Question

In textbooks on many-body quantum physics (e.g. Fetter and Walecka), Feynman diagrams are typically introduced after formulating the Dyson perturbative expansion of the Green's function using Wicks theorem. Then the Feynman diagrams follow as a way to conveniently represent the resulting integral equations.

In most literature however, I have noticed that the language is somewhat different. Typically, after introducing the Hamiltonian a certain quantity of interest will be introduced, typically the single-particle Green's function of self-energy. Then there is often a sentence like:

To evaluate this quantity of interest we sum the set of Feynman diagrams shown in Fig. X

Then Fig. X will contain the perturbative series already written in terms of Feynman diagrams. It seems that the step where this series is formulated, as I find in the textbooks, is typically skipped. I can think of two possible reasons for this:

The actual formulation of the Feynman rules from Dyson's equation and Wicks theorem is seen as trivial, and hence not repeated in typical papers. Even if the system is not a standard system treated in other literature.
There is actually a faster or more intuitive way to write down the relevant Feynman diagrams from the Hamiltonian, without having to resort to the perturbative expansion explicitly. If this is the case then I would love to see a textbook where such a procedure is explained. Currently whenever I want to understand a paper I go through the whole perturbative expansion for the respective Hamiltonian, which is a very tedious and time consuming process.
I would be very appreciative if someone more familiar with this field could tell me which of these is true. Thanks!

Answer 1

The use of Feynman diagrams is sometimes presented without the detailed formulation process in the literature, as the construction rules from the Hamiltonian and the Dyson equation are assumed to be known. However, for non-standard systems or those new to the field, full derivation may still be necessary.

Step-by-step explanation:

The question you're asking pertains to the application of Feynman diagrams in many-body quantum physics, specifically in the domain of the perturbative expansion using Dyson's equation and Wick's theorem. The primary concern seems to be the apparent omission of the step-by-step formulation of these diagrams in literature when compared to textbooks like Fetter and Walecka. It's not uncommon for literature to skip some steps in a derivation or argument that is well-established, particularly if the target audience is expected to be familiar with the fundamentals. In the case of Feynman diagrams, once the rules for their construction are known and the interaction Hamiltonian is given, experienced practitioners can often write down the relevant diagrams directly without going through the full perturbative series. This is because the Hamiltonian dictates the possible interactions and, through the Feynman rules that encode the properties of these interactions, the diagrams can be constructed to represent all the terms in the perturbative expansion.

However, for those new to the field or when dealing with non-standard systems, it might still be necessary to go through the detailed procedure to ensure the Feynman diagrams are accurately representing the quantum process of interest. While there might not be a specific textbook that dramatically simplifies this process, it's worth looking for resources or advanced texts that bridge the gap between standard textbook material and the shorthand commonly found in research papers.