# Mastubara transform conventions

1. Jul 9, 2011

### a2009

Hello,

When calculating the dynamic structure factor I need to express $$\rho(q)$$ in terms of creation and annihilation operators. I tried a direct calculation of the form

$$\rho_q = \int e^{i q x} \bar{\psi}(x) \psi(x)$$
$$= \int e^{iqx} \sum_{k k'} e^{ikx} \bar{\psi}_k e^{-ik'x} \psi_{k'}$$
$$= \sum_k \bar{\psi}_k \psi_{k+q}$$

however in all of the literature the result is stated

$$\rho_q = \sum_k \bar{\psi}_{k+q} \psi_k$$

I've thought of the following possible causes for this discrepancy:
1. I'm using the wrong sign convention for the transform: WRONG because even if I switch all of the signs of the exponents we get back the same result.
2. ONLY the sign of rho's exponent is wrong: possible, because it seems that it is arbitrary to decide that rho transforms like psi and not like psi bar.

This is where I'm at. I would really like to understand the source of the mistake.

Thanks for any help!!