Wick’s Theorem

Wick’s theorem is a way to reduce a time-ordered operator into the sum of normal-ordered operator and various contractions. Then, where is the time-ordered operator from? Why we need this reduction?

1. Where is the time-ordered operator from

Usually, it’s from perturbation expansion of Green’s function. The Green’s function is defined as

(1)   \begin{equation*} iG_{\alpha\beta}(\mathbf{x}t, \mathbf{x'}t') =\frac{\langle \Psi_0| T[\hat{\psi}_{H\alpha}(\mathbf{x}t)\hat{\psi}^{\dagger}_{H\alpha}(\mathbf{x}t)]|\Psi_0\rangle} {\langle \Psi_0|\Psi_0\rangle} \end{equation*}

where |\Psi_0\rangle is the Heisenberg ground state of the interacting system satisfying

(2)   \begin{equation*} H|\Psi_0\rangle=E|\Psi_0\rangle \end{equation*}

E depends on time if the Hamiltonian vary with time. And \hat{\psi}_{H\alpha}(\mathbf{x}t) is a field operator in Heisenberg picture

(3)   \begin{equation*} \hat{\psi}_{H\alpha}(\mathbf{x}t) =e^{iHt/\hbar}\hat{\psi}_{\alpha}(\mathbf{x})e^{-iHt/\hbar} \end{equation*}

Let’s assume

(4)   \begin{equation*} H=H_0+e^{-\epsilon|t|}H_I(t) \end{equation*}

With the help of Gell-Mann and Low theorem, it can be proved that

(5)   \begin{align*} \frac{\langle \Psi_0| O_H(t)|\Psi_0\rangle} {\langle \Psi_0|\Psi_0\rangle} \nonumber =&\frac{1}{\langle \Phi_0|S|\Phi_0\rangle} \langle \Phi_0|\sum_{0}^{\infty}\left(\frac{-i}{\hbar}\right)^{\nu}\frac{1}{\nu!}\int_{-\infty}^{\infty}dt_1\cdots\int_{-\infty}^{\infty}dt_{\nu}|\Phi_0\rangle\\ &\times e^{-\epsilon(|t_1|+\cdots+|t_{\nu}|)}T[H_I(t_1)\cdots H_I(t_{\nu})O_I(t)]|\Phi_0\rangle \end{align*}

where |\Phi_0\rangle is the ground state of H_0, S is the evolution operator

(6)   \begin{equation*} S=U_{\epsilon}(\infty,-\infty) \end{equation*}

What’s more, we can prove

(7)   \begin{align*} \frac{\langle \Psi_0| O_H(t)P_H(t')|\Psi_0\rangle} {\langle \Psi_0|\Psi_0\rangle} \nonumber =&\frac{1}{\langle \Phi_0|S|\Phi_0\rangle} \langle \Phi_0|\sum_{0}^{\infty}\left(\frac{-i}{\hbar}\right)^{\nu}\frac{1}{\nu!}\int_{-\infty}^{\infty}dt_1\cdots\int_{-\infty}^{\infty}dt_{\nu}|\Phi_0\rangle\\ &\times e^{-\epsilon(|t_1|+\cdots+|t_{\nu}|)}T[H_I(t_1)\cdots H_I(t_{\nu})O_I(t)P_I(t')]|\Phi_0\rangle \end{align*}

Let \epsilon \to 0, these results are useful for general H with interaction. As for the Green’s functions, it is

(8)   \begin{align*}  iG_{\alpha\beta}(x,y) = \frac{1}{\langle \Phi_0|S|\Phi_0\rangle} \sum_{0}^{\infty}\left(\frac{-i}{\hbar}\right)^{\nu}\frac{1}{\nu!}\int_{-\infty}^{\infty}dt_1\cdots\int_{-\infty}^{\infty}dt_{\nu}| \langle \Phi_0|T[H_I(t_1)\cdots H_I(t_{\nu})\hat{\psi}_{\alpha}(x)\hat{\psi}^{\dagger}_{\beta}(y)]|\Phi_0\rangle \end{align*}

where the notation x=(\mathbf{x},t_x) has been introduced´╝î and \hat{\psi}_{\alpha}(x) is the field operator in interaction picture.
Rewrite V(\mathbf{x, x'}) as

