Correlation functions: second quantization form

Quantum description of identical particles can be conveniently done in terms of the creation and annihilation field operators. The operator $\hat\Psi^\dagger(\r)$ puts a particle into a point $\r$, while $\hat\Psi(\r)$ destroys a particle at the same point. Field operators can be conveniently presented in terms of creation $\hat a_k$ (annihilation $\hat a^\dagger$) operator that puts (destroys) a particle in a single particle orbital $\varphi_\k(\r)$:

$$\left\{ \begin{array}{lll} \hat\Psi^\dagger(\r) &=& \sum\limits_\... ...\ \hat\Psi(\r) &=& \sum\limits_\k\varphi_\k(\r)\,\hat a_\k\end{array}, \right.$$ (1.1)

In a uniform gas occupying a volume $V$ single particle orbitals $\varphi_\k(\r)$ are plain waves $\varphi_\k(\r) = \frac{1}{\sqrt{V}}\,e^{i\k\r}$. In a system of bosons operators (1.1) commute $[\Psi(\r),\Psi^\dagger(\r')]=\delta(\r-\r')$, $[\Psi(\r),\Psi(\r')]=0$ and anticommute in a system of fermions.

Before giving the definition of the correlation functions in terms of the field operators (1.1), let us discuss how the correlation functions come naturally from the calculation of the mean values of operators. We shall start with a very general form of a Hamiltonian consisting of one- and two- body operators

$$\hat H = \hat F^{(1)}+\hat F^{(2)},$$ (1.2)

where the one-body operator $\hat F^{(1)}$ is a sum of operators $\hat f^{(1)}_i$ each acting only on one particle:

$$\hat F^{(1)} = \sum\limits_{i=1}^N \hat f^{(1)}({\vec r}_i)$$ (1.3)

$$\hat F^{(2)} = \frac{1}{2}\sum\limits_{i\ne j}^N \hat f^{(2)}({\vec r}_i,{\vec r}_j)$$ (1.4)

For example, it can be an external potential $f^{(1)}(\r) = V_{ext}(\r)$ (and, thus, the operator is diagonal the coordinate representation) or it can be the kinetic energy $f^{(1)}(p)=p^2/2m$ (the operator is diagonal in the momentum representation). A commonly used two-body operator is a particle-particle interaction is usually defined in the coordinate representation $f^{(2)}({\vec r_1},{\vec r_2}) = V_{int}({\vec r_1},{\vec r_2})$.

In the second quantization representation the one-body $\hat F^{(1)}$ and two-body $\hat F^{(2)}$ operators are conveniently expressed in terms of the field operators (1.1).

$$\hat F^{(1)} = \int\!\!\!\!\int \hat\Psi^\dagger(\r)f^{(1)}({\vec r},\r')\hat\Psi(\r')\,{d\vec r}{d\vec r}'$$ (1.5)

$$\hat F^{(2)} = \frac{1}{2}\int\!\!\!\!\int \hat\Psi^\dagger(\r)\hat\Psi^\dagger(\r') f^{(2)}({\vec r},\r')\hat\Psi(\r')\hat\Psi(\r)\,{d\vec r}{d\vec r}'$$ (1.6)

Here we assume that the one-body operator can be either local $\langle\r\vert f^{(1)}\vert\r'\rangle = f^{(1)}(\r)\delta(\r-\r')$ (like in the case of an external field), either non local (like in the case of the kinetic energy), so, in general, we have two arguments $f^{(1)}=f^{(1)}({\vec r},\r')$. Instead, for the two-body term we always assume that it is local (i.e. it has a form similar to the particle-particle interaction energy) $\langle{\vec r_1},{\vec r_2}\vert f^{(1)}\vert{\vec r_1}',{\vec r_2}'\rangle = ... ...c r_1},{\vec r_2}) \delta({\vec r_1}-{\vec r_1}')\delta({\vec r_2}-{\vec r_2}')$, so in (1.6) we have only two arguments instead of four.

The quantum averages of $\hat F^{(1)}$ and $\hat F^{(2)}$ can be extracted from $\hat f^{(1)}$ and $\hat f^{(2)}$ if the correlation functions are known^1.1:

$$\langle\hat F^{(1)}\rangle = \int\!\!\!\!\int \hat f^{(1)}({\vec r},\r')G_1({\vec r},\r')\,{d\vec r}{d\vec r}'$$ (1.7)

$$\langle\hat F^{(2)}\rangle = \frac{1}{2}\int\!\!\!\!\int \hat f^{(2)}({\vec r},\r')G_2({\vec r},\r')\,{d\vec r}{d\vec r}'$$ (1.8)

Here $G_1({\vec r},\r')$ and $G_2({\vec r},\r')$ are the non normalized correlation functions defined as

$$G_1({\vec r},\r') = \langle\hat\Psi^\dagger(\r)\hat\Psi(\r')\rangle$$ (1.9)

$$G_2({\vec r},\r') = \langle\hat\Psi^\dagger(\r)\hat\Psi^\dagger(\r')\hat\Psi(\r')\hat\Psi(\r)\rangle$$ (1.10)

The function $G_1({\vec r},\r')$ characterizes correlations existing between values of the field in two different points $\r$ and $\r'$. The total phase does not enter in the definition, but instead the relative phase between two points is important. The diagonal term $\r=\r'$ of (1.9) gives the density of the system $n(\r)=\langle\hat\Psi^\dagger(\r)\hat\Psi(\r)\rangle=G_1({\vec r},\r)$, so the trace of the matrix $G_1$ gives the total number of particles $\mathop{\rm tr}\nolimits G_1 = \int G_1({\vec r},\r)\,{d\vec r} = N$. The function $G_2({\vec r},\r')$ characterizes the density correlations between points $\r$ and $\r'$.

It is convenient to introduce dimensionless versions of functions (1.9) and (1.10):

$$g_1({\vec r},\r') = \frac{G_1({\vec r},\r')}{\sqrt{G_1({\vec r},\r)}\sqrt{G_1(\r',\r')}}$$ (1.11)

$$g_2({\vec r},\r') = \frac{G_2({\vec r},\r')}{G_1({\vec r},\r)G_1(\r',\r')}$$ (1.12)

The function (1.11) is limited to the range $[0,1]$ and can be understood as the probability to destroy a particle at ${\vec r_1}$ and put it at ${\vec r_2}$. It is always possible to put a particle to the place where it was, so $g_1({\vec r},\r)=1$. The non-diagonal long range asymptotic vanishes in trapped systems and also in homogeneous systems in the absence of Bose-Einstein condensation $g_1({\vec r},\r')\to 0, \vert{\vec r},\r'\vert\to\infty$.

A more detailed introduction to the analytic properties of the correlation functions can be found, for example, in [Gla63,NG99,GS03a]