Unitary equivalence versus Noether symmetry

Unitary equivalence and Noether symmetry are the two different ways a map like Born reciprocity can relate to a physical theory. It can be a change of description, a unitary equivalence — the same physics written in other variables, as a Fourier transform rewrites a waveform as a spectrum — or it can be a symmetry of the dynamics: an operation that leaves a specific action unchanged, and therefore constrains what the theory can do (a Noether symmetry). In ordinary quantum mechanics ρ is the first kind, because the Stone–von Neumann theorem makes all the standard ways of representing the position–momentum commutation rules equivalent. The program needs the second kind, since only a symmetry of an action can be spontaneously broken. Its referees made exhibiting such an action the first acceptance criterion for anything it proposes, and #11 records the criterion as met, in the referees’ wording, by the toy models.

Status: Established — the equivalence is textbook quantum mechanics [T], stated here from general knowledge; the program’s claim that ρ is a symmetry of an underlying action is the postulate P3 [P]. The distinction was raised by the referees and conceded in #14 §1; #11 §14 item 11 records the criterion as met in the referees’ wording by the toy models, whose invariance is machine-checked [V] (the toy-model scripts).

Two ways a map can be a symmetry#

A change of description#

Two descriptions of a quantum system are unitarily equivalent when a unitary operator U carries the states and operators of one into those of the other [T]. Every probability and every expectation value comes out the same, so nothing observable tells the two apart: they are one theory in two coordinate systems. Rewriting a wavefunction ψ(x) as its momentum wavefunction ψ̃(p) is of this kind (the Fourier transform as a phase-space rotation).

A symmetry of the dynamics#

A Noether symmetry is a transformation of the fields that leaves a specific action S unchanged [T]. It constrains the dynamics. Terms that the transformation does not leave invariant cannot appear in the action, solutions map to solutions, and lowest-energy states map to lowest-energy states. For each continuous symmetry, Noether’s theorem gives a conserved quantity [T].

Only a symmetry of this second kind can be spontaneously broken. The action has the symmetry, but its lowest-energy states do not; they form a family that the symmetry permutes, and the system settles in one member [T] (spontaneous symmetry breaking). The program’s hard core claims that the world’s preference for position is a choice of that kind, made by the vacuum. The withdrawn manuscript #14 states what that requires: “The framework’s claim is that rho is a genuine symmetry of an underlying action, spontaneously broken” (#14 §1).

The Stone–von Neumann theorem#

The commutation relation [x̂, p̂] = iℏ could in principle be represented by operators on many different spaces of states. For finitely many degrees of freedom the Stone–von Neumann theorem settles the question: every irreducible representation of the relations, in their exponentiated (Weyl) form, is unitarily equivalent to the ordinary Schrödinger representation on wavefunctions ψ(x) [T]. The exponentiated form, built from the bounded operators e^{iax̂} and e^{ibp̂}, avoids the domain problems of the unbounded x̂ and p̂, for which the uniqueness can fail [T].

Applied to ρ, the theorem works as follows.

  1. The pair (p̂, −x̂) obeys the same relation as (x̂, p̂), since [p̂, −x̂] = [x̂, p̂] = iℏ (the canonical commutator).
  2. By the theorem a unitary operator carries the first representation into the second. It is the Fourier transform: F† x̂ F = p̂ and F† p̂ F = −x̂ [T].
  3. The same argument covers every linear map of phase space that preserves the symplectic form. Each is carried out by a unitary operator, unique up to a phase (the metaplectic representation) [T]; see the symplectic form and Sp(2d, ℝ).

So in quantum mechanics with finitely many degrees of freedom, ρ and every other linear symplectic map is a change of description.

The theorem does not extend to infinitely many degrees of freedom. Quantum field theories admit unitarily inequivalent representations of the commutation relations, and distinct vacua of a spontaneously broken symmetry live in such inequivalent representations [T]. The toy models are field theories: the worklog’s criterion for quantum stability starts them in 1+1 dimensions, because “In 0+1 dimensions (quantum mechanics), tunnelling restores any symmetry” (notes/toy-lagrangian.md, “Success criteria”, S4).

Why ρ is ordinarily only a change of description#

#14 conceded the point at its outset: “We note at the outset a distinction the proposal must eventually discharge: as usually encountered, rho is a unitary equivalence between representations of the canonical commutation relations, not a Noether symmetry of a specified action. The framework’s claim is that rho is a genuine symmetry of an underlying action, spontaneously broken; the emergent structure eta below is the proposed order parameter, and the exhibition of a toy action realizing this is stated in Sec. 6 as the proposal’s first acceptance criterion.” (#14 §1).

The order parameter #14 names is the neutral metric η (the order parameter). The program postulates the second kind of symmetry: “P3 The action is invariant under the Born rotation generated by I = ω⁻¹H. [P]” (#11 §4; P3).

The referees’ demand#

The demand came out of the blind reviews of #14:

  • Round 1. The referee’s “toy-Lagrangian demand” was “adopted as acceptance criterion #1” (HANDOFF.md §6). The handoff’s first open item adds that “both reviewers independently named it” (§2). See review rounds 1 and 2.
  • #14 states the criterion: “(1) exhibition of an action - a toy model suffices - in which rho is a genuine Noether symmetry, spontaneously broken, with eta the order parameter selecting both the position polarization and the hidden sector’s electric description; deriving hypothesis H from that action would make the coupling a prediction” (#14 §6).
  • Round 3. The referees’ shared verdict became the program’s main critique (the ‘framework does no work’ critique), and the adjudication made the toy Lagrangian the next task, as the one item that could answer it: “Every referee across three rounds converges on ’the framework does no work’. The toy action is the only item that can change that, and it is acceptance criterion 1 for anything the program proposes.” (review/round3-adjudication.md, decision 4; round-3 decisions).

