QFT on non-commutative spaces

Fiore and Wess in their article ”On full twisted Poincare’ symmetry andQFT on Moyal-Weyl spaces” shows that non-commutative QFT withproperly enforced ”twisted symmetry” is equivalent to ordinary commutativeQFT. They start from enforcing space non-commutativity in most simleway:

(1)

Where is a constant matrix. So the limit gives ordinarycommutative space. Then algebra generated by is equivalent to algebra offunctions on commutative space with deformed product, called star-product.Such star product is introduced using twist :

(2)

Properly chosen gives

(3)

as required.

There are whole classes of equivalent twists, which lead to the same starproduct, so the theory doesn’t depend on the choice of particular twist. ThenFiore and Wess show how to construct QFT starting with . To obtain properself-consistent theory they have to change Poincare-invariance with invariancewith respect to deformed Poincare group, obtained from deformation of Poincarealgebra with the same twist , as was used to deform algebra of functions.

After the introduction of Wightman axioms and study of Wightman andGreen’s functions it turns out, that this functions coincide with their undeformedcounterparts at least perturbatively. So deformed Poincare symmetry works as”compensation” of space non-commutativity, and theory gives no newphysics.

But there are another possibilities. We can use more complex non-commutativespace, for example, we can start from deformation of Poincare symmetry to-Poincare, then introduce a twist on -Poincare and build non-commutativespace with twisted -Poincare symmetry.

I hope to study this case in details in the next several days.