Cumulants and their evolution.

Let us define the cumulant of the $N$ product of spatially dependent field operators $\psi({\bf x_j}), j=1,2...N$ to be the moment of order $N$ from which the appropriate combinations of products of lower order moments are subtracted so that the resulting expression has the property that it decays to zero as the separations ${\bf x_j}-{\bf x_i}$ become large. The Fourier transforms of these cumulants are therefore well defined ordinary (as opposed to generalized) functions and these are the objects $\rho (k), {{\cal{P}}}_{1234}, ...$ with which we deal. Moreover, if the statistics are exactly Gaussian (namely, Hartree-Fock like), then all cumulants of order $N$, $N>2$, are zero. Because these weakly interacting fermionic systems relax to a near Gaussian state, the cumulants are the most convenient dependent variables.

For the two and three particle functions we define the fourth and sixth order cumulants ${{{\cal{P}}}_{1234}}$ and ${{{\cal Q}}}_{123456}$ respectively by

$$N_{1234}= \rho _1 \rho _2 (\delta^{2}_{3}\delta^{1}_{4}- \delta^{2}_{4}\delta^{1}_{3})+{{\cal{P}}}_{1234}\delta ^{12}_{34}$$ (2.10)

and

$$N_{123456}= \rho _1 \rho _2 \rho _3 \cdot \left( \right. \delta ^... ...\delta ^{1}_{5}\delta ^{2}_{4}- \delta ^{1}_{4}\delta ^{2}_{5}) \left. \right)+$$

$$\rho _3[+{{\cal{P}}}_{1256}\delta ^{3}_{4}\delta ^{12}_{56}-{{\ca... ...}\delta ^{1}_{6}\delta ^{23}_{45}]\cr+ {{{\cal Q}}}_{123456}\delta ^{123}_{456}$$

The expressions are analogues to what one would obtain in the classical case. There, the symbols $a^\dagger$ and $a$ are complex numbers (or c-numbers, as opposed to operators in the quantum case) and one defines the cumulants by expanding the fourth order moments into all possible decompositions; namely

$$\left<a_1^*a_2^* a_3 a_4\right>=\{a_1^*a_3\}\{a_2^*a_4\}\delt... ...\}\delta ^1_4\delta ^2_3+ \{a_1^*a_2^*a_3a_4\}\delta ^{12}_{34}$$ (2.11)

where angle brackets denote moments and curly brackets denote the corresponding cumulants. In the classical case, it is consistent, but not necessary (one can keep the other terms and discover they play no essential role), to set all correlations such as $\{a^*_1a^*_2\}$ and $\{a_3a_4\}$ equal to zero. In the quantum case, one also decomposes $<\Phi\vert a_1^\dagger a_2^\dagger a_3 a_4\vert\Phi>$ into products of all possible decompositions. Again, it is consistent, but not necessary, to set terms such as $\{a^\dagger_1a^\dagger_2\}$ and $\{a_3a_4\}$ equal to zero. The resulting decomposition should be consistent with the anticommutation relations (2.1), from which follows that $N_{1234}=-N_{2134}$. Therefore, certain terms (for example, $\rho _1 \rho _2\delta^{2}_{4}\delta^{1}_{3}$) are negative in (2.10-2.11).

A general algorithm for the decomposition of the $N^{\rm th}$ order expectation value is given in the Appendix.

Having defined the higher order cumulants, we can now write down the evolution equations for the cumulant hierarchy. For the purpose of deriving the QKE, it is sufficient to consider only the equations for $\rho _k$ and ${{\cal{P}}}_{1234}$. To obtain the frequency corrections to order $\epsilon ^2$ ( $\epsilon , 0<\epsilon \ll 1$, is a measure of the strength of the coupling coefficient), we need to consider contributions coming from the equation for ${{{\cal Q}}}_{123456}$. In carrying out the analysis on $\rho _k$, ${{\cal{P}}}_{1234}$ and ${{{\cal Q}}}_{123456}$, one finds, just as in the classical case, that certain patterns emerge which allow one to identify the terms in the equations for the cumulants of arbitrary high order that gives rise to long time effects. Taking account of these terms gives the expansions (2.16,2.17) which will be discussed in the next section. In this section we only write down equations for $\rho _k$ and ${{\cal{P}}}_{1234}$. They are

$$\frac{d}{dt} \rho _{k_0}= 2 \rm Im\int T_{01,23}{{\cal{P}}}_{0123}\delta ^{01}_{23}d123 ,$$ (2.12)

where ${\rm Im}$ denotes the imaginary part and the symbol zero denotes ${\bf k}$, and

$$\frac{d}{dt}{{\cal{P}}}_{1'2'3'4'} = i \tilde\Delta^{1'2'}_{3'4'} {{\cal{P}}}_{1'2'3'4'}- (\dot \rho _... ...'}) (\delta ^{2'}_{3'}\delta ^{1'}_{4'}-\delta ^{1'}_{3'}\delta ^{2'}_{4'}) \cr$$

on ${\bf k_{1'}}+{\bf k_{2'}}={\bf k_{3'}}+{\bf k_{4'}}$, and the Hartree-Fock self-energy is

$$\tilde \Delta ^{1'2'}_{3'4'}=\omega _{1'}+\omega _{2'}-\omega... ...t((T_{1'1,11'}+ T_{2'1,12'}-T_{3'1,13'}-T_{4'1,14'})n_1\right).$$

