In the functorial approach to QFT, each Cauchy surface $\Sigma$ has an associated Hilbert space $\mathcal{H}_\Sigma$, and each pair of Cauchy surfaces $\Sigma,\Sigma'$ has an associated unitary $U_{\Sigma\to\Sigma'}:\mathcal{H}_\Sigma\to\mathcal{H}_{\Sigma'}$. These unitaries compose functorially. In practice, the unitaries are calculated by the path integral, at least formally.
I'm struggling to understand how the usual Heisenberg field operators are defined in this framework. Presumably, we define a commuting set of field operators on some initial slice $\Sigma$. These form an operator algebra which I'll call $\mathcal{A}_\Sigma$. In the spirit of the Heisenberg picture, we could then define the operator algebra associated to a different slice $\Sigma'$ to be $\mathcal{A}_{\Sigma'} = U_{\Sigma\to\Sigma'} \mathcal{A}_{\Sigma} U_{\Sigma\to\Sigma'}^{\dagger}$. This is nice, but it's not yet the full Heisenberg picture: we want to associate field operators to points, not just sets of field operators to Cauchy slices. Also, we'd like all of our Heisenberg field operators to act on one fixed Hilbert space, not different Hilbert spaces corresponding to each slice.
So it seems that in passing from the functorial/path integral picture to the Heisenberg picture, one must choose some way to identify points, and Hilbert spaces, on different slices. Essentially, we need to make some arbitrary choice of coordinates. However I've been unable to get such a procedure to work, since the definition of the field operators seems to end up depending on the coordinate choice, which is clearly wrong.
So: How are the Heisenberg field operators defined in the functorial/path integral QFT framework?
Best Answer
Perhaps it may help to be slightly more general, defining Hilbert spaces and operators separately from the dynamics.
So, let's suppose we have a state $|\psi\rangle$ defined on one Cauchy surface $\Sigma_0$ in a Hilbert space $\mathcal{H}_0$, and an operator $\mathcal{O}$ which acts on a different surface $\Sigma_t$ (a local operator on $\Sigma_t$, say), so it's defined on a different Hilbert space $\mathcal{H}_t$. At this point, in general we may have no canonical way to identify $\mathcal{H}_0$ and $\mathcal{H}_t$ (for example, we might have a QFT on some unknown curved spacetime, and we're just given $\Sigma_0$ and $\Sigma_t$ as some spatial manifolds with no information about the spacetime between them). There's no way yet to compute the expectation value of $\mathcal{O}$ in $|\psi\rangle$.
To do that, we need more information: the `time evolution' operator $U=U_{\Sigma_0\to \Sigma_t}$, an isomorphism $\mathcal{H}_0\to \mathcal{H}_t$. (For example, someone tells us about a spacetime bounded by $\Sigma_t$ and $\Sigma_0$, and $U$ is constructed from the path integral on that spacetime.) Now we can build something that makes sense: $$\langle \psi|U^\dagger \mathcal{O} U|\psi\rangle.$$
Now the "picture" just refers to two ways of splitting up this calculation. Schrödinger means that we "evolve the state", so we define $$ |\psi(t)\rangle = U|\psi\rangle \in \mathcal{H}_t $$ and then compute $\langle \psi(t)|\mathcal{O}|\psi(t)\rangle$ using our original representation of $\mathcal{O}$. Heisenberg means that we instead "evolve the operator", $$ \mathcal{O}(t) = U^\dagger \mathcal{O} U : \mathcal{H}_0\to \mathcal{H}_0 $$ and compute $\langle\psi|\mathcal{O}(t)|\psi\rangle$.
In conclusion, we have a collection of many possible Hilbert spaces $\mathcal{H}_\Sigma$, along with isomorphisms $U_{\Sigma\to \Sigma'}$ between them. The "Heisenberg picture" means that we make one particular choice $\mathcal{H}_0$ from this collection, and define a "Hiesenberg picture operator" simply as an operator on $\mathcal{H}_0$ (typically for operators which act locally on some other surface).
For a more concrete example and a slightly different perspective, suppose we're interested in some QFT on some static spacetime $\Sigma\times \mathbb{R}$, so $\Sigma_t =\{(x,t):x\in\Sigma\}$. We first define a Hilbert space of states on $\Sigma$, perhaps by something like Osterwalder-Schader, along with a collection of local operators $\mathcal{O}(x)$. We consider a one-parameter family $\{\mathcal{H}_t:t\in\mathbb{R}\}$ of copies of this Hilbert space. The Heisenberg operator $\mathcal{O}(x,t)$ is defined as $\mathcal{O}(x)$ acting on $\mathcal{H}_t$. Next, we introduce a one-parameter group of unitaries $U(t)$ that define isomorphisms between different $\mathcal{H}_t$. Finally, we reduce $\{\mathcal{H}_t:t\in\mathbb{R}\}$ to a single Hilbert space $\mathcal{H}$ by `quotienting by time translation', defining $|\psi_1\rangle \simeq U(t_1-t_2)|\psi_2\rangle$ for $|\psi_1\rangle \in \mathcal{H}_{t_1}$ and $|\psi_2\rangle \in \mathcal{H}_{t_2}$: a state in $\mathcal{H}$ is an equivalence class under this relation. The definition of the Heisenberg picture operator $\mathcal{O}(x,t)$ descends to $\mathcal{H}$ in the obvious way.