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Jeudy–Rouleux global label permutations

Abstract

The spin-1 three-particle exchange Hamiltonian is outside the Lie algebra generated by global permutations of the spin labels.

Definition 1.1 (The generated Lie algebra).

Formalization. D5/S3/Quantum/Algebra/JeudyRouleuxGlobalPermutationLie.L (✓ std3).

Citation. C. Jeudy; M. Rouleux (2024). Generalized exchange operators for a system of spin-1 particles. DOI: 10.1088/1742-6596/2912/1/012043. URL: https://arxiv.org/abs/2411.03952v1.

Commentary.

Section 3.1, p. 6: “At least in case of (S,N) = (1/2,N) and (S,N) = (1,3) it is easy to see that the Q(ij) span a Lie algebra L of operators on the total space ℋ and that the Cayley table for [Q(α), Q(β)] is identical to this of [Q̃(α), Q̃(β)].” Definition 4, p. 6: “We call Q-representation the product representation (S(2S+1), ℋ, Q), ℋ = ℂ^{(2S+1)^N} the N-lift of Q̃, defined on transpositions (ij) by Q(ij) = Q̃(ij) ⊗ ⋯ ⊗ Q̃(ij) (N times).” Configurations are functions Fin 3 → Fin 3. The row x, column y entry of the global action is one exactly when x = α ∘ y. LieSubalgebra.lieSpan includes all six permutations α of Fin 3. The bracket on matrices is A * B − B * A. This algebra also contains the Lie algebra generated by the three transpositions alone.

Definition 1.2 (The three-particle Hamiltonian).

Formalization. D5/S3/Quantum/Algebra/JeudyRouleuxGlobalPermutationLie.schrodingerHamiltonian (✓ std3).

Citation. C. Jeudy; M. Rouleux (2024). Generalized exchange operators for a system of spin-1 particles. DOI: 10.1088/1742-6596/2912/1/012043. URL: https://arxiv.org/abs/2411.03952v1.

Commentary.

Section 2, equation (7), p. 4: “We can then form the multi-particle (non-linear) Hamiltonian (Schrödinger Hamiltonian) H_S(S,N) = Σ_{⟨i,j⟩} P_S(S_i · S_j).” At (S,N) = (1,3), the three unordered pairs are (0,1), (0,2) and (1,2), each counted once. Section 2 identifies the Exchange Operator P_ij = P_S(S_i · S_j); its row x, column y entry is one exactly when x = y ∘ Equiv.swap i j. The formulas use zero-based particle and label indices.

Definition 1.3 (The expectation in Section 3.1).

Formalization. D5/S3/Quantum/Algebra/JeudyRouleuxGlobalPermutationLie.claim (✓ std3).

Citation. C. Jeudy; M. Rouleux (2024). Generalized exchange operators for a system of spin-1 particles. DOI: 10.1088/1742-6596/2912/1/012043. URL: https://arxiv.org/abs/2411.03952v1.

Commentary.

Section 3.1, p. 7: “Moreover since that the CSCO of the P- and Q-reps are the same in this case (see [Jin], p.70), we expect that Schrödinger Hamiltonian P_S, as well as other Hamiltonians, belongs to L.” Here claim is the membership assertion for the Schrödinger Hamiltonian at (S,N) = (1,3); no assertion about other Hamiltonians is included. The space uses all six label permutations. Section 6’s enlarged space L̂ = L₁ + L₁* is outside this statement.

Theorem 1.4 (The Hamiltonian does not belong).

Proof. Machine-checked in Lean as D5/S3/Quantum/Algebra/JeudyRouleuxGlobalPermutationLie.result (✓ std3). ∎

Resolves. Problems/jeudy-rouleux-global-permutation-lie (refuted) by D5/S3/Quantum/Algebra/JeudyRouleuxGlobalPermutationLie.result.

Source. Repository-derived.

Acknowledgement. C. Jeudy; M. Rouleux (2024). Generalized exchange operators for a system of spin-1 particles. DOI: 10.1088/1742-6596/2912/1/012043. URL: https://arxiv.org/abs/2411.03952v1.

Commentary.

Every global permutation preserves equality of the first two particle labels, so its generated Lie algebra commutes with the diagonal projector onto that equality. The Hamiltonian’s commutator with the projector has entry one at row (1,0,0), column (0,0,1). Hence the Hamiltonian is outside L. This disproves the membership expectation in Section 3.1. The source’s CSCO comparisons and the enlarged space in Section 6 are outside this result.

References