String amplitudes for finite times and string wave-functions I am trying to understand string theory. In elementary quantum mechanics, one objective is to calculate $<X_{f},T|X_{i},0>$, the propagation kernel. I imagine two ways to accomplish this: 1) by directly calculating the matrix elements of $e^{-iH/\hbar}$ between the initial and final positions ,and 2) Representing this kernel as a path integral, and then evaluating this order by order in perturbation theory.
I am trying to extend this to string theory. Therefore, I replace the eigenkets $|X^{i}>$ by more complicated kets that we understand as follows: one consider target space-times of the form $\mathcal{M}=Y\times \mathbb{R}$ where $\mathbb{R}$ represents time. Now, we consider the loop space over $Y$, denoted by $\mathcal{L}(Y)$. To each point on $\mathcal{L}(Y)$ ``i.e. loop on $Y$", we associate a vector. So, these loops span the string vector space. In fact, I think we can consider string wave-functions as sections over the complex line bundle over loop space. In this picture, we consider only closed strings. I tried to study this. However, I am not sure if the theory will be manifestly Lorentz covariant. For instance, does the Lorentz group of the target space act linearly on this Hilbert Space?
My question is: Has this picture been studied carefully by physicists? Is so, is there a reference? Whenever I open a textbook on string theory, only the S-matrix is calculated: In this language, the string sources are sent to the infinite past and the infinite future. Now, why aren't we interested in string amplitudes for finite times?
 A: Here is a partial answer and/or long comment.

Now, why aren't we interested in string amplitudes for finite times?

They are difficult to consider: the thing one would naturally do (which is not the approach you suggest, but I'll comment on that shortly) would be to put a Lorentzian metric on the worldsheet, so that there exists a globally defined time function. But most string worldsheet topologies will not admit a smooth Lorentzian metric.
But one can certainly make progress when the worldsheet is $\mathbb R\times S^1$. This apparently precludes interactions but you can derive e.g. the critical dimension through loop space techniques, which is in the spirit of what you sketch: see
String Theory as the Kahler Geometry of Loop Space by Bowick and Rajeev.
(The difference to what you suggest is that it's the worldsheet and not the target that is factorised as $\mathbb R\times \cdots$.) I think most later papers that consider loop space approaches cite this work, so might be a good place to start hunting just in case they've done interactions.

In this language, the string sources are sent to the infinite past and
the infinite future.

To some extent this is forced upon us: we can unambiguously calculate correlation functions with insertions of operators of the form
$$c\bar c {\mathcal V}(z,\bar z)$$
where ${\mathcal V}(z,\bar z)$ is a weight $(1,1)$ primary, and these look like asymptotic states (coming in from $\pm\infty$, as you said) from the target space perspective. For more general states --- which the "finite time sources" you are interested in might be --- you need to make choices of coordinate systems around the insertions, which is the ambiguity I alluded to just above. There is a nice discussion around page 9 of the review by Erler:
Four Lectures on Closed String Field Theory
EDIT: Predictably immediately after I hit "enter" I remembered a reference that discusses the string propagator in loop space:
String Propagator: a Loop Space Representation by S.Ansoldi, A.Aurilia, E.Spallucci. I haven't really absorbed much of that yet though.
A: The answer of alexarvanitakis is a good one. I just want to elaborate a little bit more from the first quantized perspective.
It is a matter of first principles that any observable in a gauge theory must be gauge invariant. Subtelties appart; in a theory of gravity all observables should be invariant under an arbitrary diffeormorphism.
The existence of local observables contradicts the expected invariance under diffeomorphisms of a theory of gravity. Something more is expected to be true; the only observables in a genuine theory of gravity are the asymptotic ones. Again, having local observables in a theory of gravity seems to contradict the expected holographic nature of quantum gravity.
String theory (which unavoidably contains gravity) respects and strength all the aforementioned expectations. Finite time computations broke worldsheet conformal invariance explicitly and make no sense, at least in a first quantized theory of strings where all amplitudes are on-shell aplitudes.
Reference: Observables in quantum gravity
