Develop a systematic subtraction/normal-ordering scheme for quantum Lax–generated conservation laws

Develop a systematic method to obtain finite, well-defined quantum conserved operators from the non-normal-ordered expressions (⟨1,ℒ^n1⟩) generated by the quantum Lax pair [HI+ℳ,ℒ]=0 in Section “Quantum Lax pair and conservation laws,” by normal-ordering and subtracting divergent lower-order terms, so that the resulting operators commute with the Gérard–Grellier Hamiltonian H for all orders and the cutoff can be removed rigorously.

Background

The authors construct a quantum Lax pair (ℒ,ℳ) for the Gérard–Grellier Hamiltonian H and observe that operators of the form (⟨1,ℒ^n1⟩) formally commute with H, suggesting an infinite family of conservation laws. However, these operators are not normal-ordered and produce divergent lower-order terms upon normal-ordering.

They have verified via computer algebra that for the first few members, divergences can be canceled by adding suitable combinations of lower-order conserved quantities, but they do not possess a general scheme to perform these subtractions at all orders. Establishing such a systematic procedure would enable effective construction of the entire hierarchy of conserved operators from the quantum Lax pair.