(9)   \begin{equation*} U(x,x')=V(\mathbf{x, x'})\delta(t-t') \end{equation*}

Take the interaction

(10)   \begin{equation*} H_I(t_i)=\frac{1}{2}\sum_{\lambda\lambda'\mu\mu'}\int d\mathbf{x}_id\mathbf{x'}_i \hat{\psi}^{\dagger}_{\lambda}(x_i) \hat{\psi}^{\dagger}_{\mu}(x'_i) U(x_i,x'_i)_{\lambda\lambda',\mu\mu'} \hat{\psi}_{\mu'}(x'_i) \hat{\psi}_{\lambda'}(x_i) \end{equation*}

into Eq.8. The numerator, which we will denote by i\tilde{G}, becomes

(11)   \begin{align*} i\tilde{G}(x,y) \nonumber =&iG^0_{\alpha\beta}(x,y) +\left(\frac{-i}{\hbar}\right)\frac{1}{2}\sum_{\lambda\lambda'\mu\mu'} \int d^4x_id^4x'_i U(x_1,x'_1)_{\lambda\lambda',\mu\mu'} \\ & \times \langle \Phi_0| T[ \hat{\psi}^{\dagger}_{\lambda}(x_1) \hat{\psi}^{\dagger}_{\mu}(x'_1) \hat{\psi}_{\mu'}(x'_1) \hat{\psi}_{\lambda'}(x_1) \hat{\psi}_{\alpha}(x) \hat{\psi}^{\dagger}_{\beta}(y) ]|\Phi_0\rangle + \cdots  \end{align*}

where iG^0_{\alpha\beta}(x,y)=\langle \Phi_0| T[\hat{\psi}_{H_0\alpha}(\mathbf{x}t)\hat{\psi}^{\dagger}_{H_0\alpha}(\mathbf{x}t)]|\Phi_0\rangle refers to the noninteracting system. The second terms refers to \nu=1 in Eq.8. Then our problem to calculate Green’s function becomes how to calculate

(12)   \begin{equation*} \langle \Phi_0| T[ \hat{\psi}^{\dagger}\cdots\hat{\psi} \hat{\psi}_{\alpha}(x) \hat{\psi}^{\dagger}_{\beta}(y) ]|\Phi_0\rangle \end{equation*}

The straight forward approach for this matrix element is very length. Instead, Wick’s Theorem provides a general procedure for evaluating it.

2. What is Wick’s Theorem

Wick’s theorem is the operator identity

(13)   \begin{equation*}  T[UVW\cdots XYZ] =N[UVW\cdots XYZ]+N[\text{sum over all possible pairs of contractions}] \end{equation*}

In order to understand it, we need to find out what is the normal ordering N[] and contractions.

2.1 Normal Ordering

This is an ordering of a product of field operators, which has vanishing ground state expectation value.

In most systems of interest, \hat{\psi}(x) can be uniquely separated into a positive part \hat{\psi}^{(+)}(x) and negative part \hat{\psi}^{(-)}(x)

(14)   \begin{equation*} \hat{\psi}(x) =\hat{\psi}^{(+)}(x)+\hat{\psi}^{(-)}(x) \end{equation*}

with the property

(15)   \begin{equation*} \hat{\psi}^{(+)}(x) |\Phi_0 \rangle =0 \end{equation*}

Correspindingly, the adjoint operator becomes

(16)   \begin{equation*} \hat{\psi}^{\dagger}(x) =\hat{\psi}^{(+)\dagger}(x)+\hat{\psi}^{(-)\dagger}(x) \end{equation*}

(17)   \begin{equation*} \hat{\psi}^{(-)\dagger}(x) |\Phi_0 \rangle =0 \end{equation*}

Take the free fermions as example. After canonical transformation

(18)   \begin{equation*} c_{\mathbf{k}\lambda} =\left\{ \begin{array}{lll} a_{\mathbf{k}\lambda} & k>k_F & particles \\b^{\dagger}_{-\mathbf{k}\lambda} & k<k_F & holes \end{array} \right. \end{equation*}

