Non-injective field redefinitions and quantum inequivalence in scalar theories

Abstract: We study scalar theories obtained by pulling a free massive multiplet back through a polynomial field redefinition with constant unit Jacobian. Our main example uses the three-variable noninjective map recently announced by Alpöge. After a linear normalization, it defines a three-scalar sigma model with a flat, unit-volume field-space metric and three isolated vacua. Each vacuum is locally described by three free modes of mass m, and the exact equations of motion reduce locally on each sheet to free Klein--Gordon equations. The global theory is nevertheless not a single free theory: the field-space metric is incomplete, the number of real preimages changes across target space, and the commuting position operators have nonconstant joint spectral multiplicity. This rules out a global unitary implementation of the field redefinition and a regular Weyl exponentiation of the formal canonical momenta. We then analyze a four-scalar map with a generic quintic fiber. It exhibits the same mechanism with an additional field that controls the fiber polynomial. The two examples separate perturbative equivalence on a chosen local sheet from global quantum equivalence of the full field space.

• The paper constructs explicit three- and four-scalar models showing that polynomial field redefinitions with unit Jacobian can remain locally equivalent to free massive fields while becoming globally quantum-inequivalent when real preimage multiplicity varies.
• The models exhibit regions with three real field preimages and others with one, producing nonconstant joint spectral multiplicity that prevents a global unitary implementation and regular Weyl representation.
• The analysis links the quantum obstruction to incomplete pullback geometry and nonproper maps, while identifying unresolved issues involving boundary conditions, self-adjoint momentum operators, and nonperturbative completion.

Overview

This paper by Bin Zhu constructs explicit scalar quantum field theories whose field-space coordinates are related to free fields by a polynomial map with constant unit Jacobian that is nevertheless not globally injective. The central result is a clean separation between two notions of equivalence: on any chosen local sheet, the theory is exactly a free massive multiplet and the equivalence theorem applies without qualification; globally, however, the quantum theories are inequivalent to a single free theory because the number of real preimages of the field map varies across target space. The paper builds two concrete models — a three-scalar model based on a recent noninjective counterexample to the Jacobian conjecture announced by Alpöge (2607.18166), and a four-scalar model with generic quintic fibers in which one coefficient of the fiber equation is promoted to a dynamical field.

The setting is deliberately chosen so that no local singularity is responsible for the effect. The Jacobian determinant equals one at every finite field value, so the local functional measure has unit density and the formal canonical commutators hold on their natural domain. The obstruction is purely global: it comes from the topology of the map's real fibers.

Pullback construction and local physics

Starting from $n$ free massive scalars with target configuration $\mathcal Q_\star$, the author pulls back through a polynomial map $\mathcal F:\mathbb R^n\to\mathbb R^n$ with $\det J_{\mathcal F}=1$. The resulting action is a nonlinear sigma model with metric $g_{ab}=(J^{\mathsf T}J)_{ab}$, which is flat with $\det g=1$, plus a mass term written directly in the composite variables. On any patch admitting a local inverse branch, the action reduces exactly to the free form, and varying the equations of motion shows that

$$(-\nabla^2+m^2)\big(\mathcal F^I(\varphi)-\mathcal Q_\star^I\big)=0,$$

using only pointwise invertibility of the Jacobian matrix rather than any global inverse. Every finite critical point of the potential is a preimage of $\mathcal Q_\star$, and each vacuum supports three (or four) normal modes of identical mass $m$.

Two structural facts carry the argument forward. First, if $(\mathbb R^n,g)$ were complete, the local isometry $\mathcal Q_\star$0 would be a covering map onto simply connected Euclidean space and hence one-to-one; an explicit multi-point fiber therefore proves geodesic incompleteness of the pullback metric. Second, under finite-action Euclidean boundary conditions the only smooth solutions are the constant vacua, so distinct preimages are never connected by a smooth finite-energy wall; branch transitions would require a boundary prescription at the incomplete ends of field space.

The global quantum diagnostic

The cotangent lift preserves the symplectic form on each sheet, and the Schrödinger operators $\mathcal Q_\star$1 and $\mathcal Q_\star$2 satisfy the canonical commutators on $\mathcal Q_\star$3, with symmetry of the momenta following from the Piola identity and the unit determinant. These are, however, only formal statements: they do not imply essential self-adjointness or regular Weyl relations.

The decisive diagnostic is the joint spectral multiplicity of the commuting position operators. Via the area formula,

$\mathcal Q_\star$4

where $\mathcal Q_\star$5 counts real preimages. A regular representation of the Weyl relations has constant spectral multiplicity by Stone–von Neumann classification. Consequently, if $\mathcal Q_\star$6 on one open target set and $\mathcal Q_\star$7 on another, no unitary operator implements the redefinition globally and no regular Weyl exponentiation of the formal momenta exists. This is the paper's core claim, and it is established without any approximation.

The three-scalar model from the Alpöge map

The algebraic input is Alpöge's three-variable polynomial map with constant Jacobian $\mathcal Q_\star$8 and an explicit three-point fiber. After the linear normalization $\mathcal Q_\star$9, the Jacobian becomes $\mathcal F:\mathbb R^n\to\mathbb R^n$0. The paper supplies the physics construction absent from the announcement: a sigma-model action for three scalars with fixed relative interaction coefficients and only two physical parameters, $\mathcal F:\mathbb R^n\to\mathbb R^n$1 and $\mathcal F:\mathbb R^n\to\mathbb R^n$2.

