Peskin and Schroder Equation 4.37 invalid

Question

In chapter 4 of Peskin & Schroeder, \begin{align} T\{\phi(x)\phi(y)\} &= N\{ \phi(x)\phi(y)+ \text{Contraction}({\phi(x),\phi(y)}) \}, \tag{4.37}\\ & = N\{ \phi(x)\phi(y)\}+ N\{\text{Contraction}({\phi(x),\phi(y)}) \}. \tag{linearity of $N$} \end{align}

If for instance $x_0>y_0$, using the definition of $T$ and Contraction we have that $$\phi(x)\phi(y)=N\{ \phi(x)\phi(y)\}+ N\{ [\phi^+(x),\phi^-(y) ] \}.$$

This makes no sense, as

\begin{align} N\{ [\phi^+(x),\phi^-(y) ] \} &= N\{ \phi^+(x)\phi^-(y) -\phi^-(y)\phi^+(x) \}, \\ &=\phi^+(x)\phi^-(y)-\phi^+(x)\phi^-(y), \\ &=0. \end{align} where it should really of course not be affected by the normal ordering.

Clearly the problem is it seems like the Contraction in (4.37) should not be normal ordered. Why is it? I know this has been asked before but the answers are not direct. Please answer the question directly.

Answer 1

Remember that the Wick contraction is defined as \begin{equation} \text{Contr}(\phi(x)\phi(y))\equiv \langle 0 |T\{\phi(x) \phi(y)\}|0\rangle. \end{equation}

From this definition, we see that the Wick contraction is a number, not an operator. So the notation $[\phi^-(x),\phi^+(y)]$ when $x_0>y_0$ is to be understood as being a number. In fact, a more concrete notation would be \begin{equation} \left\{ \begin{matrix} \langle 0 |[\phi^-(x),\phi^+(y)]| 0 \rangle \,\,\,\text{if}\,\,\, x_0>y_0, \\ \langle 0 |[\phi^-(y),\phi^+(x)]| 0 \rangle \,\,\,\text{if}\,\,\, x_0 < y_0. \end{matrix} \right. \end{equation} This shortcut in the notation is common in quantum mechanics. For example, the Heisenberg uncertainty principle is written as $[x,p]=i\hbar$, but in truth it is $[x,p]=i\hbar \times \text{Id}$, where $\text{Id}$ is the identity operator. But here we see that the number corresponding to the commutator $[\phi^-(x),\phi^+(y)]$ will go out of the expectation value, leaving just $\langle 0 | \text{Id} | 0 \rangle =1$ in factor of it.