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  Noncommutative quantum field theory based on field correlations

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Consider a set of operators $\hat{Q}_i$ and its canonical momenta $\hat{P}_i = -i \partial_{\hat{Q}_i}$ (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 

$[\hat{P_i},\hat{Q}_j] = -i \delta_{ij}$ (follows directly from my definition)

and the noncommutative relations

$[\hat{Q}_i,\hat{Q}_j] = \sum_k f^{(p)}_{ijk}\hat{P}_k$,

$[\hat{P}_i,\hat{P}_j] = \sum_k f^{(q)}_{ijk}\hat{Q}_k$.

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 $\theta$ 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)?

asked Dec 9, 2017 in Mathematics by PatrickLinker (40 points) [ revision history ]

People study noncommutative spacetimes with nonstandard commutation rules, though not of your form. Do you have a proof that Lie algebras with your kind of commutation rules exist? You'd have to provide explicit structure constants! 

Please make the title conform to your actual question - which has no connection to QFT!

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