Feynman Propagator in Peskin & Schroeder

Question

To prove Wick's Theorem, Peskin & Schroeder define the contraction of two fields: \begin{align} \text{Contract}[\phi(x)\phi(y)]\equiv \begin{cases} [\phi^+(x),\phi^-(y)] & \text{for }x^0>y^0;\\ [\phi^+(y),\phi^-(x)] & \text{for }x^0>y^0, \end{cases} \end{align} where $\phi(x)=\phi^+(x)+\phi^-(x)$. Then, they claim in Equation 4.36,

This quantity is exactly the Feynman propagator:

\begin{align} \text{Contract}[\phi(x)\phi(y)]=D_F(x-y). \end{align}

However, in Equation 2.60 they define the Feynman propagator: \begin{align} D_F(x-y)\equiv \big<0\big|T\phi(x)\phi(y)\big|0\big>, \end{align} which is a c-number. But $\text{Contract}[\phi(x)\phi(y)]$ is obviously not a c-number. Could someone please explain this apparent contradiction? Should I rightly understand the Feynman propagator as a c-number or as an operator?

Answer 1

This is explained in my Phys.SE answer here. In a nutshell, under appropriate assumptions, one may show that $$ \text{Contract}[\phi(x)\phi(y)]~=~D_F(x-y) ~{\bf 1},$$ where ${\bf 1}$ is the identity operator.

Answer 2

The Feynman progapagtor is a Green's function of a wave equation which is a c-number. Also, the commutator between the positive frequency component evaluated at x, and the negative frequency component evaulated at y i.e $[\phi^+(x),\phi^-(y)]$ is obviously a c-number. The $\phi^+$ contains $a$ and the $\phi^-$ contains $a^\dagger$, and the commuatator between $a$ and $a^\dagger$ is a delta function which is a c-number. Calculation step: using p as integration variable for $\phi^+(x)$, q as integration variable for $\phi^-(y)$ and the normaliztion convention in Peskin

$[\phi^+(x),\phi^-(y] = \int \frac{d^3pd^3q}{(2\pi)^6\sqrt(2E_p2E_q)}e^{-ipx+iqy}[a_p,a^\dagger_q]=\int \frac{d^3p}{(2\pi)^32E_p}e^{-ip(x-y)}$ which is the propagation amplitude for a KG particle created at x and destroyed at y.