A question on electronic field operators

Question

People define field operators as

$\bf{\Psi}(x)$ = $\sum_k$ $\phi_k(x)$ $c_k$
$\bf{\Psi}^{\dagger}(x)$ = $\sum_k$ $\phi^*_k(x)$ $c^{\dagger}_k$

where $\phi_k(x)$ is a single particle basis.

My question is: Are these operators parametrized by "x"? In the sense that for a fixed number $x_0 \in R^n$ $\rightarrow$ $\bf{\Psi}_{x_0}$ $\equiv$ $\bf{\Psi}(x_0)$ = $\sum_k$ $\phi_k(x_0)$ $a_k$. One analogy that comes to my mind is the one-parameter family of evolution operators, for fixed $t_0$ $\rightarrow$ $U_{t_0}$. I am asking this because when one computes the zero temperature Green's function for a fermionic gas one has the following derivation

And

Where $\hat{\psi}$(x)= $\frac{1}{\sqrt{V}}$ $\sum_k$ $e^{ikx}$ $c_k$
In Eq. 1.8 one can see $\frac{e^{i(kx-k'x')}}{V}$,where $e^{ikx}$ was pulled out of the inner product. This can happen only if $e^{ikx}$ is a complex number in the field operator $\hat{\psi(x)}$ and not a function of position,I guess.

Answer 1

Yes it is an operator on the hilbert space that is parameterized by $x$. In other words, each value of $x$ gives a different operator $\hat \psi(x): {\mathcal H}\to {\mathcal H}$. You can pull the $e^{ikx}$ out of the integral because for each chosen $x$ it is just a number --- an element of the field over which the Hilbert space is defined. Do not confuse the Hilbert space on which the $c$ and $c^\dagger$ operators act with the one we use in quantum mechanics. The inner product here does not involve integrals over $x$.

When we quantize a real scalar field $\varphi(x)$ and expand $$ \hat \varphi(x,t)= \sum_k \left\{a_k e^{i(kx-\omega_k t)}$ + a^\dagger_k e^{-i(x-\omega_kt)}\right\} $$ the $a_k$ and $a_k^\dagger$ are linear combinations of operators that act of fucntionals of $\varphi$. For example the "momentum" conjugate to $\varphi(x)$ is $$ \hat \Pi(x) = -i \hbar \frac{\delta}{\delta \varphi(x)} $$ (a functional derivative) so the inner product in this space is something like $$ <F|G>= \int d[\varphi] F^*(\varphi) G(\varphi) $$ You should think of $\varphi(x)$ as the displacement of a rubber sheet at position $x$ and the field expansion as one in the normal modes of the sheet that behave like harmonic oscillators. The Hilbert space in which the modes themselves are treated as elements of $L^2({\mathbb R}^N)$ is something entirely different --- you would have this normal mode space even for a classical syste, After you quantize your rubber sheet, you later discover that these classical normal modes can also be regarded as the wavefunctions of the resuting phonon particles, but it is still a different space than that acted on by the $a_k$ and $a^\dagger_k$