Remarks on Exactly Solvable Noncommutative Quantum Field

We study exactly the solvable noncommutative scalar quantum field models of (2n) or (2n+1) dimensions. By writing out an equivalent action of the noncommutative field, it is shown that the special condition B.θ=±I in field theoretic context means the full restoration of the maximal U (∞) gauge symmetries broken due to kinetic term. It is further shown that the model can be obtained by dimensional reduction of a 2n-dimensional exactly solvable
noncommutative Ф⁴ quantum field model closely related to the 1+1-dimensional Moyal/matrix-valued nonlinear Schrodinger (MNLS) equation. The corresponding quantum fundamental commutation relation of the MNLS model is also given explicitly.