Is The Seiberg-Witten Map Unique?

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Main Question or Discussion Point

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
 

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