Is there a formal solution to the Tomonaga-Schwinger equation for scalar fields?

Given a wave functional of a scalar field in the momentum picture $Psi_t[phi]$ it can be evolved in time using:

$$Psi_{t_2}[phi] = int K_{t_2-t_1}[phi,phi’]Psi_{t_1}[phi’]Dphi’$$

Where the evolution function is given formerly by:
$$K_t[phi,phi’] = N_texpleft( int limits_{kinmathbb{R}^3} iomega_kleft(frac{phi(k)^2+phi'(k)^2}{tan(tomega_k)} +
frac{phi(k)phi'(k)}{sin(tomega_k)} right)dk^3
right)$$
with $omega_k = sqrt{k^2+m^2}$ and normalising factor $N_t = prodlimits_{kinmathbb{R}^3}frac{1}{sqrt{sin(tomega_k)}}$. There are issues here concerning how to take the limit of these integrals, but formerly this is just the infinite product of the propagator for harmonic oscillators of every momenta.

The Tomonaga-Schwinger equation instead of having a fixed time slice, has a function $tau(x)$, i.e we are taking a non-flat Cauchy surface. (A sub-manifold of flat Minkowski space).

Therefor we seek an evolution function $K[phi,phi’;tau,tau’]$ which can evolve the wave function $Psi[phi,tau]$. Therefor we are evolving the fields between two Cauchy surfaces.

i.e. it satisfies the `functional Schrödinger
equation’:

$$left(frac{delta}{delta tau(x)} – H[phi,tau] right) K[phi,phi’;tau,tau’] = 0$$

where $H$ is the Hamiltonian for a Klein-Gordon field.

Any idea how one might try to find an expression for $K$ perhaps as a series or in closed form?