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## Homework Statement

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 k

_{F}is the Fermi wavevector.

## Homework Equations

Fourier transform of creation operator: a

^{†}(x) = 1/(2π) ∫dk e

^{ikx}a

_{k}

^{†}

Delta function: 1/(2π) ∫dk e

^{ikx}= 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 e

^{ix(k-q) }<a

_{k}

^{†}a

_{q}a

_{l}

^{†}a

_{m}>

= (1/2π)

^{4}∫∫∫∫ dk dq dl dm e

^{ix(k-q)}( <a

_{k}

^{†}a

_{q}> <a

_{l}

^{†}a

_{m}> + <a

_{k}

^{†}a

_{m}> <a

_{q}a

_{l}

^{†}>)

I know that the factor of <n>

^{2}could come from say <a

_{k}

^{†}a

_{k}> , 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 sin

^{2}term is going to come from.