Consider a set of operators ˆQi and its canonical momenta ˆPi=−i∂ˆQi (this differential operator is not commutative; in a differential geometrical point of view, it produces a torsion!) with i going from 1 to n. Then I define the commutation relations
[^Pi,ˆQj]=−iδij (follows directly from my definition)
and the noncommutative relations
[ˆQi,ˆQj]=∑kf(p)ijkˆPk,
[ˆPi,ˆPj]=∑kf(q)ijkˆQk.
Here, f(q)ijk are Lie algebra structure constants that are satisfying Jacobi identity and antisymmetry in i and j. Above commutation relations are consistent with Jacobi identity and therefore, different field operator are correlated which each other due to noncommutativity.
Question: Is such a type of noncommutative quantum theory studied? I have seen theories, where commutators between operators that are not canonical conjugate to each other are a constant, the noncommutativity parameter θ and that the theory changes by replacing ordinary product with Moyal product. In case of a Lie-algebraic noncommutativity the theory would be more complicated.
Are there already results about such a type of theory (Literature dealing with it can be posted as an answer)?