Is The Seiberg-Witten Map Unique?

[QFT25]

From my understanding the Seiberg-Witten map is a way to convert a non-commutative field theory into a commutative field theory. For example for the commutative relation between positions [x,y]=i*n, the common SW map I see in the literature for non commutative quantum mechanics is

x-> x_{c}-(1/2)*n*p_{y}

y->y_{c}+(1/2)*n*p_{x}

Where [x_{c},y_{c}]=0 and [x^{j}_{c},p_{k}]=i krockner-delta^{j}_{k}

However the transformation

x-> x_{c}-n*p_{y}

y->y_{c}

Also satisfies [x,y]=i*n

Does this mean that
x->x_{c}−n*p_{y}
y->y_{c}
is also a valid Seiberg-Witten map? If it isn't, then what properties is this transformation missing?

Thanks

 

[LightHero]

!No, this transformation is not a valid Seiberg-Witten map. The Seiberg-Witten map is a specific type of map between non-commutative and commutative theories, which preserves the commutation relations between the non-commutative fields. The transformation you have given does not satisfy this requirement, as [x,y] = 0 rather than i*n.