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We often use the Fock space as the start point for our quantum field theory. In the Fock space we have definite physical meanings for the state. For example, the state $$|k_1k_2...k_n\rangle$$ represents n particles with momentum $k_1$, $k_2$,...,$k_n$.

However if we just know the field eigenvalue, i.e. $$\hat{\phi}|\phi\rangle=\phi(x,t)$$, or $$\langle\phi|\hat{\phi}|\phi\rangle=\phi(t,x)$$ then what is the interpretation of $|\phi\rangle$ in Fock space?

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We often use the Fock space as the start point for our quantum field theory

that is not entirely true. The Fock space is well defined only for fields whose equation of motion are linear, since it emerges from the Fourier expansion of $\phi(x,t)$ as $\phi = A + A^{\dagger}$ which only holds the same at any time $t$ if the equations are as said. For general field theories there is no Fock space (e. g. the gravitational field or any interacting theory).

This said, let us assume there is such a Fock space nevertheless. Since by definition $$ \mathcal{F}= \oplus_{j}^{\infty}\,\mathcal{H}_j $$ where each $\mathcal{H}_j$ contains (so to speak) $j$ particles, any element in that space can be expanded upon a (infinitely countable) basis whose elements contain any number of particles, in turn. This is reminiscent of the fact that the field $$ \phi(x,t) = \int d^4 k\ \left(a(k)e^{ikx} + a^*(k)e^{-ikx}\right) $$ sums up any possible number of particles at a given time $t$ (creating and destroying particles, taking the terminology in quotation marks).

Reducing the above to the simple case $\phi = a + a^*$ it turns out that, given $|\psi\rangle \in \mathcal{H}_j$, then $$ \phi|\psi\rangle = (a + a^*)|\psi\rangle\in \mathcal{H}_{j+1} \oplus \mathcal{H}_{j-1} $$ is an element in the direct sum of different Hilbert spaces, each one of them containing $\pm 1$ particles.

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As for your question (what $\langle \text{vac} |\hat{\varphi}|\text{vac} \rangle = \varphi $ means), the answer is simple: $|\text{vac}\rangle $ is constructed from Fock basis states as the coherent state (here I've assumed that the volume is finite), $$ |\text{vac}\rangle \equiv e^{-\frac{\varphi}{2}}\sum_{N}\frac{\left(\varphi a^{\dagger}(\mathbf p)\right)^{N}}{N!}|0\rangle , $$ which is the eigenstate of both creation and annihilation operators.

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