Why do we only consider commutators and anticommutators in QFT?

Question

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

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 \begin{equation} [\phi(x), \phi(y)]_q := \phi(x) \phi(y) + q \phi(y) \phi(x) = i \delta (x - y). \end{equation}

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?

Answer 1

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".

The answer to your question probably isn't that interesting, it's simply that the standard commutation and anti-commutation relations are what produce accurately predicting theories.

You could explore the possibilities but I suspect a lot of your theoretical predictions will be out by various powers of $q$!

Your question as it stands is a bit broad, as most "why don't we do/consider X" type questions tend to simply be answerable with "because X doesn't work". And because this is physics that is about as far as we are interested in.

Cosmos Zachos gives a bit more information on a similar question here if you're interested in reading.