How do you transform a time evolution function to work with any Cauchy surface? If you have a time evolution function $K_t(\phi,\phi')$ which gives you the amplitude to go from a field state $\phi$ to a field state $\phi'$ in time $t$ this gives you all the information you require about the system if it just contains a scalar field. $\phi$ are scalar fields that are obey a Lorentz invariant field equation.
But we also know that we need not foliate time into regular slices like this, but we can have the input and output states on any Cauchy surface. Therefor, we should be able to find functions $K_{\Sigma, \Sigma'}(\phi,\phi')$ in which $\Sigma$ and $\Sigma'$ are the initial and final Cauchy surfaces. e.g. we could specify $t(x,y,z)$ as the clock time on a Cauchy surface in a particular frame. (In the original case the surfaces are of constant time).
Since $K_t(\phi,\phi')$ already has all the information about the quantum amplitudes, it should be possible to write $K_{\Sigma, \Sigma'}(\phi,\phi')$ in terms of $K_t(\phi,\phi')$ and vice-versa.
So my question is how do we express the general case $K_{\Sigma, \Sigma'}(\phi,\phi')$ in terms of the specific cases $K_t(\phi,\phi')$ ?
 A: This answer is conceptual rather than computational.
For any foliation of spacetime by spacelike hypersurfaces, the corresponding transition functions are implied by the set of all transition amplitudes
\begin{gather}
 \int [d\phi_A][d\phi_B]\ 
 \Phi^*[\phi_B]K_{B,A}(\phi_B,\phi_A)\Psi[\phi_A]
\hspace{2cm}
\\
\hspace{2cm}
 =
 \int [d\phi]\ 
 \Phi^*[\phi_B]e^{iS[\phi]}\Psi[\phi_B]
\tag{1}
\end{gather}
where $\Psi,\Phi$ are arbitrary initial and final states defined on Cauchy surfaces $A$ and $B$, respectively. On the left-hand side, the functional integral is over all of the field variables $\phi_A,\phi_B$ associated with those two Cauchy surfaces. On the right-hand side, the functional integral is over all of the field variables $\phi$ associated with the whole spacetime interval from $A$ to $B$.
Specifying (1) for all $\Psi,\Phi$ is equivalent to specifying the transition function $K$, and doing this for all Cauchy-surface pairs in the foliation implicitly specifies the action $S$. One way to extract $S$ is to consider the effect of shifting the initial state $\Psi$ slightly in time and using the Schrödinger equation
\begin{align}
\newcommand{\la}{\big\langle}
\newcommand{\ra}{\big\rangle}
 \la\Phi(t_B)\big|\Psi(t_A+dt)\ra 
 &= 
 \la\Phi(t_B)\big|\Psi(t_A)\ra
\\
  &- i\,dt\,\la\Phi(t_B)\big|H(t_A)\big|\Psi(t_A)\ra.
\tag{2}
\end{align}
Since we know the transition amplitudes for all $\Phi,\Psi$, this gives us the Hamiltonian $H$, from which we can recover the action $S$ in the usual way. Equation (2) allows the Hamiltonian to be time-dependent, in case the foliation does not conform to any symmetries of the metric.
After we have the action, we can use (1) to construct the transition functions for any other foliation. Altogether, this process allows us to construct the transition functions for one foliation from the transition functions for another foliation, at least conceptually.
