Consider a free Bosonic system. The Hamiltonian is given by $$ H=\sum_k \frac{k^2}{2m}a_k^\dagger a_k. $$ Since the spectrum is gapless, the ground state can be of any particle number (or even superposition of different particle numbers) as long as all particles are in state $\vert k=0 \rangle$.

Now consider a particular bosonic condensed state with $N$ particles $\vert\Phi_0\rangle=\frac{(a_{k=0}^\dagger\ \ )^N}{\sqrt{N!}}\vert vac\rangle$. What is the time-ordered density correlation function defined as the following? $$ \langle\Phi_0\vert \hat{T}[\rho(x,t)\rho(0,0)]\vert\Phi_0\rangle $$ (This is Problem 3.1.2 in Wen's book)

I have done the problem by brute force (by which I mean I spanned all the states and operators and then evaluate them one by one.) My result is: $$ \begin{eqnarray} \langle\Phi_0\vert \hat{T}[\rho(x,t)\rho(0,0)]\vert\Phi_0\rangle&=&\theta(t)\langle \Phi_0\vert e^{iHt}a^\dagger(x)a(x)e^{-iHt}a^\dagger(0)a(0)\vert\Phi_0\rangle\\\ &+&\theta(-t)\langle \Phi_0\vert a^\dagger(0)a(0)e^{iHt}a^\dagger(x)a(x)e^{-iHt}\vert\Phi_0\rangle\\\ &=&\frac{N(N-1)}{V^2}+\frac{N}{V}\int\frac{\mathrm{d}k}{2\pi}[\theta(t)\exp(ikx-it\frac{k^2}{2m})+\theta(-t)\exp(-ikx+it\frac{k^2}{2m})], \end{eqnarray} $$ which indeed shows the off-diagonal long-range order of the system.

The calculation is a little bit painful. But as implied in the book, it should be obtained with Wick's theorem. However, I have some troubles:

1. The expectation value is evaluated in state $\Phi_0$ rather than vacuum. How do I use Wick's theorem in this case?
2. The book says "$a(x,t)$ is a linear combination of $a_k$ and $a_k^\dagger$" but I failed to prove it. Is it even right? I mean, using a Fourier transform $a(x)$ should only be involved with $a_k$ but no $a_k^\dagger$, and with the time-evolution operator $a(x,t)$ is more than just linear terms of $a_k$ and $a_k^\dagger$.

Consider a free Bosonic system. The Hamiltonian is given by $$ H=\sum_k \frac{k^2}{2m}a_k^\dagger a_k. $$ Since the spectrum is gapless, the ground state can be of any particle number (or even superposition of different particle numbers) as long as all particles are in state $\vert k=0 \rangle$.

Now consider a particular bosonic condensed state with $N$ particles $\vert\Phi_0\rangle=\frac{(a_{k=0}^\dagger\ \ )^N}{\sqrt{N!}}\vert vac\rangle$. What is the time-ordered density correlation function defined as the following? $$ \langle\Phi_0\vert \hat{T}[\rho(x,t)\rho(0,0)]\vert\Phi_0\rangle $$ (This is Problem 3.1.2 in Wen's book)

I have done the problem by brute force (by which I mean I spanned all the states and operators and then evaluate them one by one.) My result is: $$ \begin{eqnarray} \langle\Phi_0\vert \hat{T}[\rho(x,t)\rho(0,0)]\vert\Phi_0\rangle&=&\theta(t)\langle \Phi_0\vert e^{iHt}a^\dagger(x)a(x)e^{-iHt}a^\dagger(0)a(0)\vert\Phi_0\rangle\\\ &+&\theta(-t)\langle \Phi_0\vert a^\dagger(0)a(0)e^{iHt}a^\dagger(x)a(x)e^{-iHt}\vert\Phi_0\rangle\\\ &=&\frac{N(N-1)}{V^2}+\frac{N}{V}\int\frac{\mathrm{d}k}{2\pi}[\theta(t)\exp(ikx-it\frac{k^2}{2m})+\theta(-t)\exp(-ikx+it\frac{k^2}{2m})], \end{eqnarray} $$ which indeed shows the off-diagonal long-range order of the system.

The calculation is a little bit painful. But as implied in the book, it should be obtained with Wick's theorem. However, I have some troubles:

1. The expectation value is evaluated in state $\Phi_0$ rather than vacuum. How do I use Wick's theorem in this case?
2. The book says "$a(x,t)$ is a linear combination of $a_k$ and $a_k^\dagger$" but I failed to prove it. Is it even right? I mean, using a Fourier transformation, $a(x)$ should only be involved with $a_k$ but no $a_k^\dagger$, and with the time-evolution operator $a(x,t)$ is more than just linear terms of $a_k$ and $a_k^\dagger$.