ as hilltop quintessence; §6 — τ as the Maslov phase; §14 item 11 — what remains open
  • papers/reciprocity-toy-models.md Abstract, §6 — the scope (model I and complex model II), the restoration, “Motivation, not result”, what is not shown; §2 — model I’s vacuum circle, ρ’s action, three Goldstone directions; §3 — model II’s action, U(1) phases, vacua, Δ(β), the real model’s discrete vacua; §4 — Mermin–Wagner for the compact SO(2), the naive and bounded potentials, the Monte Carlo; §5 — ρ on matter states as ψ → Fψ
  • notes/toy-lagrangian.md ‘Chunk 2 results’ — the structural finding and consequences (i)–(iii); ‘Chunk 1 results’ — the compact Goldstone as Λ(1), the Maslov circle; ‘Chunk 3 results’ — the pseudo-Goldstone verdict, the vacua ±K_ref, the 3+1 remark; ‘Assessment’ — the findings that were not predicted
  • notes/broken-born-theory.md §2 — τ as a compact, periodic modulus; §4 — τ as a (pseudo-)Goldstone
  • review/round4-adjudication.md ‘Revisions’ 3 — “Goldstone angle” restricted to the complex model
  • review/round4/referee/audit-notes.md Step 2(b) — the independent reproduction of Δ(β); F4, the real model’s discrete choice; Step 3 — “true for the two models chosen”; the residual group {±1, ±ρ}
  • verify/toy_coset.py — the vacuum circle, ρ’s action and the flat directions of model I; verify/toy_gw.py — the U(1) phases, the flat β and Δ(β) = Δ(0) cos β; verify/toy_s4.py — the lattice runs with and without the potential