Faddeev-Zamolodchikov algebra ($A_i({\theta }_1)A_j({\theta }_2)=S_{ij}^{kl}({\theta }_{12})A_l({\theta }_2)A_k({\theta }_1)$)

Particle operators obey a rapidity-dependent exchange rule $A_i({\theta }_1)A_j({\theta }_2)=S_{ij}^{kl}({\theta }_1-{\theta }_2)A_l({\theta }_2)A_k({\theta }_1)$, with repeated species indices summed. A consistent associative algebra must give the same coefficients when a triple product is brought to decreasing or increasing rapidity order by either sequence of adjacent exchanges. This is the Faddeev-Zamolodchikov associativity constraint. Rapidity ordering here is distinct from the creation/annihilation convention of ordinary normal ordering.

Starting with three ordered rapidities, the adjacent-exchange sequences $121$ and $212$ must give identical coefficients for every final species word in the Faddeev-Zamolodchikov algebra. The exchange parameters are $\theta$, $\theta +{\theta }^{\prime }$ and ${\theta }^{\prime }$ because rapidity differences add. These coefficient identities express the spectral Yang-Baxter equation. One scalar identity is necessary but generally does not exhaust all the constraints.