Model II: the self-dual Grosse–Wulkenhaar model on The Broken-Born program

Noncommutative field theory and the Langmann–Szabo duality

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

Noncommutative field theory places fields on a plane whose two coordinates do not commute, [x₁, x₂] = iθ: the plane behaves like the phase space of a quantum particle, with θ in the role of Planck’s constant. Fields on it are multiplied with the Moyal star product, which builds that relation in. On this plane Langmann and Szabo found a duality that exchanges a field’s description in position with its description in momentum. It is a Fourier transform of the field, relating the two descriptions as a waveform is related to its spectrum. Grosse and Wulkenhaar’s renormalizable model of a self-interacting scalar field is carried into itself by the duality at one value of its oscillator parameter, Ω = 1, and there its β-function vanishes to all orders in four dimensions. This literature supplied model II of the program’s toy models: an interacting field theory with an exact exchange of position and momentum, used to test whether a vacuum alone could break that exchange.

Model II: the matrix base and the action

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

Model II is the self-dual Grosse–Wulkenhaar model of a self-interacting scalar field on the noncommutative plane, rewritten as a model of matrices. The field is expanded in functions f_mn built from harmonic-oscillator states; in this basis the star product of two fields becomes the product of their matrices, and the integral over the plane becomes a trace. At the self-dual point Ω = 1 the quadratic part of the action is diagonal, 8π(m + n + 1)|φ_mn|², so each matrix entry carries an oscillator energy of its own, and the interaction is the trace of a fourth power of the matrix. A complex field is an arbitrary matrix and a real field a Hermitian one. Every formula was checked against an independent anchor before it was used, and the first check of the kinetic form, made with finite differences, failed until exact derivatives replaced them. The duality lemma, the vacua and the restoration argument of model II are all computed on this set-up.

The duality on the matrix base

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

The lemma states how the Langmann–Szabo duality acts on the matrix-base functions f_mn of model II: it multiplies each one by (−1)^m, a sign fixed by its left index alone. The duality is a Fourier transform of the field, which exchanges its position profile with its momentum profile, and on this basis it reduces to parity, the reflection that leaves the even oscillator levels unchanged and reverses the sign of the odd ones, applied to the left index of the field’s matrix. Because Grosse and Wulkenhaar alternate the sign of the Fourier exponent from one factor of the interaction to the next, parity acts on the left index of one factor and on the right index of the next. The result came from a direct numerical Fourier integration, which settled a clash between conventions recalled from memory. The round-4 referee confirmed it and judged it correct but not new, and the note’s second version presents it as a lemma credited to the work it follows from. The rest of model II turns on it.

Complex versus real fields: symmetry versus covariance

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

Whether the Langmann–Szabo duality is a symmetry of model II depends on whether its field is complex or real. On the matrix base the duality multiplies the rows of the field’s matrix by alternating signs. A complex field is an arbitrary matrix, so the duality takes every allowed field to another allowed field with the same action: a genuine symmetry, and one element, the half-turn, of a continuous family of phase rotations that also leave the action unchanged. A real field is a Hermitian matrix, and the duality’s image of a Hermitian matrix is in general not Hermitian, so the duality leads out of the space of real fields. The real action keeps its value only when it is rewritten in the new variable, which makes the duality a covariance of the real model, not a symmetry. The distinction decides what each version can show: a symmetry can be broken spontaneously and keeps the vacua it relates exactly degenerate, while without one a preference between vacua can arise explicitly, as it does in the real model at one loop.

Complex-model vacua and the polarization Δ(β)

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

The vacua of the complex model II are its configurations of lowest action. At the parameters the toy-model note uses, μ² = 4 and λ = 1, the two lowest oscillator levels condense: the diagonal matrix entries φ₀₀ and φ₁₁ take fixed nonzero magnitudes, and their relative phase β is free, so the vacua form a circle. The duality shifts β by π, half-way round the circle, so it maps every vacuum to a different one and leaves none unchanged: classically, the vacuum breaks the exchange of position and momentum. The polarization Δ(β) measures how much broader the vacuum’s profile is in position than in momentum. It varies as cos β, so the β = 0 vacuum is broad in position and narrow in momentum and the β = π vacuum is the reverse, as a narrow pulse has a broad spectrum and a broad pulse a narrow one. The sign was written reversed in the first draft and corrected the same day. The breaking is classical only: quantization restores the symmetry.

Quantum restoration in the complex model

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

Quantum restoration is the result that the classical breaking found in the complex model II does not survive quantization. The vacua form a circle labelled by the relative phase β of two condensed oscillator levels, and an exact symmetry turns the circle at no cost in action, so the quantum theory weights every value of β equally. A spontaneous choice can still survive in a system large enough that a vanishingly small bias, acting over its whole volume, pins it. This condensate occupies two oscillator levels and is a localized lump, so it supplies no such volume factor. The quantum average of the polarization is therefore zero, and criterion S4, survival of the breaking under quantum fluctuations (success criteria), fails for this route. Round 4 confirmed the conclusion, asked for the argument to be restated on these two grounds, and added that the bare mass must be renormalized as the truncation grows. The comparison with an exactly solved related model, the natural independent check, has not been made.

Open: the real model's one-loop preference (a lead)

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

The real model’s one-loop preference is the finding that, in the real-field version of model II, the first quantum correction (the one-loop term) gives the vacuum that is sharper in position a lower free energy than its mirror image, and so favours it, although the two have exactly the same classical action, so that before quantum effects are included neither is preferred. That version’s vacua are discrete: the field’s vacuum value occupies the two lowest oscillator levels, with either the same sign on both or opposite signs, and the duality, the model’s exchange of position and momentum, maps one choice to the other. The quantum fluctuations around the two vacua differ, and the lower free energy belongs to the opposite-sign vacuum, the one sharper in position than in momentum. That is a selection of the position-sharp polarization, but an explicit one: the real model is not symmetric under the duality, because its field is real in position, so nothing is broken spontaneously. The round-4 referee found the split, the adjudicator identified which vacuum it favours, and the toy-model note’s second version showed the value unchanged at three truncations, that is, with three different numbers of oscillator levels kept. It is recorded as a lead, not a result: it holds at a fixed bare mass, the mass parameter as written in the action, which itself has to be adjusted as more levels are kept, and what happens to the preference as the number of levels grows without limit, with the mass so adjusted, is not known.