Editing Quantum field theory (section)

===Two-point correlation function===
{{Main|Correlation function (quantum field theory)}}

In calculations, one often encounters expression like<math display="block">\lang 0|T\{\phi(x)\phi(y)\}|0\rang
\quad \text{or} \quad
\lang \Omega |T\{\phi(x)\phi(y)\}|\Omega \rang</math>in the free or interacting theory, respectively. Here, <math>x</math> and <math>y</math> are position [[four-vector]]s, <math>T</math> is the [[time ordering]] operator that shuffles its operands so the time-components <math>x^0</math> and <math>y^0</math> increase from right to left, and <math>|\Omega\rang</math> is the ground state (vacuum state) of the interacting theory, different from the free ground state <math>| 0 \rang</math>. This expression represents the probability amplitude for the field to propagate from {{math|''y''}} to {{math|''x''}}, and goes by multiple names, like the two-point [[propagator]], two-point [[correlation function (quantum field theory)|correlation function]], two-point [[Green's function]] or two-point function for short.{{r|peskin|page1=82}}

The free two-point function, also known as the [[Feynman propagator]], can be found for the real scalar field by either canonical quantization or path integrals to be{{r|peskin|zee|page1=31,288|page2=23}}
:<math>\lang 0|T\{\phi(x)\phi(y)\} |0\rang \equiv D_F(x-y) = \lim_{\epsilon\to 0} \int\frac{d^4p}{(2\pi)^4} \frac{i}{p_\mu p^\mu - m^2 + i\epsilon} e^{-ip_\mu (x^\mu - y^\mu)}.</math>
In an interacting theory, where the Lagrangian or Hamiltonian contains terms <math>L_I(t)</math> or <math>H_I(t)</math> that describe interactions, the two-point function is more difficult to define. However, through both the canonical quantization formulation and the path integral formulation, it is possible to express it through an infinite perturbation series of the ''free'' two-point function.

In canonical quantization, the two-point correlation function can be written as:{{r|peskin|page1=87}}
:<math>\lang\Omega|T\{\phi(x)\phi(y)\}|\Omega\rang = \lim_{T\to\infty(1-i\epsilon)} \frac{\left\lang 0\left|T\left\{\phi_I(x)\phi_I(y)\exp\left[-i\int_{-T}^T dt\, H_I(t)\right]\right\}\right|0\right\rang}{\left\lang 0\left|T\left\{\exp\left[-i\int_{-T}^T dt\, H_I(t)\right]\right\}\right|0\right\rang},</math>
where {{math|''ε''}} is an [[infinitesimal]] number and {{math|''ϕ<sub>I</sub>''}} is the field operator under the free theory. Here, the [[Exponential function|exponential]] should be understood as its [[power series]] expansion. For example, in <math>\phi^4</math>-theory, the interacting term of the Hamiltonian is <math display="inline">H_I(t) = \int d^3 x\,\frac{\lambda}{4!}\phi_I(x)^4</math>,{{r|peskin|page1=84}} and the expansion of the two-point correlator in terms of <math>\lambda</math> becomes<math display="block">\lang\Omega|T\{\phi(x)\phi(y)\}|\Omega\rang = 
\frac{
    \displaystyle \sum_{n=0}^\infty \frac{(-i \lambda)^n}{(4 !)^n n !} \int d^4 z_1 \cdots \int d^4 z_n \lang 0|T\{\phi_I(x)\phi_I(y)\phi_I(z_1)^4\cdots\phi_I(z_n)^4\}|0\rang}{
    \displaystyle \sum_{n=0}^\infty \frac{(-i \lambda)^n}{(4 !)^n n !} \int d^4 z_1 \cdots \int d^4 z_n \lang 0|T\{                  \phi_I(z_1)^4\cdots\phi_I(z_n)^4\}|0\rang
}.</math>This perturbation expansion expresses the interacting two-point function in terms of quantities <math>\lang 0 | \cdots | 0 \rang</math> that are evaluated in the ''free'' theory.

In the path integral formulation, the two-point correlation function can be written{{r|peskin|page1=284}}
:<math>\lang\Omega|T\{\phi(x)\phi(y)\}|\Omega\rang = \lim_{T\to\infty(1-i\epsilon)} \frac{\int\mathcal{D}\phi\,\phi(x)\phi(y)\exp\left[i\int_{-T}^T d^4z\,\mathcal{L}\right]}{\int\mathcal{D}\phi\,\exp\left[i\int_{-T}^T d^4z\,\mathcal{L}\right]},</math>
where <math>\mathcal{L}</math> is the Lagrangian density. As in the previous paragraph, the exponential can be expanded as a series in {{math|''λ''}}, reducing the interacting two-point function to quantities in the free theory.

[[Wick's theorem]] further reduce any {{math|''n''}}-point correlation function in the free theory to a sum of products of two-point correlation functions. For example,
:<math>\begin{align}
\lang 0|T\{\phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\}|0\rang &= \lang 0|T\{\phi(x_1)\phi(x_2)\}|0\rang \lang 0|T\{\phi(x_3)\phi(x_4)\}|0\rang\\
&+ \lang 0|T\{\phi(x_1)\phi(x_3)\}|0\rang \lang 0|T\{\phi(x_2)\phi(x_4)\}|0\rang\\
&+ \lang 0|T\{\phi(x_1)\phi(x_4)\}|0\rang \lang 0|T\{\phi(x_2)\phi(x_3)\}|0\rang.
\end{align}</math>
Since interacting correlation functions can be expressed in terms of free correlation functions, only the latter need to be evaluated in order to calculate all physical quantities in the (perturbative) interacting theory.{{r|peskin|page1=90}} This makes the Feynman propagator one of the most important quantities in quantum field theory.