Formal derivation of the loop expansion

2 Formal derivation of the loop expansion To construct, the loop expansion we look for the extrema of S[(p] being guided by the idea that the functional integral is dominated by the neighborhood of the minimum of L [ p], i.e. the maximum of exp (-S [ / ]). To shortcut the discussion from the outset we assume that the minimizing function is a constant 

2uo r (ATTpy The form of the last term results from Eq. (A 5.22) together with 

This should be compared to Eq. (5.52), multiplied by n and summed over n  

Grand Canonical Description of Solutions at Finite Concentration 

Using the Fourier series for y analogous to Eq. (A 5,14) we can write the quadratic terms as 

Zq derives from Zq by subtracting the terms of order -x or x - The first one is contained in S o), the second one cancels due to condition (A 5.24). 

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