The worklog fixed its success criteria before building anything, and the first of them is this page’s distinction: “S1 — symmetry of an action. An explicit action S[fields] on which ρ acts as a symmetry: S is invariant, not merely related to its image by a unitary equivalence. Invariance is verified symbolically.” (notes/toy-lagrangian.md, “Success criteria”; success criteria S0–S5).

The answer so far#

#11 §14 item 11 records the criterion as “Met in the referees’ wording by the coset model of (H, K) with a bounded ρ-preserving potential”:

  • “ρ is a symmetry of an explicit action, and η = ωK is the order parameter;”
  • “the vacuum is stable with H dynamical;”
  • “there is an ordered ℤ₂ phase in 1+1 in the frozen-H reduction (Monte Carlo).”

Still open there [O]: “a Monte Carlo with H dynamical; a derivation of the potential; locality of matter beyond the weak (Yukawa-like) form, which in the toy reduces to P4.” See acceptance criterion 1 and Open: a Monte Carlo with H dynamical. #11 does not qualify the second point; the toy note does. Its stable minimum keeps one flat direction, the squeeze line, and “It is not symmetry-protected” (§4; the symplectic form and Sp(2d, ℝ)).

The invariance is machine-checked. In model I, the coset model of (H, K), “Sp(2,ℝ) acts by H → SᵀHS, K → S⁻¹KS, and the action is invariant (checked for generic symplectic S, so for ρ)” (toy-model note §2), and the gate prints “all three terms invariant under rho” [V] (Model I: action and symmetries).

“Met” leaves two things uncovered, one clause of the referees’ wording and the round-3 critique:

  • The electric sublattice. #14’s criterion and the handoff’s open item required η to select “the hidden sector’s electric description” (or “the electric sublattice”) as well. The worklog’s criteria and #11 §14 item 11 omit that clause. The root CLAUDE.md calls the toy Lagrangian “done to the referees’ wording”, and the clause remains unaddressed.
  • The round-3 critique. The worklog’s assessment: “The round-3 critique (’the framework does no work’) is only partly answered. The toy makes the program’s structures cohere (η as order parameter; τ as the polarization angle), but matter’s x-locality still enters through the P4-type coupling.” (notes/toy-lagrangian.md, “Assessment against the predictions”; Open: forced locality and the P4 lock).

Symmetry, covariance and the twisted element#

Model II shows the same distinction between its complex and real versions. In the self-dual Grosse–Wulkenhaar model the position–momentum duality of Langmann and Szabo acts on the field φ through a matrix Π (toy-model note §3):

  • Complex fields: a symmetry. The action satisfies S[Πφ] = S[φ], and “The duality Πφ is the α = π element of e^{−iαN}” (§3); the gate prints “complex model: S[Pi phi] = S[phi] (duality is a field symmetry)”.
  • Real fields: a covariance. “Πφ is not Hermitian, so the duality maps the real field space out of itself. The real action is only covariant: its quartic term is invariant in the alternating form Tr((ψψ†)²), ψ = Πφ.” (§3; here † is the Hermitian adjoint, not the citation mark). Grosse and Wulkenhaar themselves describe the real action at Ω = 1 as invariant under the duality; “covariant only” is the toy note’s reading (the referee of review round 4, audit notes Step 1).

See complex versus real fields: symmetry versus covariance.

Even a symmetry of the action does not by itself make the breaking of ρ physical. Model I has a second symmetry, K → −K, and “the twisted element ρ∘(K → −K) maps (1, K₀) to itself. So within the full symmetry group, the vacuum keeps an element that acts on H exactly as ρ does. Breaking ρ is physical only once a coupling odd in K is present” (toy-model note §2). See the twisted element.

Standard references#

Added by the wiki from model memory (†); the program’s documents do not cite them.

  • M. H. Stone (1930)† and J. von Neumann (1931)†: uniqueness of the representation of the canonical commutation relations.
  • E. Noether (1918)†: symmetries of an action and conserved quantities.
  • G. B. Folland, Harmonic Analysis in Phase Space (1989)†: the metaplectic representation.

Used by#

Sources#

  • CLAUDE.md “Since the handoff” — the toy Lagrangian “done to the referees’ wording”.
  • papers/broken-born-checkpoint-v2.md §14 item 11 — the criterion met in the referees’ wording, and what remains open.
  • papers/broken-born-checkpoint-v2.md §4 — P3, the postulated symmetry of the action.
  • papers/reciprocity-toy-models.md §1–§3 — the explicit ρ-invariant action, the twisted element, symmetry versus covariance.
  • review/round3-adjudication.md “Decisions” — decision 4: the toy Lagrangian as the next task.
  • review/round4/referee/audit-notes.md Step 1 — Grosse–Wulkenhaar’s own wording on the real action.
  • HANDOFF.md §2, §6 — the open item with the electric-sublattice clause; round 1 and the adoption of acceptance criterion 1.
  • notes/toy-lagrangian.md “Success criteria”, “Assessment against the predictions” — criterion S1; the 0+1 remark under S4; the partial answer to round 3.
  • notes/knot-dark-matter-standalone.md §1, §6 — the concession and the criterion as #14 stated them (withdrawn paper).
  • verify/toy_coset.py, verify/toy_gw.py — the invariance checks for model I and the complex model II.