Does it make sense to speak of amplitudes of finite closed boundaries in QFT? A example of amplitude in Relativistic Quantum Mechanics or specifically in QFT is the amplitude of a field configuration on a space-like hyper-surface of space-time to "lead" to another field configuration on another space-like hyper-surface of space-time. In the path-integral picture one simply integrates over all possible field configurations on the interior, giving each a weight in the normal way. Now if one wants to generalize this to finite closed boundaries, we would get an amplitude for each field configuration on a finite closed boundary of space-time. but how would we interpret this? This question relates to interpretations of quantum mechanics, has anyone investigated this line ? 
 A: A wave functional of a field A is $\Psi[A]$. This is the amplitude that there is a certain field configuration A. (Compare to ordinary quantum mechanics where a wave function $\psi(x)$ is the amplitude for a particle to be at position x). If a wave function is highly peaked at a particular value it means that this value is most likely. Similary a wave functional of a field A can be highly peaked at a certain field configuration f. It may be a Gaussian such as 
$$\Psi[A] = \exp\left(-\int (A(x)-f(x))^2 dx^3 \right)$$
(This only works for bosons. Fermions don't have anything corresponding to a classical field).
The universe at any time is described by the wave functional of the fields (which may be highly peaked at a particular field configuration... or not). The amplitude that the universe will have a different wave functional $\Psi_{out}$ at a later time is given by the path integral:
$$\Delta[\Psi_{in},\Psi_{out}] = \int \Psi_{in}[A]\Psi_{out}[A]e^{i S[A] } D[A]$$
One can calculate this by expanding the wave functional in terms of particle amplitudes:
$$ \Psi_{in}[A] = a + \int\psi(x)A(x,t_{in})dx^3 + \int\psi(x,y)A(x,t_{in})A(y,t_{in})dx^3dy^3+...$$
Where $\psi(x,y)$ is the amplitude for particles being found at both positions x and y. In particular we have:
$$ \Delta_F(x,y) = \int A(x) A(y) e^{i S[A] } D[A]$$
which is the amplitude for a particle to travel from x to y.
As for closed boundaries, the time slice for the incoming data and the time slice for the outgoing data can be joined at the boundaries forming a nut shape.
