Functionals of quantum states in QFT

Almost every book and article I can think of represents states of QFT using the Heisenberg picture of Hilbert space vectors, but Visser in "Lorentzian wormholes" does mention that you can also represent them as a functional.

It is very briefly mentionned, and he does not mention any sources for it, just the result:

$$\Psi_0[t, \Phi(\vec{x})] \propto e^{-\frac{1}{2\hbar c}\int d^3x\Phi(\vec{x}) \sqrt{-\nabla^2} \Phi(\vec{x})} e^{-\frac{i}{\hbar}E_0t}$$

With $$\sqrt{-\nabla^2}$$ some pseudodifferential operator.

Is there any source, book or otherwise, that expands on this topic?

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• $\Psi_0$ is the ground state wave function of a free scalar field. – Slereah Jan 9 '14 at 23:40
• The ground state wave functionals (GSWF) are useful for free fields as they are Gaussian. I am not sure the wave functional approach is useful for interacting theory (as a matter of fact, I think the regularization could be a problem in the interacting case). IRCC, there is an exercise problem in Srednicki's book in which you are asked to derive the GSWF for free scalar theory. In the same vein, you can derive the GSWF for other kinds of free theories. Unfortunately I am not aware of any reference for this particular topic. – Isidore Seville Jan 9 '14 at 23:47
• While it is not exactly what you are searching, there is an interesting discussion here, see for instance chapter $5.3$ (Quantum field theory) page $132$ (the formula $5.128$ page $137$ is interesting) – Trimok Jan 10 '14 at 10:22
• some relevant answers are at physicsoverflow.org/22012 – Arnold Neumaier Aug 12 '14 at 18:46