Correlation Function

In both stage 2 and stage 3, we will encounter the following functional derivative

\[\tau(x_1,\ldots,x_n) := \left.\frac{1}{\mathrm{i}}\!\frac{\delta}{\delta J(x_1)}\cdots\frac{1}{\mathrm{i}}\!\frac{\delta}{\delta J(x_n)}W_0[J]\right|_{J=0}\]

Thus, we give this quantity a name correlation function, or \(n\)-point (Green’s) function. The origin of this name comes from the similarity of its formulation in canonical quantization and hence this name may look strange here.

It is not so straight-forward about the \(n\)-point function, and thus we hope to reduce the formula. To do this, let us first evaluate the 2-point function

\[\begin{split}\tau(x_1,x_2) &= \left.\frac{1}{\mathrm{i}}\!\frac{\delta}{\delta J(x_1)}\frac{1}{\mathrm{i}}\!\frac{\delta}{\delta J(x_2)}\exp\{-\frac{1}{2}\mathrm i\!\!\int\!\! J\Delta_FJ\}\right|_{J=0} \\ &= \left.\mathrm{i}\frac{\delta}{\delta J(x_1)}\int\!\!\Delta_F(x_2;)J\exp\{-\frac{1}{2}\mathrm i\!\!\int \!\!J\Delta_FJ\}\right|_{J=0}\\ &=\left.\mathrm i\Delta_F(x_1;x_2)W_0[J]\right|_{J=0} + \left.(\int\Delta_F J)^2W_0[J]\right|_{J=0} = \mathrm i\Delta_F(x_1;x_2)\end{split}\]

It is nothing other than \(\mathrm{i}\) times the ordinary Green’s function. Next, 3-point function

\[\tau(x_1,x_2,x_3)\sim \left.\Delta_F\int\Delta_FJ\right|_{J=0} + \left.(\int\Delta_FJ)^3\right|_{J=0} = 0\]

And 4-point function

\[\begin{split}\tau(x_1,x_2.x_3,x_4) &= \mathrm{i}\Delta_F(x_1;x_2)\mathrm{i}\Delta_F(x_3;x_4)\\ &+ \mathrm{i}\Delta_F(x_1;x_3)\mathrm{i}\Delta_F(x_2;x_4) \\ &+ \mathrm{i}\Delta_F(x_2;x_3)\mathrm{i}\Delta_F(x_1;x_4)\\ &= (1\sim2)(3\sim4) + (1\sim3)(2\sim4) + (2\sim3)(1\sim4)\end{split}\]

where we have used \((1\sim2)\) to represent \(\tau(x_1,x_2)\). Therefore, we can see the regularity — the \(n\)-point function is either \(0\) or some combination of two point functions. Specifically, we have by induction

  • The odd-point function is \(0\).
  • The even-point function is
\[\tau(x_1,\ldots, x_{2n}) = \sum_{\text{perms}}(p_1\sim p_2)\ldots(p_{2n-1}\sim p_{2n})\]

which means the summation over all the permutations \(p\). This is the so-called Wick’s Theorem — transforming the \(n\)-point function into the combination of 2-point function.

You are suggested to show yourself that the S-matrix can be written using correlation function as

\[S = \sum_{n=0}^\infty\frac{\mathrm i^n}{n!}\prod_x\varphi(x)\prod_x(\partial^2\!\!-\!m^2)\tau(x_1,\ldots,x_n)\]

where \(x\) takes over \(x_1,\ldots,x_n\). This formula is the so-called perturbative expansion of the S-matrix.