the field operator in interacting picture can be written as

(19)   \begin{align*} \hat{\psi}(x) \nonumber &=e^{iH_0t}\hat{\psi}_S(\mathbf{x})e^{iH_0t}\\ \nonumber &=e^{iH_0t}\sum_{\mathbf{k}\lambda}\psi_{\mathbf{k}\lambda}(\mathbf{x})c_{\mathbf{k}\lambda}e^{iH_0t}\\ \nonumber &=e^{iH_0t} \left( \sum_{ k>k_F,\lambda}\psi_{\mathbf{k}\lambda}(\mathbf{x})a_{\mathbf{k}\lambda} +\sum_{ k<k_F,\lambda}\psi_{\mathbf{k}\lambda}(\mathbf{x})b^{\dagger}_{\mathbf{-k}\lambda} \right) e^{iH_0t}\\ \nonumber &=\sum_{ k>k_F,\lambda}\psi_{\mathbf{k}\lambda}(\mathbf{x})e^{-i\omega_kt}a_{\mathbf{k}\lambda}(t) +\sum_{ k<k_F,\lambda}\psi_{\mathbf{k}\lambda}(\mathbf{x})e^{-i\omega_kt}b^{\dagger}_{\mathbf{-k}\lambda}(t) \\ &\equiv \hat{\psi}^{(+)\dagger}(x)+\hat{\psi}^{(-)\dagger}(x) \end{align*}

The normal ordering place all the positive operator \hat{\psi}^{(+)} and \hat{\psi}^{(-)\dagger} to the right of all the negative operator \hat{\psi}^{(-)} and \hat{\psi}^{(+)\dagger}, again including a factor of -1 for every interchange of fermion operators. For example´╝î if we deal with fermion fields,

(20)   \begin{equation*} N[\hat{\psi}^{(+)}(x)\hat{\psi}^{(-)}(y)] =-\hat{\psi}^{(-)}(y)\hat{\psi}^{(+)}(x) \end{equation*}

(21)   \begin{equation*} N[\hat{\psi}^{(+)}(x)\hat{\psi}^{(+)\dagger}(y)] =-\hat{\psi}^{(+)\dagger}(y)\hat{\psi}^{(+)}(x) \end{equation*}

Clearly, the ground state expectation value of normal-ordered operator is 0.

2.2 Contractions

The contraction of two operators U and V is denoted as \contraction{}{U}{}{V} UV
and is equal to the difference between the T product and the N product

(22)   \begin{equation*} \contraction{}{U}{}{V} UV=T[UV]-N[UV] \end{equation*}

that is

(23)   \begin{equation*} T[UV]=N[UV]+\contraction{}{U}{}{V}UV \end{equation*}

this is the Wick’s theorem for two operators. As for more operators, we have Eq.13.
Notice that

(24)   \begin{equation*} \contraction{}{U}{}{V}UV =\mp \contraction{}{V}{}{U}VU \end{equation*}

Two factors that are contracted must be brought together by rearanging the order of the operators iwthin the normal product, always keeping the standard sign conventin for interchange of operators.

As for the field operators, most of the contraction is 0, only

(25)   \begin{equation*} \contraction{}{\hat{\psi}}{_{\alpha}(x)}{\hat{\psi}} \hat{\psi}_{\alpha}(x) \hat{\psi}^{\dagger}_{\beta}(y) =iG^0_{\alpha\beta}(x,y) \end{equation*}

