arxiv: 2605.07659 · v1 · submitted 2026-05-08 · 🧮 math-ph · math.MP

Recognition: 2 theorem links

· Lean Theorem

Superintegrability in the interaction of two particles with spin: First-order pseudo-scalar integrals of motion

Fatih Turkkan, I. Yurdusen, O. Ogulcan Tuncer

Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:49 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords superintegrabilitypseudo-scalar integralsspin-1/2 particlestwo-particle systemsintegrals of motionsymmetry algebrasnucleon interactions

0 comments

The pith

A complete classification of superintegrable two spin-1/2 particle systems with first-order pseudo-scalar integrals is derived.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the classification of quantum superintegrable systems for two non-relativistic spin-1/2 particles by considering those with additional first-order pseudo-scalar integrals of motion. It begins with the most general rotationally invariant Hamiltonian and constructs the general first-order pseudo-scalar operator as a matrix polynomial in momenta. Imposing commutativity with the Hamiltonian produces determining equations that are solved to yield the full classification and the explicit integrals. This provides new families of systems with spin-dependent interactions relevant to nuclear physics models.

Core claim

Solving the commutativity conditions between the general rotationally invariant Hamiltonian for two spin-1/2 particles and the most general first-order pseudo-scalar operator yields a complete classification of the superintegrable systems and their corresponding integrals of motion, resulting in new families with spin-dependent interactions and polynomial symmetry algebras for selected cases.

What carries the argument

The first-order pseudo-scalar integral of motion, formulated as a matrix polynomial in the particle momenta, which must commute with the Hamiltonian.

Load-bearing premise

The additional integral must be a first-order pseudo-scalar expressed as a matrix polynomial in the momenta, with the Hamiltonian being rotationally invariant.

What would settle it

Identification of a superintegrable system with a first-order pseudo-scalar integral not included in the obtained classification would disprove its completeness.

read the original abstract

In recent work, we initiated a research program aimed at the systematic investigation of quantum superintegrable systems describing the interaction of two non-relativistic spin-$1/2$ particles in three-dimensional Euclidean space. In that study, we classified all such superintegrable systems admitting additional first-order scalar integrals of motion. In the present paper, we continue this program by focusing on systems that admit additional pseudo-scalar integrals of motion. Starting from the most general rotationally invariant Hamiltonian for two interacting spin-$1/2$ particles, we construct the most general first-order pseudo-scalar operator in the form of a matrix polynomial in the momenta. Imposing the commutativity of this operator with the Hamiltonian leads to a system of determining equations. By solving these equations, we obtain a complete classification of such superintegrable systems and determine the corresponding pseudo-scalar integrals of motion. The resulting classification provides new families of superintegrable systems with spin-dependent interactions. These systems enrich the class of integrable models relevant to nucleon--nucleon interactions and contribute to the broader program of classifying superintegrable quantum systems with spin. For selected cases, we further construct the associated polynomial symmetry algebras generated by the integrals of motion, providing additional insight into the algebraic structure of the systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper classifies two-particle spin-1/2 systems admitting first-order pseudo-scalar integrals by solving the commutator equations, extending the authors' prior scalar case with no major internal gaps.

read the letter

The punchline is that the authors have produced a clean algebraic classification of rotationally invariant two-particle Hamiltonians that commute with an extra first-order pseudo-scalar operator. They write the general form for that operator as a matrix polynomial in the momenta, impose [H,I]=0, and solve the resulting determining equations to list the admissible families and the corresponding integrals. For some cases they also construct the polynomial symmetry algebras generated by the integrals. This is a direct continuation of their earlier scalar-integral work, and the new families are not contained in that prior classification. The approach is the standard one for these problems and the stress-test confirms the algebraic problem is closed within the stated ansatz, with no evident holes in the enumeration or the commutator condition. The paper stays strictly within its scope and does not overclaim completeness for higher-order or non-polynomial integrals. The only real limitation is that the classification applies only to first-order pseudo-scalar integrals of exactly this matrix-polynomial type; if the broader field wants all possible integrals, this is one slice rather than the full picture. The abstract nods to nucleon-nucleon modeling but the work itself remains formal and does not include numerical checks or physical applications. This is for readers who follow the catalog of superintegrable quantum systems with spin, especially those building lists of integrable models in mathematical physics. A specialist in the area will get a usable addition to the known families and can check the solutions against the determining equations. The thinking is clear and the method reproducible, so the paper deserves a serious referee even if the result is incremental. I would recommend sending it to review.

