The reason that I have this problem is that I'm trying to solve problem 14.4 of Schwartz's QFT book, which've confused me for a long time.
The problem is to construct the eigenstates of a quantum field $\hat{\phi}(\vec{x})$, such that $$ \hat{\phi}\left(\vec{x}\right)\left|\Phi\right\rangle =\Phi\left(\vec{x}\right)\left|\Phi\right\rangle . $$ I think the eigenstate should be
$$ \left|\Phi\right\rangle =e^{-\int d^{3}x\frac{1}{2}(\Phi(\vec{x})-\hat{\phi}_{+}(\vec{x}))^{2}}\left|0\right\rangle $$ where $\hat{\phi}_{+}\left(\vec{x}\right)$ is the part of $\hat{\phi}\left(\vec{x}\right)$ that only includes creation operators. I haven't found any book talking about this problem so I'm actually not sure whether this is the correct result.
And similarly the eigenstate of $\hat{\pi}\left(\vec{x}\right)$ such that $$ \hat{\pi}\left(\vec{x}\right)\left|\Pi\right\rangle =\Pi\left(\vec{x}\right)\left|\Pi\right\rangle $$ should be $$ \left|\Pi\right\rangle =e^{\int d^{3}x\frac{1}{2}(\Pi(\vec{x})+\hat{\phi}_{+}(\vec{x}))^{2}}\left|0\right\rangle . $$ Equation 14.21 and Equation 14.22 of Schwartz's book are $$ \left\langle \Pi|\Phi\right\rangle =\exp\left[-i\int d^{3}x\Pi\left(\vec{x}\right)\Phi\left(\vec{x}\right)\right] $$ $$ \left\langle \Phi'|\Phi\right\rangle =\int\mathcal{D}\Pi\left\langle \Phi'|\Pi\right\rangle \left\langle \Pi|\Phi\right\rangle =\int\mathcal{D}\Pi\exp\left(-i\int d^{3}x\Pi\left(\vec{x}\right)\left[\Phi\left(\vec{x}\right)-\Phi'\left(\vec{x}\right)\right]\right) $$ Now my question is how do I verify these relations? I've wasted a lot of time on this problem without any success.
And why doesn't any book talk about this problem? I mean the eigenstate of a quantum field? Isn't this kind of stuff very fundamental to QFT?
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