Equations of Motion and Further Developments

Abstract

We used EOM several times (see Sections 4.3,4.4,5.1.2); now we extend the approach used in Equation (4.39) for the free propagator to interacting problems. Using the many-body Hamiltonian (1.63) one readily obtains

$$ \begin{gathered} [\psi _\alpha (x),H]\_ = h_0 (x)\psi _\alpha (x) + \hfill \\ \sum\limits_{\beta '\gamma } \smallint dy\psi _\gamma ^ + (y) v (x,y)_{\alpha \beta ',\gamma \gamma '} \psi _{\gamma '} (y)\psi _{\beta '} (x), \hfill \\ \end{gathered} $$

((10.1))

with all the operators in the Heisenberg representation (h₀ is a first-quantized one-body operator), with the notation x = (x, t_x). We multiply on the left by ψ ^†_Ρ (z) and perform an interacting ground state average:

$$ \begin{gathered} \left[ {i\frac{\partial } {{\partial t}} - h_0 (x)} \right]\left\langle {\psi _\gamma ^ + (z)\psi _\alpha (x)} \right\rangle = \hfill \\ \sum\limits_{\beta '\gamma } \smallint dy w (x,y)_{\alpha \beta ',\gamma \gamma '} \left\langle {\psi _\alpha ^\dag (z)\psi _\gamma ^\dag (y)\psi _{\gamma '} (y)\psi _{\beta '} (x)} \right\rangle ,t_z > t_x . \hfill \\ \end{gathered} $$

((10.2))

Some authors use different orderings of the arguments.