Referee Report

0 major / 3 minor

Summary. The manuscript classifies rotationally invariant Hamiltonians describing the interaction of two non-relativistic spin-1/2 particles in three-dimensional space that admit an additional first-order pseudo-scalar integral of motion. Starting from the most general form of such a Hamiltonian and constructing the most general first-order pseudo-scalar operator as a matrix polynomial in the momenta, the authors impose the commutator condition [H, I] = 0, derive the resulting system of determining equations, solve them to obtain a complete list of admissible families, and determine the corresponding integrals. For selected cases they also construct the associated polynomial symmetry algebras generated by the integrals.

Significance. If the classification is complete within the stated ansatz, the work extends the authors' prior classification of systems admitting first-order scalar integrals and supplies new families of superintegrable models with spin-dependent interactions. These are directly relevant to nucleon-nucleon interaction models. The explicit construction of the polynomial symmetry algebras for representative cases provides additional algebraic insight. The systematic use of determining equations derived from the commutator condition, together with the restriction to a well-defined first-order pseudo-scalar ansatz, constitutes a clear methodological strength.

minor comments (3)

The abstract states that the classification 'provides new families' but does not indicate which of the listed systems are genuinely novel versus extensions of known scalar cases; a short comparative remark would clarify the advance.
§2 (general setup): the explicit 4×4 matrix representation of the pseudo-scalar operator (as a linear combination of Pauli-matrix structures times momentum polynomials) is introduced only after several paragraphs; displaying the block form at the outset would improve readability.
The section presenting the solutions to the determining equations would benefit from a compact table that lists each admissible Hamiltonian, the corresponding pseudo-scalar integral, and the dimension of the symmetry algebra.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive assessment of our manuscript. The summary accurately captures the scope of our classification of rotationally invariant superintegrable Hamiltonians for two spin-1/2 particles admitting first-order pseudo-scalar integrals, and we appreciate the recognition of its relevance to nucleon-nucleon interaction models as well as the methodological strengths noted. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's derivation begins from the most general rotationally invariant two-particle spin-1/2 Hamiltonian and constructs the most general first-order pseudo-scalar integral as a matrix polynomial in the momenta. It then imposes the commutator condition [H, I] = 0 to generate a system of determining equations and solves those equations to obtain the admissible families. This is a self-contained algebraic classification within the explicit ansatz; no fitted parameters are renamed as predictions, no uniqueness theorem is imported from self-citation, and the reference to the authors' prior scalar-integral work supplies only contextual background without substituting for or forcing the present pseudo-scalar results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract indicates reliance on the standard quantum-mechanical commutator condition and the assumption that the Hamiltonian is the most general rotationally invariant two-particle operator; no explicit free parameters or invented entities are named.

axioms (2)

domain assumption The Hamiltonian is the most general rotationally invariant operator for two spin-1/2 particles.
Stated in the abstract as the starting point for constructing the pseudo-scalar operator.
domain assumption The additional integral is a first-order pseudo-scalar matrix polynomial in the momenta.
The form chosen for the operator whose commutator with H is set to zero.

pith-pipeline@v0.9.0 · 5537 in / 1444 out tokens · 37829 ms · 2026-05-11T01:49:31.203267+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
Starting from the most general rotationally invariant Hamiltonian ... construct the most general first-order pseudo-scalar operator ... Imposing the commutativity ... leads to a system of determining equations. By solving these equations, we obtain a complete classification
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
the most general Hermitian pseudo-scalar operator X, at most linear in the momenta ... [H, X] = 0 ... overdetermined system of differential equations for the radial functions

