Basic Definitions and Evolution Equation

We start from the Hamiltonian (1.1) of a spatially homogeneous system of particles with binary interactions. Here, $e_k=\hbar \omega _k$ is the energy level of momentum state ${\bf k}$ (for example, in semiconductors, the parabolic band approximation is given by $\omega _k =\alpha {\bf k}^2$) where ${\bf k}$ is a d-dimensional wave vector, and $a_k ^\dagger , a_k$ are fermionic creation/annihilation operators fulfilling the anticommutation relations,

$$a_i a_j^\dagger + a_j^\dagger a_i = \delta _{ij} a_i a_j + a_j a_i =0.$$ (2.1)

We include the size of the system in the definition of the interaction matrix element $T_{12,34}$. We introduce for convenience the short hand notation: ${\bf k_1}\equiv 1$, $\frac{V}{(2\pi)^d} d {\bf k_1} \equiv d1$, and $\frac{(2\pi)^d}{V} \delta ({\bf k_1}+{\bf k_2}-{\bf k_3}- {\bf k_4}) \equiv \delta ^{12}_{34}$, $\hbar=1$. The Hamiltonian now reads

$$H= \int d1 \omega _1 a_1^\dagger a_1 + \frac{1}{2} \int d1234... ...12,34}a^\dagger_{1} a^\dagger_{2} a_{3} a_{4} \delta^{12}_{34}.$$

If one interchanges the indices $1$ and $2$ or $3$ and $4$ in the above expression and uses the fact that the Hamiltonian is Hermitian, the following properties hold:

$$T_{12,34} =-T_{21,34}=T_{21,43}=T^*_{43,21}.$$

In the Heisenberg picture, the equations of motion are

$$\dot a_k = i[H,a]_-$$ (2.2)

which give

$$\dot a_k^\dagger = i \omega _k a_k^\dagger + i \int T_{32,10} a^\dagger_3 a^\dagger_2 a_1 \delta_{32}^{10} d123$$ (2.3)

and

$$\dot a_k = -i \omega _k a_k - i \int T_{01,23} a^\dagger_1 a_2 a_3 \delta_{32}^{10} d123.$$ (2.4)

From the Heisenberg equations of motion, one can now derive the BBGKY hierarchy of equations for the normal ordered expectations values. The first three are,

$$\frac{d}{d t} \rho _k = 2 \rm Im\int T_{01,23} N_{0123}\delta _{32}^{10}d123,$$ (2.5)

$$\frac{d}{d t} N_{1' 2' 3' 4'} = i \Delta^{1'2'}_{3'4'} N_{1'2'3'4'}+i \int d123 \left( T_{3 2, 1 ... ...2 3' 4'} \delta ^{1}_{ 2'} - N_{3 2 2' 1 3' 4'})\delta ^{3 2}_{ 1 1'} + \right.$$

$$T_{3 2, 1 2'}N_{1' 3 2 1 3' 4'}\delta ^{3 2}_{ 1 2'}-$$

$$\left. T_{3' 1, 2 3} N_{1' 2' 1 2 3 4'}\delta ^{3 2}_{ 1 3'} - T_... ...N_{1' 2' 2 3}\delta ^{3'}_{ 1} - N_{1' 2' 1 3'23})\delta ^{3 2}_{ 1 4'}\right),$$

$${\bf k_1'}+{\bf k_2'}-{\bf k_3'}- {\bf k_4'}=0,$$ (2.6)

$$\frac{d}{d t}N_{1'2'3'4'5'6'} = i \Delta^{1'2'3'}_{4'5'6'}N_{1'2'3'4'5'6'}\cr$$

$$\ {\bf k_1'}+{\bf k_2'}+{\bf k_3'}- {\bf k_4'}-{\bf k_5'}+{\bf k_6'}=0,$$ (2.7)

where

$$\Delta^{12}_{34} \equiv (\omega _{k_1}+\omega _{k_2}-\omega _{k_3}-\omega _{k_4}),$$

$$\Delta^{123}_{456} \equiv (\omega _{k_1}+\omega _{k_2}+\omega _{k_3}-\omega _{k_4}- \omega _{k_5}-\omega _{k_6}).$$ (2.8)

Here, the expectation value is taken with respect to an arbitrary initial state $\Phi$, i.e.,

$$\rho (k_1)\delta ^{1}_{2} = <\Phi\vert a_1^\dagger a_2\vert\Phi>\cr N_{1234}$$ (2.9)

In the definitions of these $2 m (m=1,2,...)$ order expectation values, the first $m$ indices correspond to creation operators and the last $m$ indices correspond to annihilation operators. The number of creation and number of annihilation operators are equal to each other because the Hamiltonian (1.1) conserves number of particles. The fact that the right hand sides of (2.9) are zero on ${\bf k_1}+{\bf k_2}-{\bf k_3}-{\bf k_4}=0$, ${\bf k_1}+{\bf k_2}+{\bf k_3}-{\bf k_4}-{\bf k_5}-{\bf k_6}=0$ respectively is a direct consequence of the spatial homogeneity of the system. This means that the $N^{th}$ order moment of the spatially dependent field operators $\psi^\dagger({\bf x_j}), \psi({\bf x_j})$, the generalized Fourier transforms of the creation and annihilation operators $a^\dagger({\bf k_j}), a({\bf k_j})$, depends only on the relative spacing; i.e. on the differences of the coordinates ${\bf x_j},\ j=1..N$.