Consider the groundstate of a one-dimensional, non-interacting system of spinless fermions. Let ##a^†(x)## and ##a(x)## be the creation and annihilation operators for a fermion at the point ##x##, so that the density operator is ##n(x) = a^†(x)a(x)##. Show that the density-density correlation function has the form $$\langle n(x)n(0) \rangle=\langle n \rangle^2(1-\sin^2(k_F x)/(k_F x)^2) + \langle n \rangle\delta(x)$$where <n> is the mean density and kF is the Fermi wavevector.
Fourier transform of creation operator: a†(x) = 1/(2π) ∫dk eikx ak†
Delta function: 1/(2π) ∫dk eikx = δ(x)
The Attempt at a Solution
<n(x)n(0)> = <a†(x)a(x)a†(0)a(0)> = (1/2π)4∫∫∫∫ dk dq dl dm eix(k-q) <ak†aqal†am>
= (1/2π)4∫∫∫∫ dk dq dl dm eix(k-q)( <ak†aq> <al†am> + <ak†am> <aqal†>)
I know that the factor of <n>2 could come from say <ak†ak> , but I don't know how to get such terms when i've got a total of four different labels (k, q, l, m). I also can't see where a sin2 term is going to come from.