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages · 1 internal anchor

[1]

Tuncer OO and Yurdu¸ sen˙I 2025 Superintegrability in the interaction of two particles with spinJ. Phys. A: Math. Theor.58245201 (41pp)

work page 2025
[2]

Phys.4166–79

Okubo S and Marshak RE 1958 Velocity dependence of the two-nucleon interactionAnn. Phys.4166–79

work page 1958
[3]

Lett.16354–6

Friˇ s I, Mandrosov V, Smorodinsky J, Uhlı˜ r M and Winternitz P 1965 On higher-order symmetries in quantum mechanicsPhys. Lett.16354–6

work page 1965
[4]

Makarov A, Smorodinsky J A, Valiev K and Winternitz P 1967 A systematic search for non-relativistic systems with dynamical symmetriesNuovo CimentoA521061–84

work page 1967
[5]

Rev.A415666–76

Evans N W 1990 Superintegrability in classical mechanicsPhys. Rev.A415666–76

work page 1990
[6]

Lett.A147483–6 51

Evans N W 1990 Superintegrability of the Winternitz systemPhys. Lett.A147483–6 51

work page 1990
[7]

Bertrand J 1873 Th´ eor` eme relatif au mouvement d’un point attir´ e vers un centre fixeC. R. Acad. Sci.17849–853

work page
[8]

Two-Dimensional HyperboloidPhys

Grosche C, Pogosyan G S and Sissakian A N 1996 Path Integral Approach to Superintegrable Potentials. Two-Dimensional HyperboloidPhys. Part. Nucl.27(3)244-278 (Fiz. Elem. Chastits At Yadra27593-674)

work page 1996
[9]

Two and three dimensional quantum systemsJ

Kalnins E G, Kress J M and Miller Jr W 2006 Second order superintegrable systems in conformally flat spacesV. Two and three dimensional quantum systemsJ. Math. Phys.47 093501 (25pp)

work page 2006
[10]

Kalnins E G, Kress J M and Winternitz P 2002 Superintegrability in a two-dimensional space of nonconstant curvatureJ. Math. Phys.43 (2)970–83

work page 2002
[11]

Rodriguez M A and Winternitz P 2002 Quantum superintegrability and exact solvability inndimensionsJ. Math. Phys.431309–22

work page 2002
[12]

Kalnins E G, Williams G C, Miller Jr W and Pogosyan G S 2002 On superintegrable symmetry-breaking potentials inN-dimensional Euclidean spaceJ. Phys. A: Math. Gen. 354755–73

work page 2002
[13]

Atomic Nuclei70no.3 545–53

Kalnins E G, Kress J M, and Miller W Jr 2007 Nondegenerate superintegrable systems in n-dimensional spaces of constant curvaturePhys. Atomic Nuclei70no.3 545–53

work page 2007
[14]

Gravel S and Winternitz P 2003 Superintegrability with third-order integrals in quantum and classical mechanicsJ. Math. Phys.435902–12

work page 2003
[15]

Tremblay F, Turbiner A V and Winternitz P 2009 An infinite family of solvable and integrable quantum systems on a planeJ. Phys. A. Math. Theor.42242001

work page 2009
[16]

Marquette I 2009 Superintegrability with third-order integrals of motion, cubic algebras, and supersymmetric quantum mechanics II: Painlev´ e transcendent potentialsJ. Math. Phys. 50095202 (18pp)

work page 2009
[17]

Tremblay F, Turbiner A V and Winternitz P 2010 Periodic orbits for an infinite family of classical superintegrable systemsJ. Phys. A. Math. Theor.43015202 (16pp)

work page 2010
[18]

Post S and Winternitz P 2015 GeneralN th-Order Integrals of Motion in the Euclidean Plane,J. Phys. A-Math. Theor.48405201 52

work page 2015
[19]

