I'm currently working through an article in this book (p. 78-114) about Schrödinger representation of quantum fields. A simplified discussion can be found here.
The author arrives at the Schrödinger eigenvalue problem for the filed wave-functional: $$ \hat{H}\Psi[\phi]=E \Psi[\phi] $$ Where for a free scalar field: $$ \hat{H}=\frac{1}{2}\int_x \left(\hat{\Pi}^2+\hat{\Phi}(-\nabla^2+m^2)\hat{\Phi} \right) =\frac{1}{2}\int_{xy} \left(-\delta_{xy} \frac{\delta^2}{\delta\phi_x\delta\phi_y}+\phi_x\omega^2_{xy}\phi_y\right) $$ For simplicity I denote $\phi(x)\equiv \phi_x$, $\delta(x-y)\equiv \delta_{xy}$ and $\omega^2_{xy}=(-\nabla^2+m^2)\delta(x-y)$. To find the eigensystem we need to solve the following functional equation: $$ \frac{1}{2}\int_{xy}\left(-\delta_{xy} \frac{\delta^2\Psi[\phi]}{\delta\phi_x\delta\phi_y}+\phi_x\omega^2_{xy}\phi_y\Psi[\phi]\right)=E\Psi[\phi] $$
The author states the solution for the lowest energy state: $$ \Psi_0={\det}^{-1/4}(\frac{\omega}{\pi}) e^{-\int\phi\omega\phi} \ \ \text{and} \ \ E_0=\frac{1}{2} tr(\omega) $$ I see that this solution is an infinite-dimensional generalization of the solution for a quantum harmonic oscillator and, when I plug it in the eigenvalue equation, it solves it. However, I have absolutely no idea how to properly obtain this or higher energy solutions. (except using creator and annihilator operators that would not help for interacting fields)
I'd be very grateful, if someone explains how to solve such equations or suggests a good reference. I'd like to apply this formalism for other systems with interactions or fermions. I would be really happy if those references are written by physicist for physicist and don't have $n$ hundred pages of functional space formalities. =)
