ququartWeyl_relation
The alias supplies the ququart Weyl commutation relation ZX = i XZ for direct reference in the OperatorCore layer. Researchers constructing four-level quantum models or coupled operator algebras in Recognition Science cite it to fix the phase factor in the Weyl algebra. It is realized as a one-line alias to the imported theorem.
claimLet $X$ and $Z$ be the ququart operators on the four-dimensional state space. Then $Z X = i X Z$.
background
QuquartState is the four-dimensional complex Hilbert space equipped with basis vectors via basisKet. The operators ququartX and ququartZ are the standard generators satisfying the Weyl algebra on this space. The module CoupledRecognitionCores defines these operators together with their compositions; the present alias imports the relation into the OperatorCore namespace for use in tensor constructions.
proof idea
The declaration is a one-line alias to the theorem ququartWeyl_relation in CoupledRecognitionCores. That theorem is proved by applying LinearMap.ext to obtain pointwise equality on states, followed by direct invocation of ququartWeyl_relation_apply.
why it matters in Recognition Science
The relation supplies the canonical phase factor for ququart operator products and is referenced by tensorWeylMonomial and localWeylMonomial. It anchors the algebraic structure of coupled recognition cores. In the Recognition framework it fixes the commutation rule required for consistent operator monomials at the foundation level.
scope and limits
- Does not derive the relation from the Recognition Composition Law.
- Does not supply explicit matrix forms of X or Z.
- Does not extend the relation to dimensions other than 4.
- Does not address continuous or field-theoretic limits.
formal statement (Lean)
19abbrev ququartWeyl_relation := IndisputableMonolith.Foundation.CoupledRecognitionCores.ququartWeyl_relation