# Expanding the generating functional $W[J]$ for connected diagrams as a power series in $\hbar$

This is a follow up of a recent post I made (Making sense of stationary phase method for the path integral), but here I will work in Euclidean space, i.e. a Wick rotation has been performed.

Let $$Z[J] = \int D[\phi] e^{\frac1\hbar(S[\phi]-\int J\phi)}$$ denote the generating functional for a source $$J(x)$$. Letting $$W[J]$$ denote the generating functional for the connected diagrams, we know that $$Z[J] = e^{\frac1\hbar W[J]}. \tag{1}$$

To compute $$W[J]$$ we perform a loop expansion, which means expanding (1) in $$\hbar$$: $$W[J] = W_0[J] + \hbar W_1[J] + \frac{\hbar^2}{2}W_2[J] + \mathcal{O}(h^3) \tag{2}.$$

I am not sure how (2) is obtained. Does this follow from (1) or is this something we would like to express $$W[J]$$ as? If it is the latter, what ensures we can always do this?

• Hi CBBAM. I removed the last subquestion (v1). Commented Mar 8 at 4:37
• @Qmechanic Should I open another post for that question? Commented Mar 8 at 5:28

OP is essentially asking about the $$\hbar$$/loop-expansion for the generating functional $$W_c[J]$$ of connected diagrams, i.e. that the power of $$\hbar$$ in a diagram is given by the number of loops. This is e.g. proven in my Phys.SE answer here.