Converting a general QFT state to another inertial of light-cone reference frame Assume, in a certain reference frame, a relativistic QFT at time $t=0$ is in the state
$$
\hat{\Psi}(t=0) |vac\rangle \quad,
$$
where
$$
\hat{\Psi}(t) = \operatorname{e}^{-i\hat{H}t} \sum \limits_{n=1}^\infty \int
\tilde{\operatorname{d}}x_1
\tilde{\operatorname{d}}x_2
\ldots
\tilde{\operatorname{d}}x_n \;
\psi_{n}(x_1,x_2,\ldots,x_n)
\hat{a}^\dagger_{x_1}
\hat{a}^\dagger_{x_2}
\ldots
\hat{a}^\dagger_{x_n}
\quad.
$$
In the latter expression, $\psi_{n}(x_1,x_2,\ldots,x_n)$ is an $n$-particle wave function, $\tilde{\operatorname{d}}$ is a Lorentz-invariant integration measure, and $\hat{a}^\dagger_{x}$ is the 'position creation operator' defined as the Fourier transformed momentum creation operator:
$$
\hat{a}^\dagger_{x} = \int \tilde{\operatorname{d}}p \operatorname{e}^{ipx}\hat{a}^\dagger_p \quad.
$$
Given that $\hat{\Psi}(t)$ is known, I would like to know how this state looks like in


*

*Another inertial reference frame with time $t'$?

*Light-cone coordinates with time $x^+=t+x$? (assume QFT is in 2d for simplicity)


The major problem arises from the fact that, as we switch to another reference frame, we also change the way we foliate the spacetime.
In the following pics I illustrate the question with somewhat opposite convention  (actually, this formulation is more useful for me). The evolution took place in the primed coordinates, and we would like to know the state in the unprimed ones:
Evolution in a boosted frame:

Evolution in light-cone coordinates:

The state of the system in known in the gray area, with all the operators defined along the corresponding time axes. I want to know state of the system along the blue lines (which would amount to partly evolving the system back/forth - using the evolution in primed time, as shown with arrows).
How do I make such a switch?
 A: To talk about a relativistic QFT, we need a Hilbert space $\mathcal{H}$ with an unitary representation of the (double cover of the connected component of the) Poincare group, which i will denote $U(S,a)$, where $S \in \text{Spin}_{1,3} \rtimes \mathbb{R}^4$.
In a relativistic QFT, one has a vacuum state $|\text{VAC}\rangle$ which is assumed to be Lorentz-invariant: 
$$ U(S,a) |\text{VAC}\rangle = |\text{VAC}\rangle \ . $$
To obtain other (normalizable) states, it is useful to consider field operators which i will denote $\psi_\ell(x)$, where $x=(\vec{x},t)$, which are supposed to be in finite-dimensional representations:
$$ U(S,a) \psi_{\ell}(x) U(S,a)^{-1} = \sum_{\ell'} D_{\ell \ell'}(S) \psi_{\ell'}(S^{-1} x -a ) \ . $$
If we are to consider a scalar boson, this operators are, if the right choice of measure is made, your operators $a_x$.
Now we get normalizable states in the Hilbert space by convoluting with some smooth function $f$(here we assume for simplicity that the particles which are created by $\psi_\ell$ are their own anti-particles):
$$ | f \rangle = \int d x_1 d x_2 \cdots d x_n f_{\ell_1 \ell_2 \dots \ell_n}(x_1,x_2,\cdots x_n) \psi_{\ell_1}(x_1) \psi_{\ell_2}(x_2) \cdots \psi_{\ell_n}(x_n) |\text{VAC}\rangle  \ . $$
Until now i have just reformulated the question in a more standard notation, but i think now it should be clear what to do: choosing different coordinates can be done in the integral by letting $x \mapsto x' = x'(x)$. Then the state is
$$ | f \rangle = \int d x'_1 d x'_2 \cdots d x'_n \left|\prod_{i} J(x_i)\right| f_{\ell_1 \ell_2 \dots \ell_n}(x_1,x_2,\cdots x_n) \psi_{\ell_1}(x_1) \psi_{\ell_2}(x_2) \cdots \psi_{\ell_n}(x_n) |\text{VAC}\rangle  \ , $$
where in the integral the variables $x_i$ are considered as dependent on $x'_i$. If your chosen coordinate transformation is a Poincare-transformation, $x = S^{-1} x' - a$, then you may use the tranformation law of the fields and get a representation of the vector $|f\rangle$ with some different function $f'$. For example, in the case $n=1$:
$$ |f\rangle = U(S,a) |f'\rangle \ , \quad f'_\ell(x) = \sum_{\ell'} D_{\ell' \ell}(S) f_{\ell'}(S^{-1} x - a) \ .$$ 
This for example allows you to compute the switch from your time coordinate to light-cone coordinates.
There is one more remark: one may take the limit that $f$ is supported only on one time-slice - the above arguments still go through. One should be more careful then though since in interacting theories this may lead to the vector being non-normalizable.