Three distinguished source points,

$\mathcal F:\mathbb R^n\to\mathbb R^n$3

all map to $\mathcal F:\mathbb R^n\to\mathbb R^n$4, giving three isolated vacua with locally free modes of mass $\mathcal F:\mathbb R^n\to\mathbb R^n$5. The fiber structure is controlled by a cubic $\mathcal F:\mathbb R^n\to\mathbb R^n$6; at the vacuum target the cubic degenerates but the fiber is verified to contain exactly these three real points. By contrast, the target $\mathcal F:\mathbb R^n\to\mathbb R^n$7 has a fiber polynomial with a single real root, yielding an open one-sheeted neighborhood. The position tuple therefore has joint spectral multiplicity three on one open region and one on another, which rules out both a global unitary implementation and regular Weyl exponentiation. An exact $\mathcal F:\mathbb R^n\to\mathbb R^n$8 symmetry ($\mathcal F:\mathbb R^n\to\mathbb R^n$9 fixes $\det J_{\mathcal F}=1$0 and exchanges $\det J_{\mathcal F}=1$1; the continuous weighted scaling of the algebraic map is broken by the massive shifted action.

The four-scalar model with quintic fibers

The second example promotes the constant parameter of a Jacobian-neutral rational frame to a fourth scalar $\det J_{\mathcal F}=1$2. The resulting map $\det J_{\mathcal F}=1$3 again has unit Jacobian, but now $\det J_{\mathcal F}=1$4 enters the first two output polynomials and controls the coefficients of the quintic fiber polynomial $\det J_{\mathcal F}=1$5; it is emphatically not a spectator. For the normalized vacuum target $\det J_{\mathcal F}=1$6, the fiber factors as a product of linear factors and a strictly increasing cubic, giving three real preimages (two rational, one algebraic with $\det J_{\mathcal F}=1$7) extending to a common three-sheeted neighborhood. A comparison target $\det J_{\mathcal F}=1$8 has a fiber whose derivative is manifestly strictly positive, hence a unique real preimage persisting under perturbation.

The same spectral mechanism follows immediately: multiplicity three versus one, no global unitary equivalence, no regular Weyl representation. The invariant slice $\det J_{\mathcal F}=1$9 yields a standalone generic-degree-five three-scalar map, treated in an appendix, where all statements — including the strict positivity bound $g_{ab}=(J^{\mathsf T}J)_{ab}$0 — are checked explicitly in three variables.

Comparison of the two mechanisms

Model	Generic complex fiber	Three-sheet target	One-sheet target
$g_{ab}=(J^{\mathsf T}J)_{ab}$1	3	$g_{ab}=(J^{\mathsf T}J)_{ab}$2	$g_{ab}=(J^{\mathsf T}J)_{ab}$3
$g_{ab}=(J^{\mathsf T}J)_{ab}$4	5	$g_{ab}=(J^{\mathsf T}J)_{ab}$5	$g_{ab}=(J^{\mathsf T}J)_{ab}$6

Three ingredients are common to both. The unit determinant removes local measure density and makes the formal momenta symmetric via the Piola identity. The maps are nonproper, so real branches can escape to infinity while their images remain finite, allowing the preimage count to change without any critical point. And the pullback metric is flat but incomplete, with the same incomplete end appearing geometrically in the sigma model and spectrally in the failure of complete momentum flows. The pullback operator $g_{ab}=(J^{\mathsf T}J)_{ab}$7 satisfies $g_{ab}=(J^{\mathsf T}J)_{ab}$8: it intertwines the representations but is neither onto nor unitary. In the path integral, continuity forces the sheet label to be constant on connected spacetime configurations, so a branch-restricted sector is a sum of $g_{ab}=(J^{\mathsf T}J)_{ab}$9 identical free sectors rather than a locally chosen factor. That the obstruction survives changing the fiber degree from cubic to quintic indicates it is controlled by preimage multiplicity, not by the number of fields or polynomial degree.

Limitations and open questions

The paper is candid about what its results do not settle. The formal canonical commutators hold only on $\det g=1$0; essential self-adjointness of the momentum operators and completeness of their flows are not established, and the Weyl failure is inferred from spectral multiplicity rather than from an analysis of self-adjoint extensions. The nonperturbative definition of the theory remains open: a path integral must specify boundary conditions at the incomplete ends of field space, and the choice among branch-blind observables, explicit sheet labels, or other completions is not determined by the unit-Jacobian condition alone. Interacting target potentials $\det g=1$1 can be pulled back within the same framework, but their spectra and classical solutions are not analyzed here. Finally, the algebraic inputs rest on very recent announcements (the Alpöge map and related families), whose full peer-reviewed verification lies outside this paper.

Conclusion

The paper demonstrates, through two fully explicit scalar models, that a nonsingular polynomial field redefinition with constant unit Jacobian can produce theories that are perturbatively free on every local sheet yet globally quantum-inequivalent to any single free theory. The invariant mechanism is the variation of real preimage multiplicity, which forbids a global unitary implementation and a regular Weyl representation while leaving the branchwise equivalence theorem untouched. What remains open is the nonperturbative completion — boundary conditions at incomplete ends of field space and the admissible class of observables — which the local algebra alone cannot fix.