Question

While studying canonical quantization in QFT, I observed that we quantize fields either by a commutation or an anticommutation relation

[ϕ(x),ϕ(y)]±:=ϕ(x)ϕ(y)±ϕ(y)ϕ(x)=iδ(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
[ϕ(x),ϕ(y)]q:=ϕ(x)ϕ(y)+qϕ(y)ϕ(x)=iδ(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?

Answer 1

We use commutation and anticommutation relations in QFT based on the spin statistics theorem, which are rooted in the physical properties and behaviors of particles with integer and half-integer spins respectively.

Step-by-step explanation:

The question pertains to the reason we use specific quantization relations in quantum field theory (QFT), namely commutators for integer spin particles (bosons) and anticommutators for half-integer spin particles (fermions), instead of more general quantization relations with an arbitrary constant q. In canonical quantization, the choice of commutation or anti-commutation is not arbitrary but is deeply rooted in the physical properties of particles and their statistics, as described by the spin-statistics theorem. This theorem states that particles with half-integer spin, like electrons (fermions), follow the Pauli exclusion principle, which is mathematically represented by anticommutation relations [ϕ(x),ϕ(y)]- = ϕ(x)ϕ(y) - ϕ(y)ϕ(x) = iδ(x-y). Conversely, particles with integer spin, like photons (bosons), are described by commutation relations [ϕ(x),ϕ(y)]+ = ϕ(x)ϕ(y) + ϕ(y)ϕ(x) = iδ(x-y). The adoption of these specific quantization relations ensures consistency with observed physical phenomena, such as the statistical behavior of particles and field theories that correctly describe particle interactions.