# Why do we only consider commutators and anticommutators in QFT?

While studying canonical quantization in QFT, I observed that we quantize fields either by a commutation or an anticommutation relation $$$$[\phi(x), \phi(y)]_\pm := \phi(x) \phi(y) \pm \phi(y) \phi(x) = i \delta (x - y).$$$$

and from this, the spin-statistics theorem tells us that we should use the commutator relation for integer spins and the anticommutator for half-integer spins.

My question is this $$-$$ why do we not consider more general quantization relations, for example of the following form $$$$[\phi(x), \phi(y)]_q := \phi(x) \phi(y) + q \phi(y) \phi(x) = i \delta (x - y).$$$$

for some arbitrary constant $$q$$? Also, if we can consider such relations, then is it possible to argue that we should have $$q = 1$$ for half-integer spins and $$q = -1$$ for integer spin?

Your argument at the bottom checks out, in principle. There's no reason you can't consider the commutators as the $$q=1$$ and $$q=-1$$ cases of your generalised "commutator".
You could explore the possibilities but I suspect a lot of your theoretical predictions will be out by various powers of $$q$$!