For example,

(26)   \begin{align*} \begin{split} T[\hat{\psi}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2) \hat{\psi}_{\gamma}^{\dagger}(x_3) \hat{\psi}_{\theta}(x_4)] &=N[\hat{\psi}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2) \hat{\psi}_{\gamma}^{\dagger}(x_3) \hat{\psi}_{\theta}(x_4) +\contraction {}{\hat{\psi}}{{}_{\alpha}(x_1)} {\hat{\psi}} \hat{\psi}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2) \hat{\psi}_{\gamma}^{\dagger}(x_3) \hat{\psi}_{\theta}(x_4)\\ &\quad+\contraction {}{\hat{\psi}}{{}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2)} {\hat{\psi}} \hat{\psi}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2) \hat{\psi}_{\gamma}^{\dagger}(x_3) \hat{\psi}_{\theta}(x_4) +\contraction {}{\hat{\psi}}{{}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2) \hat{\psi}_{\gamma}^{\dagger}(x_3)} {\hat{\psi}} \hat{\psi}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2) \hat{\psi}_{\gamma}^{\dagger}(x_3) \hat{\psi}_{\theta}(x_4) \\ &\quad+\contraction {\hat{\psi}_{\alpha}(x_1)} {\hat{\psi}}{{}_{\beta}^{\dagger}(x_2)} {\hat{\psi}} \hat{\psi}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2) \hat{\psi}_{\gamma}^{\dagger}(x_3) \hat{\psi}_{\theta}(x_4) +\contraction {\hat{\psi}_{\alpha}(x_1)} {\hat{\psi}}{{}_{\beta}^{\dagger}(x_2) \hat{\psi}_{\gamma}^{\dagger}(x_3)} {\hat{\psi}} \hat{\psi}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2) \hat{\psi}_{\gamma}^{\dagger}(x_3) \hat{\psi}_{\theta}(x_4)\\ &\quad+\contraction {\hat{\psi}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2)} {\hat{\psi}}{{}_{\gamma}^{\dagger}(x_3)} {\hat{\psi}} \hat{\psi}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2) \hat{\psi}_{\gamma}^{\dagger}(x_3) \hat{\psi}_{\theta}(x_4) +\contraction {}{\hat{\psi}}{{}_{\alpha}(x_1)} {\hat{\psi}} \contraction {\hat{\psi}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2)} {\hat{\psi}}{{}_{\gamma}^{\dagger}(x_3)} {\hat{\psi}} \hat{\psi}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2) \hat{\psi}_{\gamma}^{\dagger}(x_3) \hat{\psi}_{\theta}(x_4)\\ &\quad+\contraction {}{\hat{\psi}}{{}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2)} {\hat{\psi}} \contraction[2ex] {\hat{\psi}_{\alpha}(x_1)} {\hat{\psi}}{{}_{\beta}^{\dagger}(x_2) \hat{\psi}_{\gamma}^{\dagger}(x_3)} {\hat{\psi}} \hat{\psi}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2) \hat{\psi}_{\gamma}^{\dagger}(x_3) \hat{\psi}_{\theta}(x_4) +\contraction[2ex] {}{\hat{\psi}}{{}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2) \hat{\psi}_{\gamma}^{\dagger}(x_3)} {\hat{\psi}} \contraction {\hat{\psi}_{\alpha}(x_1)} {\hat{\psi}}{{}_{\beta}^{\dagger}(x_2)} {\hat{\psi}} \hat{\psi}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2) \hat{\psi}_{\gamma}^{\dagger}(x_3) \hat{\psi}_{\theta}(x_4) ] \end{split} \end{align*}

As for its ground state expectation value, only the full connected contractions, the last three terms above, contribute non-zero value. What’s more, the last term is also zero. Thus

(27)   \begin{equation*} \langle \Phi_0 | T[\hat{\psi}_{\alpha}(x_1) \hat{\psi}_{\beta}^{\dagger}(x_2) \hat{\psi}_{\gamma}^{\dagger}(x_3) \hat{\psi}_{\theta}(x_4)] |\Phi_0\rangle =-iG^0_{\alpha\beta}(x_1,x_2)iG^0_{\theta\gamma}(x_4,x_3) +iG^0_{\alpha\gamma}(x_1,x_3)iG^0_{\theta\beta}(x_4,x_2) \end{equation*}

Now let’s look at the time-ordered operator in the first order term in Eq.11, it’s

(28)   \begin{align*} \begin{split} &\langle \Phi_0 | T[ \hat{\psi}^{\dagger}_{\lambda}(x_1) \hat{\psi}^{\dagger}_{\mu}(x'_1) \hat{\psi}_{\mu'}(x'_1) \hat{\psi}_{\lambda'}(x_1) \hat{\psi}_{\alpha}(x) \hat{\psi}^{\dagger}_{\beta}(y) ] |\Phi_0\rangle\\ &=iG^0_{\alpha\beta}(x,y) [iG^0_{\mu'\mu}(x_1',x_1')iG^0_{\lambda'\lambda}(x_1,x_1) -iG^0_{\mu'\lambda}(x_1',x_1)iG^0_{\lambda'\mu}(x_1,x_1')] \\ &\quad + iG^0_{\alpha\lambda}(x,x_1) [iG^0_{\lambda'\mu}(x_1,x_1')iG^0_{\mu'\beta}(x_1',y) -iG^0_{\lambda'\beta}(x_1,y)iG^0_{\mu'\mu}(x_1',x_1')] \\ &\quad + iG^0_{\alpha\mu}(x,x_1') [iG^0_{\mu'\lambda}(x_1',x_1)iG^0_{\lambda'\beta}(x_1,y) -iG^0_{\mu'\beta}(x_1',y)iG^0_{\lambda'\lambda}(x_1,x_1)] \end{split} \end{align*}

Reference

[1] Fetter, Alexander L., and John Dirk Walecka. Quantum theory of many-particle systems. Courier Corporation, 2012.

[2] Peskin, Michael Edward. An introduction to quantum field theory. Westview press, 1995.

Leave a Reply