I already know that $T\,:\,\,:\,=T$ and $\,:\,\,:\,T=\,:\,\,:\,$ simply because when one operator acts after the other, play a bit with what the previous has done and do what he wants to actually do, or at least that's my interpretation, probably wrong. I also always considered that time ordering and normal ordering acts linearly on operators (acting strange when there are things like $\hat{a}\hat{a}^\dagger+1$) and also act as idempotent operators, meaning that $T^2=T$ and $\,:\,(\,:\,\,:\,)\,:\,=\,:\,\,:\,$ .
But now I'm stuck with this apparently paradoxical result using Wick's theorem in a composition of different field operators considered at equal time \begin{gather*} \overset{k}{\underset{i=1}{\circ}} \hat{\phi}_i(t,\boldsymbol{r}_i) \equiv T\left( \overset{k}{\underset{i=1}{\circ}} \hat{\phi}_i(t,\boldsymbol{r}_i) \right) \\ \,:\! \overset{k}{\underset{i=1}{\circ}} \hat{\phi}_i(t,\boldsymbol{r}_i) \!:\, \equiv \,:\! \left( \,:\! \overset{k}{\underset{i=1}{\circ}} \hat{\phi}_i(t,\boldsymbol{r}_i) \!:\, + \sum_{j=1}^{\left\lfloor\frac{k}{2}\right\rfloor} \sum\limits_{\bullet\times j} \,:\! \overset{k}{\underset{i=1}{\circ}} \hat{\phi}_i(t,\boldsymbol{r}_i) \!:\, \right) \!:\, \\ \,:\!\left( \,:\! \overset{k}{\underset{i=1}{\circ}} \hat{\phi}_i(t,\boldsymbol{r}_i) \!:\,\right) \!:\, \overset{?}{=} \,:\! \overset{k}{\underset{i=1}{\circ}} \hat{\phi}_i(t,\boldsymbol{r}_i) \!:\, \\ \Rightarrow 0 = \sum_{j=1}^{\left\lfloor\frac{k}{2}\right\rfloor} \sum\limits_{\bullet\times j} \,:\! \overset{k}{\underset{i=1}{\circ}} \hat{\phi}_i(t,\boldsymbol{r}_i) \!:\, \\ \,:\! \overset{k}{\underset{i=1}{\circ}} \hat{\phi}_i(t,\boldsymbol{r}_i) \!:\, = T\left( \overset{k}{\underset{i=1}{\circ}} \hat{\phi}_i(t,\boldsymbol{r}_i) \right) = \overset{k}{\underset{i=1}{\circ}} \hat{\phi}_i(t,\boldsymbol{r}_i) \end{gather*} The last equation seems very wrong to me and for that I can suspect the hypothesis that normal ordering was idempotent. Where is the issue?