At this stage, it is worthwhile pointing out precisely those terms, underlined in (2.14) that give rise to the various long term effects:

1. The terms that gives rise to particle number transfer in the QKE are
   
   $${2 i T_{4'3',2'1'} \rho _{3'} \rho _{4'}(1- \rho _{2'}- \rho ... ..._{4'3',2'1'} \rho _{1'} \rho _{2'}( \rho _{3'}+ \rho _{4'}-1)}.$$
   
   
   The reason is that when one solves for ${{\cal{P}}}_{1234}$, one obtains an expression which contains this term multiplied by
   
   $$A_t(\Delta ^{1'2'}_{3'4'})=\int\limits_0^t d \tau \exp[i \Del... ... \Delta ^{1'2'}_{3'4'})^{-1}(\exp{(\Delta ^{1'2'}_{3'4'}t)}-1).$$
   
   
   In the long time limit,
   
   $$\lim_{t\to\infty} A_t(x)=\pi {\rm {sgn}}t \cdot \delta (x)+i P(\frac{1}{x}).$$
   
   
   Under the operator $\rm Im$ in (2.13), the delta-function is counted twice, and the principal value term cancels. The observant reader will notice that the QKE can be effectively derived by simply ignoring all terms in the equation for ${{\cal{P}}}_{1'2'3'4'}$ (except $i\Delta ^{1'2'}_{3'4'}{{\cal{P}}}_{1'2'3'4'}$) proportional to cumulants of order greater than two. In the literature, this is called the Hartree-Fock approximation. What we show in this paper is that, for the magnitude $\epsilon$ of the coupling coefficient uniformly (in ${\bf k}$) small, the Hartree-Fock approximation is indeed self consistent when one takes proper account of the frequency renormalization to order $\epsilon ^2$.
2. The order $\epsilon$ renormalizations to the frequency comes from the decomposition of sixth order moments such as $N_{1'2'13'23}\delta _{11'}^{23}$ in (2.7). These give rise to terms in the equation for ${{\cal{P}}}_{1'2'3'4'}$ which are proportional to ${{\cal{P}}}_{1'2'3'4'}$ itself. Indeed one obtains one such contribution from each of the sixth order moments in (2.7) leading to an expressions in the equation for ${{\cal{P}}}_{1'2'3'4'}$ equal to
   
   $$2 i \int d1 (T_{1'1,11'}+T_{2'1,12'}-T_{3'1,13'}-T_{4'1,14'}) \rho _1.$$
   
   
   When added to the frequency factor $i(\omega _{1'}+\omega _{2'}-\omega _3-\omega _4)$, we obtain the term denoted by $\tilde \Delta _{3'4'}^{1'2'}$ in (2.14). It is not too difficult to see that, in the equation for every cumulant $Q_N$, there is a term proportional to
   
   $$i Q_N \sum\limits_{i=1}^N(\omega _j+2\int T_{j1,1j} \rho _1 d 1)$$
   
   
   and this gives rise to the first contribution in the renormalization of the frequency (2.22). In the literature, $\tilde \Delta _{3'4'}^{1'2'}$ is called Hartree-Fock self energy for fourth order averages.
3. The order $\epsilon ^2$ terms in the renormalization to the frequency arise from the terms
   

   $$i\int \left( T_{32,11'}Q_{322'13'4'}\delta ^{32}_{11'}+ T_{32,12'}Q_{1'3213'4'}\delta ^{32}_{12'}\right.$$

   $$\left. + T_{3',23}{{{\cal Q}}}_{1'2'1234'}+ T_{3'1,23}Q_{1'2'13'23}\delta ^{32}_{13'}\right) d 123$$

   
   
   containing the sixth order cumulants in (2.14). The equations for the sixth order cumulant contains, in addition to terms proportional to a product of lower order particle number densities, terms proportional to ${{\cal{P}}}_{1'2'3'4'}$ with a factor containing $\rho _k(1- \rho _k)$.
4. All other terms are integrals which contain highly oscillatory factors which, because of the Riemann-Lebesgue lemma, contribute nothing in the long time limit.