Imagine we have a QFT (not a CFT) with some operator $\phi(\vec{x},t)$. Assume that an OPE exists and compute $\langle \phi(\vec{x},t)\phi(\vec{y},t)\phi(\vec{y}+\vec\epsilon,t)\rangle$ in the $\vec{\epsilon}\to 0$ limit.
A Contradiction
My confusion comes from the following steps where I seem to find a contradiction. At least one step is wrong, but I'm not sure which.
1) Since all operators are the same, their ordering in the expectation value is irrelevant: $\langle\phi(\vec{x},t)\phi(\vec{y},t)\phi(\vec{y}+\vec\epsilon,t)\rangle=\langle\phi(\vec{y},t)\phi(\vec{y}+\vec\epsilon,t) \phi(\vec{x},t)\rangle$, in particular.
2) Assume the the OPE contains a term of the following form: $\phi(\vec{y},t)\phi(\vec{y}+\vec\epsilon,t) \xrightarrow{\epsilon\to 0} C(\vec y,t)(\partial_\mu\phi(\vec y,t))^{2n}+\ldots$ for some $n$.
3) Compute the correlator starting from the two orderings shown above:
$\langle\phi(\vec{x},t)\phi(\vec{y},t)\phi(\vec{y}+\vec\epsilon,t)\rangle\to C(\vec y,t)\langle \phi(\vec{x},t)(\partial_\mu\phi(\vec y,t))^{2n}\rangle$
$\langle\phi(\vec{y},t)\phi(\vec{y}+\vec\epsilon,t)\phi(\vec{x},t)\rangle\to C(\vec y,t)\langle (\partial_\mu\phi(\vec y,t))^{2n}\phi(\vec{x},t)\rangle$
4) To me, 3) seems to contradict step 1) since $\langle (\partial_\mu\phi(\vec y,t))^{2n}$ involves factors of $\partial_t \phi(\vec y,t)$ which do not commute with $\phi(\vec{x},t)$, so the two-expressions in step 3) aren't strictly equal.
Why am I finding a contradiction?
Possible reasons I see:
- The factors of $\partial_t \phi(\vec y,t)$ and $\phi(\vec{x},t)$ only fail to commute at $\vec{x}=\vec{y}$. So the two expressions in 3) only fail to be equal at $\vec{x}=\vec{y}$. Since the OPE assumes that the operators which are taken to collide are far separated from all other operators in the theory, the OPE simply doesn't apply at this point and there's no contradiction.
- Since $\langle (\partial_\mu\phi(\vec y,t))^{2n}$ is really a composite operator, there's some subtlety and I shouldn't be thinking about it as a bunch of factors involving $\partial_t \phi(\vec y,t)$'s which don't commute with $\phi(\vec{x},t)$.
