I'm going through Mahan's Many-Particle Physics, and I'm a bit confused on what Wick's theorem is supposed to be. He has an intuitive explanation of it, where starting from an electron-electron interaction $$\langle0|T\hat{C}_{\alpha}(t)\hat{C}_\beta^\dagger(t_1)\hat{C}_\gamma(t_2)\hat{C}_\delta^\dagger(t')|0\rangle$$ He then (essentially) says that Wick's theorem states that "in making all the possible pairings between creation and annihilation operators, each pairing should be time ordered". He then gives $$\begin{align*} &\langle0|T\hat{C}_{\alpha}(t)\hat{C}_\beta^\dagger(t_1)\hat{C}_\gamma(t_2)\hat{C}_\delta^\dagger(t')|0\rangle\\ &=\langle0|T\hat{C}_\alpha(t)\hat{C}_\beta^\dagger(t_1)|0\rangle\langle0|T\hat{C}_\gamma(t_2)\hat{C}_\delta^\dagger(t_1)|0\rangle\\ &-\langle0|T\hat{C}_\alpha(t)\hat{C}_\delta^\dagger(t')|0\rangle\langle T\hat{C}_\gamma(t_2)\hat{C}_\beta^\dagger(t_1)|0\rangle \end{align*}$$ However,
- I don't see how you get to this from what I've found of Wick's theorem online, which seems to be an equation for rearranging a product of operators s.t. all the creation operators are to the left and all the annihilation operators are to the right. I've followed this to get $$\begin{align*} \hat{C}_{\alpha}(t)\hat{C}_\beta^\dagger(t_1)\hat{C}_\gamma(t_2)\hat{C}_\delta^\dagger(t')&=-\hat{C}_\beta^\dagger(t_1)\hat{C}_\delta^\dagger(t')\hat{C}_{\alpha}(t)\hat{C}_\gamma(t_2)\\ &+\delta_{\alpha\beta}:\hat{C}_\gamma(t_2)\hat{C}_\delta^\dagger(t')\\ &+\delta_{\alpha\delta}\hat{C}_\beta^\dagger(t_1)\hat{C}_\gamma(t_2)\\ &+\delta_{\beta\gamma}:\hat{C}_\alpha(t)\hat{C_\delta}^\dagger(t'):\\ &+\delta_{\gamma\delta}:\hat{C}_\alpha(t)\hat{C}_\beta^\dagger(t_1): \end{align*}$$ plus four delta terms. I used the fermionic/bosonic operator contractions to get the delta functions, and the other combinations yield zero. However, when dealing with the vacuum state the normal order of the creation/annihilation operators gives $0$. This would only leave the four delta functions, but this puts my answer off from Mahan's by a factor of two (generously interpreting the delta functions as pairs in the form $\delta_{ij}=\hat{C}_i\hat{C}_j^\dagger$)
- Even if I use the definition for fields, this still doesn't explain how Mahan can insert $|0\rangle\langle0|$ into the product. The only thing I can think of is the identity that the sum of all projection operators is the identity, but this would only explain it if the only state was the ground state, which would contradict the existence of the creation\annihilation operators. His intuitive explanation makes sense, where each excitation must be returned to the ground state before $\langle0|$ so they can be paired off into individual $C$ numbers, but I'm looking for a more rigorous explanation
- I don't think I can get to the field version of Wick's theorem by applying the time-ordering operator to the "usual" Wick's theorem. This makes it seem like they are two distinct theorems. Is this the case? If it is distinct, where can I find a proof for the QFT version of Wick's theorem, most proofs seem to be of the combinatorical Wick's theorem.
I'm trying to learn field theory from Mahan, and I think a text such as Weinberg's seems to be focused more on building to QED with tensors and the Dirac equation, so I think it would be a massive diversion to learn QFT "propertly". Can this all be explained without a full QFT course?