Exotic Quantum PotentialsJ

Marquette I, Sajedi M and Winternitz P 2017 Fourth Order Superintegrable Systems Separating in Cartesian Coordinates I. Exotic Quantum PotentialsJ. Phys. A-Math. Theor. 50315201

work page 2017
[20]

Escobar-Ruiz A, Winternitz P and Yurdu¸ sen ˙I 2018 GeneralN th-Order Superintegrable Systems Separating in Polar CoordinatesJ. Phys. A-Math. Theor.5140LT01

work page 2018
[21]

Miller W Jr., Post S and Winternitz P 2013 Classical and Quantum Superintegrability with ApplicationsJ. Phys. A-Math. Gen.46423001

work page 2013
[22]

Dorizzi B, Grammaticos B, Ramani A and Winternitz P 1985 Integrable Hamiltonian Systems with Velocity Dependent PotentialsJ. Math. Phys.263070-3079

work page 1985
[23]

B´ erub´ e J and Winternitz P 2004 Integrable and Superintegrable Quantum Systems in a Magnetic FieldJ. Math. Phys.451959-1973

work page 2004
[24]

Geom.16015

Marchesiello A and Snobl L 2020 Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian CoordinatesSymmetry Integr. Geom.16015

work page 2020
[25]

Phys.451169264

Kubu O, Marchesiello A, and Snobl L 2023 New classes of quadratically integrable systems in magnetic fields: The generalized cylindrical and spherical casesAnn. Phys.451169264

work page 2023
[26]

Hoque M F and Snobl L 2023 Family of nonstandard integrable and superintegrable clas- sical Hamiltonian systems in non-vanishing magnetic fieldsJ. Phys. A-Math. Theor.56 165203

work page 2023
[27]

Hoque M F, Marchesiello A, and Snobl L 2024 Integrable systems of the ellipsoidal, paraboloidal and conical type with magnetic fieldJ. Phys. A-Math. Theor.57225201

work page 2024
[28]

Ekelchik T, Marchesiello A 2026 Superintegrable 2D systems in magnetic fields with a parabolic type integral arXiv:2604.20465v1

work page internal anchor Pith review Pith/arXiv arXiv 2026
[29]

Winternitz P and Yurdu¸ sen ˙I 2006 Integrable and superintegrable systems with spinJ. Math. Phys.47103509 (10pp)

work page 2006
[30]

Winternitz P and Yurdu¸ sen ˙I 2009 Integrable and superintegrable systems with spin in three-dimensional Euclidean spaceJ. Phys. A: Math. Theor.42385203 (20pp)

work page 2009
[31]

D´ esilets J-F, Winternitz P and Yurdu¸ sen˙I 2012 Superintegrable systems with spin and second-order integrals of motionJ. Phys. A: Math. Theor.45475201 (26pp) 53

work page 2012
[32]

Yurdu¸ sen˙I, Tuncer OO and Winternitz P 2021 Superintegrable systems with spin and second-order tensor and pseudo-tensor integrals of motionJ. Phys. A: Math. Theor.54 305201 (37pp)

work page 2021
[33]

Pronko G P 2007 Quantum superintegrable systems for arbitrary spinJ. Phys. A. Math. Theor.401333–36

work page 2007
[34]

Nikitin A G 2012 Matrix superpotentials and superintegrable systems for arbitrary spin J. Phys. A-Math. Theor.45225205 (13pp)

work page 2012
[35]

Nikitin A G 2013 Superintegrable systems with spin invariant with respect to the rotation groupJ. Phys. A-Math. Theor.46265204 (15pp)

work page 2013
[36]

D’Hoker E and Vinet L 1985 Constants of motion for a spin- 1 2 particle in the field of a DyonPhys. Rev. Lett.55no.10

work page 1985
[37]

Feher L and Horvathy P A 1988 Non-relativistic scattering of a spin-1 2 particle off a self-dual monopoleMod. Phys. Lett.A31451–60

work page 1988
[38]

Yurdu¸ sen˙I and Tuncer OO 2022 Symmetrization of the product of Hermitian operators Comput. Phys. Commun.274108301 (8pp) 54

work page 2022