What are the orthogonal eigenstates of the field operator?

Question

In Peskin & Schroeder section 9.2, they derive the two-point function in the path integral formalism:

$$\langle \Omega | \mathcal{T} \left\{ \hat{\phi}(x_1)\hat{\phi}(x_2)\right\} | \Omega \rangle $$ $$= \frac{\int \mathcal{D}\phi \ \phi(x_1) \phi(x_2) e^{ i\int d^4 x\ \mathcal{L} }}{\int \mathcal{D}\phi \ e^{ i\int d^4 x\ \mathcal{L} }}. \tag{9.18}$$

The trick to derive this is to insert the identity

$$1 = \int \mathcal{D}\phi\ |\phi\rangle \langle \phi|$$

between the operators $\hat{\phi}(x_i)$. Then we can change the operator for regular functions using:

$$ \hat{\phi}(x_i) |\phi_i\rangle = \phi(x_i) |\phi_i\rangle.$$

My first question is: what are the states that form the complete orthogonal basis $|\phi\rangle$? The authors never seem to specify it. It cannot be just any complete orthogonal basis since these states seem to be eigenstates of the field operator

$$\hat{\phi}(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_p}}(\hat{a}_p e^{i p\cdot x} + \hat{a}_p^\dagger e^{-i p\cdot x}).\tag{2.25+47}$$

From what I understand, the states $|\phi\rangle$ represent all the possible classical field configurations (classical as in well-defined at all points in space at a given time, with no uncertainty) over which we integrate between two boundary states. But I don't see how these classical states are the eigenstates of $\hat{\phi}(x)$. Is there a simple expression for $|\phi\rangle$ in terms of e.g. creation/annihilation operators?

Actually what bothers me is that eigenstates of the field operator are supposed to be coherent states, which form an overcomplete set. Which means that if the $|\phi\rangle$'s are coherent states we cannot write the identity as the combination above since the states are not orthogonal (see section 8.1.3 of this document). My second question is: is it possible tha coherent states might be the eigenstates of another type of "field operator", not the one above? If so, what is this other operator? (Solved: see edit below) Note that in the given link they don't seem to define the operator for which the coherent state is an eigenstate.

(Related: 148200 and 109343. The answer in the first link doesn't really answer the question "what is $|\phi\rangle$?" and the second link mentions coherent states only, which as I mentioned are not orthogonal and therefore cannot be the states used in the derivation by Peskin & Schroeder)

EDIT: As @Mane.andrea suggested in the comments, I checked out section 4.1 of Condensed Matter Field Theory by Altland & Simons. It seems that they define the coherent state as the eigenstate of the annihilation operators $\hat{a}_i$ specifically, i.e. the positive-frequency part of the field above. So the answer to my 2nd question seems to be Yes, coherent states are the eigenstates of a different "field operator".

Answer 1

I think I found a solution with the help of @CosmasZachos's links. The state given by OP here didn't seem to match exactly my definition of the field operator. My guess is because it comes from the Schrödinger wavefunctional formalism. I'm not totally sure about my answer and I haven't found anything like it on the web so if anyone could double-check this, I'd appreciate it.

Let's define our operators carefully:

$$\hat{\phi}(x) = \hat{\phi}_+(x) + \hat{\phi}_-(x),$$

where $$ \hat{\phi}_+(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_p}}\hat{a}_p e^{-i p\cdot x};\quad \hat{\phi}_-(x) = \left(\hat{\phi}_+(x)\right)^\dagger.$$

It is useful to introduce the momentum:

$$\hat{\pi}(x) = \frac{\partial\hat{\phi}(x)}{\partial t } = \hat{\pi}_+(x)+\hat{\pi}_-(x),$$

where $$ \hat{\pi}_+(x) = -i \int \frac{d^3p}{(2\pi)^3} \sqrt{\frac{E_p}{2}}\hat{a}_p e^{-i p\cdot x};\quad \hat{\pi}_-(x) = \left(\hat{\pi}_+(x)\right)^\dagger.$$

The equal-time canonical commutation relations are:

$$[\hat{\phi}(\vec{x}),\hat{\pi}(\vec{y})] = i \delta^{(3)}(\vec{x}-\vec{y})\qquad [\hat{a}_\vec{p},\hat{a}_\vec{q}^\dagger] = (2\pi)^3 \delta^{(3)}(\vec{p}-\vec{q}).$$

From these we can find: $$ [\hat{\phi}(x)_+,\hat{\pi}_-(y)] = \frac{i}{2}\delta^{(3)}(\vec{x}-\vec{y}); \qquad [\hat{\phi}_+(\vec{x}),\hat{\phi}_-(\vec{y})] = \int \frac{d^3p}{(2\pi)^3}\frac{1}{2E_p}e^{i\vec{p}\cdot(\vec{x}-\vec{y})}.$$

Note that the last relation is not a Dirac delta function (it is in fact the propagator of the field), which is precisely why the state given by OP in the link above is not an eigenstate of the field operator defined here.

In fact, through trials and errors, I think I've found a state $|\phi\rangle$ that satisfies $\hat{\phi}(x)|\phi\rangle = \phi(x)|\phi\rangle$ (again, feel free to double-check):

$$|\phi\rangle \equiv \mathcal{N} \exp\left\{i\int d^3y\ \left(\hat{\phi}_-(y) - 2 \phi(y)\right)\hat{\pi}_-(y)\right\}|0\rangle,$$

where $\mathcal{N}$ is a normalization factor (which I haven't computed). I'm not even sure if those states are orthogonal, but at least I know what an eigenstate of the field operator might look like.

Also to answer my 2nd question, I think coherent states are defined as eigenstates of the positive-frequency field operator $\hat{\phi}_+(x)$ only (the part containing the annihilation operators). One should be careful not to confuse the two operators. Also, through my researches, I noticed that some references only deal with coherent states or field eigenstate in a non-relativistic theory or the Schrödinger wavefunctional formalism, where the definitions might be different from the convention of Peskin & Schroeder.